András Szilágyi
email: and.szilagyi@gmail.com

Géza Meszéna
email: geza.meszena@elte.hu

Coexistence in a fluctuating environment by the effect of relative nonlinearity: A minimal model

Keywords: niche, limiting similarity, stochastic population dynamics

The minimal model of the "relative nonlinearity" type fluctuation-maintained coexistence is investigated. The competing populations are affected by an environmental white noise. With quadratic density dependence, the long-term growth rates of the populations are determined by the average and the variance of the (fluctuating) total density. At most two species can coexist on these two "regulating" variables; competitive exclusion would ensue in a constant environment. A numerical study of the expected time until extinction of any of the two species reveals that the criterion of mutual invasibility predicts the parameter-range of long-term coexistence correctly in the limit of zero extinction threshold. However, any extinction threshold consistent with a realistic population size will allow only short-term coexistence. Therefore, our simulations question the biological relevance of mutual invasibility, as a sufficient condition of coexistence, for large density fluctuations. We calculate the average and the variance of the fluctuating density of the coexisting populations analytically via the moment-closure approximation; the results are reasonably close to the simulated behavior. Based on this treatment, robustness of coexistence is studied in the limit of infinite population size. We interpret the results of this analysis in the context of necessity of niche segregation with respect to the regulating variables using a framework theory published earlier.

Introduction

The role of environmental fluctuations in maintenance of species diversity is one of the most frequently investigated topics in theoretical ecology.

The classical reference point is equilibrium coexistence theory. The principle of competitive exclusion [START_REF] Gause | The struggle for existence[END_REF][START_REF] Hardin | Competitive exclusion[END_REF] states that only the most successful species can survive in a given niche. That is, coexistence requires ecological (niche) segregation; limiting similarity of the coexisting species is expected. The principle is often stated in a narrower form: the number of populations coexisting at a fixed point attractor cannot exceed the number of resources [START_REF] Armstrong | Coexistence of species competing for shared resource[END_REF]). More generally, instead of the resources, one should count all of the environmental variables involved in the feedback loop of population regulation [START_REF] Levin | Community equlibria and stability, and an extension of the competitive exclusion principle[END_REF]. In line with e.g. [START_REF] Krebs | Ecology. The experimental analysis of distribution and abundance[END_REF], p. 288 and [START_REF] Meszéna | Competitive exclusion and limiting similarity: a unified theory[END_REF], we will refer to these variables as regulating factors/variables. Following the lead of [START_REF] May | Stability and Complexity in Model Ecosystems[END_REF], [START_REF] Vandermeer | Interspecific competition: A new approach to the classical theory[END_REF] and [START_REF] Abrams | The theory of limiting similarity[END_REF], [START_REF] Meszéna | Competitive exclusion and limiting similarity: a unified theory[END_REF] proved in a model-independent way that increasing similarity between the populations makes their coexistence less likely, i.e. more sensitive to the perturbation of external parameters. For robust coexistence, the populations should differ in their relationship to the regulating variables -i.e. in their impact on, and sensitivity towards, these variables. Therefore, a population's (differential) impacts and sensitivities constitute the proper model-independent generalization of the resource utilization function as a descriptor of the population's niche [START_REF] Meszéna | Competitive exclusion and limiting similarity: a unified theory[END_REF]. [START_REF] Szabó | Limiting similarity revisited[END_REF], [START_REF] Barabás | When the exception becomes the rule: the disappearance of limiting similarity in the Lotka-Volterra model[END_REF] studied the loss of robustness with disappearing niche segregation in the Lotka-Volterra model. Szilágyi and Meszéna (2009a,b) applied the theory to spatial niche segregation in a heterogeneous environment. Here we investigate usefulness of this approach in fluctuating environment. [START_REF] Hutchinson | The paradox of the plankton[END_REF] had already questioned the universal validity of the equilibrium coexistence theory and raised the question whether the high diversity of plankton communities could be explained by either the fluctuating nature of the environment or the internal dynamics of the system. Indeed, [START_REF] Armstrong | Coexistence of species competing for shared resource[END_REF] demonstrated that the number of coexisting species can exceed the number of resources even in a constant environment if the population dynamical attractor is not a fixed point. During the '80s it has become a widely held opinion that the Hutchinson-MacArthur-era overestimated the relevance of competitive exclusion compared with nonequilibrium processes [START_REF] Huston | A general hypothesis of species diversity[END_REF], 1994[START_REF] Begon | Ecology[END_REF]. On the other hand, [START_REF] Abrams | The theory of limiting similarity[END_REF], [START_REF] Chesson | A need for niches?[END_REF], [START_REF] Chesson | The roles of harsh and fluctuating conditions in the dynamics of ecological communities[END_REF], [START_REF] Chesson | Resource pulses, species interactions, and diversity maintenance in arid and semi-arid environments[END_REF] stressed that fluctuations do not alleviate the need for ecological segregation. Even in a fluctuating environment the species having the largest long-term rate of increase (calculated via proper averaging) will outcompete the others. Therefore, competitive exclusion remains the default behavior.

According to Chesson (2000b), coexistence must be stabilized by a negative feedback that gives a boost to any of the populations that become rare. In a constant environment, niche segregation of the classical kind provides this feedback: the rare species will have an abundant supply of resources. For fluc-tuating environments [START_REF] Chesson | Multispecies competition in variable environment[END_REF] enlists two additional mechanisms that are specifically related to fluctuations: the "storage effect" and the "effect of relative nonlinearity". The first one is essentially a temporal niche-segregation, made possible by the fluctuations [START_REF] Christiansen | Theories of populations in biological communities[END_REF]Fenchel 1977, p. 69, Amarasekare 2003), while the second one means that the statistical moments of the fluctuating regulating variables emerge as additional regulating variables via averaging of non-linear dependencies (cf. Levins 1979, Kisdi and[START_REF] Kisdi | Density dependent life history evolution in fluctuating environment[END_REF]. [START_REF] Chesson | Quantifying and testing coexistence mechanisms arising from recruitment fluctuations[END_REF] provided estimations for the strength of the two mechanisms, allowing to asses their relative importance. The spatiality-related coexistence mechanisms, described in Chesson (2000a), [START_REF] Amarasekare | Competitive coexistence in spatially structured environments: a synthesis[END_REF], [START_REF] Amarasekare | Mechanism of coexistence in competitive metacommunities[END_REF], Szilágyi and Meszéna (2009a,b), are outside of our interest here. [START_REF] Turelli | Re-examination of stability in randomly varying versus deterministic environments with comments on stochastic theory of limiting similarity[END_REF] formulated the necessity of a stabilizing feedback as a sufficient condition for coexistence: if either of the two populations is able to grow when rare enough, then they are able to coexist. This criterion of mutual invasibility is widely used to simplify theoretical investigations of coexistence.

The principal goal of the present paper is to study the relation between the theory of fluctuation-mediated coexistence and the classical notion of niche, as formulated by [START_REF] Meszéna | Competitive exclusion and limiting similarity: a unified theory[END_REF]. While we aim for the general picture, here we use the simplest model that we could conceive to demonstrate the principal connection between equilibrium and nonequilibrium ecology (see [START_REF] Parvinen | Disturbance-generated niche-segregation in a structured metapopulation model[END_REF] for a more realistic model of disturbance-generated coexistence in the same vein). For this purpose we use a slightly generalized version of the relative nonlinearity-type coexistence model by [START_REF] Kisdi | Density dependent life history evolution in fluctuating environment[END_REF]. There are only two regulating variables in this model, the long-term average and the variance of the total density. Therefore, it is ensured that at most two species can coexist.

Because of the simplicity of the model, an approximate analytical treatment is possible via the method of moment closure. We will compare these approximate results with numerical simulations. The analytic solution allows us to calculate the populations' impact and sensitivity vectors, which are the descriptors of the population's niche in the framework theory. This way we will be in the position to consider the robustness of coexistence in terms of segregation of the niches. After introducing our model in Section 2, we describe the basic analytic and numerical results on coexistence in Sections 3 and 4. Section 5 discusses the moment closure approximation. Finally, Section 6 provides the niche-theoretical analysis.

Building the model

Since we wish to study principal issues, we look for a model of fluctuationmediated coexistence as close to analytical tractability as possible. Continuous time is preferred, as discrete-time population dynamics tends to exhibit more complicated behavior. The simplest source of fluctuations is the uncorrelated external noise. We expect competitive exclusion to operate in the absence of fluctuations. This is ensured if the dependence on the total density is the only regulating feedback in the model (density-dependent selection, MacArthur 1962). If density-dependence were linear, fluctuations would not affect the behavior [START_REF] Chesson | The roles of harsh and fluctuating conditions in the dynamics of ecological communities[END_REF]. Quadratic dependence is the minimalist route to the effect of relative nonlinearity.

Therefore, the fluctuations will be driven by a Gaussian uncorrelated (white) noise ξ(t) with zero mean (ξ = 0, the overline denotes time averaging). Its autocorrelation function can be written as

ξ(t 1 )ξ(t 2 ) = δ(t 1 -t 2 ), (1)
where δ(t 1t 2 ) is the "Dirac-delta function", that is zero for t 1 = t 2 , but has an integral of 1, specifying the normalization. Then, our model is defined by the stochastic differential equation (SDE)

d dt n i (t) = ri(t) σ i ξ(t) -a i (n(t) -K i ) -b i (n(t) -K i ) 2 n i (t), (2)
where n i (t) is the density of Species i at time t, n(t) = i n i (t) is the total density. The expression in the square bracket, denoted by r i (t), is the instantaneous growth rate of Species i. K i is the carrying capacity of the ith Species, i.e. the equilibrium density without noise; a i , b i and σ i are positive constants.

The ratio b i /a i characterizes the nonlinearity of population regulation, while the noise-intensity coefficient σ i specifies the dependence of the population on the external fluctuations. To avoid complications arising from an Allee-effect, we want the deterministic part of the growth rate to decrease monotonously with increasing density. To this end, the parameter values are chosen to satisfy the inequality

a i 2b i > K i . ( 3
)
We use the Stratonovich interpretation of the SDE (see e.g. May 1973, p. 204;[START_REF] Braumann | Applications of stochastic differential equations to population growth[END_REF][START_REF] Braumann | Itô versus stratonovich calculus in random population growth[END_REF] and Appendix for the different interpretations of SDEs). The intuitive advantage of this choice is that the time-average of the instantaneous growth rate should be zero for any species that survives for a long period of time, if time averaging (denoted by overline) is defined via Stratonovich integration [START_REF] Braumann | Itô versus stratonovich calculus in random population growth[END_REF]). In our model this time-average reads as

r i = -a i (n -K i ) -b i (n -K i ) 2 + σ i ξ = -a i (n -K i ) -b i (n -K i ) 2 -b i V, (4)
where we denote the variance of the total density n by V = n 2n 2 . Observe the detrimental effect of the fluctuations on the long-term growth rate through the quadratic term. The "equilibrium" condition is r i = 0.

Without fluctuations, V = 0. Then r i becomes zero when the time-averaged total density n reaches the carrying capacity K i . By the well-known argument of [START_REF] Macarthur | Some generalized theorems of natural selection[END_REF], competition is won by the species with the highest carrying capacity in this case.

In the presence of external fluctuations the densities will also fluctuate, implying V > 0. The average growth rates are receiving feedback through two "regulating" variables: n and V . Each surviving population provides one equilibrium equation (r i = 0 for Species i) for these variables. The generic solution exists only if the number of equations does not exceed the number of unknowns. That is, coexistence of two, but not more, species is allowed in this model.

Since this bound on diversity is clear from the onset of our investigations, it is instructive to write down the dynamics (2) again just for two species:

d dt n 1 (t) = σ 1 ξ(t) -a 1 (n(t) -K 1 ) -b 1 (n(t) -K 1 ) 2 n 1 (t), ( 5
) d dt n 2 (t) = σ 2 ξ(t) -a 2 (n(t) -K 2 ) -b 2 (n(t) -K 2 ) 2 n 2 (t). ( 6
)
The special case b 2 = σ 2 = 0, i.e. when the underlined terms are deleted, will be referred to as the simplified model, first published by [START_REF] Kisdi | Density dependent life history evolution in fluctuating environment[END_REF].

Since we are interested in limiting similarity here, we need the full model that will allow the two species to become equivalent when their parameters are equal. However, as the simplified model has a fewer number of parameters, it is more convenient for the study of the coexistence of two species that are distinct.

Elementary analytic results

In this Section we summarize the analytical results that do not rely on the moment closure approximation; the latter will be considered in Section 5. The calculations presented here are direct consequences of the equilibrium conditions. While they are valid for arbitrarily large fluctuations, we will often use the approximation of small fluctuations to keep the formulae transparent.

Single species

First we consider the long-term equilibrium of a single species. The equilibrium condition can be written as

r = -a(n -K) -b(n -K) 2 -b V = 0 (7)
(cf. Eq. ( 4)), where n and V denote the average and the variance of the population density, respectively, for the case of a lone population. In a constant environment V = 0, implying n = K. As density variance has a negative effect on the growth rate and there is no Allee effect (monotonicity condition (3)), the average density must decrease with increasing density fluctuation.

For small fluctuations we expect a small V implies a small value of n-K characterizing the departure of the average density from the constant-environment value. In this limit we can neglect the second order small term in Eq. ( 7) leads to the relation n

-K = - b a V . ( 8
)
That is, the difference n -K is proportional to V . In the small fluctuation approximation we will neglect the terms beyond the linear order in either quantity.

Note that the terms, like the second one in Eq. ( 7), will always be negligible for small fluctuations. Therefore, the essential role of the nonlinear terms -b i (n i -K i ) 2 in dynamics ( 7) is to implement a sensitivity towards the fluctuations on the density. The ratio b/a that measures the nonlinearity of the density dependence also characterizes the sensitivity of the average density towards density fluctuations. In the case of linear density dependence (b = 0), fluctuations of the density average out; the average density will not be affected by the fluctuations.

Mutual invasion

The condition of mutual invasibility of two species reads

1 = -a 1 (n 2 -K 1 ) -b 1 (n 2 -K 1 ) 2 -b 1 V2 > 0, 2 = -a 2 (n 1 -K 2 ) -b 2 (n 1 -K 2 ) 2 -b 2 V1 > 0. ( 9
)
Here i denotes the "boundary" growth rate of Species i, i.e. its growth rate, when it is rare and the other species fluctuates in a stationary manner. While these conditions are exact, the quantities ni and Vi cannot be calculated analytically without the moment closure approximation of Section 5.

A more concise condition can be derived for the simplified model with b 2 = σ 2 = 0. In this case Species 2 is affected by the fluctuations only through its interaction with Species 1. When alone, Species 2 assumes an equilibrium density determined by the condition

r 2 = -a 2 (n 2 -K 2 ) = 0, implying n2 = K 2 .
Then the boundary growth rate of Species 1 is

1 = -a 1 (K 2 -K 1 ) -b 1 (K 2 -K 1 ) 2 . ( 10
)
It is positive iff

K 2 < K 1 , ( 11
)
where inequality (3) was used. On the other hand, Species 2 has a positive boundary growth rate against the established population of Species 1 iff

2 = -a 2 (n 1 -K 2 ) > 0. ( 12
)
The combination of Eqs. ( 10) and ( 12) provides the necessary and sufficient condition for mutually invasibility as [START_REF] Kisdi | Density dependent life history evolution in fluctuating environment[END_REF]. There is no exact analytic way to determine n1 ; it has to come either from simulations or from moment closure approximation applied to a single species.

n1 < K 2 < K 1 (13) (cf.

Advantage of rarity

Advantage of rarity (negative frequency dependence in the terminology of population genetics) is an essential component of coexistence. Following [START_REF] Chesson | Quantifying and testing coexistence mechanisms arising from recruitment fluctuations[END_REF], we measure it by the quantity

= 1 a 1 + 2 a 2 . ( 14
)
Mutual invasibility implies > 0, but not vice versa. Obviously, this would be true for any linear combination of 1 and 2 with positive coefficients. We justify the normalization by the a i s via an argument modified from [START_REF] Chesson | Scale transition theory with special reference to species coexistence in a variable environment[END_REF].

In case of linear density dependence

1 /a 1 = K 2 -K 1 and 2 /a 2 = K 1 -K 2 .
Therefore, the chosen normalization arranges that = 0 for the reference case of linearity, when relative nonlinearity cannot operate. The essential issue behind this normalization is that the quantity r 1 /a 1r 2 /a 2 is independent of the total density n in the linear case.

We will use the approximation of small fluctuations. Straightforward calculation leads to

= b 1 a 1 - b 2 a 2 + 2 b 1 b 2 a 1 a 2 ΔK ( V1 -V2 ) - b 1 a 1 + b 2 a 2 (ΔK) 2 > 0, ( 15
)
where the notation

ΔK = K 1 -K 2 (16)
was introduced. For ΔK = 0 it simplifies to the following transparent formula

b 1 a 1 - b 2 a 2 V1 -V2 > 0. ( 17
)
Condition ( 17) represents the coexistence-stabilizing mechanism and corresponds to the last term of Eq. ( 7) in Chesson (2000b), see also Eq. ( 68) in Chesson (1994) and the explanation thereafter. It has an intuitive meaning pointing to a close analogy between fluctuation-dependent and fluctuation-independent mechanisms to maintain coexistence. As mentioned above, the ratio b i /a i measures the nonlinearity of density-dependence of Species i; their difference in the first factor is the "relative" nonlinearity [START_REF] Chesson | Multispecies competition in variable environment[END_REF](Chesson , 2000b)). Phrasing it differently, the first factor represents the difference between the species in their dependence on the two regulating variables, the average n and the variance V of the total density (cf. Section 6). On the other hand, the second term is the difference between the species in their strength of making the total population size fluctuates. For coexistence, the species must differ in both respects. Moreover, they must differ in these aspects in the same way. The species that makes n more fluctuating must be the one that is more sensitive to the fluctuations of n. The species that affects the regulating variable V must be the one that is also more sensitive toward V . Like in constant-environment resource partitioning, this arrangement ensures rare advantage and stabilizes coexistence. We will revisit this expectation in Section 6.2 in a different way.

Obviously, conditions (15-17) provide only the necessary, and not the sufficient conditions for mutual invasibility. They represent the conditions for having a tendency for rare advantage. If the parameter choices (esp. of ΔK) are such that one species has a large advantage, then it is possible that the inferior species will be unable to invade despite its advantage gained from rarity.

Coexistence

Assume now that the two species coexist. Then their long-term growth rates satisfy the equilibrium equations

r 1 = -a 1 (n -K 1 ) -b 1 (n -K 1 ) 2 -b 1 V = 0, r 2 = -a 2 (n -K 2 ) -b 2 (n -K 2 ) 2 -b 2 V = 0. ( 18
)
These (quadratic) conditions determine the equilibrium values of the regulating variables n and V via a 4 th order algebraic equation. We write down the solution for small fluctuations, i.e. in linear order in V . For V = 0 coexistence is possible only if K 1 = K 2 ; the solution is characterized by n = K 1 = K 2 . Therefore, we expect that small V implies K 1 ≈ K 2 and then n ≈ K 1 ≈ K 2 . Therefore in the small fluctuation limit -as in Eq. ( 8) -the second terms in both equations are of second order small; neglecting them leads to

n = a 1 b 2 K 1 -a 2 b 1 K 2 a 1 b 2 -a 2 b 1 (19) and V = K 1 -K 2 b1 a1 -b2 a2 . ( 20
)
Note the consistence of these formulae with the assumptions we made: smallness of V implies smallness of K 1 -K 2 as a condition for coexistence. The analogue of Eq. (8),

n -K i = - b i a i V (21)
remains valid for the two-species case by the same argument from which it was derived in Section 3.1. For the simplified model,

n = K 2 (22) and V = a 1 b 1 ΔK (23)
applies.

Coexistence: simulation results

For simulation purposes, the continuous dynamics (2) is discretized as

n i,t+Δt -n i,t = -a i (n t -K i ) -b i (n t -K i ) 2 + 1 2 σ 2 i n i,t Δt + σ i n i,t √ Δtξ t ,
(24) where n i,t and n t denote the density of Species i and the total density, respectively, at time t. The noise ξ t is an i.i.d. process of variance 1 for time steps of length Δt. The scaling factor √ Δt in the random term ensures that the normalization (1) is obeyed in the Δt → 0 limit. The last term in the square bracket is the correction characteristic of Stratonovich integration (see Appendix; [START_REF] Sethi | A comparison of the Itô and the Stratonovich formulations of problems in finance[END_REF][START_REF] Stratonovich | Some Markov methods in the theory of stochastic processes in nonlinear dynamical systems[END_REF][START_REF] Braumann | Itô versus stratonovich calculus in random population growth[END_REF].

A simulated time series of two coexisting species is plotted in Fig. 1 for a set of parameters satisfying the mutual invasibility condition. In line with the invasibility prediction, they coexist for an arbitrary long period of time. Nevertheless, one should note that the densities fluctuate by many orders of magnitude. The populations experience extremely low densities.

Excursions to low densities warrants introduction of an extinction threshold. This way extinction of one of the species becomes a probability 1 event; the possibility of coexistence for infinite time is lost. Then the appropriate question is the expected length of coexistence.

Fig. 2 presents the average time until one of the densities goes below the threshold as a function of K 2 for different threshold values. Since the simplified model is used, condition (13) provides the parameter range for mutual invasibility, depicted by vertical dashed lines. Note that the plateau on the top of the curves is an artifact of terminating the simulations at T max = 10 7 time steps. Fig. 3 shows the average length of coexistence as a function of the extinction threshold. The two curves represent two parameter sets that do and do not allow for mutual invasibility. In the first case the length of coexistence -after a transient period -increases linearly as a function of the inverse of the extinction threshold. In contrast, the expected time until extinction increases very slowly and remains essentially bounded when the condition of mutual invasibility is not met.

The presented results confirm the prediction of the mutual invasibility criterion unequivocally for sufficiently low extinction thresholds. The expected length of coexistence is bounded only by the simulation time T max where the condition is met, but falls off by orders of magnitude outside the range of mutual invasibility. On the other hand, one should note that the "sufficiently low" extinction threshold is unrealistically low from biological point of view. The extinction threshold 10 -20 , i.e. population size 10 20 is required for coexistence length 10 6 even in case of mutual invasibility. For realistic population sizes the period of coexistence is very short.

The extreme level of density fluctuations are presumably related to the lowfrequency components of the white noise. Therefore, introduction of a lowfrequency cutoff of the power spectrum would make our model a good candidate for describing real species coexistence. Note that bounded periodic drive would result in bounded oscillation of the densities; therefore they would never become exponentially small.

However, for the rest of the current model, we remain interested in the analytic study of the white-noise model in the hope for insights that remain valid for more parameter-rich models.

Moment closure approximation

The equilibrium equations ( 18) are insufficient to fully determine the statistics of the population fluctuations. When a population is considered alone (Section 3.1), this condition establishes a relation between the average and the variance of the density, but does not allow us to calculate these quantities separately. For two populations, the average and variance of the sum of the densities are obtainable (Section 3.4), but not the statistics of the two populations separately. For more detailed analytic results the moment closure approximation will be used: we assume small fluctuations and neglect the third and higher statistical moments of the fluctuating variables.

The long-term average of the time-derivative of any quantity, which remains bounded, should be zero. Applying this principle to the logarithmic density ln n i (time derivative of which is the growth rate r i ) leads to the equilibrium condition r i = 0. We have exploited this condition already in Section 3.1. We need an additional relation to proceed further. For this purpose here we consider the average of the time derivative of n i , instead of ln n i :

d dt n i (t) = r i n i = σ i ξ(t) -a i (n(t) -K i ) -b i (n(t) -K i ) 2 n i = 0. ( 25
)
We introduce the deviation m i = n in and use the notation

m = m 1 + m 2 = n -n. Note that m(t) = m i (t) = 0 ( 26
)
and

m 2 = m 1 m + m 1 m = V. ( 27
)
Moreover, we will use the relation

m i ξ = σ i 2 n i , ( 28
)
which is proven in the Appendix. We define the moment closure approximation by neglecting the moments of m beyond m 2 . It is consistent with the small fluctuation approximation according to the relation ( 27). When the relationships (26-28) are taken into account, the moment closure version of condition (25) reads as

σ 2 i 2 n i -a i n i (n-K i )-a i m i m-b i n i (n-K i ) 2 -b i n i V -2b i m i m(n-K i ) = 0. ( 29
)
Now we substitute Eq. ( 21) into the expressions (n -K i ) and according to the small fluctuation limit after neglecting all second order terms in V we arrive at the equality

σ 2 i 2 n i = a i 1 - 2b i a i V m i m. ( 30
)
Neglecting the term V m i m, which is of order m 4 , leads to

σ 2 i 2a i n i = m i m. ( 31
)
With summation of this formula for i = 1, 2 one arrives to

V = σ 2 1 2a 1 n 1 + σ 2 2 2a 2 n 2 = v 1 n 1 + v 2 n 2 , ( 32
)
where the notation

v i = σ 2 i 2a i ( 33
)
was introduced. Note the intuitive meaning of v i : it is the per capita contribution of Species i to the variance of the total population size. Equation ( 32), together with the trivial relation

n = n 1 + n 2 (34)
establish the connection between the average densities of the two species and the two regulating variables. One can solve the system of equations (32,34) for the average densities:

n 1 = V -v 2 n v 1 -v 2 , n 2 = -V + v 1 n v 1 -v 2 . ( 35
)
These results, together with equations (19-20), provide all the interesting quantities as a function of the model parameters.

For the simplified model, substitution of b 2 = σ 2 = 0 and Eqs. (22-23) results in

n 1 = 2a 2 1 b 1 σ 2 1 ΔK = n1 K 1 -n1 ΔK, n 2 = K 2 - 2a 2 1 b 1 σ 2 1 ΔK = K 2 - n1 K 1 -n1 ΔK.
(36)

For one species a similar moment closure calculation leads to the relations

n = 1 - σ 2 b 2a 2 + σ 2 b K = 2a 2 K 2a 2 + σ 2 b , ( 37
) and V = σ 2 aK 2a 2 + σ 2 b . ( 38
)
Fig. 4 presents the comparison of this one-species result with numerical simulations. The coincidence of the predicted and simulated average densities is remarkable; the fluctuations need not to be extremely small for this. The moment closure method turns out to be reliable as long as the noise intensity parameter σ is not larger than the parameters a and b in the regulation terms. Observe that the equilibrium density is a monotonically decreasing function of the strength of the external fluctuations. With very high fluctuations the average density can be made arbitrary small, provided that the extinction threshold is small enough.

The analogous comparison between moment closure (Eq. ( 36)) and simulation is presented for two species in Fig. 5. The departure of the moment closure prediction from the simulated values is larger than in case of a single species. Still, the agreement is reasonable. The moment closure approximation seems to capture the essential behavior of the system. Observe that the deviation is the same in magnitude, but opposite in direction, for the two species. That is, the total density is still provided accurately; the moment closure method is less precise in predicting the ratio of the two densities. This situation is understandable in the light of high sensitivity of the relative densities for the fluctuations.

Niche segregation and robustness of coexistence

Based on the analytic results above, in this section we reinvestigate the model in the context of the theory of limiting similarity by [START_REF] Meszéna | Competitive exclusion and limiting similarity: a unified theory[END_REF]. First we briefly reiterate how it works in a constant environment; then we consider the current model.

Reference theory: constant environment

Here we summarize the theory of limiting similarity by [START_REF] Meszéna | Competitive exclusion and limiting similarity: a unified theory[END_REF]. It is a model-independent implementation of the proposal by [START_REF] Abrams | The theory of limiting similarity[END_REF] that limits to similarity must be considered in conjunction with the relative competitiveness of the coexisting species; see also Chesson (2000b) in the same vein. The key statement of Meszéna et al. is that species should differ in their way of regulation for robust coexistence; the more similar they are, the more narrow is the range of competitive parameters allowing for their coexistence. More specifically, one should consider the species' impact on, and sensitivity towards the regulating factors. Strength of competition between the populations is reduced and their coexistence is robust if they differ in both respects.

Write the dependence of the growth rate of the ith species (i = 1, 2, . . . , L) on the densities n 1 , . . . , n L , as

r i = r i (I(n 1 , n 2 , . . . , n L ), E) (i = 1, 2, . . . , L), ( 39
)
where the vector I = (I 1 , I 2 , . . . , I D ) collects the D number of regulating variables; E denotes the vector of "external" variables that affect the population growth rates but are not affected by the population densities. Then, the elements of the competition matrix are

α ij = - ∂r i ∂n j = D k=1 S ik C jk = S i • C j , ( 40
)
where the partials

C ik = ∂I k ∂n i , S ik = - ∂r i
∂I k (41) measure the ith species' impact on, and sensitivity towards, the kth regulating factor, respectively. The vectors C i = {C i1 , C i2 , . . . , C iD } and S i = {S i1 , S i2 , . . . , S iD } will be referred to as the impact and sensitivity niches of the ith species. The sensitivity of the equilibrium population sizes towards the additional mortalities can be determined by implicit differentiation of the equilibrium equation r i = 0:

∂n e i ∂E = - L j=1 (α -1 ) ij ∂r j ∂E = - 1 J L j=1 adj(α) ij ∂r j ∂E , ( 42
)
where adj(α) denotes the adjunct of the community matrix. The determinant J = det(α ij ) measures the strength of regulation at the community level. As it appears in the denominator, a weakly regulated community is sensitive towards the extra mortalities. Therefore a small |J| implies that only a small range of the external parameters can be tolerated without extinction of one of the species. Robust coexistence requires large |J|. Note that if the sensitivity niches of two different species coincide, the corresponding two rows of matrix α are the same and J = 0. A similar argument holds for the impact vectors and the columns of the matrix. Therefore, the requirement of large |J| translates to the requirement of sufficiently different niche vectors.

Impact and sensitivity niches in our system

The sensitivity analysis in Section 6.1 is based on the assumption of equilibrium in an essential way. Its applicability for our model is not trivial.

A part of the framework is easy to translate. The growth rate r i of the constant environment model should be replaced by the averaged growth rate r i because the "equilibrium" equation r i = 0 holds even in our variable environment. As we discussed already, the averaged total density n and the variance of the total density V play the role of regulating variables: these quantities determine the r i s unequivocally. Therefore, the vector of regulating factors can be written as

I = n V , ( 43
)
while the sensitivity vectors are

S i = - ∂ri ∂n ∂ri ∂V = a i + 2b i (n -K i ) b i . ( 44
)
Omitting n -K i ∼ V the formula simplifies to

S i ≈ a i b i .
In this sense one can say that the coefficients of the linear and the quadratic density dependence measure the sensitivity towards the average and the variance of the total population size, respectively. Note however, that here we neglect a term which is linear in V , because the leading term is independent of V . The nontrivial part is the interpretation of the equilibrium density n e i . The average density n i is the obvious candidate. However, the average density of the species alone does not determine the regulating vectors. Therefore, the impact vectors cannot be defined, and the formalism cannot be applied without further considerations.

To apply the framework we have to restrict the possibilities of the stochastic dynamics to a two-parameter sub-family that can be parameterized by n 1 and n 2 . We do this by picking up the carrying capacities K 1 and K 2 as competitive parameters [START_REF] Abrams | The theory of limiting similarity[END_REF], playing the role of the vector E above. All other model parameters are regarded as fixed; robustness of coexistence is considered with respect to changes of the carrying capacities. This way both the average densities n 1 , n 2 and the regulating variables n, V become functions of the carrying capacities K 1 , K 2 . Inversion of these relationships leads to an unequivocal definition of the dependence of the regulating variables on the average densities. Note that a similar reduction of dimensionality was applied when population structure caused an analogous problem (Szilágyi and Meszéna 2009a,b).

Then the impact niche vector of Species i is calculable from the relations (32) and (34) as

C i = ∂n ∂ni ∂V ∂ni = 1 v i . ( 45
)
From the vectors S i and C j we can obtain the competiton matrix:

α ij = - ∂r i ∂n j = S i • C j . ( 46
)
As explained in Section 6.1, the determinant of this matrix is the measure of the robustness we are looking for:

det(α) a 1 a 2 = b 1 a 1 - b 2 a 2 + 2 b 1 b 2 a 1 a 2 ΔK (v 1 -v 2 ), ( 47
)
where Eqs. ( 44) and ( 45) were used. Observe resemblance of this measure of regulatedness to the quantity in the l.h.s. of the necessary condition (15) in Section 3.3 for coexistence. While invasion from rarity was considered in Section 3.3, here we studied the effect of a small perturbation. This difference in the approaches explains the discrepancies between the results. First, while the per capita fluctuation-producing effect of the species (v 1 and v 2 ) appears here, the analogous quantities for the whole populations ( V1 and V2 ) were used in Section 3.3. Second, as here we considered robustness with respect to ΔK, the dependence on ΔK is different from the one in Section 3.3.

These differences notwithstanding, the intuitive meaning of the robustness measure [START_REF] Braumann | Itô versus stratonovich calculus in random population growth[END_REF] is the same as of in Section 3.3: the species must differ both in their fluctuation-maintaining effect and their sensitivities towards fluctuations. The differential approach here has the advantage of implementing the modelindependent connection between robustness and niche segregation [START_REF] Meszéna | Competitive exclusion and limiting similarity: a unified theory[END_REF]. This formalism makes the analogy between fluctuation-related and fluctuation-unrelated coexistence, which we explained after Eq. ( 17), quantitative.

Robustness of coexistence

Here we demonstrate the loss of robustness of coexistence when the two species becomes similar. For this purpose the simplified model is rewritten as

d dt n 1 (t) = μσξ(t) -a(n(t) -K 1 ) -μb(n(t) -K 1 ) 2 n 1 (t), ( 48
) d dt n 2 (t) = -a(n(t) -K 2 )n 2 (t), ( 49
)
where μ is the parameter measuring dissimilarity. The two species are identical for μ = 0 and become dissimilar with increasing μ. The robustness of coexistence is determined by varying the carrying capacities. Fig. 6 plots the ratio of the parameter region (K 1 , K 2 ) ∈ [0, 0.3] × [0, 0.3], which allows coexistence, as a function of μ. Observe that robustness is lost gradually when the species become more and more similar. Coexistence becomes structurally unstable for μ = 0, when only the case

K 1 = K 2 supports coexistence.
This result, which is in line with theoretical expectations, is comparable with Fig. 6.4-6.6 of [START_REF] May | Stability and Complexity in Model Ecosystems[END_REF], or with Figs. 4-5 of Szilágyi and Meszéna (2009b).

Discussion

We aimed at studying the simplest possible model of fluctuation-mediated coexistence as deeply as possible in the context of a mathematical niche theory published earlier [START_REF] Meszéna | Competitive exclusion and limiting similarity: a unified theory[END_REF], Szilágyi and Meszéna 2009a,b, Parvinen and Meszéna 2009). In line with this approach, simplicity is measured by the number of regulating variables. As a single regulating factor cannot support coexistence, we constructed a model with two of them (see [START_REF] Kisdi | Density dependent life history evolution in fluctuating environment[END_REF] for the original version). While all "strategic" (Czárán 1998, p. xii) model studies pay a price in terms of realism, our sacrifice turned out to be higher than usual. The coexistence, which is maintained in our model in the infinite population size limit, requires astonishingly large population sizes. Its practical irrelevance notwithstanding, our model provided a test-bed for developing moment-closure treatment of problems and developing niche analyses in line with our theoretical context. The model supported the idea that sufficient niche differentiation with respect to the way populations are regulated is the requirement for robust coexistence in an environment that displays stationary fluctuations -just like in a constant one.

Our wider context is classical coexistence theory. It was conceived in the framework of the Lotka-Volterra model and disregarded stochasticity. The resulting picture was simple and intuitive: coexistence was based on ecological niche differentiation leading to reduced interspecific competition and rare advantage (Case 2000, p. 368). However, as no clear lower bound of similarity was found [START_REF] Abrams | The theory of limiting similarity[END_REF], further model studies blurred this simplicity. Strength of competition and niche have become terms of unclear meaning, terms to be defined separately in every specific situation. The goal of [START_REF] Meszéna | Competitive exclusion and limiting similarity: a unified theory[END_REF] was to reestablish the intuitive as well as the mathematical clarity of coexistence theory in a general, model-independent way. Competition coefficients were defined differentially; therefore they became independent of the assumption of linearity in the Lotka-Volterra model. Niche was defined as the species' differential impact on, and differential sensitivity towards the regulating variables, a generalization of the concept of the resource utilization function. Clear general connection between niche segregation and robustness of coexistence was established. It was asserted that complications like population structure and environmental fluctuations can be tackled within this framework via time-scaling arguments [START_REF] Meszéna | Competitive exclusion and limiting similarity: a unified theory[END_REF]. The theory was extended by Szilágyi and Meszéna (2009a,b) for structured populations in a constant environment. The goal of the current paper was to apply the same framework for the minimal model of fluctuation-maintained coexistence.

There are two distinct sources of stochasticity in population dynamics: environmental noise and demographic stochasticity (Case 2000, p. 30). While both of them are present in almost any real ecological situation, it is meaningful to consider the approximations under which either or both of them are negligible. Demographic stochasticity is the more difficult problem. Neglecting it is equivalent to assuming infinite population size, i.e. describing population dynamics in terms of continuous density variables. In this case, even if environmental noise is present, the dynamics is deterministic at least in the conditional sense, i.e. for a given realization of the environmental process. In contrast, finite population size leads to demographic stochasticity unavoidably. (Even local finiteness of a spatially extended infinite population results in local demographic stochasticity, see [START_REF] Oborny | Dynamics of populations on the verge of extinction[END_REF][START_REF] Oborny | Survival of species in patchy landscapes: percolation in space and time[END_REF]) Analytic treatment of the stochastic dynamics of finite populations is mainly restricted to density-independent growth (cf. branching process theory, [START_REF] Haccou | Branching Processes: Variation, Growth and Extinction of Populations[END_REF]; but see Klebaner (2005), p. 366;[START_REF] Jagers | A plea for stochastic population dynamics[END_REF], Klebaner et al. (in press) for intriguing progress beyond this limitation.

In many situations environmental stochasticity maintains coexistence that would not be otherwise possible. In contrast, demographic stochasticity is invariably detrimental for coexistence, because it causes rare disadvantage and extinction. While analytic studies of fluctuation-maintained coexistence often assume infinite population size, numerical investigations can never completely disregard the consequences of finiteness. Accordingly, we discuss fluctuationrelated coexistence first for infinite populations then with considering finiteness.

In the limit of infinite population size it is self-evident that mutual invasion implies long-term coexistence; numerical experimentation with our model supported this principle unequivocally. It is the basis of the comprehensive theory of fluctuation-mediated coexistence by [START_REF] Chesson | Multispecies competition in variable environment[END_REF] (see also [START_REF] Chesson | Scale transition theory with special reference to species coexistence in a variable environment[END_REF] for a very readable summary). Chesson's formalism and [START_REF] Meszéna | Competitive exclusion and limiting similarity: a unified theory[END_REF] share the feature of being framework theories: both of them represent a common mathematical structure behind a class of models. [START_REF] Chesson | Multispecies competition in variable environment[END_REF] stresses the distinction between fluctuation-independent and fluctuation-dependent mechanisms for coexistence and concentrates on the latter. In particular, he studies emergence of coexistence-maintaining effects from the process of time averaging. Averaging implement a "scale transition" [START_REF] Chesson | Scale transition theory with special reference to species coexistence in a variable environment[END_REF] between the original time scale and the long time scale of stationarity. His basic conclusion is that coexisting species should average the environmental process differently -otherwise the species with the highest average growth rate would win.

In contrast, the complementary approach by [START_REF] Meszéna | Competitive exclusion and limiting similarity: a unified theory[END_REF] concentrates on the commonality of all kinds of coexistence-maintaining mechanisms (fluctuating, or not) on the time scale of stationarity. At this level the common cause for coexistence is segregation with respect to the "regulating" variables. Any fluctuation-dependent maintenance mechanism must be represented by proper regulating variables at the long time scale. On one hand, our model is consistent with Chesson's framework. Our formulas often directly correspond to his one. On the other hand, the current work is an initial exercise to apply the theory by [START_REF] Meszéna | Competitive exclusion and limiting similarity: a unified theory[END_REF] for fluctuation-mediated coexistence.

The central issue of commonality of coexistence-maintaining mechanisms is loss of robustness of coexistence with increasing similarity of the species. In equilibrium resource competition the species, which depletes a specific resource more than its competitor, must depend on that specific resource more strongly to establish a coexistence-stabilizing rare advantage. Weak differentiation results in weak robustness against environmentally induced relative advantage/disadvantage between the species -a well known, but rarely stressed, phenomenon on the Lotka-Volterra model [START_REF] Vandermeer | Interspecific competition: A new approach to the classical theory[END_REF]May 1973, Fig. 6.4 on p. 158). In clear analogy, the relative nonlinearity mechanism for coexistence requires the species that causes the larger fluctuations of the total density to be more sensitive towards the fluctuations as well. If the populations become similar in either respect, then the populations must be fine tuned for coexistence. This is expressed by Eq. ( 68) in [START_REF] Chesson | Multispecies competition in variable environment[END_REF] or Eq. (2) in [START_REF] Chesson | Scale transition theory with special reference to species coexistence in a variable environment[END_REF]; the same issue is encountered in our Eqs. ( 17) and [START_REF] Braumann | Itô versus stratonovich calculus in random population growth[END_REF].

In this paper we turned the analogy between fluctuation-independent and -dependent mechanisms into an essential mathematical connection. We were able to determine the impact/sensitivity niche vectors for our model and demonstrate their expected connection to robustness of coexistence. For the analytic results we relied on the moment closure approximation, i.e. we assumed small fluctuations. However, the basic definitions, as well as of the underlying theory, are independent of these approximations; therefore, the principal connection we advocate is also exact. Note the similar conclusions in other models of fluctuation-mediated coexistence. [START_REF] Parvinen | Disturbance-generated niche-segregation in a structured metapopulation model[END_REF] studied the case of repeated local catastrophes, while Barabás et al. (in prep.) investigated periodic environments. Now we turn our attention to the unavoidable consequences of demographic stochasticity. Any population with bounded population size is destined to extinction with probability 1. Mutual invasibility has the tendency to increase the time-span of coexistence, but it cannot prevent eventual extinction. When demographic stochasticity is relatively small, it remains true that mutual invasibility (and the niche-segregation behind it) can maintain coexistence on the biologically relevant time-scale. However, as our example shows, the interaction between environmental and demographic stochasticity may lead to large density fluctuations that makes the diversity-stabilizing effect of mutual invasibility irrelevant. Therefore, we should caution against the uncritical use of mutual invasibility as a sufficient condition for coexistence, a point that was already made by [START_REF] Proulx | What can invasion analyses tell us about evolution under stochasticity in finite populations?[END_REF]. [START_REF] Johansson | Will sympatric speciation fail due to stochastic competitive exclusion?[END_REF], [START_REF] Johansson | The risk of competitive exclusion during evolutionary branching: Effects of resource variability, correlation and autocorrelation[END_REF] studied evolutionary consequences of finiteness, while [START_REF] Mágori | Adaptive dynamics in cellular automaton: Role of spatiality in competition, coexistence and evolutionary branching[END_REF][START_REF] Habets | Spatial structure inhibits the rate of invasion of beneficial mutations in asexual populations[END_REF] dealt with the coexistence-restricting effect of local demographic stochasticity.

The observed unreliability of mutual invasibility, as a predictor of coexistence, does not imply uselessness of the principle. As it is exact in the limit of infinite population size, it remains the correct tool to study whether environmental stochasticity has a coexistence-maintaining effect in a given ecological situation. However, one has to be aware that demographic stochasticity always has an opposite effect. This opposite effect can be negligible in some cases, can completely eliminate coexistence in others, or it may have an intermediate importance. It is entirely possible that our model will turn out to be extremely unfortunate in this respect. As the effect of relative nonlinearity is a weak force for coexistence, it requires large environmental fluctuations for robust coexistence, causing large density fluctuations. On the other hand, storage effect can be less vulnerable to demographic stochasticity: "storage" actually means buffering at rarity. The bottom line is that studies of mutual invasibility are useful to be complemented by investigating time-to-extinction with extinction threshold.

Within the confines of the current model, the extremely large density-fluctuations are presumably related to the low-frequency end of the white-noise spectrum of the environmental fluctuation: extremely low densities are consequences of unfavorable environmental conditions experienced for an extended period of time. Therefore we conjecture that the same model with a different noise spectrum, while complicates the analytic treatment and increases the number of parameters, will turn out to be a biologically more relevant description of fluctuation-mediated coexistence. Bored by the growing volume of the Itô-Stratonovich literature, [START_REF] Smythe | Itô versus stratonovich revisited[END_REF] established the validity of the Stratonovich approach for describing continuous stochastic phenomena experimentally. They observed also that the naive numerical discretization, corresponding to the Itô-interpretation, leads to an incorrect result.

a 1 = a 2 = b 1 = 0.1; b 2 = 0.02; σ 1 = 0.1; σ 2 = 0.05; K 1 = 0.3; K 2 = 0.
Consider now the discretization of our model defined by the SDE (2). Naively, one would arrive at the iteration n i,t+Δtn i,t = -a i (n t -K i )b i (n t -K i ) 2 n i,t Δt + σ i n i,t √ Δtξ t . (A.7) (The factor √ Δt ensures compliance with Eq. (A.3) when the i.i.d. ξ t has a variance 1.) However, the Δt → 0 limit of this iteration would lead to the Itô, instead of the Stratonovich-interpretation of the stochastic dynamics (2). One has to take into account the correction (A.6).

Only the stochastic term of the iteration (A. The expectation value of this correction leads to the correction term 1 2 σ 2 i n i in the iteration (24).

We proceed to prove Eq. ( 28). Averaging is defined via integration, so the considerations above apply. The Itô average of the product mξ is zero: value of m in each time step depends only on the values of ξ of the previous steps, which is independent from the current ξ. Therefore the Stranovich average mξ comes entirely from the Stratonovich-Itô correction. As now g(t) corresponds to m(t) and Δm = Δn, Eq. (A.8) leads to the relation (28).

Figure 1 :

 1 Figure 1: Time course of the densities of the two coexisting species. The left pane with logarithmic vertical scale demonstrates the regular occurrence of extremely low densities, incompatible with realistic population sizes. A part of the same curve is presented with linear scale on the right pane. Parameters: a 1 = a 2 = b 1 = 0.1; b 2 = 0.02; σ 1 = 0.1; σ 2 = 0.05; K 1 = 0.3; K 2 = 0.298. The same parameters are used for the rest of the Figures unless indicated otherwise.

Figure 2 :

 2 Figure2: Expected time until extinction of one of the co-occurring species in the presence of an extinction threshold (LIM) as a function of the carrying capacity K 2 (simplified model, average of 50 runs). Vertical (time) scale is linear on the left plot and logarithmic on the right one. The simulations were terminated at time T = 10 7 if no extinction occurred. The region of mutual invasibility (n 1 , K 1 ) is shown by the vertical dashed lines. For extremely low extinction thresholds the parameter region of long-term coexistence coincides with that range. For more realistic thresholds, the possibility for coexistence is more restricted. Parameters are the same as in Fig.1except that now b 2 = σ 2 = 0. This parameter choice is maintained in later figures of the simplified model when not indicated otherwise. The equilibrium density of Species 1, when alone, is measured to be n1 = 0.286.

Figure 3 :

 3 Figure 3: Expected time until extinction as a function of the extinction threshold (simplified model, average of 500 runs). Continuous line: K 2 = 0.29 satisfying the condition for mutual invasibility. The coexistence time diverges for low thresholds and becomes inversely proportional to the threshold. Dotted line: K 2 = 0.19, no mutual invasibility. Coexistence time remains essentially bounded.

Figure 4 :

 4 Figure 4: Average density (n) and variance ( V ) of a single population as a function of the fluctuation strength σ. The moment closure method (dashed line) approximates the simulation results (solid line) very well for σ < 0.1 and reasonably well for 0.1 < σ < 0.2. Parameters: a = 0.1, b = 0.1, K = 0.3.

Figure 5 :

 5 Figure 5: Average densities of two coexisting species as a function of a 1 in the simplified model. The difference between the moment closure (dashed lines) and the simulated (solid lines) densities are significantly larger than in the single species case of Fig. 4. Still, moment closure remains a reasonable approximation. K 2 = 0.295, within the range of mutual invasion.

  298. The same parameters are used for the rest of the Figures unless indicated otherwise.

		1e+7						LIM=1e-120 LIM=1e-80		1e+7						LIM=1-e120 LIM=1e-80
	average time until extinction	2e+6 4e+6 6e+6 8e+6							LIM=1e-40	average time until extinction	1e+5 1e+6							LIM=1e-40
		0 0.25	0.26	0.27	0.28	0.29	0.3	0.31	0.32	0.33	0.34	0.25 1e+4	0.26	0.27	0.28	0.29	0.3	0.31	0.32	0.33	0.34
						K	2									K	2

  7) is of interest. Then ΔW (t) corresponds to √ Δtξ t and g(t) = σ i n it ; therefore Δg(t i ) = σ i Δn i (t i ). However, Δn i (t i ) = σ i n i (t i )ΔW (t i ) ( A . 8 ) by the iteration; only the stochastic term was taken into account. Therefore, Δg(t i ) = σ 2 i n i (t i )ΔW (t l )

					( A . 9 )
	and a term in the correction (A.6) reads as
	1 2	σ 2 i n i (t i )(ΔW (t l )) 2 =	1 2	σ 2 i n i (t i )Δt.	(A.10)

Acknowledgements

We thank György Barabás, Carlos Braumann, Géza Györgyi, Éva Kisdi, Dina Lantos, Peter Jagers, Hans Metz, Liz Pásztor and Zoltán Rácz for discussions and two anonymous referees for useful comments on the original verision of the ms. This work was financed by OTKA grants K81628 and NK73047.

Appendix A. Stratonovich and Itô interpretation of stochastic dynamics

Caution is needed in defining a "stochastic" integral, like

where ξ(t) is the white noise. Here W (t) denotes the Wiener process, the derivative of which is the white noise:

a quite nontrivial feature of the white noise. The integral (A.1) can be approximated by two different sums, referred to as "Stratonovich" and "Itô" types: Were g and W smooth functions, each term of (A.6) would be proportional to (Δt) 2 and the difference between the two approximations would disappear in the limit Δt → 0. However, because of the property (A.3), if Δg is proportional to ΔW , then the expectation of a term is proportional to (ΔW ) 2 = Δt. Then, the difference (S-I) remains finite in the limit; the sums (S) and (I) converge to two different values. These two limits constitute two different definitions of the stochastic integral (A.1). For stochastic differential equations (SDE) the two different ways of integration establishes two different solutions, i.e. two inequivalent interpretations of the SDE.

We assume that in the real word the fluctuations are correlated, i.e. the environment cannot change arbitrarily fast. Then, we assume that the correlation time is short compared to the timescale of population dynamical effects. Therefore, we study the limit of zero correlation time, as specified by Eq. ( 1), leading to Stratonovich-interpretation of Eq. ( 2) [START_REF] Braumann | Applications of stochastic differential equations to population growth[END_REF][START_REF] Braumann | Itô versus stratonovich calculus in random population growth[END_REF]. It is something different than considering the continuous-time process as a limiting case of a discrete dynamics with independent and identically distributed (i.i.d.) random variables as noise, e.g. Itô-interpretation [START_REF] Feldman | A population's stationary distribution and chance of extinction in a stochastic environment with remarks on the theory of species packing[END_REF].


Anneke Bühler
email: buehler@ift.de

Charlotte Kötter

Stefanie Jaursch

Friedrich Lösel

Prevention of familial transmission of depression: EFFEKT-E, a selective program for emotionally burdened families

Keywords: selective prevention, family-oriented, depression, behavioral problems, children, depressed mothers

Prevention of familial transmission of depression:

EFFEKT-E, a selective program for emotionally burdened families

3 About 17% of German citizens develop a depressive disorder during their lifetime [START_REF] Jacobi | Prevalence, comorbidity and correlates of mental disorders in the general population: results from the German Health Interview and Examination Survey (GHS)[END_REF]). Emotional problems can be observed as early as in pre-school age. A prevalence rate of 6, 4% has been estimated for 3 to 6 year old children in Germany [START_REF] Groen | Verhaltensauffälligkeiten bei Kindern und Jugendlichen[END_REF]). During childhood, anxiety and depressive mood rates are equally distributed between girls and boys, but in adolescence a gender difference occurs: Female adolescents report more emotional problems than boys (10,8% vs. 6,6%). In addition, prevalence rates depend on socio-economic and migrant status. More children from families with lower SES or from migrant families suffer from anxiety and depressive mood [START_REF] Groen | Verhaltensauffälligkeiten bei Kindern und Jugendlichen[END_REF]).

Depression produces substantial economic costs [START_REF] Paykel | Size and burden of depressive disorders in Europe[END_REF]. Besides the financial load, the individual burden of a child with anxious and depressive behavioral tendencies on his or her way through adolescence to adulthood calls for action. Depressed children have poorer communication and problem-solving skills, and are less supportive and assertive than nondepressed children [START_REF] Garber | Depression in children and adolescents: Linking risk research and prevention[END_REF]. They have less intensive relationships with peers and family members, experience rejection, have less academic skills, and more school problems [START_REF] Groen | Verhaltensauffälligkeiten bei Kindern und Jugendlichen[END_REF]. Thus, internalizing behaviors impede the accomplishment of nearly all adolescence-specific developmental tasks increasing the probability of lifelong persistent maladaptive development [START_REF] Groen | Verhaltensauffälligkeiten bei Kindern und Jugendlichen[END_REF].

Parental depression is one of the strongest risk factors for depression in children [START_REF] Garber | Depression in children and adolescents: Linking risk research and prevention[END_REF].

Compared to children of non-depressed parents, girls and boys in emotionally burdened families are about three to four times more likely to develop a mood disorder, and have an increased risk for initiation of other internalizing behaviors, conduct disorders, school problems, substance abuse disorders and lower overall functioning [START_REF] Garber | Depression in children and adolescents: Linking risk research and prevention[END_REF]. Depressive episodes of children of depressed parents start earlier in life, last longer, are more severe and reoccur more often than those from children without familial vulnerability [START_REF] Goodman | Depression in mothers[END_REF]. Behavioral-genetic and environmental studies have provided evidence of both genetic and environmental factors contributing to development of depression [START_REF] Garber | Depression in children and adolescents: Linking risk research and prevention[END_REF].

Evidence of environmental risk factors guides behavioral prevention efforts. Families with a depressed parent have been described as having more problems with attachment, communication, cohesion, social support, childrearing practices, chronic criticism, harsh discipline, and inappropriate peer-like relationships [START_REF] Garber | Depression in children and adolescents: Linking risk research and prevention[END_REF]. Cognitive tendencies such as less parental self-efficacy, more negative views of themselves as parents and more critical perceptions of their children are characteristics of depressed mothers [START_REF] Goodman | Depression in mothers[END_REF] and play an important role in the interaction between depressed mother and her child [START_REF] Teti | Parenting Sensitivity, Parental Depression and Child Health: The Mediational Role of Parental Self-Efficacy[END_REF]. [START_REF] Goodman | Depression in mothers[END_REF] reviewed studies that tried to disentangle the mechanisms mediating between parental depression and emotional or other behavioral problems in childhood. They showed that the psychosocial pathway from parental depression to child maladaptation includes specific impairments in parenting behavior: low levels of parental warmth, high levels of maternal hostility, harsh and inconsistent discipline, dysfunctional communication, less social support, worse interaction quality and less maternal social skills [START_REF] Goodman | Depression in mothers[END_REF]. Recently, Garber and Cole (2010) confirmed a "launch and grow" type of cascade in which maternal depression is conceptualized as launching a set of risk factors which then in turn predict the growth of children's depression development during adolescence. Garber concludes: "An important next step in the development of depressionprevention programs would be to explicitly target parenting behaviors that are most likely to contribute to depression in children (e.g. harsh, inconsistent discipline, criticism and rejection, withdrawal, intrusiveness). This then could supplement the child-focused components of the intervention that more directly address children's cognitions and coping strategies" (Garber 2006, p.S114).

Existing prevention programs for families with a depressed parent target adolescents. Beardslee and his colleagues (1997;[START_REF] Beelmann | Wirksamkeit eines sozialen Problemlösetrainings bei entwicklungsverzögerten Vorschulkindern[END_REF] developed a selective prevention program for adolescent offspring of depressed parents. The program showed to be more effective in reducing adolescents' depression than a solely psycho-educative approach with small to moderate effect-sizes at end of intervention and 18 month follow-up. An indicative program for offspring of depressed parents was evaluated by [START_REF] Clarke | A randomized trial of a group cognitive intervention for preventing depression in adolescent offspring of depressed parents[END_REF]. The effect sizes for this program were moderate at shortand long-term. Recently, [START_REF] Compas | Randomized controlled trial of a family cognitive-behavioral preventive intervention for children of depressed parents[END_REF] tested a family cognitive-behavioral preventive intervention for parents with a history of depression and their 9-15-year-old children. The authors compared it with a self-study written information condition in a randomized clinical trial. The family-based measure led to a moderate effect in favourable outcomes for child behaviour. In line with the intervention model, changes in adolescents' coping and positive parenting mediated the effects of the intervention on depressive, internalizing, and externalizing symptoms [START_REF] Compas | Coping and parenting: Mediators of 12month outcomes of a family group cognitive-behavioral preventive intervention with families of depressed parents[END_REF]. Thus the family-oriented approach is promising. However, there is a lack of research on the effectiveness of family-oriented programs for younger children of selective target groups. Early prevention at preschool age could be particularly promising because these years are crucial for socio-emotional development, which is influenced by familial context [START_REF] Reichle | Childhood social and emotional development in family contexts[END_REF].

The impact of a prevention program does not only depend on effectiveness but also on reach of the target group. Even the most effective approach is useless if motivating the target group to participate fails. Recruitment is an issue in most selective and indicative prevention domains, and due to the nature of the disorder may be even more a problem among depressive populations (Beeber et al. 2007, Pauschardt et al. in press). In order to be successful, an intervention setting needs to attract emotionally burdened families and provide the necessary time and space requirements for a family-oriented program. In Germany, mother-child rehabilitation centers fulfill these conditions. These clinics offer prevention and/or treatment for mothers and fathers with or without diagnosis of somatic or psychosomatic disorders. Studies show that in up to 50% of women seeking support from these institutions clinically relevant psychological symptoms can be observed [START_REF] Meixner | Externe Qualitätsmessung in Mutter-Kind und Müttereinrichtungen[END_REF]). More than 80% of these mothers report depressive mood at the beginning of their rehabilitation stay. A regular stay lasts three weeks, in which psychosocial and effectiveness of the program can be found elsewhere [START_REF] Kötter | EFFEKT-E: Forschungsprojekt zur Prävention von Depression in Mutter-Kind-Kliniken[END_REF][START_REF] Kötter | Mütterliche Depressivität, Erziehung und kindliche Erlebens-und Verhaltensprobleme[END_REF][START_REF] Bühler | Prävention der familialen Transmission der Depression bei Kindern depressiver Mütter[END_REF]Kötter et al. submitted).

Method Development of the program

EFFEKT-E is an adaptation of a universal program (EFFEKT ® , www.effekt-training.de) which trains parenting behavior and children's social competence in order to prevent general child behavioral problems. EFFEKT ® was chosen as a basis for the selective program because it is based on theoretically sound concepts, evaluated in controlled studies, and showed desirable effects in universal and selective prevention [START_REF] Kötter | Mittelfristige Effekte des Präventionsprogramms EFFEKT-E für emotional belastete Mütter und ihre Kinder unter besonderer Berücksichtigung psychosozialer Risikofaktoren[END_REF][START_REF] Lösel | Universal prevention of antisocial development: Short-and long-term effects of a child-and parent-oriented program[END_REF][START_REF] Runkel | EFFEKT-Interkulturell: Implementierung und Evaluation eines präventiven Eltern-und Kindertrainings an Grundschulen mit einem hohen Anteil von Kindern mit Migrationshindergrund[END_REF][START_REF] Stemmler | Improving parenting practices in order to prevent child behavior problems: A study on parent training as part of the EFFEKT[END_REF][START_REF] Beelmann | Wirksamkeit eines sozialen Problemlösetrainings bei entwicklungsverzögerten Vorschulkindern[END_REF].

Adaptation was necessary in terms of content and delivery. In order to identify specific risk factors for development of child behavior problems in families with a depressed parent a literature search was conducted. It resulted in reviews and single studies on development of anxiety and depression in children and adolescents, on parental depression as well as on depression prevention approaches Most studies stem from US-research groups. There was no need to add elements with regard to development of externalizing problems as the original program EFFEKT ® focusses on this domain. Adapting the program to the setting of a mother-child clinic included a survey among 15 initially interested clinics which informed about realistic and sustainable implementation conditions (number of sessions, length and timing of sessions, mediators of program).

Implementation and Evaluation

An evaluation study was carried out in 13 mother-child-clinics in Germany. A quasi-experimental design was implemented that contained a pre-post and follow-up comparison with an equivalent control group (cohort design, see figure 2 Mothers completed questionnaires at arrival and end of their three week's stay at the mother-child health resorts (t0 and t1) as well as six months later back home (t2). Short German versions of standardized instruments were used for assessing mediating and outcome variables (SBQ Social Behavior Questionnaire, [START_REF] Tremblay | A prosocial scale for the Preschool Behavior Questionnaire: Concurrent and predictive correlates[END_REF]. Mothers reported on parenting behavior (APQ Alabama Parenting Questionnaire [START_REF] Shelton | Assessment of Parenting Practices in Families of Elementary School-Age Children[END_REF] and cognitions (PSOC Parenting Sense of Competence Scale [START_REF] Johnston | A measures of parenting satisfaction and efficacy[END_REF]; PSI Parenting Stress Index [START_REF] Abidin | Parenting Stress Index[END_REF]) and on children's social behavior and skills (VBV Verhaltensbeobachtungsbogen für Vorschulkinder [START_REF] Döpfner | Verhaltensbeurteilungsbogen für Vorschulkinder (VBV)[END_REF]. For process evaluation purposes, mothers filled in evaluation forms for each training session as well as global ratings at the end of the training. Trainers of the children's program rated children's on-task and off-task behavior during each session. Trainers of both courses also self-rated the quality of implementation of each session. Methodological details are reported in [START_REF] Kötter | EFFEKT-E: Forschungsprojekt zur Prävention von Depression in Mutter-Kind-Kliniken[END_REF], [START_REF] Bühler | Prävention der familialen Transmission der Depression bei Kindern depressiver Mütter[END_REF], and Kötter et al. (in press).

The EFFEKT-E program

The new EFFEKT-E-program aims at improving specific risk-factors for development of internalizing and externalizing behaviors in families with a depressed parent. In line with [START_REF] Garber | Depression in children and adolescents: Linking risk research and prevention[END_REF] and [START_REF] Goodman | Depression in mothers[END_REF] EFFEKT-E consists of (a) a parenting module which focuses on

• positive parenting and psycho-educative elements to ameliorate mother-role-related depressive cognitions and perceptions, • adequate discipline techniques,

• coping in depression-specific stress situations including seeking for social support

The EFFEKT-E parent training:

The training contains six sessions to be held twice weekly within three weeks (see Table 2 for a short description of each session). Four sessions seek to improve general parenting skills, i.e. principles of positive parenting, requesting cooperation, adequate discipline, and enhancement of the family cohesion and social relations. Two sessions deal with depression-specific content: coping with stressful parenting situations, especially during periods of depression, and dysfunctional thoughts about the maternal role. In order to enhance learning effects, the training incorporates various methods (presentations, group discussions, role plays, home-work, etc.).

The EFFEKT-E child training:

Due to setting-specific reasons, the 15 session long original children's training ("I can problem solve", German adaptation by Beelmann et al. 2007) had to be reduced to 6 sessions. The training is a manualized program targeting social problem-solving skills. The content of the training was slightly changed for the selective approach (addition of the emotion "guilt"). A mother-child session was added to exercise problem solving and parenting behavior in the mother-child dyad and facilitate transfer into daily life. Through means of roleplays, question-answer-rounds, active games, pictures and hand puppets, children learn socialcognitive problem-solving skills with an emphasis on the identification of emotions as well as causes and reasons for behavior (see table 3 for a short description of each session).

Insert Table 1 and 2 about here

Results

Within the evaluation study, 31 EFFEKT-E courses were implemented with numbers of participants ranging from 4 to10 mothers and children respectively. The trainings were held by ten different teams who belonged to the staff of the particular mother-child clinic.

Mediation analysis of the baseline data was used to test the prevention model of the program [START_REF] Kötter | Mütterliche Depressivität, Erziehung und kindliche Erlebens-und Verhaltensprobleme[END_REF]. The model was supported by the finding that the association between maternal depression and children's emotional problems was fully mediated by perceived parental stress.

Perceived stress partly mediated the relationship between maternal depression and hyperactivity of the child. Unexpectedly, maternal depression did not correlate with parenting behaviors (Kötter et al., 2010).

With regard to reach of the target group, using mother-child-clinics as an access strategy was successful. On average, participating mothers were 35 years old and had two kids (TG M=35,8 years, of age 1,97 children; CG M=36,2 years of age, 1,99 children, see [START_REF] Bühler | Prävention der familialen Transmission der Depression bei Kindern depressiver Mütter[END_REF] for details). About one third were single mothers (TG: 32%, CG: 32,8%), more than two thirds were in part-or full-time work (TG: 72,4%, CG: 70,2%) and had a monthly income of more than 1.500 Euro (TG: 67,8%, CG: 69,5%).About every tenth family had migrant background (TG: 8,3%, CG: 12,2%). Mean depression CES-D score was high (TG: M=28,8, CG: M=26,3) at entry of the clinic stay but only about 25% of patients had already been identified to have a clinical diagnosis of a mood disorder (TG: 25,5%, CG: 26,5%). Children's mean age was 5,3 years (TG: M=5,3, CG:M=5,32) and 45% were female (TG: 44,5%, CG: 45,8%). Nearly all children were in child care or first grade of school (TG: 99,3%, CG: 98,4%). More than 75% of the mothers reported that their child had not yet needed treatment due to psychosocial problems (TG: 81,4% CG:76,3%). There were no significant differences between the treatment and control sample. Thus due to the pretest-equivalence of the groups and the clinically relevant level of depression of the sample far beyond the critical value of the CES-D [START_REF] Radloff | The CES-D scale: A self-report depression scale for research in the general population[END_REF] testing the effectiveness hypothesis seems appropriate.

Implementation and acceptance of the training content was satisfying. Contents were nearly completely realized in the parent and child component. With regard to implementation quality, 77% of the trainers stated on average, that they could implement the sessions very successfully or successfully. Children's training was seen as being implemented very successfully or successfully by 97% of program facilitators [START_REF] Kötter | EFFEKT-E: Forschungsprojekt zur Prävention von Depression in Mutter-Kind-Kliniken[END_REF]. Process evaluation data showed that mothers were satisfied with the content and composition of the parent training. In general they rated the program content as very helpful. Children actively participated and enjoy their seven EFFEKT-E sessions [START_REF] Kötter | EFFEKT-E: Forschungsprojekt zur Prävention von Depression in Mutter-Kind-Kliniken[END_REF].

Outcome evaluation indicated an intervention effect on the main outcome criterion, i.e. children's emotional disruption at the end of the clinic stay (posttest) and six months later (follow-up) [START_REF] Bühler | Prävention der familialen Transmission der Depression bei Kindern depressiver Mütter[END_REF]. At follow-up, there was an intervention effect on internalizing and hyperactive behavior. Children whose mothers took part in the prevention program were rated as heving less emotional problems and less hyperactivity and inattention. Development of externalizing behaviors did not differ between the EFFEKT-E and the TAU comparison group.

Regarding risk factors for maladaptive development, there was a short-term effect on problematic parenting but no effect on positive parenting. Parental sense of competence was influenced to the largest extent. EFFEKT-E mothers reported to be more self-efficacious and content in their parental role at posttest and follow-up. After the training, disturbed interaction -one factor of perceived parental stress -was significantly lower than in the comparison group. We did find a short-term intervention effect for children's social competence [START_REF] Bühler | Prävention der familialen Transmission der Depression bei Kindern depressiver Mütter[END_REF]Kötter et al. in press;Kötter et al. submitted). Effect sizes for children's behavior and disturbed interaction between mother and child are small and large for maternal sense of competence (Kötter et al. submitted).

Conclusions

A new prevention program for children of depressive mothers was developed on a sound theoretical and empirical basis. The family oriented approach focuses on important malleable risk factors for child development such as parenting behavior and beliefs, familial stress exposure, and children's social-cognitive skills [START_REF] Goodman | Transmission of risk to children of depressed parents: Integration and conclusions[END_REF][START_REF] Garber | Depression in children and adolescents: Linking risk research and prevention[END_REF]. A mediation analysis with the evaluation study baseline data supported the prevention model, i.e. perceived parenting stress and competence mediated the relationship of maternal depression and children's emotional and behavioral problems. The program has proven to attract the target group, to be accepted by mothers, children and trainers, and to be implementable in the routine practice of mother-childclinics. Finally, the program has shown various desirable effects. When mothers and children participated in the EFFEKT-E program offspring on average showed less emotional and hyperactive behavior problems. Conclusions are limited insofar as effects are based on motherreported data and cover a follow-up observation period of six months. In addition, the average effect sizes were not large. However, one must take into account that EFFEKT-E is a short intervention that was added to the much more intensive usual treatment in mother-child rehabilitation centers. Because the program is relatively easy to deliver in a group format, the findings of our evaluations are also encouraging with regard to cost-benefit issues. But at the end, an even longer-term follow up study would be necessary to prove whether a short term intervention like EFFEKT-E in a special setting without aftercare in daily life is able to favourably shape the developmental trajectory of a depressive disorder in children of depressed mothers.

Depressed mother

Heritability of depression

  medical treatment are administered. Mothers are accompanied by their child(ren) which in most cases have not yet developed behavioral or emotional problems. Within this context we aimed to develop and evaluate a family-oriented program for depressed mothers and their pre-school children. The program should prevent the offspring's emotional and behavioral problems by promoting maternal parenting behavior and children's social-cognitive problem solving skills during their stay in mother-child rehabilitation clinics. This work was supported by a grant from the prevention research program of the German Federal Ministry of Education and Research (BMBF). The project is a cooperation between the IFT Institut für Therapieforschung, Munich, the national welfare organization "Bundesverband Arbeiterwohlfahrt" and their clinics, and the Institute of Psychology at the University of Erlangen-Nuremberg. This paper focuses on the development of the prevention program EFFEKT-E and briefly sums up evaluation research findings. Detailed information about basic assumptions, implementation, and

  ) Data of the control group of mother-child-dyads were collected in clinics first (TAU treatment as usual condition). During their three week's stay, mother-specific rehabilitation included various medical and psychosocial treatment but no family-oriented parenting program. Children did not receive special treatment but spent their time in day care. Afterwards clinic staff was trained to deliver the program in the subsequent treatment phase. Mothers and children in the training group participated in the usual clinic procedures plus the EFFEKT-E-training for mothers and children. A total of 2379 mothers were admitted to the clinics during the study interval in 2007 (treatment group (TG) and control group (CG)). At arrival, a mother was asked to enroll in the study if she (a) had a child aged between four and seven years accompanying her and if she (b) met criteria for depression (critical value on CES-D (above 16, Radloff 1977) or documented diagnosis of depressive disorder). Screening revealed a subgroup of 33% depressive mothers, of which 70% agreed to participate in the study. A retention rate of 69% was reached over the pretest-posttest and 6 months follow-up period. The sample consisted of 375 families that provided data during the clinic stay. 31 training courses were delivered with numbers of participants ranging from 4 to10 mothers and children respectively.

  and (b) a child training which targets problem solving in order to enhance social-cognitive skills. Our proposed prevention model is based on Goodman and Gotlib's Integrative Model for the Transmission of Risk to Children of Depressed Mothers (1999, 2002) considering selected environmental risk-factors which are highlighted in Figure 1. Insert Figure 1 about here EFFEKT-E is implemented by trainer teams. Clinical psychologists run the parent training, certified social workers are in charge of the child component. Teams participate in two-day training workshops. Materials are available at the University of Erlangen-Nürnberg (www.effekttraining.de).

Figure 1 :

 1 Figure 1: Integrative Model for the Transmission of Risk to Children of Depressed Mothers[START_REF] Goodman | Risk for Psychopathology in the Children of Depressed Mothers: a Developmental Model for Understanding Mechanisms of Transmission[END_REF], 2002) as a basis for development of EFFEKT-E

Table 2 :

 2 Content of the EFFEKT-E child training

	Session 1	Preliminary, word-concepts and definitions
	Session 2	Emotions (happy, sad)
	Session 3	Emotions (anxious, angry, guilty)
	Session 4	Causality of actions (What happens next?)
	Session 5	Alternative solutions (What else could you try?)
	Session 6	Consequences of actions/ Evaluation of solutions
	Session 7	Exercising problem solving and communication in mother-child dyads

Acknowledgements

The study was funded by the German Federal Ministry of Education and Research. We wish to thank the Bundesverband Arbeiterwohlfahrt and clinic staff for their cooperation and mothers and children for study participation.