domain: math sdv

text:
Philippe Michel

Suman Kumar Tumuluri

Suman Kumar

A NOTE ON A NEURON NETWORK MODEL WITH DIFFUSION

Keywords: 2010 Mathematics Subject Classification. Primary: 35A01, 35A02, 35K20, 35K55, 92D25 Renewal equation with diffusion, existence of solution, asymptotic behavior, General relative entropy

We study the dynamics of an inhomogeneous neuronal network parametrized by a real number σ and structured by the time elapsed since the last discharge. The dynamics are governed by the parabolic PDE which describes the probability density of neurons with elapsed time s after its last discharge. We prove existence and uniqueness of a solution to the model. Moreover, we show that under some conditions on the connectivity and the firing rate, the network exhibits total desynchronization.

1. Introduction. The presence of variability in neural activity (in vivo and in vitro) is well known [START_REF] Faisal | Noise in the nervous system[END_REF]. For example, in the case of individual cortical neurons it is observed (in vivo) that their spike trains are highly irregular [START_REF] Shadlen | Noise, neural codes and cortical organization[END_REF][START_REF] Softky | The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs[END_REF]. Besides the randomness observed in spontaneous neuronal activity, in vitro experiments also confirmed irregular behavior of neural activity [START_REF] Gerstner | Spiking neuron models[END_REF]. The recent development in theoretical neuroscience focus on understanding the essence of these variability. The random influences on the neuronal firing activity is termed as 'neuronal noise' [START_REF] Longtin | Neuronal noise[END_REF] and the sources of noise are broadly classified as extrinsic and intrinsic [START_REF] Gerstner | Spiking neuron models[END_REF][START_REF] Longtin | Neuronal noise[END_REF]. The extrinsic noise is due to the network effects and signal transmission over cells whereas the intrinsic noise is generated at cell level.

In the extrinsic noise case, one adds the noise term explicitly while modeling the evolution of state of the nervous cells to get stochastic differential equations. One of the classical models in this context is the Noisy leaky integrate-and-fire model which is structured by the membrane potential [START_REF] Abbott | Lapique's introduction of the integrate-and-fire model neuron[END_REF][START_REF] Bressloff | Stochastic models of intracellular transport[END_REF][START_REF] Brunel | Lapicque's 1907 paper: from frogs to integrate-and-fire[END_REF][START_REF] Brunel | Fast global oscillations in networks of integrate-and-fire neurons with low firing rates[END_REF][START_REF] Caceres | Analysis of nonlinear noisy integrate & fire neuron models: Blow-up and steady states[END_REF][START_REF] Izhikevich | Dynamical Systems in Neuroscience[END_REF][START_REF] Knight | Dynamics of encoding in a population of neurons[END_REF]. To account for intrinsic variability and randomness in the firing of neurons, the main assumption in mathematical models is that a firing event can take place at any time according to a certain probability rate [START_REF] Gerstner | Time structure of the activity in neural network models[END_REF]. The implementation of intrinsic noise influences the firing time and the process can be completely characterized by the amount of time elapsed since the last spike (action potential). Motivated by this, many neuronal models in the literature neglect the mechanisms responsible for spike generation and instead describe the dynamics of neurons structured by the discharge times [START_REF] Pakdaman | Dynamics of a structured neuron population[END_REF][START_REF] Pakdaman | Relaxation and self-sustained oscillations in the time elapsed neuron network model[END_REF][START_REF] Pakdaman | Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation[END_REF][START_REF] Mischler | Relaxation in time elapsed neuron network models in the weak connectivity regime[END_REF]. Numerical experiments [START_REF] Pham | Activity in sparsely connected excitatory neural networks: effect of connectivity[END_REF][START_REF] Pham | Noise-induced coherent oscillations in randomly connected neural networks[END_REF] suggest that the population models provide insight to the mechanisms which regulate various forms of activities in neuronal population [START_REF] Pakdaman | Dynamics of a structured neuron population[END_REF][START_REF] Pakdaman | Relaxation and self-sustained oscillations in the time elapsed neuron network model[END_REF]. In [START_REF] Perkel | A computer program for simulating a network of interacting neurons. I. Organization and physiological assumptions[END_REF][START_REF] Perkel | A computer program for simulating a network of interacting neurons. III. Applications[END_REF][START_REF] Perkel | A computer program for simulating a network of interacting neurons. II. Programming aspects[END_REF], neuronal population dynamics models are based on occurrence times of events. In [START_REF] Dumont | Theoretical connections between mathematical neuronal models corresponding to different expressions of noise[END_REF], analytical links between Fokker-Planck equation which arises in membrane potential-structured models and McKendrick-von Foerster equations which are age-structured systems were established. A microscopic model corresponding to the models described in [START_REF] Pakdaman | Dynamics of a structured neuron population[END_REF][START_REF] Pakdaman | Relaxation and self-sustained oscillations in the time elapsed neuron network model[END_REF][START_REF] Pakdaman | Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation[END_REF][START_REF] Pham | Activity in sparsely connected excitatory neural networks: effect of connectivity[END_REF][START_REF] Pham | Noise-induced coherent oscillations in randomly connected neural networks[END_REF] for finite number of neurons is given in [START_REF] Quiñinao | A microscopic spiking neuronal network for the age structured model[END_REF]. The work presented in this paper is based on partial differential equations models of evolution developed in [START_REF] Kanga | Dynamics of time elapsed inhomogeneous neuron network model[END_REF][START_REF] Pakdaman | Dynamics of a structured neuron population[END_REF][START_REF] Pakdaman | Relaxation and self-sustained oscillations in the time elapsed neuron network model[END_REF][START_REF] Pakdaman | Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation[END_REF]. Homogeneous neural networks are those networks in which neural activities are driven by the same dynamic, all the neurons are excitatory and the interactions between the neurons are taken in account through a global neural activity. In [START_REF] Pakdaman | Relaxation and self-sustained oscillations in the time elapsed neuron network model[END_REF], the authors model homogeneous neural networks as the time evolution of neurons structured in 'time after the last discharge' (or age) s. More precisely, the dynamics of neurons are described by the probability density n(s, t) of finding a neuron in state s, i.e., time elapsed since the last discharge, at time t. Let X represent the environment due to the global neural activity in a network of connectivity J and p(s, X) represent the firing rate of neurons in the state s and environment X. Then the density n satisfies a partial differential equation of evolution type (in particular McKendrick-Von Foerster type [START_REF]Perthame Transport Equations in Biology[END_REF])

                   n t (s, t) + n s (s, t) + p(s, X(t))n(s, t) = 0, s > 0, t > 0, n(0, t) = ∞ 0 p(s, X(t))n(s, t)ds, t > 0, X(t) = J ∞ 0 p(s, X(t))n(s, t)ds, t > 0, n(s, 0) = n 0 (s), s > 0. (1)
The boundary condition, at state s = 0, makes the neurons reenter the cycle after firing. We notice that the authors assume that the firing rate p is increasing with respect to s, increasing with respect to the global activity variable X and asymptotically converges to 1 as s and X go to infinity. In fact, these assumptions model the biological property of neurons to discharge in response to a stimulation and to recover their excitability in time. Mathematical difficulties such as proving existence, uniqueness of solution and studying the longtime behavior of the solution to [START_REF] Abbott | Lapique's introduction of the integrate-and-fire model neuron[END_REF], are due to the nonlinear coupling between X and n. Under some properties of the function p and the connectivity, authors prove the existence and uniqueness through a Banach-Picard fixed point theorem. They use an entropy method [START_REF] Michel | General relative entropy inequality: an illustrations on growth models[END_REF] to study the convergence, as time goes to infinity, of the network to a steady state.

Later, the authors of [START_REF] Kanga | Dynamics of time elapsed inhomogeneous neuron network model[END_REF] consider an infinite inhomogeneous networks parametrized by a real number σ with different refractory periods. All these sub-networks are coupled by their mean activity. In this model, the population of neurons are described by the probability density n(s, t, σ) of finding a neuron in state s, where s is the time elapsed since the last discharge, at time t in the sub-network σ. Let p(s, X) denote the firing rate of neurons in the state s, in environment X. The main assumption in their model is that the synaptic weights are the same for all the networks. This implies that the density n is governed by the partial differential equation of evolution (which is similar to that of (1))

                   n t (s, t; σ) + n s (s, t; σ) + p(s, X(t); σ)n(s, t; σ) = 0, s > 0, t > 0, σ ∈ R, n(0, t; σ) = ∞ 0 p(s, X(t); σ)n(s, t; σ)ds, t > 0, σ ∈ R, X(t) = J R ∞ 0 p(s, X(t); σ)n(s, t; σ)dsdσ, t > 0, n(s, 0, σ) = n 0 (s; σ), s > 0, σ ∈ R. (2)
The mathematical difficulties in this case are the same as that of (1) and the authors of [START_REF] Kanga | Dynamics of time elapsed inhomogeneous neuron network model[END_REF] use the same tools to investigate the long time behavior of the solution to [START_REF] Bressloff | Stochastic models of intracellular transport[END_REF].

In this paper we consider an infinite inhomogeneous neuronal networks parametrized by a real number σ ∈ R. Moreover, we assume that each network has specific intrinsic dynamics and refractory period and all the networks are coupled by means of their mean activity. Further, we assume that the synaptic weights are given by the same constant J which models the connectivity. As in the earlier papers, p (the instantaneous firing rate) and X describe the post discharge recovery of the neural membrane and the global neural activity, respectively. We denote by n(s, t; σ) the probability density of neurons at time t in the state s (where s, as in the earlier works, is the time elapsed since the last discharge). We take into account the variability/randomness in the phenomena that cause firing of neurons by incorporating a diffusive effect through the second order derivative (with respect to s) of the probability density of neurons n(s, t; σ) in system [START_REF] Bressloff | Stochastic models of intracellular transport[END_REF]. Thus, we would like to study the following model for σ ∈ R,

                   n t (
s, t; σ) + n s (s, t; σ) + p(s, X(t); σ)n(s, t; σ) = n ss (s, t; σ), s, t > 0, n(0, t; σ) -n s (0, t; σ) = ∞ 0 p(s, X(t); σ)n(s, t; σ)ds, t > 0,

X(t) = J R ∞ 0 p(s, X(t); σ)n(s, t; σ)dsdσ, t > 0, n(s, 0, σ) = n 0 (s; σ), s > 0. (3)
When the diffusion constant = 0 then the model given in (3) reduces to the time elapsed inhomogeneous neuron network model [START_REF] Kanga | Dynamics of time elapsed inhomogeneous neuron network model[END_REF]. It is worth noticing that the quantity X, which is coupled with the partial differential equation satisfies the nonlinear integral equation. In fact, we make some technical assumptions on p to establish the existence of solution to this problem. In fact, the role that X plays in the model is similar to that of the weighted population (see [START_REF] Kakumani | On a nonlinear renewal equation with diffusion[END_REF], [START_REF] Perthame | Nonlinear renewal equations, in Selected Topics On Cancer Modelling Genesis-Evolution-Immune Competition-Therapy[END_REF]). Now, on integrating (3) with respect to s, we get

d dt ∞ 0 n(s, t; σ)ds = 0, i.e., ∀t > 0, ∞ 0 n(s, t; σ)ds = ∞ 0 n 0 (s; σ)ds =: g(σ). (4)
Here g(σ) denotes the probability density of neural networks parameterized by σ.

Therefore it is natural to assume that n 0 satisfies

R ∞ 0 n 0 dsdσ = R g(σ)dσ = 1.
As before, difficulties to prove existence, uniqueness and convergence as time goes to infinity come from the nonlinear coupling between the environment (macroscopic variable) and the neuron density n (microscopic variable). In Section 2, we give assumptions on the firing rate of neurons p to have existence and uniqueness of solution to (3) through Theorem 2.1 and its long time behavior to a steady state through Theorems 2.2 and 2.3. Theorem 2.1 focuses on the existence and uniqueness of solution to (3) and its proof is based on the Schauder fixed point theorem and the L 1 stability that we establish. It is very important to notice that the assumptions to prove existence and uniqueness result in this article are weaker than that of those used in [START_REF] Pakdaman | Dynamics of a structured neuron population[END_REF] where the Banach fixed point theorem is employed. Theorem 2.2 provides the existence of a steady state and the convergence of solutions to the steady state is given by Theorem 2.3. The proof of the convergence is based on entropy results and compactness. Moreover, we add Theorem 3.1, which, under stronger assumptions, gives the exponential convergence to the steady state. For the sake of readability, the proofs of Theorems 2.1 and 2.3 are given in Section 3 and the most technical part of the proofs of these theorems is given in Annex 5. Conclusions are provided in Section 4.

Assumptions and Main Results

. Throughout the article we assume that p is a nonnegative continuous function with 0 ≤ p ≤ 1. Furthermore, we assume that

(p(s, 0; σ) -1) ∈ L 2 (R + ), a.e. σ ∈ R, (5)
and that for every (s, σ) ∈ R + × R the map

x → p(s, x; σ) ∈ C 0 (R + ), and is increasing.

We notice that assumptions ( 5)-( 6) mean that the firing rate of neurons, i.e., the probability of firing p, in the 'state s' (age after firing) and in an environment X, increases with the environment and when the age after firing goes to infinity, almost surely, a new discharge happens. We generalize assumptions given in [START_REF] Pakdaman | Dynamics of a structured neuron population[END_REF][START_REF] Pakdaman | Relaxation and self-sustained oscillations in the time elapsed neuron network model[END_REF][START_REF] Pakdaman | Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation[END_REF][START_REF] Kanga | Dynamics of time elapsed inhomogeneous neuron network model[END_REF], which model the neurons discharge behavior. In fact, ( 5)-( 6) are enough to prove the existence of a solution to [START_REF] Brunel | Fast global oscillations in networks of integrate-and-fire neurons with low firing rates[END_REF]. Moreover, we assume that

κ := J sup x≥0 R ∞ 0 | ∂ ∂x p(s, x, σ)|dsdσ < 1. ( 7
)
This assumption means that the variation of the discharge probability, with respect to the environment, is small enough. Assumption ( 7) is indeed enough to prove uniqueness of solution to (3).

We now define the steady state equations corresponding to (3) as

                     N s (s; σ) + p(s, X * ; σ)N (s; σ) = N ss (s; σ), s > 0, σ ∈ R, N (0; σ) -N s (0; s) = ∞ 0 p(s, X * ; σ)N (s; σ)ds, σ ∈ R, X * = J R ∞ 0 p(s, X * ; σ)N (s; σ)dsdσ, ∞ 0 N (s; σ)ds = g(σ), σ ∈ R. (8)
The steady state which is defined as a solution to (8) plays a crucial role in the study of the longtime behaviour of the solution to (3).

We begin with the statement of existence and uniqueness result in Theorem 2.1.

Then we focus on the existence of the steady state (8) in Theorem 2.2. Finally in Theorem 2.3, we discuss the convergence of the solution to equation ( 3) to the steady state as t tends to infinity.

Theorem 2.1. Assume that 0 ≤ p ≤ 1, (5) and ( 6) are satisfied. Furthermore, we assume that the initial condition n 0 satisfies

(1 + s 2 )n 0 (s; σ) ∈ L 2 (R + ) ∩ L 1 (R + ), σ ∈ R, and R ∞ 0 n 0 dsdσ = 1. ( 9
)
Then there exists a solution n

∈ L 2 [0, T ], H 1 (R + ) ∩ L ∞ [0, T ], L 1 (R + ) to (3) with R ∞ 0 n(s, t; σ)dsdσ = 1, t > 0.
Moreover, if [START_REF] Faisal | Noise in the nervous system[END_REF] holds then a solution to (3) is unique.

Proof. (Outline) We prove the existence of a solution n to the problem, for σ ∈ R

                         n t (s, t; σ) + n s (s, t; σ) + p(s, X(t); σ)n(s, t; σ) = n ss (s, t; σ), s, t > 0, n(0, t; σ) -n s (0, t; s) = ∞ 0 (p(s, X(t); σ) -1)n(s, t; σ)ds + ∞ 0 n 0 (s; σ)ds, t > 0, σ ∈ R, X(t) = J R ∞ 0 p(s, X(t); σ)n(s, t; σ)dsdσ, t > 0, n(s, 0, σ) = n 0 (s; σ), s > 0. ( 10
)
Later we show that ( 10) is equivalent to (3) by establishing [START_REF] Mischler | Stability in a nonlinear population maturation model[END_REF]. The advantage of this form is that, we can work with the Hilbert space L 2 (R + ×R). As p-1 ∈ L 2 (R + ), we seek the solution n to (10) in a time dependent L 2 -space so that the boundary term in ( 10) is well defined. The proof is quite classical and uses a fixed point argument (in particular the Schauder fixed point theorem). Let T > 0,

X := L 2 ([0, T ] × R + × R) ∩ Y , where Y = f : [0, T ] × R + × R → R : f (t, .; .) ∈ L 1 (R + × R) for a.e. t ∈ [0, T ] .
We define the operator

T : X → X, n → T (n) := n
where n is a solution to

         n t (s, t; σ) + n s (s, t; σ) + pn(s, t; σ) = n ss (s, t; σ), s > 0, t > 0, σ ∈ R, n(0, t; σ) -n s (0, t; s) = ∞ 0 (p -1)n(s, t; σ)ds + ∞ 0 n 0 (s; σ)ds, t > 0, σ ∈ R, n(s, 0, σ) = n 0 (s; σ), s > 0, σ ∈ R, ( 11
) and p is a solution to p(s, t, σ) = p s, J R ∞ 0 (p(s, t, σ) -1)ndsdσ + J R ∞ 0 n 0 dsdσ, σ . (12)
We show that T has a fixed point which is indeed a solution to [START_REF] Izhikevich | Dynamical Systems in Neuroscience[END_REF]. In order to prove uniqueness we establish [START_REF] Perkel | A computer program for simulating a network of interacting neurons. I. Organization and physiological assumptions[END_REF]. Details of the proof are given in Section 3.1.

We now state the existence result for the steady state.

Theorem 2.2. Assume that 0 ≤ p ≤ 1, (5), ( 6) and ( 7) then there exists a unique solution N to (8) in L 2 (R + × R).

Proof. We notice that

Γ : N ∈ L 2 (R + × R) → M ∈ L 2 (R + × R),
where M is the solution to

             M s (s; σ) + pM (s; σ) = M ss (s; σ), s > 0, σ ∈ R, M (0; σ) -M s (0; σ) = ∞ 0 pM (s; σ)ds, σ ∈ R, p = p(s, J R ∞ 0 pN dsdσ; σ), ∞ 0 M (s; σ)ds = g(σ), σ ∈ R. (13)
The map Γ is a strongly positive and compact operator (assumption (7) implies uniqueness of p). Therefore, using the Krein-Rutman theorem, we have the existence of N , a positive eigenfunction of Γ. Integrate the equation Γ(N ) -λN = 0 (where λ is the eigenvalue of Γ associated to N ), we find that λ N = 0, which means that λ = 0 (and N is a solution to ( 8)). Since N (s, σ)ds = g(σ), N is uniquely defined.

Finally, we state the result pertaining to the asymptotic behavior of the solution to [START_REF] Brunel | Fast global oscillations in networks of integrate-and-fire neurons with low firing rates[END_REF]. In fact, we establish desynchronization.

Theorem 2.3. Assume that 0 ≤ p ≤ 1, (5), ( 6), [START_REF] Faisal | Noise in the nervous system[END_REF] and

sup σ ∞ 0 |1 -p(s, 0, σ)|ds ≤ 1 -κ 1 + κ , ( 14
) then R ∞ 0 |n(s, t; σ) -N (s; σ)|dsdσ → 0, as t → ∞.
The proof is given in Section 3.2.

Moreover, under some technical assumptions on p, L and J, we prove that n converges exponentially to the steady state N (see Theorem 3.1).

3. Proof of Main Theorems. This section is subdivided in two subsections. In the first subsection we prove Theorem 2.1 whereas in the second one we prove the long time behavior result, i.e., Theorem 2.3.

3.1.

Existence and Uniqueness Results. The objective of this subsection is to establish the existence and uniqueness of solutions to (3). In particular, we provide the details of the proof of Theorem 2.1. Interesting parts of the proof is the existence of p satisfying (12) for a given p and the L 1 -stability estimate for solutions.

Proof. of Theorem 2.1: The proof is divided into four main steps.

Step 1: In this step we show that T is a well defined map.

To this end, we first prove the existence of a unique solution p to [START_REF] Kanga | Dynamics of time elapsed inhomogeneous neuron network model[END_REF]. For a given n ∈ X, we define a sequence (p k ) as

p0 = 0, pk+1 (s, t, σ) = p s, J R ∞ 0 pk (s, t, σ)n(t, s, σ)dsdσ, σ , k ≥ 0.
Since p is nonnegative, p 1 ≥ 0 = p 0 . As p is increasing with respect to x, by induction one can show that pk+1 (s, t, σ) ≥ pk (s, t, σ), s > 0, t > 0, σ ∈ R, and 0 ≤ pk ≤ 1. Therefore the sequence (p k ) converges to some function, say p. Moreover by the Lebesgue dominated convergence theorem we have

R ∞ 0 pk (s, t, σ)n(t, s, σ)dsdσ → R ∞ 0 p(s, t, σ)n(t, s, σ)dsdσ,
and finally by the continuity of p we obtain that p is a solution to [START_REF] Kanga | Dynamics of time elapsed inhomogeneous neuron network model[END_REF]. Using the standard arguments it is straightforward to show the existence and uniqueness of a solution n ∈ X, to [START_REF] Kakumani | On a nonlinear renewal equation with diffusion[END_REF] (see [START_REF] Kakumani | On a nonlinear renewal equation with diffusion[END_REF], for instance). Hence the map T is well defined.

Step 2: In this step we prove that T is a compact operator.

To prove the compactness of T , we use the Lions-Aubin lemma ( [START_REF] Lions | Equations differentielles operationnelles et problemes aux limites[END_REF]). Indeed, if n is a solution to (11) then we have the following bounds (see Annex 5.1 for a proof):

for t ∈ [0, T ], σ ∈ R, ∞ 0 n 2 (s, t; σ)ds ≤ ∞ 0 n 2 0 (s; σ)ds + 2T exp 2 [0,T ]×R + (p -1) 2 dsdt , (15)
[0,T ]×R+

n 2 s (s, t; σ)dsdt ≤ T + ∞ 0 n 2 0 (s; σ)ds + [0,T ]×R+ (p -1) 2 dsdt ∞ 0 n 2 0 (s; σ)ds + 2T exp 2 [0,T ]×R+ (p -1) 2 dvdτ , ( 16
) ∞ 0 s 2 n(s, t; σ)ds ≤ ∞ 0 s 2 n 0 (s; σ)ds + T max s (-s 2 p(s, 0; σ) + 2s + 2 ), (17) 2
∞ 0 sn(s, t; σ)ds ≤ ∞ 0 (s 2 + 1)n 0 (s; σ)ds + T max s (-s 2 p(s, 0; σ) + 2s + 2 ), ( 18
) ∞ 0 nds = ∞ 0 n 0 ds, R ∞ 0 ndsdσ = R ∞ 0 n 0 dsdσ = 1, (19)
and

∀φ ∈ H 1 , [0,T ] ( ∞ 0 φ(s, σ)n t (s, t; σ)ds) 2 dt ≤ C T ( , p, n 0 ) φ H 1 , (20)
where C T is a constant which depends only on T, , p and n 0 .

Therefore from the estimates ( 15)-( 20) it follows that T is a compact operator. Moreover, since the constant map which takes one is a super solution to [START_REF] Izhikevich | Dynamical Systems in Neuroscience[END_REF], we have the following inequality

n(s, t; σ) ≤ 1, s > 0, t > 0, σ ∈ R. ( 21
)
Step 3: In this step we prove the existence of a solution to [START_REF] Izhikevich | Dynamical Systems in Neuroscience[END_REF].

Since the operator T is compact on X, thanks to the Schauder fixed point theorem, there exists a fixed point of T which is a solution to [START_REF] Izhikevich | Dynamical Systems in Neuroscience[END_REF] satisfying all the estimates given in ( 15)-( 21). This proves the existence of solution to [START_REF] Izhikevich | Dynamical Systems in Neuroscience[END_REF].

Step 4: We now turn our attention towards the uniqueness of solution to (3).

The key ingredient in the proof of uniqueness result is the L 1 -stability estimate that we prove here. Let (m, q) and (n, p) be solutions to [START_REF] Kakumani | On a nonlinear renewal equation with diffusion[END_REF] with m(0, s; σ) = m 0 (s; σ), n(0, s; σ) = n 0 (s; σ),

∞ 0 n 0 (s; σ)ds = ∞ 0 m 0 (s; σ)ds, a.e. σ ∈ R, R ∞ 0 n 0 dsdσ = R ∞ 0 m 0 dsdσ = 1, p(s, t; σ) = p s, J R ∞ 0
p(s, t; σ)n(s, t; σ)dsdσ; σ , and q(s, t; σ) = p s, J R ∞ 0 q(s, t; σ)m(s, t; σ)dsdσ; σ .

By the mean value theorem we obtain

|q -p| = J| ∂ ∂x p(s, u, σ)| R ∞ 0 (q -p)ndsdσ + R ∞ 0 q(m -n)dsdσ , for some u ∈ [J min( R ∞ 0 qmdsdσ, R ∞ 0 pndsdσ), J max( R ∞ 0 qmdsdσ, R ∞ 0 pndsdσ)]. In view of (21) we obtain R ∞ 0 |q -p|ndsdσ ≤ R ∞ 0 (q -p)ndsdσ + R ∞ 0 q(m -n)dsdσ × R ∞ 0 J| ∂ ∂x p(s, u, σ)|dsdσ.
From assumption [START_REF] Faisal | Noise in the nervous system[END_REF], and

R ∞ 0 ndsdσ = R ∞ 0 mdsdσ = 1, we find R ∞ 0 |q -p|ndsdσ ≤ κ 1 -κ R ∞ 0 |1 -q||m -n|dsdσ. (22)
Moreover, from [START_REF] Pakdaman | Dynamics of a structured neuron population[END_REF] we readily get

R ∞ 0 (qm -pn)ds dσ ≤ R ∞ 0 (q -p)nds dσ + R ∞ 0 q(m -n)ds dσ ≤ κ 1 -κ R ∞ 0 |1 -q||m -n|dsdσ + R ∞ 0 |1 -q||m -n|dsdσ ≤ 1 1 -κ R ∞ 0 |1 -q||m -n|dsdσ. (23)
On the other hand, in view of Lemma 5.1 (the choice of the convex function is

H(v) = |v -1|) we readily get ∂ ∂t ∞ 0 |n -m|ds ≤ ∞ 0 |n -m|[-q] + sgn(n -m)(q -p)n ds -sgn(n -m)(s = 0) ( n m )(s = 0) ∞ 0 qmds - ∞ 0 pnds +|n/m -1|(s = 0) ∞ 0 qmds ≤ ∞ 0 |n -m|[-q] + sgn(n -m)(q -p)n ds +sgn(n -m)(s = 0) ∞ 0 pnds - ∞ 0 qmds .
Therefore using ( 22) and ( 23) we have that

∂ ∂t R ∞ 0 |n -m|dsdσ ≤ 1 + κ 1 -κ R ∞ 0 |m -n|dsdσ.
Thus we get the following L 1 -stability estimate, thanks to the Gronwall lemma

R ∞ 0 |n -m|dsdσ ≤ e γt R ∞ 0 |n 0 -m 0 |dsdσ, (24)
where γ = 1+κ 1-κ . If m 0 = n 0 then from stability estimate [START_REF] Perkel | A computer program for simulating a network of interacting neurons. I. Organization and physiological assumptions[END_REF], we immediately get m(s, t; σ) = n(s, t, σ), t > 0 s > 0, σ ∈ R a.e. This proves uniqueness of a solution to [START_REF] Izhikevich | Dynamical Systems in Neuroscience[END_REF].

Long Time Behavior.

In this subsection, we present results pertaining to the asymptotic behavior of the solution to [START_REF] Brunel | Fast global oscillations in networks of integrate-and-fire neurons with low firing rates[END_REF]. GRE inequality (see Annex) together with the La Salle principle give us the convergence of the solution to (3) to the corresponding steady state.

Proof. of Theorem 2.3. We define u = n -N . Then u solves for s > 0,

t > 0, σ ∈ R                        u t + u s + p(s, X(t); σ)u + [p(s, X(t); σ) -p(s, X * ; σ)]N = u ss , u(0, t; σ)-u s (0, t; σ) = ∞ 0 p(s, X(t); σ)u +[p(s, X(t); σ) -p(s, X * ; σ)]N ds, u(s, 0, σ) = n 0 (s; σ) -N (s; σ), s > 0, X(t) = J R ∞ 0 p(s, X(t); σ)n(s, t; σ)dsdσ, t > 0, X * = J R ∞ 0 p(s, X * ; σ)N (s; σ)dsdσ. ( 25
)
We multiply equation ( 25) by sgn(u)Ψ(s) (Ψ ≥ 0) and integrate with respect to s, σ, to find

d dt R ∞ 0 |u|Ψdsdσ -Ψ(0) R |u|(0, t; σ)dσ + R ∞ 0 p|u|Ψdsdσ + R ∞ 0 [p-p * ]N sgn(u)Ψdsdσ = R ∞ 0 u ss sgn(u)Ψdsdσ + R ∞ 0 |u|Ψ s dsdσ,
where p * = p(s, X * ; σ). We now use the Kato inequality, integration by parts, and the boundary condition to obtain that

∂ t R ∞ 0 |u|Ψdsdσ ≤ Ψ s (0) R |u|(0, t; σ)dσ + R ∞ 0 |u|(Ψ s + Ψ ss -pΨ)dsdσ + Ψ(0) R ∞ 0 (pn -p * N )ds sgn(u)(0, t; σ)dσ - R ∞ 0 [p -p * ]N sgn(u)Ψdsdσ, is satisfied. Let Ψ : s → 1 - 1 + κ 1 -κ s 0 1 -e w-s |1 -p(w, 0, σ)|dw,
then, by assumption ( 14), Ψ ≥ 0. Moreover, we have

Ψ(0) = 1, Ψ s (0) = 0 and Ψ s + Ψ ss = - 1 + κ 1 -κ |1 -p(s, 0, σ)|.
In view of inequalities ( 22) and ( 23) (and the decay of x → 1 -p(., x, .)) it follows that

R ∞ 0 (Ψ s + Ψ ss )|u|dsdσ = - R ∞ 0 1 + κ 1 -κ |1 -p(s, 0, σ)||u|dsdσ ≤ - R ∞ 0 |p -p * |N dsdσ - R ∞ 0 pnds - ∞ 0 p * N ds dσ.
Since Ψ s ≤ 0, s > 0, we get Ψ ≤ 1 and

R ∞ 0 (Ψ s + Ψ ss )|u|dsdσ ≤ -Ψ(0) R ∞ 0 pnds - ∞ 0 p * N ds dσ - R ∞ 0 |p -p * |N Ψdsdσ.
Therefore we find that

∂ t R ∞ 0 |u|Ψdsdσ ≤ - R ∞ 0 pnds - ∞ 0 p * N ds (-1 + sgn(u(0, t; σ)))dσ - R ∞ 0 [|p -p * | -(p -p * )sgn(u)]N Ψdsdσ + R ∞ 0 |u|(-pΨ)dsdσ ≤ - R ∞ 0 |u|(pΨ)dsdσ. ( 26
)
Therefore the quantity

R ∞ 0 |u|Ψdsdσ decreases as long as R ∞ 0 |u|(pΨ)dsdσ
does not vanish. To prove the convergence, we use the La Salle principle. The proof is similar to the one given in [START_REF] Mischler | Stability in a nonlinear population maturation model[END_REF]. Let (t k ) k be a sequence of positive numbers such that t k → ∞, n k (s, t; σ) := n(s, t + t k ; σ) and p k (s, t; σ) := p(s, t + t k ; σ). Using inequality [START_REF] Perkel | A computer program for simulating a network of interacting neurons. II. Programming aspects[END_REF] and by integration we find the uniform

L 1 bound ∞ 0 R |n k -N |p k Ψdsdσ ≤ ( ∞ 0 R |n k -N |Ψdsdσ)(t = 0) ≤ 2. Since ∞ 0 R |n k -N |p k Ψdsdσ = ∞ t k R |n(s, t; σ) -N |pΨdsdσ, we have that ∞ 0 R |n k -N |p k Ψdsdσ → t k →∞ 0.
We now recall that (n k ) k belongs to a compact subset of L 2 (see the proof of Theorem 2.1). Moreover, since 1 -p k is uniformly bounded in L 2 and using the Banach Alaoglu theorem, we have (1 -p k ) k belongs to a weak compact set of L 2 and there exists a subsequences (which are still denoted by p k , n k ) such that

∞ 0 R |n k -N |(p k -1)Ψdsdσ → t k →∞ ∞ 0 R |n * -N |(p * -1)
Ψdsdσ, and

∞ 0 R |n k -N |Ψdsdσ → t k →∞ ∞ 0 R |n * -N |Ψdsdσ. Hence ∞ 0 R |n k -N |p k Ψdsdσ = ∞ 0 R |n k -N |(p k -1)Ψdsdσ + ∞ 0 R |n k -N |Ψdsdσ converges to ∞ 0 R |n * -N |p * Ψdsdσ with p * = p(s, X * ; σ) with X * = J p(s, X * ; σ)n * (s; σ)dsdσ.
Thus we have that n * = N as long as p * does not vanish, i.e., p * = p(s, X * ; σ) with X * = J p(s, X * ; σ)N (s; σ)dsdσ. By uniqueness of the stationary solution (see Theorem 2.2), we find that n * = N everywhere. Therefore, we have the convergence of |n k -N | to 0 as t → ∞. To conclude, it suffices to notice that R ∞ 0 |u|Ψdsdσ is decreasing by [START_REF] Perkel | A computer program for simulating a network of interacting neurons. II. Programming aspects[END_REF], which implies the convergence of |u| to 0 as t → ∞.

In the next result, we provide a condition which ensures us the convergence of the solution to (3) to the steady state with an exponential rate. Theorem 3.1. Assume that there exist positive numbers L, m σ and M σ such that

|p(s, X 1 ; σ) -p(s, X 2 ; σ)| ≤ L|X 1 -X 2 |, σ ∈ R, X 1 , X 2 ≥ 0, ( 27
) 0 < m ≤ m σ ≤ p(s, X; σ) ≤ M σ ≤ 1, s > 0, X > 0, σ ∈ R. (28)
Furthermore, assume that there exists β > 0 such that

β < m σ - 2LJM σ 1 -JL , σ ∈ R, (29) then
R ∞ 0 |n(s, t; σ) -N (s; σ)|dsdσ ≤ e -βt R ∞ 0 |n 0 (s; σ) -N (s; σ)|dsdσ.
Proof. We begin with the setting u = n -N . Then u is a solution to

         u t + u s + p(s, X(t); σ)u + [p(s, X(t); σ) -p(s, X * ; σ)]N = u ss , u(0, t; σ) -u s (0, t; σ) = ∞ 0 p(s, X(t); σ)u + [p(s, X(t); σ) -p(s, X * ; σ)]N ds, u(s, 0, σ) = n 0 (s; σ) -N (s; σ),
where s > 0, t > 0 and σ ∈ R. Multiplication with sgn(u) and integration with respect to s, σ leads to

d dt R ∞ 0 |u|dsdσ ≤ 2 R ∞ 0 |p(s, X(t); σ) -p(s, X * ; σ)|N dsdσ + R ∞ 0 p(s, X(t); σ)uds dσ- R ∞ 0 p(s, X(t); σ)|u|dsdσ. Since ∞ 0 uds = 0, t > 0, σ ∈ R, we obtain d dt R ∞ 0 |u|dsdσ ≤ 2 R ∞ 0 |p(s, X(t); σ) -p(s, X * ; σ)|N dsdσ + R ∞ 0 p(s, X(t); σ) -m σ uds dσ - R ∞ 0 p(s, X(t); σ)|u|dsdσ ≤ 2 R ∞ 0 |p(s, X(t); σ) -p(s, X * ; σ)|N dsdσ - R ∞ 0 m σ |u|dsdσ ≤ 2L R ∞ 0 |X(t) -X * |N dsdσ - R ∞ 0 m σ |u|dsdσ = 2L|X(t) -X * | - R ∞ 0 m σ |u|dsdσ. ( 30
)
We now estimate |X(t) -X * |. For, we consider

|X(t) -X * | ≤ J R ∞ 0 p(s, X(t); σ)|u|dsdσ +J R ∞ 0 |p(s, X(t); σ) -p(s, X * ; σ)|N dsdσ ≤ J R ∞ 0 M σ |u|dsdσ + JL|X(t) -X * |.
Therefore we obtain

|X(t) -X * | ≤ J 1 -JL R ∞ 0 M σ |u|dsdσ. (31)
Combining [START_REF] Quiñinao | A microscopic spiking neuronal network for the age structured model[END_REF] and [START_REF] Pham | Noise-induced coherent oscillations in randomly connected neural networks[END_REF] we get

d dt R ∞ 0 |u|dsdσ ≤ R ∞ 0 2LJM σ 1 -JL -m σ |u|dsdσ. (32)
Thanks to assumption [START_REF] Pham | Activity in sparsely connected excitatory neural networks: effect of connectivity[END_REF], we find

d dt R ∞ 0 |u|dsdσ ≤ - R ∞ 0 β|u|dsdσ.
Now the Gronwall lemma gives us the promised result.

Conclusions.

In this paper we have proposed a neural network model which describes the dynamics of inhomogeneous neuronal networks with diffusive effect due to variability/randomness. Our model is an extension of the nonlinear renewal model proposed in [START_REF] Pakdaman | Relaxation and self-sustained oscillations in the time elapsed neuron network model[END_REF][START_REF] Kanga | Dynamics of time elapsed inhomogeneous neuron network model[END_REF] where the density of neurons follows a nonlinear renewal partial differential equation in which the randomness is modeled by a diffusive effect. We have proved that, under certain technical conditions on the firing rate and connectivity, the problem is well posed. In fact we have established an L 1 -stability estimate to our model. Proves are slightly different from those proposed in [START_REF] Pakdaman | Relaxation and self-sustained oscillations in the time elapsed neuron network model[END_REF][START_REF] Kanga | Dynamics of time elapsed inhomogeneous neuron network model[END_REF] as we have used compactness arguments to prove the existence instead of a 'contractant' argument. In fact, we could establish the existence of a solution to (3) under weaker assumptions on the firing rate p. Moreover, using the La Salle principle, we have proved, the convergence of the density of neurons solution to our neural network model converges to the steady of the model when the synaptic weight (connectivity) is sufficiently small. In order to study the asymptotic behavior of the solution of our model, we have used the General Relative Entropy inequality which is natural in structured population models. Finally, we show that the convergence of the neuron density to the steady state is exponential whenever the connectivity is very small.

In a future work, it would be interesting to extend the existence and stability results to periodic solutions to our model [START_REF] Brunel | Fast global oscillations in networks of integrate-and-fire neurons with low firing rates[END_REF]. We anticipate that the compactness argument to prove the existence and uniqueness, and the entropy result to prove convergence in this article are well suited to the periodic problems also. Another open and interesting question is the optimality of assumptions to prove the convergence (and the existence of a steady state).

5.

Annex. In this section we prove a priori estimates stated in Section 3 and present General relative entropy (GRE) estimate.

Proof of Estimates (A Priori Bounds).

In order to prove [START_REF] Longtin | Neuronal noise[END_REF], we first multiply [START_REF] Kakumani | On a nonlinear renewal equation with diffusion[END_REF] by n and integrate with respect to s to have

∞ 0 (n 2 ) t (s, t; σ)ds + ∞ 0 2(n s ) 2 (s, t; σ)ds = -n(0, t; σ) 2 -2 ∞ 0 pn 2 (s, t; σ)ds + 2n(0, t; σ) ∞ 0 (p -1)n(s, t; σ)ds + ∞ 0 n 0 (s; σ)ds . We notice that -z 2 + 2Az ≤ A 2 , A, z ∈ R, hence it follows that ∞ 0 (n 2 ) t (s, t; σ)ds + ∞ 0 2(n s ) 2 (s, t; σ)ds ≤ ∞ 0 (p -1)n(s, t; σ)ds + 1 2 . (33)
Therefore, using the Cauchy inequality, we find that

∞ 0 (n 2 ) t (s, t; σ)ds ≤ 2 + 2 ∞ 0 (p -1) 2 ds ∞ 0 n 2 (s, t; σ)ds.
Owing to the Gronwall lemma, we have the first bound [START_REF] Longtin | Neuronal noise[END_REF]. On integrating [START_REF] Softky | The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs[END_REF] with respect to time t ∈ [0, T ] and using estimate [START_REF] Longtin | Neuronal noise[END_REF], we find that (16) holds. We now turn our attention towards [START_REF] Mischler | Stability in a nonlinear population maturation model[END_REF]. Let ψ ∈ H 1 (R + ), which will be chosen later. On multiplying [START_REF] Kakumani | On a nonlinear renewal equation with diffusion[END_REF] by ψ, integrating with respect to s, and using the integration by parts we obtain

∞ 0 ψ(s)n t (s, t; σ)ds + ∞ 0 ψ(s)pn(s, t; σ)ds = - ∞ 0 ψ (s)n s (s, t; σ)ds + ∞ 0 ψ (s)n(s, t; σ)ds + ψ(0)[ ∞ 0 (p -1)n(s, t; σ)ds + ∞ 0 n 0 (s; σ)ds].
After rearranging the terms we get

∞ 0 ψ(s)n t (s, t; σ)ds = - ∞ 0 ψ (s)n s (s, t; σ)ds + ∞ 0 ψ (s)n(s, t; σ)ds + ∞ 0 (ψ(0) -ψ(s))(p -1)n(s, t; σ)ds - ∞ 0 ψ(s)n(s, t; σ)ds + ψ(0) ∞ 0 n 0 (s; σ)ds. (34) d dt ∞ 0 s 2 n(s, t; σ)ds ≤ max s (-s 2 p(s, 0, σ) + 2s + 2 ) < ∞.
We integrate the previous equation with respect to t to prove that (17) holds. Using the estimate 2s ≤ 1 + s 2 , ( 17) and [START_REF] Mischler | Stability in a nonlinear population maturation model[END_REF] it is straightforward to obtain inequality [START_REF] Michel | General relative entropy inequality: an illustrations on growth models[END_REF]. In other words we have

2 ∞ 0 sn(s, t; σ)ds ≤ ∞ 0 (1 + s 2 )n(s, t; σ)ds ≤ ∞ 0 (s 2 + 1)n 0 (s; σ)ds + T max s (-s 2 p(s, 0; σ) + 2s + 2 ) < ∞, t < T.
For the remaining estimate, i.e., for [START_REF] Mischler | Relaxation in time elapsed neuron network models in the weak connectivity regime[END_REF], we multiply the first equation of ( 3) with φ and integrate with respect to s to get

∞ 0 φ(s, σ)n t (s, t; σ)ds = ∞ 0 pn(s, t; σ)[φ(0, σ) -φ(s, σ)]ds + ∞ 0 φ s (s, σ)n(s, t; σ)ds - ∞ 0 φ s (s, σ)n s (s, t; σ)ds. Since we have |φ(0, σ) -φ(s, σ)| ≤ √ s s 0 φ s (v, σ) 2 dv, σ ∈ R, it follows that ( ∞ 0 φ(s, σ)n t (s, t; σ)ds) 2 ≤ 2[ ∞ 0 (s 2 + 1)n 0 (s; σ)ds + T max s (-s 2 p(s, 0; σ) + 2s + 2 ) + 1] ∞ 0 φ s (s, σ) 2 ds + 2 2 ( ∞ 0 φ s (s, σ) 2 ds)( ∞ 0 n s (s, t; σ) 2 ds).
Then integrating with respect to time and using inequalities ( 15)-( 19) then we find (20).

GRE Inequality.

In this subsection we prove GRE inequality which plays crucial role in the study of the long time behavior of the solution to (3). GRE inequalities for structured population models can be found in [START_REF] Kakumani | On a nonlinear renewal equation with diffusion[END_REF][START_REF] Michel | General Relative Entropy in a nonlinear McKendrick model[END_REF][START_REF] Michel | General entropy equations for structured population models and scattering[END_REF][START_REF] Michel | General relative entropy inequality: an illustrations on growth models[END_REF][START_REF]Perthame Transport Equations in Biology[END_REF][START_REF] Perthame | Nonlinear renewal equations, in Selected Topics On Cancer Modelling Genesis-Evolution-Immune Competition-Therapy[END_REF].

Lemma 5.1. Let H ∈ C 2 (R, R), (m, q) and (n, p) be solutions to [START_REF] Kakumani | On a nonlinear renewal equation with diffusion[END_REF] with the initial data n(0, s; σ) = n 0 (s; σ) and m(0, s; σ) = m 0 (s; σ) respectively. Let Using the first equation in [START_REF] Kakumani | On a nonlinear renewal equation with diffusion[END_REF] Again, using the boundary condition, we find that (36) is satisfied.

n 0

 0 dsdσ, σ , q(s, t, σ) = p s, J R ∞ 0 (q(s, t, σ) -1)mdsdσ + J

  which n and m solve, we find that On integrating the above equation with respect to s, we obtain

	∂ ∂t		0	∞	(H(	n m	)m)ds=
	H (	n m	)(	∂ ∂t	n -(	n m	)	∂ ∂t	m) = H (	n m	)(-	∂ ∂s	n -pn +	∂ 2 ∂s 2 n)
																							+ H (	n m	)(	∂ ∂s	m + qm -	∂ 2 ∂s 2 m)	n m	,
														= H(	n m	)	∂ ∂s	m -	∂ ∂s	[H(	n m	)m] + H (	n m	)(-pn +	∂ 2 ∂s 2 n)
																							+ H (	n m	)(qm -	∂ 2 ∂s 2 m)	n m	.
	Therefore we get
	∂ ∂t	(H(	n m	)m) = H(	n m	)[-qm +	∂ 2 ∂s 2 m] -	∂ ∂s	[H(	n m	)m] + H (	n m	)(-pn +	∂ 2 ∂s 2 n)
																							+ H (	n m	)(qm -	∂ 2 ∂s 2 m)	n m	.
	A straightforward computation yields
												H (	n m	)(			∂ 2 ∂s 2 n) =	∂ ∂s	[H (	n m	)(	∂ ∂s	n)] -	∂ ∂s	n	∂ ∂s	[H (	n m	)],
			-H (	n m	)(	∂ 2 ∂s 2 m)	n m	= -	∂ ∂s	[H (	n m	)(	∂ ∂s	m)	n m	] +	∂ ∂s	m	∂ ∂s	[H (	n m	)	n m	],
	and
														H(	n m		)	∂ 2 ∂s 2 m =	∂ ∂s	[H(	n m	)	∂ ∂s	m] -	∂ ∂s	mH (	n m	)	∂ ∂s	n m	.
	By adding the previous three equations we obtain
	H (	n m	)(	∂ 2 ∂s 2 n) -			n m	H (	n m	)(	∂ 2 ∂s 2 m) + H(	n m	)	∂ 2 ∂s 2 m =
																	∂ ∂s	[H (	n m	)(	∂ ∂s	n m	)m] +	∂ ∂s	[H(	n m	)	∂ ∂s	m] -H (	n m	)(	∂ ∂s	n m	) 2 m.
	Thus we get
	∂ ∂t	(H(	n m	)m) = H(	n m	)[-qm] + H (	n m	)(q -p)n +	∂ ∂s	[H (	n m	)(	∂ ∂s	n m	)m]
																							+	∂ ∂s	[H(	n m	)	∂ ∂s	m] -	∂ ∂s	[H(	n m	)m] -H (	n m	)(	∂ ∂s	n m	) 2 m.

Acknowledgments The authors are grateful to the anonymous reviewer for his/her valuable suggestions which indeed helped to improve the quality of the manuscript. The corresponding author would like to acknowledge the financial support received from UGC DSA-1, SAP at School of Mathematics and Statistics, University of Hyderabad, India.

We now fix R > 0 and choose ψ such that 0 ≤ ψ ≤ 1, ψ(x) = 1, if 0 ≤ x < R, and ψ(x) = 0, if x > R + 1.

By setting

A straightforward computation gives

Thus we have

Again, we use Hölder's inequality to find

In view of ( 15) we obtain G R (τ ) → R→∞ 0, a.e.

Therefore by the Lebesgue dominated convergence theorem, we have t 0 G R (u)e u-t du → 0, as R → ∞. Now using estimate [START_REF] Michel | General Relative Entropy in a nonlinear McKendrick model[END_REF] and letting R → ∞, in (35) we get [START_REF] Mischler | Stability in a nonlinear population maturation model[END_REF]. In order to prove [START_REF] Michel | General entropy equations for structured population models and scattering[END_REF], we multiply the first equation in (3) with s 2 and integrate to find Again, by integration by parts we get


text:
Dr Philippe Michel
email: philippe.michel@ec-lyon.fr

( Researcher

Model of neo-malthusian population anticipating future changes in resources

Keywords:

In this paper we develop a class of models to study a population and resource dynamical system in which the decision to give birth is based on a rational far-sighted cost-benefit analysis on what the future of the ressource level will be. This leads to consider a system in which a time forward population/resource dynamical system is coupled with a time backward Bellman's equation (which models the choice of having a child). We construct, from a population model with food consumption, an example, to study the change in time of the fertility rate when a catastrophic change in ressource is announced at a given moment, when a birth control policy is announced and we compare these two announcements in case nothing happens. Moreover, we provide, mathematical tools to theoretically and numerically study this complex coupling of time forward and time backward equations.

Introduction

Basically, living organisms are born, consume resources (organisms), give birth and finally become resources, in a competitive environment. Therefore, the survival of a specie depends on the ability to produce well fitted newborns and, obviously, on the level of available resources. The relationship between consumers and their resources is a core study in ecology and in demography. There exists numerous theoretical models, which deal with different cases of relationship between consumers and their resources [START_REF] Freedman | Deterministic mathematical models in population ecology[END_REF][START_REF] Getz | Consumer-resource dynamics: quantity, quality, and allocation[END_REF][START_REF] May | Stability and complexity in model ecosystems[END_REF][START_REF] Van Opheusden | Competition for resources: complicated dynamics in the simple tilman model[END_REF][START_REF] Vance | A general dynamical model of one consumer-one resource interactions[END_REF][START_REF] Volterra | Variations and fluctuations of the number of individuals in animal species living together[END_REF][START_REF] Terry | A predator-prey model with generic birth and death rates for the predator[END_REF], where consumer is a predator and resource is a prey. The foundation of the mathematical modeling approach is based on the Lotka-Volterra equations [START_REF] Volterra | Variations and fluctuations of the number of individuals in animal species living together[END_REF] of coupled predator-prey (or consumer-resource) dynamics. Moreover, in some works (for instance [START_REF]Large-Scale Scientific Computing[END_REF] p277, [START_REF] Stanková | Joining or opting out of a lotka-volterra game between predators and prey: does the best strategy depend on modelling energy lost[END_REF][START_REF] Stukalin | Some applications of optimal control in sustainable fishing in the baltic sea[END_REF]), the authors have introduced the optimization of a gain functional, to optimize the number of prey and predators at, or during, a given time. This approach leads to adding an adjoint equation to the Lotka-Volterra system of equations (Bellman's equation [START_REF] Bellman | Dynamic Programming[END_REF]).

Since there is a close correlation between economic and human population growth [START_REF] Nielsen | Scientifically unacceptable established knowledge in demography and in economic research[END_REF], application to human populations arises in demography and economics. In this context, the Lokta-Volterra equations therefore describe the demography of human populations [START_REF] Basener | Dynamics of a discrete population model for extinction and sustainability in ancient civilizations[END_REF]. The same type of equations is suggested to model a human population and a resource is given in [START_REF] Anderies | On modeling human behavior and institutions in simple ecological economic systems[END_REF][START_REF] Anderies | Economic development, demographics, and renewable resources: a dynamical systems approach[END_REF][START_REF] Basener | Dynamics of a discrete population model for extinction and sustainability in ancient civilizations[END_REF][START_REF]Evolution of population-resource dynamics models[END_REF][START_REF] Roman | The dynamics of humanenvironment interactions in the collapse of the classic maya[END_REF][START_REF] Uehara | Integrating economics and systemdynamics approaches for modelling an ecological-economic system[END_REF] (close to Lotka-Volterra model). There has been much recent activity in the modeling of human civilizations using this formalism. In [START_REF] Brander | The simple economics of easster island: A ricardo-malthus model of renewable resource use[END_REF], Brander and Taylor study the collapse of Easter Island. Robert Axtel, in [START_REF] Axtell | Population growth and collapse in a multiagent model of the kayenta anasazi in long house valley[END_REF], uses these models to study the collapse of the Kayenta Anasazi civilization. In [START_REF] Roman | The dynamics of humanenvironment interactions in the collapse of the classic maya[END_REF], authors propose a model to study the dynamics of Human/Environment interactions in the collapse of classic Maya. All of these models consider that the population undergoes a lack of resources without birth control (or with a fixed birth control that does not anticipate the lack of resources). In [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF], the resources used are modeled by the food ratio which is a function of the size of the population and could be considered as a steady state of the resource equation.

In these models, individuals adapt their fertility rate to the current level of resources which is implicitly modeled by the choice of the birth/fertility rate and its variations with respect to the level of resources. Nevertheless, they do not anticipate their behavior with future variations in the amount of resources. Along with the question of population growth and its impact on the environment, arise the question of birth control and the anticipation of the fertility rate in the ecological debate [START_REF] Obaid | Footprints and milestones: Population and environmental change. The state of World Population[END_REF]. More precisely, population growth is seen as the main detriment to the environment: some two billion people already lack food security and water supplies, and agricultural land is under increasing pressure. These figures alarm many people, who take it for granted that population growth will imply famine, economic backwardness, more pollution and a faster depletion of natural resources in the world [START_REF] Collins | Population growth the scapegoat? rethinking the neo-malthusian debate[END_REF]. Consequently, the child can be considered as a cost for the environment (environmental cost1 ) and the expected difficulties of life of the child in the future (due to the environmental problems) could be a reason for parents not to have children (personal cost). Both of these costs decrease with the level of resources and the value ( ) of having a child depends on the predictions (see figure 1) on the future states of the resources which is given by the dynamic programming via Bellman equations [START_REF] Bellman | Dynamic Programming[END_REF][START_REF] Stukalin | Some applications of optimal control in sustainable fishing in the baltic sea[END_REF] (and, for instance, in [START_REF] Flaig | Canonical modelling of anticipatory vaccination behavior and long term epidemic recurrence[END_REF][START_REF] Flaig | Cost effectiveness and policy announcement: The case of measles mandatory vaccination[END_REF] in epidemiological models).

In this paper, we develop a class of toy models to take into account the value of having a child due to ressource level assessment, using backward/forward ordinary differential systems close to those developed in [START_REF] Flaig | Canonical modelling of anticipatory vaccination behavior and long term epidemic recurrence[END_REF][START_REF] Flaig | Cost effectiveness and policy announcement: The case of measles mandatory vaccination[END_REF][START_REF] Stanková | Joining or opting out of a lotka-volterra game between predators and prey: does the best strategy depend on modelling energy lost[END_REF][START_REF] Stukalin | Some applications of optimal control in sustainable fishing in the baltic sea[END_REF]. These models are developed in Section 2. In Section 3, we compare the behavior of a population which adapts its birth rate to the current level of resources and of the same population which anticipates the variation of resources (here food ratio). Finally, we conclude our work. 2

Population Change in Time with Anticipation and Adaptive Behavior

In the classical Lotka-Volterra equations [START_REF] Volterra | Variations and fluctuations of the number of individuals in animal species living together[END_REF], the level of resources (represented by ( ) at time ) and the level of consumers (represented by ( ) at time ) satisfy the following Ordinary Differential Equations, for ≥ 0,

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ( ) = ( ( ), ( )) ( ) ( ) = ( ( )) -( ( ), ( )) (0) = 0 ≥ 0, (0) = 0 ≥ 0 , ( 1
)
where is the functional gain in the absence of consumer, the function governing the consumption of resources and the growth function (birth minus death) in the population of consumers as a function of resources [START_REF] Getz | Consumer-resource dynamics: quantity, quality, and allocation[END_REF][START_REF] Basener | Dynamics of a discrete population model for extinction and sustainability in ancient civilizations[END_REF]. The growth rate could be decomposed into a birth rate and a death rate (see [START_REF] Terry | A predator-prey model with generic birth and death rates for the predator[END_REF]) and we focus on (1) of the form

⎧ ⎪ ⎨ ⎪ ⎩ ( ) = ( ( )) -( ( )) ( ) ( ) = ( ( )) -( ( ), ( )) , (2)
with the initial data (0) = 0 ≥ 0, (0) = 0 ≥ 0. In [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF], the authors use a discrete time evolution equation for the population level equation and a resource level (food ratio) which is expressed directly as a function of the population level. This could be understood as if the resources in (1) are in a stable steady state, i.e., ( ) = 0, ( ( )) = ( ( ), ( )), with and define, not only, as ( ) = ( -) and

( , ) = (1 -(1-- ) )
where and are two constants, i.e., = (1-- ) (see [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF] for the equation of the food ratio). This behavior can be approximated in (1) by taking

( ) = 1 [ ( ( )) -( ( ), ( ))],
with ≪ 1 (corresponding to a time scale difference : slow variation for population and rapid variation for resources (agriculture)) and corresponds to the following system of coupled population/resource time evolution equations

⎧ ⎪ ⎨ ⎪ ⎩ ( ) = ( ( )) -( ( )) ( ) 0 = ( ( )) -( ( ), ( )) . (3)
In Section 2.1, we complete the population and resources co-evolution model with equations describing decision-making process of individuals to have children.

Construction of Anticipation Models

At time , the size of the population ( ) and the level of resources ( ) are driven by a time evolution equation such as (2) (resp. (3)). Now we consider that the decision to give birth is assumed to be based on a rational and far-sighted cost-benefit analysis on what the future will be. If the parents anticipate that, in the near future, there will be war or famine (due to overpopulation) they are not having children at present (for their own safety and for the environment). On the other hand, if the parents anticipate that the future will be safe, they can choose to have a child (see figure 1). Let be a probability/wish to have a child, then (2), becomes

⎧ ⎪ ⎨ ⎪ ⎩ ( ) = ( ) -( ( )) ( ) ( ) = ( ( )) -( ( ), ( )) , ( 4
)
where is the maximum birth rate. We notice that (3), respectively, becomes

⎧ ⎪ ⎨ ⎪ ⎩ ( ) = ( ) -( ( )) ( ) 0 = ( ( )) -( ( ), ( ))
.

(

) 5
If we follow [START_REF] Stanková | Joining or opting out of a lotka-volterra game between predators and prey: does the best strategy depend on modelling energy lost[END_REF][START_REF] Stukalin | Some applications of optimal control in sustainable fishing in the baltic sea[END_REF], in a certain sense, by choosing this probability of having a child, the parents want to maximize a gain functional of the Bolza type (from current time 0 to time where is large enough)

∫ 0 [ ( ( )) ( ) - ( ) ( )] - , ( 6
)
where is a utility function depending on the resource level , is a constant cost of having a child (related to the time (or money) parents have to spend raising a child : one of the reasons why some adults do not want children [START_REF] Gillespie | Childfree and feminine: Understanding the gender identity of voluntarily childless women[END_REF]) 3 and is a time discount factor (simplifying assumption modeling the forgetting process). This approach was used in [START_REF]Large-Scale Scientific Computing[END_REF] p277 to optimize the number of predators and prey at a given time by choosing the best hunting rate function of predators. The gain (or cost function) is similar to those proposed in [START_REF] Stanková | Joining or opting out of a lotka-volterra game between predators and prey: does the best strategy depend on modelling energy lost[END_REF][START_REF] Stukalin | Some applications of optimal control in sustainable fishing in the baltic sea[END_REF] for the fishing optimization problem and in [START_REF] Flaig | Canonical modelling of anticipatory vaccination behavior and long term epidemic recurrence[END_REF][START_REF] Flaig | Cost effectiveness and policy announcement: The case of measles mandatory vaccination[END_REF] for rational far-sighted cost-benefit analysis of vaccination in the case of epidemiology. Moreover in [START_REF] Stukalin | Some applications of optimal control in sustainable fishing in the baltic sea[END_REF], the authors show how to obtain the adjoint equation (Bellman's equation [START_REF] Bellman | Dynamic Programming[END_REF]) to solve the optimal control problem with a gain function close to [START_REF] Brander | The simple economics of easster island: A ricardo-malthus model of renewable resource use[END_REF]. More precisely, optimized gain function

̃ ( , ) = max ∫ [ ( ( )) - ( )] ( ) -( -) ,
where 0 ≤ ( ) ≤ 1 for all , satisfies the time backward equation ( [START_REF] Stukalin | Some applications of optimal control in sustainable fishing in the baltic sea[END_REF][START_REF] Bellman | Dynamic Programming[END_REF])

- ̃ ( , ) = -̃ + max( ̃ -, 0) + ( ( )) ( ) -( ( )) ( ) ̃ ,
and

( ) = ( ̃ -) = ⎧ ⎪ ⎨ ⎪ ⎩ 1, ̃ -> 0 0, ̃ -< 0 .
Therefore, by setting,

̃ ( , ) = ( ) ,
we have which satisfies

- = ( ( )) + ( )( -) -( + ( ( ))) . (7)
Hence, the value ( ) (at time ) that the individual expects to have a child, depends on the variation of the level of resources and the immediate cost of raising a child. Note that the value of having a child depends on predictions about future resource states. In order to ease numerical methods and yet certainly as a realistic assumption, we use the concept of smoothed best response [START_REF]The Theory of Learning in Games[END_REF] that uses logistic functions, i.e.

↦ 1 (1 + -) ,
which is a smooth approximation of the Heaviside step function (with which parametrizes the slope of the function at the origin). Moreover we choose the following form for the utility function (exponential)

∶ ↦ 1 --+ , 3 Therefore ( ( )) - ( ) is the individual's gain function ( ) = + d d d d d d d d d 1 - 0 1 - 0 ( ) ( + ) ( + ) ⏟⏞⏞ ⏟⏞⏞ ⏟ ℎ + ( ( + ) -) ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ 1 -( )
Figure 2: Probability tree for individuals for Equation 7. During a step time at time , we evaluate ( ) as ( ( )) (the utility of the resources during time ) plus a value, depending on events happening to individuals and the value at time + :

-when death happens (with probability ) the value is zero -when forgetting (with probability ) the value is zero -if the parents do not have child (with probability 1 -) the value is ( + ) -if the parents want a child and have one (with probability ) the value is ( + ) + ( ( + ) -) (value to the child and value to parents minus the cost of raising the child). Therefore, to evaluate the value at time , we compute the mean of the value, which depends on the value at time + :

( ) = ( ( + )) + ( )( ( + ) -) + (1 -( + ) ) ( + ) + ( 2 ),
therefore by dividing by and passing to the limit in to zero we find [START_REF]Interest rate models-theory and practice[END_REF].

where and are reals. The equations describing the decision making process by individuals are given in the Equation 7and can be understood using the decision trees [START_REF] Zwanziger | Evaluating the benefits of increasing measles immunization rates[END_REF][START_REF] Hinman | An economic analysis of the current universal 2-dose measlesmumps-rubella vaccination program in the united states[END_REF] which is a list of scenarios in which an individual is confronted with their probabilities, and individual and societal costs.

We modify the value equation by adding an altruism parameter which models a biased assessment of the value for the child when the decision to have a child is made 4 (see figure 2 for the change in probability tree and figure 6 to see the variation in birth rate due to this parameter : when resources are low (resp. high) the birth rate can be higher or lower (resp. lower or higher) from one population to another) and the value when finally, the value function ( ), for individual, from 'having a child' at time follows the time backward equation

- ( ) = ( ( )) + ( )( ( ) -) -( ( ( )) + ) ( ), ( ) = , ( 8
)
where is an altruism parameter (we add to the value when having the child), the cost of having a child, ) the value is zero -when forgetting (with probability ) the value is zero -if the parents do not have child (with probability 1 -) the value is ( + ) -if the parents want a child and have one (with probability ) the value is ( + ) + ( ( + ) -) (biased value to the child and value to parents minus the cost to raise the child). Therefore, to evaluate the value at time , we compute the mean of the value, which depends on the value at time + :

Model of Neo-Malthusian Population Anticipating Future Changes in Resources

( ) = + d d d d d d d d d 1 - 0 1 - 0 ( ) ( + ) ( + ) ⏟⏞⏞⏞⏟⏞⏞⏞⏟ ℎ + ( ( + ) -) ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ 1 -( )
( ) = ( ( + )) + ( )( ( + ) -) + (1 -( + ) ) ( + ) + ( 2 ),
therefore by dividing by and passing to the limit in to zero we find [START_REF]Theory of Ordinary Differential Equations[END_REF].

the time discount factor, a utility function for available resources and

( ) = 1 (1 + -( ( )-) ) , (9)
with ∈]0, ∞[ the slope of the sigmoid and ∈]0, ∞[ the maximal number of birth rate. Therefore, we can compute the individual and societal costs of a single case of 'having a child' as the net present value over lifetime of the expected cost of all scenarios (see figure 3). The complete model of anticipation for (4) (resp. ( 5)) is then given by the coupled system of equations

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ( ) = ( ) -( ( )) ( ) ( ) = ( ( )) -( ( ), ( )) - ( ) = ( ( )) + ( )( ( ) -) -( ( ( )) + ) ( ) , (10)
(resp. 0 = ( ( )) -( ( ), ( )) for the resources equation in the adaptation model of ( 5)) with initial data (0) = 0 , (0) = 0 ≥ 0 and final data ( ) = (mathematical results are in Appendix A). To model the change in the case where individuals adopt an adaptive behavior, i.e. do not anticipate the variation of the level of resources at an aggregate level, even if they evaluate the resources for their own survival, we set the value equation at its equilibrium ( ( ) = 0) :

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ( ) = ( ) -( ( )) ( ) ( ) = ( ( )) -( ( ), ( ) 0 = ( ( )) + ( )( ( ) -) -( ( ( )) + ) ( )
, [START_REF] Flaig | Canonical modelling of anticipatory vaccination behavior and long term epidemic recurrence[END_REF] with (0 = ( ( )) -( ( ), ( )) for the resources equation to the adaptive behavior of ( 5))

( ) = 1 (1 + -( ( )-) ) , ( 12
)
and initial data (0) = 0 , (0) = 0 ≥ 0 (here dynamic is only time forward).

Discussion and Parameters Estimation

In this Section, we discuss the model and its parameters. The probability tree given in figure 3 shows that the first time event of 'having' a child depends on . We give its law in Section 2.2.1 and we give some examples when is fixed. Then we show, in the Sections 2.2.2-2.2.3, we explain how to understand and estimate parameters. In the Section 2.2.4, we discuss about the model and its limits.

Choice and its Related Law

The personal choice to have a child, during [ , + ], is modeled by ( ) which represents the probability of choosing to have a child (during [ , + ]). Therefore, the first time the event 'have a child' occurs follows a Cox process ([7] p763) of intensity ∫ 0 ( ) , and its density is given by

( ) = 0, ≤ 0 ( ) -∫ 0 ( ) ,
which is, when is constant over time, nothing more than an exponential law. In order to understand it in demographic terms [START_REF] Hayford | Spacing, stopping, or postponing? fertility desires in a sub-saharan setting[END_REF], we give two examples, when is constant over time and when is a Heaviside step function .

1. Case = 0 means to stop or never have children and case ( ) = 0 > 0, for all , means that time between two events 'having a child' is, on average, 1∕( 0 ) and so decay of 0 involves spacing two events 'having a child'. 2. When is an Heaviside step function

( ) = 0, ≤ 0 , > ,
then the first time the event 'have a child' happens follows an exponential law of parameter 0 translated, i.e.

postponed, of time .

Time Discount Factor and its Estimation

First, we study the dynamics of in the case of constant resources (which is valid in a short time analysis), i.e.

̄ ,

= ̄ ≥ 0, and so satisfies

- ( ) = ̄ + ( )( ( ) -) -( ̄ + ) ( ).
Under assumption ≫ and ̄ , we have that and so, the probability of having a child, converges to an equilibrium in a long time and with

∼ 1 1 + -( ̄ -)
, and so , , and are linked with the probability of having a child in a long time

( ̄ -) ∼ log( 1-) . ( 13
)
The value after having a child could be measured by the probability of wanting a child immediatly after having a child (figure 1 [START_REF] Hayford | Spacing, stopping, or postponing? fertility desires in a sub-saharan setting[END_REF]) : 0 . The dynamics of when ( = 0) = 0 is given by

( ) ∼ 0 ( (1-0 ) (1-) 0 ) 1-- 1 -0 + 0 ( (1-0 ) (1-) 0 ) 1-- ,
and so, at time 1∕2 such that

log( 1 - )( 1∕2 ) = [log( 0 1 -0 ) + log( 1 - )]∕2, we have -1∕2 = 1∕2, i.e., ∼ log (2)
1∕2

. The time discount factor can be related to the time necessary to reach half of the equilibrium (of log( 1-)) in growth, i.e., in a sense, 'forgeting' the cost of having a child. For instance, in [START_REF] Hayford | Spacing, stopping, or postponing? fertility desires in a sub-saharan setting[END_REF], the authors study the desire of women (in Sub-Saharan) childbearing and by identifying, in figure 1 [START_REF] Hayford | Spacing, stopping, or postponing? fertility desires in a sub-saharan setting[END_REF], the probability that a woman want to delay childbearing, as 1 -we can plot the ↦ log(

Adaptive Equation and its link with Classical Time Evolution Equations

Since individuals evaluate the resources for their own survival : the time evolution Equation ( 2) is, in fact, the adaptive Equation ( 11)- [START_REF] Flaig | Cost effectiveness and policy announcement: The case of measles mandatory vaccination[END_REF]. By identifying ( 11)-( 12) and (2) we have is equal to the birth rate function ↦ ( ), i.e. for well-chosen parameters ( , , , and function ) we should have for all : solution to

⎧ ⎪ ⎨ ⎪ ⎩ 0 = ( ) + ( -) -( ( ) + ) = 1 (1+ -( -) )
, (

) 14
satisfies = ( ). Therefore, ( 2) and ( 11)-( 12) have the same solutions.

Discussion

Since we do not model the variation of age in (1), it is impossible to model the change of childbearing with respect to age nor the 'peak' around year three in figure 4. To cope with these two effects, it is necessary to introduce age of individuals and age of the last birth. It might be possible to introduce age structure using that models continuous aging of population or a discrete age model (with a vector of population such as [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF]) for the population time evolution equation and adapt the value equation (time backward) with these age structure. Nevertheless, this could increase complexity of the model and make it less readable. The choice of a linear relationship between the value given to the child and the value, through the 'altruism' parameter, is done to add more complex behavior without being too complex : this could be improved in the future with a more complex relationship.

Adaptive Versus Anticipation Over an Example

In this Section, we adapt a model given in [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF] of time evolution of the population with food consumption (Section 3.1) where the size of the population is represented by a vector structured by age with a discrete time to a continuous time model with a single age group. Then, in Section 3.2, we give the anticipation model and compare with the dynamics of the adaptive model. This is an application of the construction of adaptive and anticipation model from equation type (3)5 .

Basic Model : an Adaptive Behavior

At time the size of the population is given by ( ), the consumption is represented by the food ratio ( ) and is computed by the food produced over the food consumed :

( ) = ( ) (1 -- ( ) ̂ ∕ ) ( ) ̂ ,
where is the maximum area in active cultivation, the hours worked per day, ̂ part of the population that produces food, a constant that converts hour of labour to area, the caloric yield per area cultivated and the calorie requirement (per individual) modulated by a consumption rate (variable with age of individuals) ̂ (see [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF] for more details). The survival probability of the population depends on the food ratio ∶ ↦ ( ), that we take as a piecewise linear function close to the curves given in figure 3 [28] (ages 5 or 65) and we see in figure 5, the death rate, i.e.

( ) = 1 -( ).

The fertility rate, named in [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF], is also an increasing function of the food ratio

∶ ↦ ( ) ∈ [0, .135].
We notice that the birth rate (and the death rate) is depending on food ratio, i.e., the birth rate is 'adapted' to the level of this variable. The food ratio depends on the maximum area cultivated and a sudden change in this value could be related to one of the expected difficulties predicted by neo-malthusianists : the depletion of natural resources in the world [START_REF] Collins | Population growth the scapegoat? rethinking the neo-malthusian debate[END_REF]. Finally, a low food ratio is an indicator of a famine or an overpopulation. Therefore, we choose this variable as the key variable, for parents, to decide whether or not to have a child in order to optimize the well-being of the population (particularly their child) in the future.

The size of the population follows the main ordinary differential equation

⎧ ⎪ ⎨ ⎪ ⎩ ( ) = ( ( )) -( ( )) ( ), ( ) = (1-- ( ) ̂ ∕ ) ( ) ̂ ∕ . ( 15
)
Since individuals value resources for their own survival : the time evolution Equation ( 15) is, in fact, the adaptive equation : the fertility rate ↦ ( ( )) is computed as ↦ .135 ( ) where .135 is the maximal fertility rate and is the probability/wish to have a child (adaptative behavior)

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ( ) = .135 ( ) -( ( )) ( ) ( ) = (1-- ( ) ̂ ∕ ) ( ) ̂ ∕ 0 = ( ( )) + .135 ( )( ( ) -) -( ( ( )) + ) ( ) , ( 16
)
Figure 5: Up : In blue the curve of death rate, i.e. one minus the survival probability, given in [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF]. Down : Plot of ↦ ( ) (plain line, is the name of the fertility rate in [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF]) and .135 (dashed line) solution to [START_REF] Hinman | An economic analysis of the current universal 2-dose measlesmumps-rubella vaccination program in the united states[END_REF] with parameters given in table 1.

with

( ) = 1 (1 + -( ( )-) ) , ( 17
)
and so by identifying ( 16)-( 17) and ( 15) we choose parameters (see table 1) to have, for all ∈ [0, 1] : .135 is close to ( ) where satisfies

⎧ ⎪ ⎨ ⎪ ⎩ 0 = ( ) + .135 ( -) -( ( ) + ) = 1 (1+ -( -) ) . ( 18
)
For well chosen parameters (see table 1) we have .135 close to given in figure 4. [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF] (see figure 5).

Remark on Parameters and Sensitivity of Fertility Rate to Personal Cost and Altruism

The evaluation of the cost parameter , the altruism factor and the parameters of the utility function is not easy and depends on countries, age, social category, religion [START_REF] Pearce | Explaining religious differentials in family size preferences: Evidence from nepal in 1996[END_REF]... We notice that childfree movement [START_REF] Gillespie | Childfree and feminine: Understanding the gender identity of voluntarily childless women[END_REF], claiming that having children is not costless in particular for women who lose

Model of Neo-Malthusian Population Anticipating Future Changes in Resources

Figure 6: Variation of 0.135 ↦ 0.135 ( ) solution to [START_REF] Hinman | An economic analysis of the current universal 2-dose measlesmumps-rubella vaccination program in the united states[END_REF] with respect to and . Down, we set parameters in table 1 (except ) and we plot ↦ .135 ( ) for = 0.001 → 0.101. The birth rate is decreasing with respect to . Up, we set parameters in table 1 (except ) and we plot ↦ .135 ( ) for = 0.0319 → 0.2867.

their freedom and career to raise their(s) child(ren) [START_REF] Maier | No kid: quarante raisons de ne pas avoir d'enfant[END_REF], surely have an impact on the parameters of cost and altruism.

We use a genetic algorithm to find parameters such that

.135 is sufficiently close to the fertility rate given in [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF], i.e. a prehistoric population and therefore, these parameters must be re-evaluated for another fertility rate data (see figure 5). In order to see the sensitivity of the fertility rate to these parameters, we plot, in figure 6, the variation of .135 solution to [START_REF] Hinman | An economic analysis of the current universal 2-dose measlesmumps-rubella vaccination program in the united states[END_REF] with respect to the personal cost (with other parameters fixed in table 1) and the altruism factor (with other parameters fixed in table 1). We notice that, as the personal cost increases, .135 decreases and therefore the equilibrium ̂ also increases. The variation with respect to the altruism factor is more complicated. Note that the shape of .135 changes from nearly linear to sigmoid and is increasing with respect to when is large enough ( > ̄ , with ( ̄ ) = 0) and decreasing with respect to when is small enough ( < ̄ ). Since the death rate function

↦ (1 -( ))
intersects the curves in the first case ( large enough), ̂ decreases with respect to .

First Simulation

We have directly that ( ( ), ( )) reaches a stable equilibrium when goes to infinity (due to the growth of + 16)- [START_REF] Hayford | Spacing, stopping, or postponing? fertility desires in a sub-saharan setting[END_REF].

with respect to and decreasing of with respect to ), and this equilibrium ( ̂ , ̂ ) satisfies

( ̂ ) = ( ̂ ), ̂ = (1 --̂ ̂ ∕ ) ̂ ̂ ∕ ,
which is (0.5606; 9786) for the parameters given in the table 1 (same as parameters give in table 1 [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF]) (see figure 7 for a numerical simulation of ( 16)-( 17). The equilibrium found in [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF] is (0.680; 4752). The difference with [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF] is due to the approximation of the death rate, fertility rate and loss of age structure : indeed, by taking only one age group, we choose a fertility rate of one age group that overestimates the fertility rate of the whole population, the same goes for the death rate which underestimates the death rate of the whole population.

To find an equilibrium close to that given in [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF], it might be possible to modify average effective workers/person parameter ̂ and average age structure weighted consumption ̂ to take into account the simplification to an age class. However we have chosen to keep parameters as given in [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF], to be more consistent with the original model and its functions definitions.

The Anticipation Model and Simulations

Now we add the anticipation, following [START_REF]Mathematical Analysis and Numerical Methods for Science and Technology: Volume 3 Spectral Theory and Applications[END_REF], with the backward equation of the value and ( ) = 1 1+ -( ( )-) with parameters given in the table 1. We give three numerical simulations to analyze the difference in behavior between an adaptive and an anticipating individual. In the first case ('nothing change'), we compare both models when there is no change in the future : no change in the agriculture parameter such as the area of arable land, the agricultural potential... and no birth rate policy. In the second case ('agriculture catastrophe'), we see what is changed by announcing that at a given time the total arable land has its area will be divided by two. In the third case ('birth control policy'), we see the effect of announcing of a birth control policy that would be modeled by fixing the birth rate at a certain value at a given time.

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ( ) = .135 ( ) -( ( )) ( ) ( ) = ( ) (1-- ( ) ̂ ∕ ) ( ) ̂ - ( ) = ( ( )) + .135 ( )( ( ) -) -( ( ( )) + ) ( ) , (19)

Case 1. Nothing Change

Since the adaptive model corresponds to an equilibrium for the value equation in the anticipation model, equilibria of the adaptive model ( 16) are the same as the anticipation model [START_REF]Fixed Point Theory[END_REF]. The stability of steady states can be studied as in classical (forward) way by keeping the backward/forward structure.

Theorem 1. (Stability) The only steady state of (16) (resp. (19)) : ( ̂ , ̂ ) is locally stable for both model.

The proof is direct for the adaptive model (due to the growth of + with respect to and the decay of with respect to

). For the adaptation model, we have to deal with the Jacobian at ( ̂ , ̂ ) = 0 12 [START_REF]Large-Scale Scientific Computing[END_REF]

22

, with 12 = .135 > 0, 21 = ( ′ ( )+ ′ ( ) ) < 0 and 22 = .135 ( ( ( ) -)) -(1 -( ) + ) < 0 (by numerical computation). Since ( ) > 0 and ( ) < 0, eigenvalues are negative and the steady state is locally asymptotically stable stable. In blue (plain line) we see time evolution of the population that adapts its behavior and in red (dashed line) time evolution of the population that anticipates. There is a small difference on the fertility rate due to the adaptation to the variation of the food ratio . Population, food ratio and death rate are similar.

Therefore, we do not expect much difference between the two models when the resource condition (here, for instance, : area of arable land) does not change (see figure 8) during of temporal evolution.

Case 2. Agriculture Catastrophe

Nevertheless, differences between anticipation and adaptative behavior appear when adding information such as ecological catastrophe (modeled here by a decay in very short time of the total area of arable land).

To illustrate a difference in behavior, we assume that the area of arable land is divided by two at a given time : = 1000, ∈ [0, 500] 500, ≥ 500 , where = 500 is chosen to compare behavior of adaptive and adaptation when population has reached its equilibrium before area change happens. The information of area decay is known by the population. We see, in figure 9, that the food ratio only changes for a relatively short time (time 500 to time 510) due to the short evolution over time of the area of arable land and the high death rate during this period to reach a new steady state in population and food ratio.

Note that in the adaptive model (the classical model), individuals do not change their birth rate before the loss of area of arable land = 500.

In the adaptation model, we see, in figure 9, that after time 490 (therefore before any change of the arable land area), the birth rate decreases due to the anticipation of the decay of the food ratio . True or not, the information of the decay of the area of arable land has an impact on the behavior on the fertility rate and therefore on the dynamics Model of Neo-Malthusian Population Anticipating Future Changes in Resources Figure 9: Anticipation VS Adaptation case 2. In blue (plain line) we see time evolution of the population that adapts its behavior and in red (dashed line) time evolution of the population that anticipates. In the four up plots : we show simulations on time interval 0..1000. Note the effect of the change of area at time = 500 : the population changes from an equilibrium to an another with a large death rate and a decay of fertility. In the four down plots we magnify the change around time = 500 in order to observe more precisely the difference of behavior.

of the population before the moment it could (or not) occur (neo-Malthusian behavior).

Case 3. Birth Control Policy

Here we illustrate an another difference in behavior in the case of a birth control announcement that would be applied after a given time.

We assume that at time = 470 is announced that the birth rate will be limited to 0.0420 (which corresponds to half of the case 1. at equilibrium) thirty years after the announcement, i.e. at time = 500.

We observe, in the figure 10, that between time = 470 and = 500 the birth rate increases due to the fact that individuals anticipate an increase in the food ratio under the effect of limitation of the birth rate after time = 500. Consequently, the effect of announcing a birth control policy to a population that anticipates change of resources may have a counter effect just before the application of the policy.

Conclusion

In this work we have given a construction scheme to model the anticipation behavior in a dynamic population and resource model. This construction leads to a coupling of a forward-in-time system of equations which models the interaction between population and resources and a backwardin-time equation which models the valuation of the value of childbearing. On an example using a simple dynamic on the time evolution of population, we show the construction of the anticipation model and we apply it on different scenarios. In a catastrophic scenario of a loss of arable area (announcement), we see that an adaptive population, i.e., that evaluates the value but does not anticipate its variations, does not adapt their fertility rate to the loss. Whereas in the anticipation model, the evolution of the fertility rate shows an anticipation of the loss. Since, the coupling is not classic in analysis we have given general tools to prove the existence, the uniqueness and the numerical analysis in appendix. To go further, it should be interesting to adapt the construction to an age structured model such as the model given in [START_REF] Puleston | Population and prehistory ii: Space-limited human populations in constant environments[END_REF] (or a partial differential equation model) and to adjust parameters to more recent population fertility rate. Moreover, we consider that and are constant in time, an inflation of these parameters could have a non-zero impact on the evolution of the birth rate over time.

A. Appendix

There are several and interesting mathematical problems arising in [START_REF]Mathematical Analysis and Numerical Methods for Science and Technology: Volume 3 Spectral Theory and Applications[END_REF]. Due to the coupling of time forward (for and ) and time backward (for ), the existence and uniqueness of a solution to [START_REF]Mathematical Analysis and Numerical Methods for Science and Technology: Volume 3 Spectral Theory and Applications[END_REF] is not a trivial application of the Cauchy-Lipshitz theorem. Moreover, numerical approximation scheme of this problem is not easy. Trying to inverse time in the third equation (in ) and using a shooting method to find (0) such that ( ) = fails due to the fact that the equation in is unstable for positive time, i.e. a small variation of (0) implies a large variation of ( ). A direct Banach-Picard fixed point algorithm fails also. In this Section, we first give results on the existence and the uniqueness of solution. Moreover, we explain the reason of the convergence of a relaxed Banach-Picard scheme to the solution (the method has been used in previous works [START_REF] Flaig | Canonical modelling of anticipatory vaccination behavior and long term epidemic recurrence[END_REF][START_REF] Flaig | Cost effectiveness and policy announcement: The case of measles mandatory vaccination[END_REF] but not explained).

A.1. Mathematical Results

In this Section, we give two main results. The first theorem concerns the existence and uniqueness of a solution to [START_REF]Mathematical Analysis and Numerical Methods for Science and Technology: Volume 3 Spectral Theory and Applications[END_REF] and the second gives an approximation scheme which converges to the solution. In both theorems we use the function defined as follows

∶ ( , ) ∈ ([0, ], [0, 1] 2 ) ↦ ( , ) ∈ ([0, ], [0, 1] 2 ), (20) with ∶ ( , ̃ ) ↦ ( 1 (1 + -( -) ) , ) (21)
where ( , ) is solution to -( ( ( )) + ) ( ), [START_REF] Maier | No kid: quarante raisons de ne pas avoir d'enfant[END_REF] with initial data (0) = 0 ∈]0, ∞[, (0) = 0 ∈ ]0, 1], ( ) = .

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ( ) = ( ) -( ̃ ( )) ( ), ≥ 0
Theorem 2. Assuming that

, ∈ 1 (ℝ) ∈ 1 (ℝ 2 ), (23) with
(1) = (0) = ( , 0) = 0, (0) > > (1), [START_REF] Nielsen | Scientifically unacceptable established knowledge in demography and in economic research[END_REF] and

′ < 0 < 0. ( 25
)
Then, there exists an unique solution to [START_REF]Mathematical Analysis and Numerical Methods for Science and Technology: Volume 3 Spectral Theory and Applications[END_REF]. 6 We approach this solution can be by the following algorithm : let 0 = 0, and define by induction

+1 = (1 -) + ( ), (26)
with ∈]0, 1[ small enough. More precisely, we have the following convergence result.

Theorem 3. Under assumptions ( 23)-( 25) then = ( , ) the solution to ( 10) is a locally asymptotically stable state to the equation where is the time step, i.e., the relaxed algorithm [START_REF] Van Opheusden | Competition for resources: complicated dynamics in the simple tilman model[END_REF]. An another scheme, given by a semi-implicit Euler approximation of ( 27) could be +1 ( ) = ( ( ) + ( ( , .))( ))∕(1 + ), where is the time step.

A.2. Proof of Existence and Uniqueness

We first give the existence result which comes directly from the compactness of the operator and the Schauder fixed point theorem (see [START_REF]Fixed Point Theory[END_REF]). Then we prove uniqueness using the decay of and Ordinary Differential Equations tools (upper/lower solutions).

Existence. This is a direct application of Schauder fixed point theorem (see [START_REF]Fixed Point Theory[END_REF]). Indeed, for all ( , ̃ ) which belongs to ([0, ], [0, 1] 2 ), ( , ,

) is the solution to the Ordinary Differential system of Equations ( 22) which existence and uniqueness is a consequence to the Cauchy Lipschitz theorem (see [START_REF]Theory of Ordinary Differential Equations[END_REF]). Moreover ( , ,

) are regular functions, i.e. ( , ) ∈ 1 ([0, ], [0, 1]) and is a 1 ([0, ], [0, 1] 2 ). Now, using that

↦ 1 1 + -( -)
∈ 1 (ℝ, ℝ),

6 1 assumptions could be replaced by Lipschitz, ′ < 0 by decreasing and < 0 by decreasing with respect to .

The same computation holds for -and we find

Figure 1 :

 1 Figure 1: Population/Resource coupling in an anticipation dynamics. The population changes due to death and birth. Resources change due to their own dynamics and their consumption by the population. If the population anticipates a problem of resource in the future, from a neo-Malthusian point of view, it does not give birth and vice versa, if the resources are sufficiently important, it gives birth.

Figure 3 :

 3 Figure 3: Probability tree for individuals Equation 8. During a step time at time , we evaluate ( ) as ( ( )) (the utility of the resources during time ) plus a value, depending on events happening to individuals and the value at time + : -when death happens (with probability) the value is zero -when forgetting (with probability ) the value is zero -if the parents do not have child (with probability 1 -) the value is ( + ) -if the parents want a child and have one (with probability ) the value is ( + ) + ( ( + ) -) (biased value to the child and value to parents minus the cost to raise the child). Therefore, to evaluate the value at time , we compute the mean of the value, which depends on the value at time + :

Figure 4 :

 4 Figure 4: In blue : log( 1-) where 1is given by the probability than a woman want to delay childbearing in Fig1. [17]. In red, the contant line [log( 0 1-0 ) + log( 1-)]∕2 which cuts the blue line in different points (around 3 years and around 7 years).

Figure 7 :

 7 Figure 7: Numerical solution of (15) with parameter given in table 1. Time change of the population size ↦ ( ). Plain line : solution to Equation (15) and dashed line : solution to (16)-(17).

Figure 8 :

 8 Figure 8: Anticipation VS Adaptation case 1.In blue (plain line) we see time evolution of the population that adapts its behavior and in red (dashed line) time evolution of the population that anticipates. There is a small difference on the fertility rate due to the adaptation to the variation of the food ratio . Population, food ratio and death rate are similar.

Figure 10 :

 10 Figure 10: Anticipation of a birth control policy. In the four up plots : we show simulations on time interval 0..1000. Note the effect of the change of area at time = 500 : the population moves from one equilibrium to an another with a decrease in fertility (except a peak around time = 500). In the four down plots we magnify the change around time = 500 to observe the fertility and the behavior of anticipating population.

(Remark 1 .

 1 , ) = ( ( , .))( )-( , ), ∀ ∈ [0, ], ≥ 0. (27)Indeed, an Euler approximation of (27) is given by+1 ( ) = ( ) + [ ( ( , .))( ) -( , )] = ( ( , .))( ) + (1 -) ( , ),

∫

  -( , ) ∫ 0 ( ′ ) ′ Φ( , ) = ∫ ( , ) ∫ 0 ( ′ ) ′ [ * ( -( ( , .))Φ( , .))- * (Φ( , .) -( ( , )))( )] . Now using that  = -( + ) and  * = -( * + ), we have∫ + ( , ) ∫ 0 ( ′ ) ′ Φ( , ) = ∫ ( , ) ∫ 0 ( ′ ) ′ [ * ( + ( ( , .))Φ( , .))- * (Φ( , .) + ( ( , )))( )] ,and∫ -( , ) ∫ 0 ( ′ ) ′ Φ( , ) = -∫ ( , ) ∫ 0 ( ′ ) ′ [ * ( -( ( , .))Φ( , .))- * (Φ( , .) -( ( , )))( )] .Therefore, using that | ( ,)| = + + -we obtain ∫ | ( , )| ∫ 0 ( ′ ) ′ Φ( , ) = -∫ ( , ) ∫ 0 ( ′ ) ′  * (Φ( , .)) + ∫ | ( , )| ∫ 0 ( ′ ) ′  * (Φ( , .)) .Using thatΦ + ( -1)Φ =  * Φ, we have ∫ | ( , )| ∫ 0 ( ′ ) ′ Φ( , ) = -( -1) ∫ ( , ) ∫ 0 ( ′ ) ′ Φ + ( -1) ∫ | ( , )| ∫ 0 ( ′ ) ′ Φ -∫ [ ( , ) -| ( , )|] ∫ 0 ( ′ ) ′ Φ ,and so we find∫ | ( , )| ∫ 0 ( ′ ) ′ Φ( , ) = -( -1) ∫ ( , ) ∫ 0 ( ′ ) ′ Φ + ( -1) ∫ | ( , )| ∫ 0 ( ′ ) ′ Φ + 2 ∫ -( , ) ∫ 0 ( ′ ) ′ Φ .

Table 1

 1 Parameters definition.

	Definition	Variable Value
	max. food agricultural potential total arable land labor contribution conversion from time to annual area cultivated Average effective workers/person Average age structure weighted consumption	̂ ̂	2, 785 Kcal/day 13, 100 Kcal/ha/day 1000 ha 5 hours/indiv./day 0.0944 ha-days/hour/yr 0.723 0.827
	parameters of the utility function slope of the sigmoid cost altruism factor discount time factor	( ; )	(15.5347; 3.4672) 3.1826 0.6102 0.1659 15∕100

environmental degradation, dwindling fisheries, shrinking forests, decreasing biodiversity

We give in Appendix A, mathematical proves of existence, uniqueness and numerical tools used to compute solutions.

We choose a linear evaluation : ℎ = .

J o u r n a l P r e -p r o o f

We give an example of construction of adaptive and anticipation model from equation type[START_REF] Anderies | Economic development, demographics, and renewable resources: a dynamical systems approach[END_REF] in appendix Section B

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Journal Pre-proof Model of Neo-Malthusian Population Anticipating Future Changes in Resources then ( , ) is bounded in 1 ([0, ], [0, 1] 2 ). Therefore is compact on the convex set ([0, ], [0, 1] 2 ) and so, using Schauder theorem (see [START_REF]Fixed Point Theory[END_REF]), there exists a fixed point to .

Uniqueness. Let ( 1 , 1 ) and ( 2 , 2 ) two solutions to the fixed point ( , ) = ( , ), i.e., solutions to [START_REF]Mathematical Analysis and Numerical Methods for Science and Technology: Volume 3 Spectral Theory and Applications[END_REF] then +( ( ( )) + ) ( ), with the initial data (0) = 0 ∈]0, ∞[, (0) = 0 ∈]0, 1], (0) = 0 and 0 = ( -log(1∕ (0) -1)∕ )∕ . If 0 1 = 0 2 , by Cauchy Lipschtiz theorem 1 = 2 . We suppose that 1 (0) > 2 (0) (and so 1 (0) > 2 (0)), then we have 1 (0) > 2 (0) and so

), in a neighborhood of = 0. Then, using assumption (25), we have 1 ( ) < 2 ( ), and so we have

which implies that 1 ( ) > 2 ( ) and so 1 ( ) > 2 ( ) in a neighborhood of 0. Therefore Ω = { ∈ [0, ] . . 1 ( ) > 2 ( ) ∀ ∈ [0, [} is a nonempty set. On Ω∕{0}, we have by comparison principle 1 > 2 and 1 < 2 and so Ω = Ω = [0, ] : which is impossible since 1 ( ) = 2 ( ) = and so 1 ( ) = 2 ( ). □

A.3. Proof of Convergence Result

The most difficult is to prove the convergence of the algorithm [START_REF] Van Opheusden | Competition for resources: complicated dynamics in the simple tilman model[END_REF]. Ideed, since is not a contractant operator, the Banach fixed point theorem (see [START_REF]Fixed Point Theory[END_REF]) cannot be used directly. Nevertheless, we can prove that is decreasing (antitone, see [START_REF] Sommariva | Computing positive fixed-points of decreasing hammerstein operators by relaxed iterations[END_REF]), its differential is a strongly negative (see [START_REF]Mathematical Analysis and Numerical Methods for Science and Technology: Volume 3 Spectral Theory and Applications[END_REF]) and compact operator and so we can construct a dynamical system which converges to the solution. The algorithm [START_REF] Van Opheusden | Competition for resources: complicated dynamics in the simple tilman model[END_REF] can be seen as a numerical approximation of this dynamical system.

By differentiation of = ( ) -with respect to we have directly that = ( ) -, where is defined by ( 21)- [START_REF] Maier | No kid: quarante raisons de ne pas avoir d'enfant[END_REF] and solution to [START_REF] Pearce | Explaining religious differentials in family size preferences: Evidence from nepal in 1996[END_REF] satisfies

We show that ( ,.) is a strongly negative (see [START_REF]Mathematical Analysis and Numerical Methods for Science and Technology: Volume 3 Spectral Theory and Applications[END_REF]) and compact operator (uniformly along a trajectory ). Lemma 4. Let defined by ( 21)-( 22) then ( ,.) an integral operator with the following properties :

1.

( ,.) ∶ ℎ ↦ ( ,.) (ℎ) is a linear, regular (continuous) and strongly negative operator, i.e., -( ,.) is strongly positive (see [START_REF]Mathematical Analysis and Numerical Methods for Science and Technology: Volume 3 Spectral Theory and Applications[END_REF]), 2.

∶ ℎ ∈ 0 ([0, ]×ℝ + ) ↦ ( ,.) (ℎ) ∈ 0 ([0, ]× ℝ + ) is a compact operator, 3. inf ∈{ ( ,.), ≥0}

, sup ∈{ ( ,.), ≥0} are linear, regular (continuous) and strongly negative operator.

And finally we prove that, for all dynamical system of this form (with a strongly negative (see [START_REF]Mathematical Analysis and Numerical Methods for Science and Technology: Volume 3 Spectral Theory and Applications[END_REF]) and compact operator), we have the convergence of the solution to 0 exponentially.

Lemma 5. Let ( , ) solution to

where  is a compact strongly positive operator on the function set 0 ([0, ], [0, 1] 2 ) then we have 1. the spectral radius of  , named ( ), is a single eigenvalue, there exists (., ) ≥ 0 eigenfunction associated to this eigenvalue 2. there are no other eigenvalue of modulus ( ) More precisely there exists ( , , ), with > 1, solution to

with  = -( + ) and  * its dual operator. Moreover, we have the convergence of to 0 and

Therefore we have proved that solution to [START_REF] Pearce | Explaining religious differentials in family size preferences: Evidence from nepal in 1996[END_REF] satisfies

Proof of Lemma 4 Let , and 1 , 2 then compute ( , , ), with the input functions ( , ) and ( , , ) with the input functions ( + 1 , + 2 ). We have, at first order,

Moreover, we have

with (0) = 0, i.e., by integration we have

and finally

Now, for , we have

with ( ) = 0 and ( ) ∶=

. Therefore, by integration, we obtain

. Finally, we find )]( )

which could be written, using Fubini, in the following form

Therefore, we have

with ,′ ) .

11 ( , ′ ) = ′ ( ( ′ )) ( )

. We notice that , < 0 for all , and using assumptions ( 23) and ( 25) we have that inf , ′ , , , ( , ′ ) < 0. Therefore, -is a strongly positive operator (and the same holds for sup and inf of -). Since is an integral operator with continuous with respect to and 1 with respect to ′ we have compactness of this operator on 0 ([0, ] × ℝ + ). □

Proof of Lemma 5

Using Krein Rutmann theorem [START_REF]Mathematical Analysis and Numerical Methods for Science and Technology: Volume 3 Spectral Theory and Applications[END_REF],

we have existence of ( , , ) solution to [START_REF] Roman | The dynamics of humanenvironment interactions in the collapse of the classic maya[END_REF]. By computation, we find

and so we have the conservation law

Let + ( , ) ∶= max( ( , ), 0) and -( , ) ∶= max(-( , ), 0) then we have

))Φ( , .))

Using [START_REF] Roman | The dynamics of humanenvironment interactions in the collapse of the classic maya[END_REF], we find

J o u r n a l P r e p r o f

Journal Pre-proof

Model of Neo-Malthusian Population Anticipating Future Changes in Resources (using Krein Rutmann), we have

and, using Gronwall inequality, we obtain

and ( 30) is satisfied. □

B. An Exemple of Predator -Prey Model with Anticipation

We follow here the work of J. Terry in [START_REF] Terry | A predator-prey model with generic birth and death rates for the predator[END_REF], in which, the growth rate of predators is decomposed into a birth rate and a death rate . At time , predators ( ) and preys ( ) follow the master system of equations

with (Holling type II) defined as ( , positive constants)

Paper gives as example,

and

with 1 , 2 , 1 , 2 , (maximal birth rate), (maximal death rate) and (minimal death rate) positive constants. Following the construction of Section 2.1, we have

and

, for the anticipation model and

and

, for the adaptive model. The author studies the existence, uniqueness and stability of steady states depending of values of 1 , 2 , 1 , 2 , , , , , , et . For 1 = 1 =

.05, 2 = 2 = .1250, = .15, = .0375, = .0125, = .0125, = 9, = .0250 et = .1, there exists a locally asymptotically stable non-trivial steady state. The birth and the death rates are given in figure 11, we notice that for parameters [ , , ℎ , , ] well chosen, the birth rate can be approximated by as in [START_REF]The Theory of Learning in Games[END_REF]. To compare both dynamics (adaptive versus adaptation), we simulate a sudden decrease in the growth rate of the prey : which is divided by two after time = 4500. In the figure 12, we see that adaptation involves less oscillations (in amplitude and more rapidly converging) than for the adaptive behavior.