A Stahl

Existence of a stationary distribution for multi-dimensional infinite volume forest-fire processes

Consider the following forest-fire process on a connected graph. Each site of the graph can be either occupied or vacant. A vacant site becomes occupied with rate 1. A site is ignited with rate λ, and its whole occupied cluster burns instantaneously.

The purpose of this paper is to show the existence of a stationary distribution for forestfire processes on Z d , for d ≥ 2. We define a distribution on {0, 1} Z d as a limit of a sequence of invariant distributions of finite volume forest-fire processes, and then show it is a stationary distribution for forest-fire processes on Z d , d ≥ 2.

Introduction

Forest-fire models have been introduced within the study of self-organised criticality ( [START_REF] Bak | A forest-fire model and some thoughts on turbulence[END_REF]). The idea behind self-organised criticality is the evolution of the system toward a critical behavior, only driven by its own interaction rules (see [START_REF] Jensen | Emergent complex behavior in physical and biological systems[END_REF] for example). In the nineties, a forest-fire model in discrete time and on a finite graph, introduced by Drossel and Schwabl [START_REF] Drossel | Self-organized critical forest-fire model[END_REF], has been studied in the physics literature. In 2005 in [START_REF] Van Den Berg | On the asymptotic density in a one-dimensional selforganized critical forest-fire model[END_REF] van den Berg and Jarai studied the asymptotic behavior of a continuous-time version of the Drossel-Schwabl forestfire model, on an infinite graph : Z. Then this model has been studied on Z, on Z d and on connected graphs with a vertex degree bounded by d, for d ≥ 2.

Let G = (V, E) be a connected graph with vertex degree bounded by d ≥ 2. V is the set of vertices and E the set of edges of the graph. In the model, each site of V can be vacant or occupied by a tree. Two sites of V are neighbors if they are linked by an edge in E. The evolution is driven by the initial configuration and two families of Poisson processes, one governing the birth of trees and the other the ignition of a site. Throughout this paper "birth" of a tree and "growth" of a tree have the same meaning. An empty site becomes occupied if and only if there is a birth attempt. An occupied site becomes empty either if the site is ignited or if a site of its connected cluster of occupied sites is ignited. The second case models the propagation of fire. As the burning is instantaneous here, the speed of the fire propagation is infinite.

More precisely, the configuration at time t of the forest is the process ((η t,x ) x∈V ) t≥0 where η t,x takes the value 1 if the site x is occupied at time t and 0 otherwise. If the site x is occupied at time t, its connected cluster C t,x is the set of all occupied sites connected to x by a path of occupied sites.

Definition 1. (open path and connected cluster of a site)

1. A path between 2 sites x and y in V is a set of sites {z i } 1≤i≤n of V such that z 1 = x, z n = y and for all integers i ∈ {1, . . . , n -1} there is an edge of E linking z i and z i+1 .

The path is open at time t if all its sites are occupied at time t.

2. The connected cluster C t,x of a site x at time t is the set of all sites of V linked to x with an open path.

To each site x ∈ V are associated two Poisson processes, (G t,x ) t≥0 and (I t,x ) t≥0 , with intensity 1 and λ > 0 respectively. The process G x represents the birth attempts on the site x whereas I x represents ignition attempts on the site x. These two processes are independent and moreover independent of each Poisson process of all the other sites. The following definition gives a rigourous definition of the forest-fire process. Definition 2. Let S be a subset of V and λ a non negative real number. A forest-fire process on S with parameter λ > 0 is a process (η t ) t≥0 where η t = (η t,x ) x∈S = (η t,x , G t,x , I t,x ) x∈S , with values in ({0, 1} × N × N) S , that has the following properties.

1. For all sites x in S, the processes (G t,x ) t≥0 and (I t,x ) t≥0 , are independent Poisson processes with parameter 1 and λ respectively.

2. For all sites x in S, the process (η t,x , G t,x , I t,x ) t≥0 is right continous and left limited.

3. For all x ∈ S and all t ≥ 0, (a) if there is a growth of a tree at the site x at time t, then the site

x is occupied at time t, G t -,x < G t,x ⇒ η t,x = 1,
(b) if the site x becomes occupied at time t, then a tree must have grown at the site

x at time t η t -,x < η t,x ⇒ G t -,x < G t,x , (c)
if the site x is ignited at time t, then all the sites of its connected cluster get vacant at time t, I t -,x < I t,x ⇒ ∀y ∈ C t -,x , η t,y = 0, (d) if the site x gets vacant at time t, then the connected cluster of the site x has been ignited at time t, η t -,x > η t,x ⇒ ∃y ∈ C t -,x : I t -,y < I t,y .

The notation η(t) will be used to denote the configuration at time t : {η t,x } x∈S .

On a finite graph, the forest-fire process is uniquely determined by the following contruction. At time zero, the initial configuration of the forest is given. Then, the evolution is governed by the Poisson processes. Due to the finite number of sites, all the Poissonian events can be ordered in time. Then, four situations can occur. Imagine that there is a growth attempt on the site x. If the site x is vacant, then it becomes occupied. But if x is already occupied, nothing happens. Imagine now that there is an ignition attempt on the site y. If the site y is vacant, nothing happens. But if y is occupied, then its whole connected cluster burns down immediately. At each Poissonian event, the configuration is updated using these rules.

However on an infinite graph, infinitely many Poissonian events can occur in a finite interval of time. Thus the construction given above does not work anymore. Does this forest-fire process exist on infinite graphs? The answer is yes. It can easily be seen on Z using a graphical construction ( [START_REF] Liggett | Interacting particle systems[END_REF]) but it is harder on Z d for d ≥ 2. On Z, if an initial configuration has infinitely many empty sites, then at a time t > 0 there are almost surely infinitely many empty sites that have been empty during the whole interval of time [0, t]. An empty site blocks the propagation of fire on Z. So, Z can be divided into finite pieces and the previous construction can be used on each piece. But on Z d for d ≥ 2, an empty site alone cannot block fires. By using a sequence of finite-volume processes and finding a control of the influence of the state of long-distance sites, Dürre has shown in [START_REF] Dürre | Existence of multi-dimensional infinite volume self-organized critical forestfire models[END_REF] the existence of forest-fire processes on Z d for all initial configurations that contain no infinite clusters. This result can be extended to connected graphs with vertex degree bounded by d ≥ 2. However, nothing ensures that given an initial configuration and all the Poisson processes, the forest-fire process is unique. This uniquenesss result is proven for all parameter λ > 0 in [START_REF] Dürre | Self-organized critical phenomena: Forest fire and sandpile models[END_REF], with an assumption on the size of the clusters in the initial configuration. The question of uniqueness for all initial configurations is still open.

In this paper, we will be interested in stationary distributions of the forest-fire process. The forest-fire process is a Markov process but not a Feller process. Thus the usual arguments used with interacting particles systems will not work here (see for example [START_REF] Liggett | Interacting particle systems[END_REF]). Brouwer and Pennanen have shown in [START_REF] Brouwer | The cluster size distribution for a forest-fire process on Z[END_REF] the existence of at least one stationary distribution for the process on Z. In [START_REF] Bressaud | On the invariant distribution of a one-dimensional avalanche process[END_REF] Bressaud and Fournier have shown that the stationary distribution is unique when the parameter λ is equal to 1.

The purpose of this note is to prove the existence of a stationary distribution for forest-fire processes on Z d with d larger than 2. The method of Brouwer and Pennanen combined with some tools introduced by Dürre in his thesis [START_REF] Dürre | Self-organized critical phenomena: Forest fire and sandpile models[END_REF] will be used. Since the existence of forest-fire processes on Z d is known yet for initial configurations that contain no infinite clusters, we will consider only such forest-fire processes in this paper.

Theorem 1. A forest-fire process on Z d with d ≥ 2, with parameter λ > 0 has at least one stationary translation-invariant distribution.

This paper is divided into three parts. The first section presents the main notations and the tools introduced by Dürre in [START_REF] Dürre | Self-organized critical phenomena: Forest fire and sandpile models[END_REF] that will be used in this paper. Then, the construction of the candidate for the stationary distribution is explained in the second part. Finally the last section is devoted to the proof of Theorem 1.

Notations and tools

The purpose of this section is to present the main notations and the tools introduced by Dürre in [START_REF] Dürre | Self-organized critical phenomena: Forest fire and sandpile models[END_REF].

Notations

Unless stated otherwise, in this section, G will denote a connected graph with a vertices set V and an edges set E. A bound on the vertex degree of the graph G will be denoted by d G .

For x = (x 1 , . . . , x d ) ∈ Z d , |x| ∞ = sup {|x i |, i = 1 . . . d} and |x| 1 = d i=1 |x i |.
The expression the graph Z d will refer to a graph with vertices set V (d) = Z d and edges set E (d) corresponding to the hypercubic lattice :

E (d) = {x, y} ∈ Z d × Z d : |x -y| 1 = 1 .
The site (0, . . . , 0) ∈ Z d will be called the origin. We will work with specifc finite subsets of Z d called boxes, and defined by

B k = {x ∈ Z d : |x| ∞ ≤ k}.
Here k is called the radius of the box B k .

The set of neighbors of a finite set of sites S is called the boundary of S :

N (S) = {x ∈ V \ S, ∃ y ∈ S linked by an edge in E to x}.
We can define the set S = S ∪ N (S).

Let X = {0, 1} Z d be the set of all possible configuations and B(X) the associated borel σ-field. Let µ be a measure on (X, B(X)). For a finite subset J of Z d we denote by µ |J the restriction of µ to J. For a forest-fire process η with an initial configuration with law µ, let us denote by P µ (η t ∈ •) its distribution at time t.

Dürre's tools

In this model, the problem is to control the influence of long-distance sites. Two notions introduced by Dürre in his thesis will be used in this paper. The first one is a condition on the tail of the distribution of cluster size, named CCSB. The second one is a process, called the blur-process, used to study the influence on a given site of long-distant sites. Throughout this section, forest-fire processes on connected graphs will be considered.

CCSB condition

An important quantity to consider here is the size of clusters. In his thesis, Dürre restricted to forest-fire processes with initial configurations satisfying what he called the conditionned cluster size bound condition to prove the uniqueness. In order to use his results, we will need to use this conditionned cluster size bound condition.

This condition is a constraint on the cluster size distribution. It implies a uniform bound on the cluster size distribution. It is stronger in the sense that the result still holds when conditionning by the event that a specific configuration of a finite number of sites outside the cluster occurs. Definition 3. (Conditionned cluster size bound, CCSB) For all s ≥ 0, δ ≥ 0 and m ∈ N, η has CCSB(s, δ, m) if it has the following property. Let B an D be two finite subsets of V and x ∈ V \ D. Then conditionned on the occurence of the event ∪ y∈B C s,y = D, the probability that the size of the cluster at x is larger than m at time s is smaller than or equal to δ. And almost surely the cluster at x is finite at time s.

More formally, for all finite B, D ⊂ V , for all x ∈ V \ D,

P |C s,x | > m y∈B C s,y = D ≤ δ P y∈B C s,y = D ( 1
)
and P |C s,x | = ∞ = 0. ( 2
)
For example, consider a configuration where the size of all the clusters is smaller than an integer m, or a configuration obtained by sub-critical or critical site percolation on Z d . Then, a forest-fire process with one of these initial configurations satisfy a CCSB(0, δ, m) condition.

A natural question to ask is whether a forest-fire process satisfies a CCSB condition. If the forest-fire process has a CCSB at time 0, then the following theorem states that after a certain time the answer is yes. The uniformity in time and in δ will be useful in the proof of the result.

Theorem 2. (Theorem 2 in [START_REF] Dürre | Self-organized critical phenomena: Forest fire and sandpile models[END_REF]) Consider a forest-fire process η on G with parameter λ > 0 that satisfies CCSB(0, λ 4d 2 G , m) for some integer m. Then for all real number γ > 0, for all δ ∈]0, 1], there exists an integer m γ,λ,d G (δ) such that for all time s ≥ γ, η has CCSB(s, δ, m γ,λ,d G (δ)).

In [START_REF] Dürre | Self-organized critical phenomena: Forest fire and sandpile models[END_REF] an explicit formula for the parameter m γ,λ,d G (δ) appearing in the CCSB condition is given.

Blur process

We turn now to the blur-process. The aim of this process is to control the influence of long-distant sites on a given site. This process is associated with a forest-fire process η, and defined from an initial time t 0 and a finite set S. The blur process indicates if the state of a site x ∈ S at time t ≥ t 0 might be influenced by the state of the sites located outside the set S at time t 0 . If it might, the site is said to be (t 0 , S)-blurred at time t. The word "might" is important here. It means that the blur-process is a domination of the process of the real influence from outside S. If the site x is not (t 0 , S)-blurred at time t, its state at time t can be recovered by using only the configuration at time t 0 and the Poisson processes of the sites in the set S.

The value of the (t 0 , S)-blur process at a site x ∈ S at a time t is either 2 if the site is blurred or 0 otherwise. Once a site is blurred, it is blurred forever. The set of blurred sites at t 0 is N (S) and the clusters in η(t 0 ) intersecting the boundary of S. As time evolves, this set grows toward the site x, due to births of trees that create new occupied paths. Here is the formal definition of the blur process. Definition 4. (blur process) Let t 0 ≥ 0 and S ⊂ V a finite subset. The (t 0 , S)-blur process (β s,x ) s≥t 0 ,x∈S is a process with values in {0, 2} S , right continuous that has the following properties. For all site x in S,

• the site x is (t 0 , S)-blurred at time t 0 if and only if its cluster is connected to the boundary of S :

β t 0 ,x = 2 if C s,x ∩ N (S) = ∅ 0 else,
• once a site is (t 0 , S)-blurred, it remains (t 0 , S)-blurred forever :

if β s,x = 2, then ∀s ≥ s, β s ,x = 2,
• the site x is (t 0 , S)-blurred at time s > t 0 if and only if the set C s,x contains a site that has been (t 0 , S)-blurred before time s :

{β s,x = 2} = ∃z ∈ C s,x ∩ S : β s -,z = 2 .
Like for the configuration process, the notation β(t) will be used for {β t,x } x∈Z d .

It seems that the bigger the set S is, the longer it takes for the site x to be S-blurred. What happens when the size of the set S tends to infinity? Is there a non empty interval during which a site is not influenced by "infinity"? The answer is given by the following proposition, assuming a CCSB condition is satisfied by the forest-fire process. Proposition 3. (Proposition 2 in [START_REF] Dürre | Self-organized critical phenomena: Forest fire and sandpile models[END_REF]) For all m ∈ N, there exists m > 0 with the following property. Let t ≥ 0 and suppose that the forest-fire process has CCSB(t, λ 4d 2 G , m). Then for all sites x in V , as n tends to infinity, the probability that the site x is (t, B n )-blurred at time t + m tends to zero :

lim n→∞ P (β t+ m,x = 2) = 0.
Remark. The real number m > 0 is chosen such that P (G 0, m > 0) < 1 4md G , where G 0, m is the event that there is at least an occurrence in [0, m ] of a Poisson process with intensity 1.

Remark. This proposition is slightly different from the one written in [START_REF] Dürre | Self-organized critical phenomena: Forest fire and sandpile models[END_REF]. The additionnal hypothesis of almost-sure convergence at time t is omitted here. Actually Dürre has shown that it is always satisfied for a forest-fire process having a CCSB condition at time zero.

A candidate distribution

To prove the existence of a stationary distribution, we exhibit a distribution and then show that it is stationary. The aim of this section is to explain how the candidate distribution is obtained. The idea is to use forest-fire processes on finite graphs. To get the translationinvariance property of the stationary distribution, a modified version of the boxes B k will be used.

Definition of "finite volume" forest-fire processes

Let k be a non negative integer.

The first step is to define a graph. Contrary to what was mentionned above, the graph defined below is infinite. The fact that the process defined on it will behave like a finitevolume forest-fire explains why it is called "finite volume".

Define a graph G k = (V (d) , E k ). The set of sites is V (d) = Z d . The set of edges E k is bigger than the set of edges of the graph Z d . Some edges constructed from the box B k defined in Section 1.1 will be added. The goal is to make opposite sites of the interior boundary of B k neighbors.

Recall that E (d) is the set of edges corresponding to the hypercubic lattice of Z d (see Section 1.1).

Define now the additionnal set of edges A k . For all indices j in {1, . . . , d}, consider a site with the j-th coordinate equal to k and the other coordinates in the set {-k, . . . , k}. It is an interior boundary point. Then consider the opposite boundary site, i.e. the site with its j-th coordinate equal to -k and the other coordinates exactly the same as the first site. Then A k consists in the edges between such pair of points for the whole interior boundary of B k . More formally, for each j ∈ {1, . . . , d} the edges between the following couples of sites are in A k :

(i 1 , . . . , i j , . . . , i d ); (i 1 , . . . , -i j , . . . , i d ) with

i l = k if l = j i l ∈ {-k, . . . , k} if l = j.
Then the set E k is the union of the sets E (d) and A k . The extra edges are not necessary to prove the existence of a stationnary distribution but there are used to prove that the stationnary distribution is translation-invariant. Now the graph is defined, the second step is to consider a particular forest-fire process on this graph. Definition 5. A G k -forest-fire process ηk with parameter λ > 0 is a forest-fire process on the graph G k with parameter λ > 0 with all the sites outside B k always empty : for all times t ≥ 0, for all sites x ∈ B c k , η k t,x = 0.

This forest-fire process behaves like a forest-fire process on B k where the opposite sites of the interior boundary are neighbors. The dynamic does not take into account the Poisson processes corresponding to the sites located outside B k . However, its configurations take values in {0, 1} Z d . The G k -forest-fire process can be seen as a forest-fire process on a discrete torus of dimension d.

Construction of the candidate measure

For each non negative integer k, let ηk be a G k -forest-fire process. The configuration process η k is a finite state-space Markov chain. Consider two possible states of this Markov chain ξ 1 and ξ 2 and study the probability that the state ξ 1 leads to ξ 2 . Consider the following event: all the occupied sites in ξ 1 are ignited and then a tree grows on each site which is occupied in ξ 2 . As the growths and ignitions are driven by independent Poisson processes, this event occurs with a non negative probability. Therefore, this Markov chain is irreducible, recurrent and aperiodic. Thus it has a unique invariant distribution, which will be denoted by µ k .

Since the space {0, 1} Z d is compact, the sequence (µ k ) k≥0 has a weakly convergent subsequence. Let µ be its limit and K the set of indices of the subsequence. Theorem 4. Let η be a forest-fire process on Z d with parameter λ > 0. Then µ is an invariant and translation-invariant probability measure for η.

As mentionned above, the G k -forest-fire process behaves exactly as a forest-fire process on a discrete torus. Thus each distribution µ k can be seen as a measure on a discrete torus, where we have the rotation-invariance property. Here a translation on the graph G k corresponds to a rotation on the discrete torus with (2k + 1) d points, due to the G kforest-fire process behavior. It follows that the weak limit µ is translation-invariant.

Proof of the result

The goal of this section is to prove Theorem 1. It follows from Theorem 4 stated above. This section is divided in three parts. The first part contains the proof of Theorem 4 assuming two lemma. The lemma are then proved in the second and third part.

Proof of Theorem 4

To show that the distribution µ is stationary, the idea is to couple a forest-fire process with initial configuration with law µ with G k -forest-fire processes with initial configuration with law µ k using the same Poisson processes. Then, the evolution of the processes will be compared using Dürre's tools.

A diagram of the main steps of the proof is given at the end of this first subsection. Let λ and γ be two non negative real numbers. For simplification, as λ and d f are fixed here, let us denote by m γ the real m γ,λ,d f ( λ

4 d 2 f ) of Theorem 2 (recall that d f = 3d, see Section 2.1). Let > 0 be a real number such that P (G 0, > 0) < 1 4mγ d 2 f
, where G 0, is the event that there is at least an occurrence in [0, ] of a Poisson process with intensity 1.

The aim is to prove the following : for all times t < , for all cylinder events A,

P µ (η(t) ∈ A) = µ(A). (3)
Let t be a non negative real number such that t < , and let A be a cylinder event.

Throughout this proof, two specific sets of sites will be used.

Since A is a cylinder event, it is determined by the configuration on a finite number of sites. Let I be the smallest box centered at the origin such that the event A can be described using only sites in I. There exists r I such that I = B r I .

We will need to work with a set containing I but contained in B k . Let L be a non negative integer and J be the box B r I +L . In order to simplify the notation, we will not write the dependence of J on L. Using the construction of the candidate measure µ, a first upper bound is obtained :

|P µ (η(t) ∈ A) -µ(A)| ≤ |P µ (η(t) ∈ A) -µ k (A)| + |µ k (A) -µ(A)|. ( 4
)
Since µ k is an invariant distribution for a G k -forest-fire process, the equality ( 3) is satisfied for µ k . Turn now to study of the difference

|P µ (η(t) ∈ A) -P µ k (η k (t) ∈ A)|.
Consider a forest-fire process on Z d and a G k -forest-fire process, driven by the same Poisson processes on each site, and with initial configurations which coincide on J. Is it possible to find a time t when the two configurations on a site x in I are not the same? The answer is yes if (and only if) the configuration outside the box J influences the configuration in the box I during the time interval [0, t] (through some fires outside J). Which events can allow influence on I from outside J? If at a time t 0 in [0, t] there exists a path of occupied sites linking a site in I to a site outside J, then a lightning falling in J c can burn trees in I. That means, using the blur process, the existence of a site in the set I that is (t, J)-blurred.

However, in the difference we are interested in, the initial configuration of the two processes have the respective distribution µ and µ k . In order to use our previous argument, we are going to optimally couple the measures µ |J and µ k |J . The coupling of µ |J and µ k |J is said to be optimal here if it maximizes the weight of the diagonal of the product of the spaces of configurations. For two configurations ξ 1 with law µ |J and ξ 2 with law µ k |J , the optimal coupling ν satisfies ν(ξ

1 = ξ 2 ) = inf π∈M π(ξ 1 = ξ 2 )
, where M is the set of all possible couplings of µ |J and µ k |J .

Now we couple a forest-fire process η on Z d with an initial configuration of law µ and the G k -forest-fire process ηk with an initial configuration of law µ k , by using the same Poisson processes and by optimally coupling µ |J and µ k |J . Then the following bound is obtained. Lemma 5. Let us denote by β the (0, J)-blur process associated with η. Then,

|P µ (η(t) ∈ A) -P µ k (η k (t) ∈ A)| ≤ |I| sup x∈I P µ (β t,x = 2) + 2 d T V (µ |J , µ k |J ). (5)
This lemma will be proved in the following section. The idea of the proof is to use the event "the initial configurations of the two processes coincide on J". When it occurs, an upper bound is found using the blur process. Otherwise, the upper bound is given by the total variation between the two measures µ |J and µ k |J . Using this lemma, the upper bound of the desired quantity is :

|P µ (η(t) ∈ A) -µ(A)| ≤ |I| sup x∈I P µ (β t,x = 2) + 2 d T V (µ |J , µ k |J ) + |µ k (A) -µ(A)|. (6)
The next step is to show that the upper bound of (6) tends to zero when k and L tend to infinity.

The distribution µ is the weak limit of a subsequence of (µ k ) k≥0 . We first let k go to infinity along the set of indices K. Since k > radius(J) = r I + L, L does not depend on k and J is a finite set of sites, so :

lim k→∞ |µ k (A) -µ(A)| = 0 (7) lim k→∞ d T V (µ |J , µ k |J ) = 0. (8)
Recall that β is a (0, J)-blur process. So the event {β t,x = 2} depends on L. To conclude we need to show that lim L→∞ P µ (β t,x = 2) = 0. This is given by Proposition 3 applied with t = 0 and δ = λ 4d 2 f , provided the CCSB condition is satisfied. The purpose of the following lemma is to show that this hypothesis holds true. Lemma 6. For all non negative real number δ, a forest-fire process on Z d with parameter λ > 0, with an initial configuration with distribution µ, has CCSB(0, δ, m γ,λ,d f (δ)).

Remark. In the last parameter of the CCSB condition, the bound used for the vertex degree of Z d is d f = 3d and not 2d which is the optimal bound. This is due to our argument to prove this lemma, which uses the G k -forest-fire process. Nevertheless this is sufficient here to show our theorem.

Using Lemma 6 with δ = λ 4d 2 f , we claim that a forest-fire process with an initial configuration with law µ has CCSB(0, λ

4d 2 f , m γ,λ,d f ( λ 4d 2 f )).
By Proposition 3, there exists mγ such that for all site x in Z d , as L tends to infinity, the probability that the site x is (0, J)-blurred at time mγ tends to zero. That is to say, with the choice of made here, that for all t < , for all x ∈ Z d lim

L→∞ P µ (β t,x = 2) = 0.
Since |I| < ∞, this concludes the proof.

We conclude this part with a diagram of the main steps of the proof. As mentionned in the proof of Theorem 4, the idea to prove this lemma is to use the event "the initial configurations of the two processes coincide on J". Then, the study differs depending on whether this event occurs or not. A diagram of the main steps of the proof is given at the end of the section.

goal : ∀t < , |P µ (η(t) ∈ A) -µ(A)| = 0 use of the subsequence of {µ k } k |P µ (η(t) ∈ A) -µ(A)| ≤ |P µ (η(t) ∈ A) -µ k (A)| +|µ k (A) -µ(A)|
Recall that we coupled a forest-fire process η on Z d with an initial configuration of law µ and the G k -forest-fire process η k with an initial configuration of law µ k , by using the same Poisson processes on each site and by optimally coupling µ |J and µ k |J . Let ν be the distribution of the optimal coupling. We have

|P µ (η(t) ∈ A) -P µ k (η k (t) ∈ A)| = |P ν ((η(t), η k (t)) ∈ A × X) -P ν ((η(t), η k (t)) ∈ X × A)|.
Consider now the event "the initial configurations of the two processes coincide on J". Let C J = {η(0) |J = η k (0) |J } be this event. Denote by C c J the complement of this event. Then we can upper bound the difference we are interested in :

|P µ (η(t) ∈ A) -P µ k (η k (t) ∈ A)| = |Dif + Eq| ≤ |Dif | + |Eq| (9)
with

Dif = P ν (η(t), η k (t)) ∈ A × X ∩ C c J -P ν (η(t), η k (t)) ∈ X × A ∩ C c J (10)
Eq = P ν (η(t), η k (t)) ∈ A × X ∩ C J -P ν (η(t), η k (t)) ∈ X × A ∩ C J . (11)
Study of Dif : Since µ |J and µ k |J are optimally coupled,

P ν (η(0) |J = η k (0) |J ) ≤ d T V (µ |J , µ k |J )
and we get

|Dif | ≤ 2 P ν (η(0) |J = η k (0) |J ) ≤ 2 d T V (µ |J , µ k |J ). (12)
Study of Eq : After using the Bayes formula in [START_REF] Liggett | Interacting particle systems[END_REF], we are interested in the difference

Eq = P ν (η(t), η k (t)) ∈ A × X C J -P ν (η(t), η k (t)) ∈ X × A C J • P (C J ). (13)
We can use indicator functions to rewrite the difference in the righthand term :

Eq = E ν (1 X×A (η(t), η k (t)) -1 A×X (η(t), η k (t)) | C J ) • P (C J ).
When is the difference 1 X×A (η(t), η k (t)) -1 A×X (η(t), η k (t)) non zero? It occurs only when (η(t), η k (t)) ∈ A c × A or when (η(t), η k (t)) ∈ A × A c . These events occur when there exists a time t 1 > 0 and a site x in I where the two configurations are not the same, i.e. η t 1 ,x = η k t 1 ,x . The two forest-fire processes are driven by the same Poisson processes and we conditionned by the event "their initial configurations coincide on J". So this non equality must be the consequence of the influence of at least one cluster outside J on the box I. This is equivalent to the existence of a site in I whose state is influenced by the state of a site outside J. More precisely, there exists a site x in I which is (0, J)-blurred at time t 1 .

Firstly we give some notations associated with the blur processes. We denote by β the (0, J)-blur process associated with η and by β k the (0, J)-blur process associated with ηk . To simplify the notation, we do not write the dependence in J (neither in L) of the blur processes. Let N B be the set of configurations where all the sites of the box I have the value 0 and N B x be the set of configurations where the site x has the value 0.

Then with these notations we can write

|1 X×A (η(t), η k (t)) -1 A×X (η(t), η k (t))| ≤ 1 N B c ×N B c (β(t), β k (t)). So |Eq| ≤ E ν (1 N B c ×N B c (β(t), β k (t)) | C J ) P ν (C J ) ≤ P ν ( (β(t), β k (t)) ∈ N B c × N B c | C J ) • P ν (C J )
The (0, J)-blur process associated with a forest-fire process is defined only with the Poisson processes on the sites of the set J and the initial configuration of the sites of the box J. We know that the two initial configurations coincide on J and that the two processes are driven by the same Poisson processes on each site. Thus, the blur processes β and β k are the same. So

|Eq| ≤ P µ (β(t) ∈ N B c ).
The complements of the events N B and N B x satisfy the relation

N B c = x∈I N B c x , so |Eq| ≤ |I| sup x∈I P µ (β(t) ∈ N B c x ). (14)

Conclusion

Since P µ (β(t) ∈ N B c x ) = P µ (β t,x = 2), using equations ( 12) and ( 14) we obtain the desired result.

|P µ (η(t) ∈ A) -P µ k (η k (t) ∈ A)| ≤ 2 d T V (µ |J , µ k |J ) + |I| sup x∈I P µ (β t,x = 2). goal : |P µ (η(t) ∈ A) -P µ k (η k (t) ∈ A)| ≤ |I| sup x∈I P µ (x is (0, J)-blurred at t) + 2 d T V (µ |J , µ k |J )
optimal coupling of µ and µ k either the 2 initial processes coincide on J (Eq) or they are different (Dif )

|P µ (η(t) ∈ A) -P µ k (η k (t) ∈ A)| ≤ |Eq| + |Dif | study of the influence of sites outside J |Eq| ≤ P µ (∃x ∈ I : x is (0, J)-blurred at t) ≤ |I| sup x∈I P µ (x ∈ I is (0, J)-blurred at t) coupling property |Dif | ≤ 2 d T V (µ |J , µ k |J )
conclusion : the desired upper bound is obtained Figure 4: Scheme of the proof of Lemma 5

Proof of Lemma 6

The construction of the distribution µ as a weak limit of the sequence {µ k } k∈K is used to prove this lemma. We will need the CCSB condition for a G k -forest-fire process, which is the result of the following lemma.

Lemma 7. Let n be in N and δ be a non negative real number. Consider a G n -forest-fire process α n with parameter λ, with an initial configuration with distribution µ n . Then α n has CCSB(0, δ, m γ,λ,d f (δ)).

Proof. (of Lemma 7)

Let n and m be in N.

A forest-fire process having a CCSB condition at time 0 satisfies one for all times t > γ by Theorem 2. Thus to show (16), we will use this result combined with the ergodic theorem.

Consider a G n -forest-fire process β n with parameter λ > 0, with an initial configuration with law ρ n and having CCSB(0, λ Recall that γ was set in Section 3.1 and take t > γ.

By Theorem 2, for all times s larger than γ, the forest-fire process β n has CCSB(s, δ, m γ,δ ), so ∀s ≥ λ, P Finally, we let t tend to infinity and use (17) twice to get (16). This concludes the proof.

We can now show the lemma concerning the distribution µ.

Proof. (of Lemma 6)

Let δ > 0.

The idea of the proof is to use the fact that the measure µ is a weak limit of a subsequence of (µ k ) k≥0 , combined with the previous lemma.

To

Figure 1 :

 1 Figure 1: Exemple of additional edges for the graph G 2

Figure 2 :

 2 Figure 2: Sets of sites

Lemma 5 :Figure 3 :

 53 Figure 3: Scheme of the proof

  The goal is to show a CCSB condition. Therefore let B and D be two subsets of B n , x a site in B n \ D and considerCond B,D = y∈B C y = D and Size x,m = |C x | > m .To simplify the notations, we denote m γ,λ,d f (δ) by m γ,δ in this section.The G n -forest-fire process α n has CCSB(0, δ, m γ,λ,d f (δ)) ifP µ n (α n (0) ∈ Size x,m γ,δ ∩ α n (0) ∈ Cond B,D ) ≤ δP µ n (α n (0) ∈ Cond B,D ). (15)Since µ n is the stationary distribution of a G n -forest-fire process, (15) can be written :µ n (Size x,m γ,δ ∩ Cond B,D ) ≤ δµ n (Cond B,D ).

t t 0 P

 0 Since µ n is the unique stationary distribution of a G n -forest-fire process, the ergodic theorem givesµ n (Size x,m γ,δ ∩ Cond B,D ) = lim t→∞ 1 ρ n (β n s ∈ Size x,m γ,δ ∩ β n (s) ∈ Cond B,D )ds.(17)

  show the CCSB condition, consider two finite subsets B and D of Z d , a site x in Z d \ D and Cond B,D = y∈B C y = D and Size x,m γ,δ = |C x | > m γ,δ .For all finite subsets B and D of Z d and all sites x in Z d \ D, the sets Cond B,D and Size x,m γ,δ are defined by a finite number of sites. Therefore we can writeµ(Size x,m γ,δ ∩ Cond B,D ) = lim k→∞ µ k (Size x,m γ,δ ∩ Cond B,D ) (18) µ(Cond B,D ) = lim k→∞ µ k (Cond B,D ). (19)By the previous lemmaµ k (Size x,m γ,δ ∩ Cond B,D ) ≤ δ µ k (Cond B,D ). (20)The result is obtained by letting k tend to infinity : µ(Size x,m γ,δ ∩ Cond B,D ) ≤ δ µ(Cond B,D ).

  ρ n (β n s ∈ Size x,m γ,δ ∩ β n (s) ∈ Cond B,D ) ≤ δP ρ n (β n s ∈ Cond B,D).P ρ n (β n s ∈ Size x,m γ,δ ∩ β n (s) ∈ Cond B,D )ds ≤ Size x,m γ,δ ∩ β n (s) ∈ Cond B,D )ds ≤

	Therefore,
	1 t	γ	t		1 t		γ	t	δ P ρ n (β n s ∈ Cond B,D )ds
					≤ δ	1 t
	Since	1 t			γ t	,
	we get a bound for the integral :
	1 t	0	t	P ρ n (β n s ∈ γ t	+ δ	1 t

t 0 P ρ n (β n s ∈ Cond B,D )ds. γ 0 P ρ n (β n s ∈ Size x,m γ,δ ∩ β n (s) ∈ Cond B,D )ds ≤ t 0 P ρ n (β n s ∈ Cond B,D )ds.


LEMEL Etienne Sirot
email: sirot@univ-ubs.fr

Olivier Pays

On the dynamics of predation risk perception for a vigilant forager

Keywords: anti-predatory behaviour, information, predation, risk, vigilance

Foraging animals often raise their head to scan for predators. Scanning intervals have variable durations, and occur more or less frequently, depending on ecological conditions. Our study relies on the assumption that temporal patterns of vigilance depend on the speed with which information concerning the likelihood of a predator's presence in the neighbourhood is gathered when an animal is vigilant, and lost when it is not. Using an analytical model, we study how the perceived level of risk progressively decreases, when the individual is vigilant and detects no predator, then increases again, when it lowers its head to feed, thereby losing most of its detection abilities. The speed of these variations is affected by the likelihood of the presence of a predator in the whole environment, by the mobility of this predator, and by the detection capacities of the prey. We show how, combined with the range of risk levels tolerated by this animal, this dynamics determines the frequency and the duration of its scanning intervals. The dynamics of risk perception can also explain particular behavioural patterns, such as the progressive decrease of vigilance that may occur after the arrival into a novel environment, and the central tendency in the distribution of interscan durations reported by many studies. Next, we use the model to compute optimal vigilance strategies, taking into account the trade-off between feeding and limiting exposure to predators. The model predicts that a forager will scan more often, and for longer periods, when the likelihood a predator's presence in the surrounding environment is increased. A similar response is expected when the mobility of the predator is increased. By contrast, when the detection capacities of the prey are reduced, it will increase its vigilance by scanning for longer periods, but scanning intervals will be separated by longer interscans.

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INTRODUCTION

Time-sharing between anti-predatory vigilance and other activities, such as foraging, reproduction, or sleep, implies that animals have to trade some component of fitness (e.g. energy gain through feeding) for better security against predators [START_REF] Elgar | Predator vigilance and group size in mammals and birds[END_REF][START_REF] Quenette | Functions of vigilance in mammals: a review[END_REF]. This trade-off has been studied by mathematical models that derived optimal or evolutionarily stable vigilance strategies as a function of ecological conditions (e.g. [START_REF] Pulliam | The scanning behaviour of juncos: a gametheoretical approach[END_REF][START_REF] Mcnamara | Evolutionarily stable levels of vigilance as a function of group size[END_REF]. The assumption of vigilance models is that individuals with high vigilance levels have more chance of detecting an approaching predator, hence more chance of escaping when under an attack. The level of vigilance is represented by a behavioural variable; namely, the scanning rate (e.g. [START_REF] Pulliam | On the advantages of flocking[END_REF][START_REF] Pulliam | The scanning behaviour of juncos: a gametheoretical approach[END_REF][START_REF] Lima | Vigilance while feeding and its relation to the risk of predation[END_REF][START_REF] Rodriguez-Gironés | Evolutionary stability of vigilance coordination among social foragers[END_REF], or the proportion of time spent vigilant (e.g. [START_REF] Packer | Should co-operative groups be more vigilant than selfish groups?[END_REF][START_REF] Mcnamara | Evolutionarily stable levels of vigilance as a function of group size[END_REF][START_REF] Sirot | Social information, antipredatory vigilance and flight in bird flocks[END_REF].

Concurrently, temporal patterns of vigilance have been studied over a wide range of animal taxa, making the topic of anti-predatory vigilance a particularly fruitful example of joint development between theoretical and field work [START_REF] Bednekoff | Randomness, chaos and confusion in the study of antipredator vigilance[END_REF]. Field studies do indeed report that the proportion of time dedicated to vigilance varies with ecological conditions, but, moreover, that the duration of both scanning intervals and non vigilant intervals (interscans) may be affected by these conditions (e.g. [START_REF] Metcalfe | The effect of habitat on the vigilance of shorebirds : is visibility important ?[END_REF]Whittingham et al., 2004;[START_REF] Pays | Coordination, independence or synchronization of individual vigilance in eastern grey kangaroo[END_REF]. Measuring scans and interscans allows to calculate the proportion of time dedicated to vigilance, but also conveys more precise information, as different scanning rates may lead to the same proportion of time spent vigilant (e.g. [START_REF] Pöysä | Group foraging, distance to cover and vigilance in the teal, Anas crecca[END_REF][START_REF] Pays | Coordination, independence or synchronization of individual vigilance in eastern grey kangaroo[END_REF]. A relevant question is thus to ask, not only what proportion of time should be allocated to vigilance, but also how total vigilance time should be apportioned among consecutive scans. This question is particularly important because predatory attacks are very sudden events, and the exact posture of the individual at the onset of an attack may determine its chance of escape [START_REF] Hilton | Intra-flock variation in the speed of response on an attack by an avian predator[END_REF]. This is probably the reason why animals obey short-term decision rules that dictate the precise moments of head raising and lowering, as a response to immediate conditions [START_REF] Bekoff | Vigilance, flock size and flock geometry : information gathering by western evening grosbeaks (Aves, Fringillidae)[END_REF][START_REF] Pays | The effect of social facilitation on vigilance in the eastern gray kangaroo, Macropus giganteus[END_REF].

In the present study, we use a behavioural model to study the alternation of scanning and interscan intervals for an isolated forager on a food patch. The model addresses the following questions: (1) how long should a forager that does not spot any predator stay vigilant before resuming food search? and (2) how long should it feed after resumption, before switching back to vigilance?

Our study relies on the assumption that the behaviour of the animal is dictated by its instantaneous perception of predation risk, which fluctuates as a function of its activity. In the first part of the article, we describe the dynamics of risk perception over a foraging period during which no detection occurs, successively considering what happens during feeding and scanning intervals. In the second part, we consider the trade-off between feeding and limiting exposure to predators, and derive optimal vigilance strategies.

THE MODEL

The model considers an isolated forager on a food patch, which shares its time between feeding and vigilance, considered as two mutually exclusive activities. The duration of the whole foraging process, T, is fixed. Feeding allows the individual to increase its level of energy reserves, while scanning is used to detect a potential predator. The forager's perception of risk at time t, μ(t), corresponds to the estimated probability of the predator's presence in the neighbourhood (i.e. within detection range). P 0 , initially assumed to be a constant, denotes the probability of the predator's presence in the whole environment, which encompasses both the neighbourhood of the forager and the surrounding places, which are not observable by the forager, but from which the predator may arrive. Parameters s and S denote the areas of the forager's neighbourhood, and of the whole environment, respectively. The initial estimated probability of the predator's presence in the neighbourhood, when the forager arrives on the patch and has not yet had the opportunity to scan, is : μ 0 = P 0 x s/S. The level of risk that prevails in the forager's neighbourhood may nevertheless change with time, because the predator, if present, may move. During a small time interval dt, the predator moves with probability mdt, where m is a measure of its mobility. When moving, the predator arrives in the neighbourhood of the forager with probability s/S.

Interval [μ 1 ,μ 2 ] corresponds to the range of risk levels tolerated by the animal. This means that the forager switches from vigilance to feeding when its perception of risk μ(t) reaches threshold μ 1 during a scan, and from feeding to vigilance when μ(t) reaches threshold μ 2 during an interscan.

We first determine the fluctuations of predation risk perception during successive scans and interscans, when no detection occurs. Then, we consider the trade-off between searching for food and scanning for the predator, and compute optimal vigilance strategies.

Variations of predation risk estimation during interscans

Here, we consider what happens when the forager keeps its head at ground level to feed, after a vigilant period. When the animal begins to feed, its estimation of predation risk is μ 1 .

Thereafter, due to its incapacity to track changes in the environment, its estimated level of risk changes, as a function of the predator's tendency to move and approach close by. t denotes the time elapsed since the animal began feeding, and μ f (t,μ 1 ) is its estimation of predation risk at time t. The initial condition is : μ f (0,μ 1 ) = μ 1 . μ f (t+dt,μ 1 ) estimates the probability of the predator's presence in the neighbourhood of the forager at time t+dt. Following the rules for predator movement described above, three different reasons may explain this presence. First, the predator may already be present at time t, and be immobile during interval dt. This happens with probability μ f (t,μ 1 ) x (1 -mdt).

Second, the predator may already be present, move, and arrive again in the forager's neighbourhood. This happens with probability μ f (t,μ 1 ) x mdt x s/S. Third, the predator may initially be present in another part of the environment, and move into this neighbourhood. This event occurs with probability (P 0 -μ f (t,μ 1 )) x mdt x s/S. Thus, we finally have :

μ f (t+dt,μ 1 ) = μ f (t,μ 1 )
x (1 -mdt) + μ f (t,μ 1 ) x mdt x s/S + (P 0 -μ f (t,μ 1 )) x mdt x s/S (1).

or :

μ f (t+dt,μ 1 ) = μ f (t,μ 1 ) x (1 -mdt) + μ 0 x mdt, as μ 0 = P 0 x s/S (2).

This leads to the differential equation :

dμ f /dt x 1/(μ f -μ 0 ) = -m (3).
The solution of Eq. ( 3) is : μ f (t,μ 1 ) = (μ 1 -μ 0 ) x exp(-mxt) + μ 0 (4).

Eq. ( 4) describes the dynamics of predation risk perception when the animal feeds. It

shows how the level of risk perceived progressively rises, tending towards its basic level μ 0 , when the animal keeps its head down. Since the animal switches back to vigilance when the level of risk perceived reaches threshold μ 2 , we have :

μ 2 = μ f (t f ,μ 1 ) (5),
where t f denotes the duration of the current feeding interval.

Information gathering during vigilance

Here we study how the level of risk perceived changes over one scanning interval, still under the assumption that no detection occurs. μ v (t,μ 2 ) is the estimation of predation risk at time t, the time elapsed since the individual started being vigilant. μ 2 is the perceived level of risk at the beginning of the vigilant period, so we have : μ v (0,μ 2 ) = μ 2 . When considering predator movements only, predation risk follows the same dynamics as during interscans. Thus, the estimated probability of the predator's presence at time t+dt is given by : μ v (t,μ 2 ) x (1 -mdt) + μ 0 x mdt (see Eq. ( 2)). However, the level of risk perceived by the forager at time t+dt is also influenced by the fact that it did not detect the predator during interval dt. We must then compute the probability of the predator's presence, conditional on the absence of a detection.

Parameter D, which measures the detection abilities of the forager, is the per time unit probability of detection when a predator is present. Thus, the predator, when present, is detected during small interval dt with probability Ddt, and remains undetected with probability 1 -Ddt. When the predator is not present, the absence of detection occurs with probability 1. Hence we get, using Bayes' formula :

μ v (t+dt,μ 2 ) = 1 mdt)) μ mdt) (1 ) μ (t, (μ (1 Ddt) (1 mdt) μ mdt) (1 ) μ (t, (μ Ddt) (1 mdt) μ mdt) (1 ) μ (t, (μ x x 0 x 2 v x x 0 x 2 v x x 0 x 2 v + - - + - + - - + - (6).
Eq. ( 6) is built under the hypothesis that detection prevails over movements, that is, assuming that the predator has the same chance of being detected if it moves to or from the forager's neighbourhood during interval dt as if it had spent the whole interval within this neighbourhood. This approximation, which concerns only rare events, was made to insure coherence with the dynamics of risk perception during interscans, where detection ability is cancelled (when D = 0, Eq. ( 6) collapses to Eq. ( 2)). Re-arranging Eq. ( 6) leads to the following differential equation : dμ

v /dt x 1/(μ v 2 xD -μ v x(m+D) + μ 0 xm) = 1 (7).
The solution of Eq. ( 7) is :

t) D a) - exp(-(b c 1 t) D a) - exp(-(b c b a ) μ (t, μ x x x x x x x 2 v + + = (8), where D 2 D) m μ 4 - D) ((m - D) (m a x x x 0 x 2 + + = , D 2 D) m μ 4 - D) ((m D) (m b x x x 0 x 2 + + + = , and b - μ a - μ c 2 2 =
. Eq. ( 8) shows how the level of risk perceived progressively decreases when the forager is vigilant and does not detect the predator. The forager switches back to feeding when the level of risk perceived reaches threshold μ 1 , so we have :

μ 1 = μ v (t v ,μ 2 ) (9),
where t v denotes the duration of the current vigilant interval.

Updating the perception of the overall level of environmental risk

Here, we explore the possibility that the observations made by the forager in its immediate neighbourhood also convey information about the likelihood of the predator's presence in the surrounding places. Thus, we study the process whereby, if the predator, which is mobile, has not been detected for a certain period of time in the neighbourhood, it becomes more likely that it is indeed absent from the whole environment.

In the model, this means that parameter P 0 , which estimates the probability of the predator's presence in the whole environment, also becomes the object of an updating process.

We assume that this updating process takes place at the end of each scanning interval. P n denotes the estimated probability of the predator's presence in the whole environment immediately after scan n, still conditional on the absence of any detection. Q(t v ,P n ) denotes the probability that the predator is present in the environment, but remains undetected during a scanning interval of duration t v , when the probability of the predator's presence in the environment is P n (see Appendix A). Using Bayes formula, we have :

P n+1 = Q(t v ,P n )/(Q(t v ,P n ) + 1 x (1-P n )) (10).
Eq. ( 10) is used to update the value of P at the end of each scanning interval. The dynamics of risk perception during the following interscan and scan is then computed using Eqs. ( 4) and ( 8), respectively, with this updated value.

The situation where parameter P is the object of an updating process yields relatively complex vigilance behaviour, with scans and interscans that progressively vary in duration (see Results). In the following section, which is dedicated to the influence of ecological parameters on vigilance strategies, we do not consider this effect of time. P 0 is thus treated as a constant.

Optimal vigilance

Here we define a fitness measure which takes into account the advantages of vigilance, in terms of reduced exposure to the predator, and its drawbacks, in terms of lost feeding time. In the framework of a constantly fluctuating level of perceived risk, the vigilance strategy of the forager is associated with the range of tolerated risk levels, [μ 1 ,μ 2 ]. The analysis of risk perception dynamics presented above shows that, when μ 1 is not too low and μ 2 not too high (i.e. μ 1 > a and μ 2 < μ 0 ), we can find one single set of durations for scans and interscans, hereafter denoted by t v and t f , respectively, for which the level of risk perceived by the forager describes interval [μ 1 ,μ 2 ] during each scan/interscan cycle. These durations verify Eqs. ( 5)

and ( 9). The range of risk levels tolerated by the forager thus induces a single vigilance strategy, characterized by feeding and vigilant intervals of constant durations.

We assume that the forager always escapes if it detects the predator while vigilant. In this situation, the total time lost in the current scan and in the escape, T E , only impedes its overall feeding rate. The probability that the predator is detected during a scan is : P D = P 0 -Q(t v ,P 0 ), since Q(t v ,P 0 ) is the probability that the predator is present in the environment, but remains undetected during the scan (see Appendix A).

For the forager, the total level of exposure to predation risk thus only depends on the number and duration of feeding periods, and on the probabilities of the predator's presence during these periods. This simplifying assumption conserves the advantages of vigilance, namely, the fact that the forager has more chance of escape if attacked while vigilant than while feeding, and the fact that it endures a reduced level of risk when returning to feed after a vigilant period during which no detection occurred.

We consider a given feeding period. t denotes the time elapsed since the forager resumed feeding, and P NE (t,μ 1 ) is the probability that the forager has not been exposed to the predator at time t. We make the per time unit probability of exposure to predation risk equal to the estimated probability of the predator's presence. Since μ f (t,μ 1 ) is the probability of the predator's presence in the neighbourhood at time t, we have : P NE (t+dt,μ 1 ) = P NE (t,μ 1 ) x (1-μ f (t,μ 1 )dt) (11), which leads to the differential equation : dP NE /dt x 1/P NE = -(μ 1 -μ 0 ) x exp(-mxt) -μ 0 (12).

The solution of Eq. ( 12) is : P NE (t,μ 1 ) = exp(-μ 0 xt) x exp((μ 1 -μ 0 ) x (exp(-mxt)-1) / m) (13).

Hence, the probability of not being exposed to the predator's presence during a feeding period of duration t f is P NE (t f ,μ 1 ), and the probability of not being exposed to the predator's presence during the whole foraging bout is P(t f ,t v ) = (P NE (t f ,μ 1 ))

n

, where n = T/(t f + (1-P D )xt v +P D xT E ) is the total number of feeding periods.

The expected energy gain for the whole foraging bout, provided the individual survives, is :

E(t f ,t v ) = n x (t f x g -e)
, where g is the energy gain per time unit of feeding, and e the energy cost of switching from vigilance to feeding, then from vigilance to feeding.

To compute optimal strategies in a way that takes into account both the advantages and the drawbacks of vigilance, we use the following quantity to measure fitness :

F(t f ,t v ) = P(t f ,t v ) x E(t f ,t v ) (14).
The individual does indeed increase its probability of not being exposed to the predator, P(t f ,t v ), by being more vigilant, but, at the same time, it reduces its energy gain E(t f ,t v ). Fitness as defined by Eq. ( 14) thus allows to study the response to the trade-off between feeding and limiting exposure to predators (see [START_REF] Packer | Should co-operative groups be more vigilant than selfish groups?[END_REF][START_REF] Brown | Vigilance, patch use and habitat selection : Foraging under predation risk[END_REF]. The optimal vigilance strategy is the set of values (t f ,t v ) for which F(t f ,t v ) is maximized. It is derived numerically.

RESULTS

Fluctuations of risk perception

After a relatively long initial scan following the arrival on the patch, the level of risk perceived by the animal begins to fluctuate in a periodic way. It increases during each interscan, then decreases again during the following scan, with dynamics respectively given by Eqs. ( 4) and ( 8) (Fig 1). Excepted for the long initial scan, both scans and interscans have constant durations, which depend on this dynamics and on the range of risk levels [μ 1 ,μ 2 ] tolerated by the forager.

Compared to a control situation (dashed line, t v = 5.99, t f = 5.03), the detection capacity of the forager is increased by increasing the value of parameter D, which measures this capacity (full line). As a consequence, the level of risk perceived decreases faster during scans, while the dynamics of risk perception does not change during interscans. To maintain the level of risk perceived within the same interval, the forager thus shortens its scans, while the duration of interscans remains the same (full line, t v = 2.14, t f = 5.03). By contrast, if the level of danger is decreased by reducing the overall likelihood of the predator's presence in the environment (i.e. parameter P 0 , dotted line), the dynamics of risk perception is affected during both scans and interscans, because the arrival of the predator becomes less likely during both kinds of intervals. The level of risk perceived thus decreases more rapidly during scans, and increases more slowly during interscans (compare dotted and dashed line). The forager thus responds to this situation by shortening its scans, and increasing its interscans (dotted line, t v = 3.5, t f = 8.11). Qualitatively similar results are obtained when the level of danger is increased by an increased mobility of the predator (data not shown).

Updating the perception of the overall level of environmental risk

Here we explore the situation where the observations made by the forager in its immediate neighbourhood contribute to update its estimate of the probability of the predator's presence in the whole environment.

The effect of this updating process is that, over a foraging bout during which the predator is not detected, the estimated probability of the predator's presence in the whole environment (i.e. P) progressively decreases with time. As a consequence of this declining perception of the overall level of risk, the level of risk perceived locally decreases more rapidly during scans, and increases less rapidly during interscans, as predicted when the value of P 0 is decreased (see Logically, additional simulations show that when s/S, the proportion of the whole environment that is observable by the forager, gets smaller, vigilance decreases more slowly.

Optimal vigilance

Here we study how the optimal vigilance strategy varies with ecological conditions. The level of environmental danger increases when either the likelihood of the predator's presence in the whole environment, P 0 , or its mobility, m, increases (Fig 3a,b), and when the detection ability of the forager, D, decreases (Fig 3c).

In the three situations, the durations of both scans and interscans change as a response to an increased level of danger, and the proportion of time dedicated to vigilance increases.

The effects of increasing the likelihood of the predator's presence in the environment, or the mobility of the predator, are similar. In both cases, scans get longer, and interscans shorter (Fig. 3a,b). By contrast, when the level of exposure increases because the detection capacities of the prey are reduced, the increase in vigilance only results from the lengthening of scans, which, as they become longer, also become separated by longer feeding intervals (Fig 3c).

DISCUSSION

In the present article, we study how the alternation of anti-predatory vigilance and feeding intervals determines the levels of risk successively perceived by a forager, in an environment potentially occupied by a predator. The model dedicates special attention to the process of information acquisition during scans, as information gathering is the primary function of vigilance, and considers the loss of this information during feeding intervals.

In the first part of the study, we compute the dynamics of predation risk perception, in the common situation where no detection occurs. We show how the level of risk perceived continuously vary with the activity of the animal, decreasing during scans, as the animal accumulates information about safety in the environment, then increasing again during interscans, when it is no longer able to detect the possible arrival of the predator. The speed of these variations, combined with the range of risk levels tolerated by the animal, finally determines its vigilance strategy, which means that the range of risk levels accepted by the forager and its vigilance behaviour are both tightly connected to the dynamics of risk perception. The model shows that this dynamics should tend to make the duration of both scans and interscans relatively constant. However, the initial scan should always be longer than the following ones, and, if we take into account the possibility for the forager to learn about surrounding places through its local observations, the proportion of time dedicated to vigilance should decline with time. Such progressive decrease in vigilance following the arrival into a novel environment has been reported in harbour seals, Phoca vitulina, attending haul out sites [START_REF] Terhune | Harbour seal vigilance decreases over time since haul out[END_REF].

In real conditions, however, the regular dynamics of predation risk perception predicted by our model will be disturbed by accidental changes, coming from external events (such as unidentified shapes or noises; Ruxton and Roberts 1999), or observation of companions, whose behaviour may convey information about potential predators [START_REF] Fernández | Effect of group size on individual and collective vigilance in greater Rheas[END_REF], and influence individual predation risk in several ways [START_REF] Sirot | Coordination and synchronization of vigilance in groups of prey : the role of collective detection and predators' preference for stragglers[END_REF]. As a consequence, the individual will reach the critical switching values for predation risk perception at different moments, which will induce some variability in the temporal patterns of vigilance (see [START_REF] Ruxton | Are vigilance sequences a consequence of intrinsic chaos or external changes ?[END_REF].

Nevertheless, the mechanism described here, whereby predation risk perception progressively increases during interscans, until it triggers a switch to vigilance, and progressively decreases during vigilance, until the level of safety allows the animal to resume feeding, should contribute to reduce the variability of the duration of both scans and interscans, and the instantaneous probability of switching should increase with time, during both kinds of intervals. The model thus proposes an alternative, based on mechanistic and adaptive reasoning, to the original hypothesis of vigilance studies that scans should be initiated with a constant rate (that is, with a constant per time unit probability; [START_REF] Pulliam | On the advantages of flocking[END_REF].

In accordance with our results, several studies demonstrate that the per time unit probability of looking up increases as an animal feeds, instead of being constant [START_REF] Hart | Vigilance and scanning patterns in birds[END_REF][START_REF] Sullivan | Vigilance patterns in downy woodpeckers[END_REF][START_REF] Lendrem | Scanning for predators in the purple sandpiper: a time-dependent or time-independent process?[END_REF][START_REF] Beauchamp | Nonrandom patterns of vigilance in flocks of the greater flamingo, Phoenicopterus ruber ruber[END_REF][START_REF] Pays | How unpredictable is the individual scanning process in socially foraging mammals ?[END_REF]. As a consequence, both short and long interscans become more rare than under the original hypothesis of a constant scan initiation rate. Scan duration has received much less attention than interscan duration [START_REF] Bednekoff | Randomness, chaos and confusion in the study of antipredator vigilance[END_REF]. The present model suggests that it should also be relatively constant. The fact that the efficiency of vigilance declines with time, because attention cannot be sustained for long periods, should reinforce this tendency [START_REF] Dukas | Sustained vigilance and animal performance[END_REF].

In the second part of the study, we derive optimal strategies for the alternation of scanning and interscan intervals. Doing so, we make several simplifying assumptions. First, we assume that a feeding individual cannot detect predators, whereas an animal may retain limited detection capacities when it searches for food [START_REF] Lima | Back to basics of anti-predatory vigilance: can non-vigilant individuals detect attacks ?[END_REF]. However, even animals with such ability also possess characteristic vigilance attitudes, with erect postures of the neck or the whole body [START_REF] Bednekoff | Testing for peripheral vigilance: do birds value what they see when not overtly vigilant ?[END_REF][START_REF] Ebensperger | On the relationship between herbaceous cover and vigilance activity in degus (Octodon degus)[END_REF]. This situation could thus be studied under the present framework. Including a limited detection ability for feeding individuals would quantitatively affect the results of the model, by slowing the resilience of risk estimation during interscans, but not its general qualitative predictions. Reciprocally, overtly vigilant individuals may sometimes continue to feed. For example, birds swallowing seeds raise their heads, which allows them to scan [START_REF] Baker | Vigilance and the functional response of granivorous foragers[END_REF], and large herbivores may also become vigilant while chewing [START_REF] Fortin | Foraging costs of vigilance in large mammalian herbivores[END_REF]). In these conditions, feeding does only partially, or not at all, impede vigilance, and the temporal pattern of vigilance is dependent on the feeding method. We can hypothesize that this process should contribute to make interscans more variable in length, as the time necessary to gather food on the ground is variable, and scans more constant, as the handling time for one particular type of food should be relatively constant.

Second we chose, for the sake of mathematical tractability, to consider the probability of the predator's presence as a measure of risk, thus assuming that minimizing predation risk is equivalent to limiting exposure to the predator's presence. In reality, predation risk also depends on the behaviour of the predator, which could in principle adapt its hunting behaviour to the vigilance displayed by the prey, thus initiating a game between predator and prey. The outcome of such a game on temporal patterns of prey vigilance has been studied by [START_REF] Scannell | Prey scan at random to evade observant predators[END_REF] and [START_REF] Bednekoff | Why are scanning patterns so variable? An overlooked question in the study of anti-predator vigilance[END_REF], who identified two situations. In the first one, predators are prevented from timing their attack according to the vigilance schedule of the prey, for example because they have to cross a large portion of open space when attacking. In this situation, scanning at regular intervals is an advantage. By contrast, scanning at irregular intervals could be the best strategy against stalking predators that launch their attacks from close distances [START_REF] Scannell | Prey scan at random to evade observant predators[END_REF][START_REF] Bednekoff | Why are scanning patterns so variable? An overlooked question in the study of anti-predator vigilance[END_REF]. The mechanism described in the present study could thus, in some situations, be reinforced by the necessity to counter the hunting strategy of the predator. In others, the prey should simultaneously compromise with an efficient handling of the information garnered through vigilance, and an appropriate response to the tactic of the predator.

The levels of risk accepted by an animal are also expected to depend on the level of its energy reserves, which may fluctuate in a non-deterministic way. For example, an animal may be prevented from feeding for a part of the day. It will thus have low levels of reserves, and should become relatively risk tolerant, especially if the end of the foraging period is getting near [START_REF] Mcnamara | The common currency for behavioral decisions[END_REF]. Reciprocally, an animal with important levels of reserves could afford to invest more in anti-predatory vigilance [START_REF] Beale | Behavioural response to human disturbance: a matter of choice ?[END_REF].

Across a foraging bout, the variations of individual levels of reserves and the Bayesian updating process concerning the prevailing level of risk could thus have conflicting effects on the level of vigilance. A model incorporating both a state-dependent approach and a description of the dynamics of risk perception should allow to study how the range of tolerated levels of risk could vary, according to current level of reserves of the animals and the time of the day.

In accordance with other models (e.g. [START_REF] Mcnamara | Evolutionarily stable levels of vigilance as a function of group size[END_REF][START_REF] Brown | Vigilance, patch use and habitat selection : Foraging under predation risk[END_REF], our model predicts higher vigilance in dangerous environments. This increase in vigilance is achieved by longer and more frequent scans when the likelihood of the predator's presence in the environment is high, and when the predator is highly mobile. By contrast, scan should increase in length, but not in frequency, when danger originates from decreased detection capacities for the prey. Additional simulations show that, when the cost of switching e is reduced and tends towards zero, the durations of both scans and interscans also tend towards zero, while the proportion of time spent vigilant does not change. Thus, we can draw the prediction that an animal will prefer to alternate between feeding and vigilance at the maximum rate that is compatible with the efficiency of both feeding and vigilance activities.

In this way, it will avoid sporadic periods of high risks, corresponding to high values of μ 2 .

Field studies confirm that vigilance rises during periods of increased predator activity (e.g. [START_REF] Caraco | Avian flocking in the presence of a predator[END_REF][START_REF] Scheel | Watching for lions in the grass: the usefulness of scanning and its effects during hunts[END_REF][START_REF] Devereux | Predator detection and avoidance by starlings under different scenarios of predation risk[END_REF]. Vigilance is often measured as the proportion of time spent vigilant, but the effects of environmental conditions on scan and/or interscan lengths have also been considered in several studies. The level of risk endured by the animals generally corresponds to the amount of obstructive cover near the feeding place, which hinders predator detection [START_REF] Harkin | The influence of visual obstructions on the vigilance and escape behaviour of house sparrows, Passer domesticus[END_REF]. In accordance with the predictions of the model, the proportion of time spent vigilant generally increases with the proximity of obstructive cover (e.g. [START_REF] Underwood | Vigilance behaviour in grazing African antelopes[END_REF][START_REF] Lazarus | Contrasting effects of protective and obstructive cover on avian vigilance[END_REF][START_REF] Watson | Vigilance and fitness in grey partridges Perdix perdix : the effects of group size and foraging-vigilance trade-off on predation mortality[END_REF]; but see [START_REF] Scheel | Watching for lions in the grass: the usefulness of scanning and its effects during hunts[END_REF], and scanning intervals become longer [START_REF] Metcalfe | The effect of habitat on the vigilance of shorebirds : is visibility important ?[END_REF][START_REF] Goldsmith | Vigilance behaviour of pronghorns in different habitats[END_REF][START_REF] Pöysä | Group foraging, distance to cover and vigilance in the teal, Anas crecca[END_REF]. Interestingly, scans may become more frequent when they get longer [START_REF] Mcvean | Vigilance schedules among house sparrows Passer domesticus[END_REF][START_REF] Metcalfe | The effect of habitat on the vigilance of shorebirds : is visibility important ?[END_REF][START_REF] Goldsmith | Vigilance behaviour of pronghorns in different habitats[END_REF], but this is not always the case [START_REF] Pöysä | Group foraging, distance to cover and vigilance in the teal, Anas crecca[END_REF]Whittingham et al., 2004). Reciprocally, increases in scanning rate with no concomitant changes of scan duration have also been reported, although these results concern circumstances that are not considered in the present study (in [START_REF] Lendrem | Predation risk and vigilance in the blue tit (Parus caeruleus)[END_REF] study on house sparrows Passer domesticus, an individual could leave its feeding place to join a better observatory, and in [START_REF] Bertram | Vigilance and group size in ostriches[END_REF]

  Fig 1). The proportion of time dedicated to vigilance thus progressively declines (see Fig 2).

LEGENDSFig. 1 Fluctuations

 1 Fig.1

Fig. 2

 2 Fig.2

Fig. 3

 3 Fig.3

  Fig. 2

Acknowledgments

We thank Joël Brown, Peter Jarman, Frédéric Mathéus, Franz Weissing and two reviewers for helpful comments on previous versions of the model.

Appendix A. Probability of not detecting the predator during a scanning interval.

Here we compute Q(t v ,P n ), the probability of not detecting the predator during a scanning interval of duration t v , when the probability of the predator's presence in the whole environment is P n . Two cases must be considered.

First, the predator may initially be present in the neighbourhood of the forager. This happens with probability μ 1 , which corresponds to the level of risk perceived at the end of the vigilant interval. At time t, the predator is still present and has remained undetected with probability exp(-(m+D)xt), since both movements and detection are Poisson processes. If still present, the predator will be detected during small interval dt with probability Ddt. Thus, the probability of not detecting the predator over the whole scanning interval of duration t v when it is initially present in the neighbourhood, is :

Second, the predator may initially be present in the environment, but not in the neighbourhood of the forager. This happens with probability P n -μ 1 . The arrival of the predator in the neighbourhood of the forager is then a Poisson process of parameter m' = mxs/S. The probability that the predator arrives between time t and t + dt is thus exp(m'xt)xm'dt, and the probability that it is detected before the end of the scan, that is, during the time that remains, is given by : 1 -Q 1 (t v -t) (see Eq. ( A1)). Thus, the probability of not detecting the predator if it is initially present in the environment, but not in the neighbourhood of the forager, is : We finally have : Q(t v ,P n ) = Q 1 (t v ) x μ 1 + Q 2 (t v ) x (P n -μ 1 ) (A4).