Evolution of cross-diffusion and self-diffusion

Abstract

This article is concerned with the evolution of certain types of density-dependent dispersal strategy in the context of two competing species with identical population dynamics and same random dispersal rates. Such density-dependent movement, often referred to as cross-diffusion and self-diffusion, assumes that the movement rate of each species depends on the density of both species and that the transition probability from one place to its neighbourhood depends solely on the arrival spot (independent of the departure spot). Our results suggest that for a one-dimensional homogeneous habitat, if the gradients of two cross- and self-diffusion coefficients have the same direction, the species with the smaller gradient will win, i.e. the dispersal strategy with the smaller gradient of cross- and self-diffusion coefficient will evolve. In particular, it suggests that the species with constant cross- and self-diffusion coefficients may have competitive advantage over species with non-constant cross- and self-diffusion coefficients. However, if the two gradients have opposite directions, neither of the two dispersal strategies wins as these two species can coexist.

Keywords:

• density-dependent dispersal
• competition
• coexistence

AMS Classification :

• 35K57
• 92D25

1. Introduction

Dispersal is probably one of the most common yet fascinating features that we often witness in this world: birds fly across continents to breed, herds run over wild lands in searching for food, etc. How did they adopt their dispersal behaviours? How would these behaviours change in the future? The evolution of dispersal is of keen biological interest and importance and has been investigated extensively for decades in biological literature (see, e.g. 1 12 13,17–19,28 38 42 45,49–51). One of the major mathematical modelling approaches is to use reaction-diffusion models (see, e.g. 4–8,11 21 22 24 36). While substantial progress has been made via this mathematical approach in understanding random spatial movement of species, much less is known on the role of population density-dependent dispersal in the evolution of dispersal. In reality, species do not always move randomly. Instead, they can often sense local environment and may adopt different dispersal strategies under different circumstances. This paper is concerned about certain kind of density-dependent dispersal strategies.

To be more specific, we are interested in how cross-diffusion and self-diffusion can affect the dynamics of two competing species. Cross-diffusion and self-diffusion models are originally proposed by Shigesada et al. 46 to model the spatial segregation of two competing species. The main feature of such model is that the movement rate of each species depend on the density of both species, and one key underlying biological assumption is that the transition probability from one place to its neighbourhood depends solely on the arrival spot and is independent of the departure spot. The goal of Shigesada et al. 46 was to show that if random diffusion alone cannot give rise to spatial segregation of two competing species, such nonlinear dispersal strategy may be able to do so. For later work on competition models with cross- and/or self-diffusion, see 9 23 30 31 33 34 37,39–41 and references therein.

While we will use the cross-diffusion and self-diffusion model in this paper, our motivations are different. Our goal here is to understand how density-dependent dispersal strategies will evolve in spatially homogeneous environment. Following Shigesada et al. 46, we consider the following strongly coupled parabolic system

where u(x, t) and v(x, t) are the densities of two competing species and , are their density-dependent dispersal rates 46, respectively. We assume that Ω is a domain in R ^N with smooth boundary and ν is the outward unit normal vector on . The boundary conditions in Equation (1) means that there is no individuals crossing the boundary. In 46, D _i (x, u, v) assumes the form of . Throughout this paper, we consider the case . More precisely, we assume that

where the parameter μ is positive and can be regarded as the random dispersal rate; γ, τ are two nonnegative constants, and they together with functions ρ _i (x) measure the degree of density dependence in the movement of the species. We shall focus on the situation when τ is small, i.e. the two species are almost identical and their cross-diffusion and self-diffusion coefficients are all close to the same constant γ. The biological motivation behind is that we envision that some mutation occurs and both species are adopting biased movement strategies that avoid the crowding of the whole population. One natural question is: what kind of ρ _i (x) can convey some competitive advantage?

For notational convenience, we will refer ρ _i (x) as cross- and self-diffusion coefficients throughout this paper. Also for technical reasons, we assume that , are non-negative smooth functions and that they satisfy the boundary condition

We first consider the case when one of ρ _i is a constant function.

Theorem 1.1

Suppose that at least one of ρ₁, ρ₂ is a constant. Then, for every μ>0, there exists τ₀ such that for every the system (1) does not admit any nonconstant positive steady state.

If both ρ₁ and ρ₂ are constants, then Equation (1) has a continuum family of constant positive steady states, which are all neutrally stable and as a whole may be the global attractor of system (1). Also, both semi-trivial steady states are neutrally stable. Therefore, if both species adopt cross-diffusion and self-diffusion strategy with a constant coefficient, neither one wins or loses.

If ρ₁ is a nonconstant and ρ₂ is a constant, we can further show that the semi-trivial steady state (u, 0) is locally unstable and (0, v) is neutrally stable for τ small. We conjecture that for this case, (0, v) is the global attractor. If this were the case, biologically this would mean that the species with nonconstant cross- and self-diffusion coefficients always lose and thus constant cross- and self-diffusion will evolve.

What happens if both ρ₁ and ρ₂ are nonconstant? It turns out that this case is much more delicate and there may or may not exist any stable positive steady state. Thus, we are going to be looking for positive solutions of

We first state some abstract results for the coexistence of two species and then give some more explicit conditions under which coexistence of two species is possible or impossible. For small positive τ, the conditions for existence/nonexistence of coexistence states can be expressed in terms of two functions G(μ) and H(μ). Define as

where , p>N. We define

Using Fourier series expansion, it is easy to prove that if is a nonconstant then . Therefore, if is a nonconstant, we have that for every μ. Thus, given any μ, at least one of G(μ) and H(μ) is strictly positive.

Theorem 1.2

Suppose that

1. If system (3) has no positive solutions provided that τ>0 is small. Furthermore, if the semi-trivial steady state (u, 0) is stable and (0, v) is unstable; the stability is switched if

2. If , system (3) has a unique positive solution, which is also locally stable, provided that τ>0 is small. Furthermore, both semi-trivial steady states are unstable.

It is unknown whether smooth solutions of system (1) exist for all time and it is even more difficult to predict its dynamics in general. In view of Theorem 1.2, for small positive τ the simplest possible scenario for (1) would be that one of the two semi-trivial states is globally asymptotically stable if and the unique positive steady state is globally asymptotically stable if . The global existence of classical solutions to time-dependent cross-diffusion and self-diffusion systems is a challenging problem and less is known for the existence of the global attractors for such strongly coupled parabolic systems (see 9 10,25–27,32 43 47 and references therein for some recent progress).

Since the biological meanings of functions G and H are not quite clear to us in general situations, the assumptions and results in Theorem 1.2 seem to be difficult to interpret. On the other hand, for one-dimensional habitat and monotone ρ _i (x) we have better understanding of functions G and H, e.g. if one of ρ’s is monotone increasing and the other monotone decreasing, then both G(μ) and H(μ) are positive for all μ>0. More precisely, as an application of Theorem 1.2, we have the following corollary.

Corollary 1.1

Suppose that and are nonconstant functions.

1. If or for all x∈(0, 1), then for any μ>0, there exists τ₁ such that Equation (3) has no positive solution provided that Furthermore, the semi-trivial steady state (u, 0) is locally unstable and (0, v) is stable.

2. If or for all x∈(0, 1), then for any μ>0, there exists τ₂ for which the system (3) admits a unique positive solution, which is also locally stable provided that

Part (1) suggests that the semi-trivial steady state (0, v) may be globally asymptotically stable, which biologically would mean that if the gradients of cross- and self-diffusion coefficients have the same direction, then the one with the smaller gradient will evolve. This is also consistent with our discussions following Theorem 1.1, where we conjecture that the species with constant cross- and self-diffusion coefficients always wins.

Part (2) implies that if the two gradients have opposite directions, these two species can coexist. Intuitively, one would reason that neutral competitions often lead to competitive exclusion since similar species will likely compete for similar resources. So it is interesting to see from our model that two similar competing species can coexist under fairly general conditions. Recently there have been some studies on the dynamics of two similar competing species in various contexts. For example, in 3 16 20 the two species are identical except their intrinsic growth rates; in 35, the two species are identical except their inter-specific competition coefficients. While the general analytical approach adopted in this paper shares some similarity with those in previous works, both mathematical results and technical details are rather different, so are biological motivations, interpretations (of mathematical results) and predictions.

This article is organised as follows: Theorem 1.1 follows from Propositions 2.2 and 3.3 that are established in Sections 2 and 3, respectively; Theorem 1.2 follows from Theorem 3.1 (existence part), Propositions 4.1 and 4.2 and Theorem 4.1 (stability part). In fact, Theorem 3.1 gives a global bifurcation diagram for positive steady states of Equation (1) and is more general than what is stated in Theorem 1.2. Corollary 1.1 follows from Theorem 1.2 and Corollaries 3.3 and 3.4. Finally in Section 5 we will give some discussions of our assumptions and results.

2. Asymptotic behaviour of solutions as τ→0

In our first statement we will prove that any positive solution of Equation (3) converges to , with s∈[0, 1] as τ→0.

Proposition 2.1

Suppose that for every τ sufficiently small system (3) admits a positive solution Then, after passing to a subsequence if necessary, we have that in , as τ→0, for some s∈[0, 1].

Proof

We set , , which satisfy the equation

Using the maximum principle 44, it is easy to prove that there exists a constant C>0 such that . Hence, by elliptic regularity and Sobolev embedding theorems 15, after passing to a subsequence if necessary, , with , in W ^{1, p} (Ω) as τ→0, for any p>1.

Observe that with , for V an open subset of , and a smooth diffeomorphism. Therefore, , with , in W ^{1, p} (Ω) as τ→0. Observe that by definition and .

Using Equation (6) we obtain that in and consequently in . Observe that satisfies the equation

which has only two nonnegative solutions w≡ 0 and w≡ 1. To verify this assertion, suppose that w≥0 and . By writing and then applying the maximum principle 44 on w, we can deduce that w≡ 1. Therefore, either or .

Suppose that , i.e. . In this case we define for . It is easy to see that converges, except possibly for a subsequence, to a nontrivial nonnegative solution of

which is a contradiction.

Hence, and since

we have that in with s∈[0, 1].   ▪

The following proposition states that if τ is small enough and ρ₁, ρ₂ are constants, then Equation (3) does not admit any nonconstant positive solution.

Proposition 2.2

Suppose that are constants. For every μ>0, there exists τ₃ such that if system (3) does not admit nonconstant positive solutions.

Proof

We proceed by contradiction. Suppose that we have a sequence of nonconstant positive solutions, with τ→0. Then, we can write , , with , and in . After considering a subsequence if necessary, we have that as τ→0, with s∈[0, 1]. Set

which, by some simple computations, satisfy

where since

For any s∈[0, 1] the linear operator , defined as

satisfies and . Hence, from Equation (7) and elliptic L ^p estimate we obtain that

which contradicts .   ▪

3. Coexistence states

In this section we will study the existence of positive steady-state solutions of Equation (1) which are close to the surface

for some . Observe that for every μ>0,

is the set of solutions of Equation (3) when τ=0. Following the proof of Proposition 2.1 we can show that there exists τ₃>0 such that for any positive solution of Equation (3) is close to Σ provided that τ<τ₃. We define the spaces

where p>N so that . We start with the following result that establishes the existence and expansion of the semitrivial solutions of Equation (3).

Proposition 3.1

Consider the equation

with for and positive satisfying Equation (2). Then there exists τ₀ such that for all Equation (10) admits a unique positive solution Moreover can be expanded as

Proof

The proof is a direct application of the implicit function theorem using a priori estimates similar to the ones shown in the proof of Proposition 2.2.   ▪

Set The main result of this section is as follows.

Theorem 3.1

Suppose that in Ω.

1. If GH>0 in (μ₁, μ₂) then there exists a neighbourhood U of Σ and δ>0 such that for the set of nontrivial solutions of Equation (3) consists of the semi-trivial solutions and with where

   Here are smooth functions and s (μ, τ) is smooth and satisfies

2. If GH<0 in system (3) has no positive solutions for provided that τ>0 is small.

Proof

For τ small, each solution (u, v) of Equation (3) near Σ can be written as

where in a neighbourhood of (0, 0). We define the map by

with L _{μ, s} as in Equation (8) and

Clearly F is smooth and is a solution of Equation (3) if and only if . The two semitrivial equilibria of Equation (3) can be expressed as

where for i=1, 2 and . Thus we have that F satisfies

We define the projection

which satisfies and . Following the Lyapunov–Schmidt procedure, we need to solve

where I is the identity map.

Observe that and that . Thus, we can apply the implicit function theorem to solve Equation (18) through the map with . Moreover, there exists a neighbourhood V of (0, 0) in X ₂ and δ₁ such that satisfies if and only if and it solves Equation (17). We observe that

We define by the formula

and using Equation (16), we have that it admits the following decomposition:

It is easy to check that

therefore . Then,

To find the nontrivial solutions of Equation (18), we need to solve

with s∈(0, 1), which will be done by using the implicit function theorem, expressing the solutions of Equation (20) as with being solutions of .

Differentiating the above equation, we obtain that

After a simple computation, we have that

where and . If we differentiate Equation (18) with respect to τ and evaluate it at τ=0 we obtain that satisfy

Adding the two equations of Equation (23), we obtain that w=y ₁+z ₁ is the unique solution of

Define w ₁ and w ₂ the unique solutions of

respectively. Adding Equations (25) and (26) we obtain that w=sw ₁+w ₂. Moreover, replacing y ₁+z ₁ by w in Equation (23) we obtain that y ₁=su ₁ and where u ₁ and v ₁ satisfy

Subtracting Equation (28) from Equation (27), we obtain that

where . Since y ₁+z ₁=w we have that

which together with Equation (29) gives that

Replacing the expressions for u ₁ and v ₁ in Equations (21) and (22), and using we obtain that

Thus, to have bifurcation s should satisfy

Since and , thus if we have

Therefore, by the implicit function theorem there exists δ>0 such that we can find a solution s(μ, τ) of Equation (20) satisfying

Using the functions G(μ) and H(μ), we can rewrite the above condition as

and we observe that in order to have we must have that

To conclude Equations (11) and (12) we set

  ▪

Remark 3.1

Using Equation (19) it is easy to prove that when Equation (12) holds in Theorem 3.1, then the solution can be expressed as

where are smooth functions.

As a consequence of this remark, we can refine alternative (1) of Theorem 3.1.

Proposition 3.2

Suppose that the hypothesis of Theorem 3.1(1) holds and that μ₁ is a simple zero of GH. Then, there exists τ₁>0 and a smooth function such that and for the steady states of Equation (3) as given in Equations (11) and (12) are positive whenever and semitrivial if where δ is taken as in Theorem 3.1(1).

Proof

We consider the case G(μ₁)=0 and we prove the existence of the function . The case H(μ₁)=0 is similar. Observe that by Equation (39) the functions u(μ, τ) and v(μ, τ) are positive when and . Since , we just have to prove our result when . By assumption , and since we have , hence we can use the implicit function theorem to find τ₁>0 and a function satisfying . Since we have that for whenever .   ▪

Corollary 3.1

Suppose that

Then, there exists μ₀>0 such that for every 0<μ<μ₀ there is a τ₄ for which system (3) admits a unique positive solution provided that

Proof

To apply the above theorem, we just need to prove that for μ small enough we have that G(μ)>0 and H(μ)>0. By doing a standard blow-up argument for elliptic equations (see 14 for such argument for the zero Dirichlet boundary condition and 29 for the zero Neumann boundary condition), it is easy to show that if then as μ→0 we have uniformly in Ω. Therefore as μ→0,

and integrating by parts we obtain from Equation (40) that H(μ)>0 and G(μ)>0 if μ is small enough.   ▪

Corollary 3.2

Suppose that for i=1, 2,

Then, there exists μ₁>0 such that for every μ>μ₁ there is a τ₅ for which system (3) admits a unique positive solution provided that

Proof

As in the previous corollary, we just have to check that G(μ)>0 and H(μ)>0 for μ large enough. We claim that, as ,

uniformly in Ω, where . To establish this assertion, set . Then z satisfies the zero Neumann boundary condition and

Hence, in as 2. Thus, as

After a simple expansion, it is easy to verify that the condition (41) is equivalent to H(μ)>0 and G(μ)>0.   ▪

Observe that the hypothesis in Corollaries 3.1 and 3.2 hold if Ω=(0, 1) and ρ₁ increasing and ρ₂ decreasing in (0, 1), or ρ₁ decreasing and ρ₂ increasing in (0, 1). Indeed, one can prove that in this case G(μ)>0 and H(μ)>0 for every .

Corollary 3.3

Suppose that or for all x∈(0, 1), and Then G(μ)>0 and H(μ)>0 for all

Proof

We only consider the case ρ₁ increasing and ρ₂ decreasing in (0, 1). Note that because the operator is self-adjoint we have that

Since is decreasing and nontrivial, the solution of

is decreasing with f′<0 in (0, 1). Therefore, using integration by parts

Similarly, we can show that G(μ)>0.   ▪

Similar to previous corollary, we have the following corollary.

Corollary 3.4

Suppose that or for all x∈(0, 1), and Then for all μ>0.

The following proposition, which is a generalisation of Proposition 2.2, states that a necessary condition to have nonconstant positive steady states for small τ is that both ρ₁ and ρ₂ must be nonconstant.

Proposition 3.3

Suppose that ρ₂≥0 is a constant. Then for every μ>0, there exists τ₆ such that Equation (3) does not admit any nonconstant positive solutions for any

Proof

We just need to consider the case ρ₁≥0 nonconstant. We argue by contradiction. Suppose that we have a sequence of nonconstant positive steady states, with τ→0. Then, we can write , , with and in . In this situation we have that H(μ)>0 and G(μ)=0. Therefore, by Equations (20) and (37) we obtain that and as τ→0. Moreover, using that we can deduce that there exists a constant C>0 such that

After doing some expansions, and using the inequality above we can prove that

with . By Equation (43), and using the fact that , we conclude by elliptic regularity that

for some constant C>0. Define and . Using the above estimate, considering possibly a subsequence, we have that and as τ→0. Doing some expansions we find that [ytilde] and [ztilde] are solutions of

Hence, we have that , where ¯ρ₁ denotes the average in Ω of ρ₁. If we expand in Fourier series , where , for k≥0, are the eigenfunctions and eigenvalues of

we have that

Now, if we integrate the first equation of Equation (3) and divide it by to obtain that

as τ→0, where the last inequality holds in view of Equation (42). On the other hand, by Equation (46) we obtain that

which contradicts Equation (47).   ▪

4. Stability of steady states

In this section we study the stability of steady states including both positive steady states constructed in Theorem 3.1 and two semi-trivial steady states of Equation (1). The main results of this section are Propositions 4.1 and 4.2 and Theorem 4.1.

For each solution (u, v) of Equation (3), we have the following eigenvalue problem:

where . Observe that Equation (48) corresponds to the linearisation of Equation (1) at the equilibrium (u, v). It follows from 48 that if all the eigenvalues of Equation (48) have negative real parts, then the equilibrium (u, v) is asymptotically stable, while if Equation (48) has eigenvalues with positive real parts, then a local unstable manifold arises.

After some computations it is easy to obtain the following relation:

We start by studying the stability properties of the semitrivial steady states.

4.1. Stability of the semitrivial steady states

We consider the semitrivial steady-state solution of Equation (1) for τ>0 small. The stability properties of this steady state are given by the sign of the principal eigenvalue of

In this situation, formula (49) can be simplified as

We can expand the functions θ₁ and ψ as

where ω₁ and ψ₁ are smooth functions. Thus, we obtain from Equation (51) that

where

In our first result, we show that the stability of can be characterised in terms of the function H(μ) for τ small.

Lemma 4.1

We have

Proof

Since τ is small and M is smooth, we can write the expansion , where

It can be easily seen that and satisfy

By some computations, we obtain that

since . Using that as τ→0 we conclude the desired result.   ▪

The above result establishes the stability of the semi-trivial steady state (θ₁, 0) at μ for which H(μ)≠0. We now turn to the case and for some μ*. In this situation, by Theorem 3.1 there exists a branch of nontrivial positive solutions bifurcating from (θ₁, 0). More precisely, by Proposition 3.2 there exists a smooth function , defined for small τ≥0 with , such that , i.e. positive solutions bifurcating from (θ₁, 0) at . In this situation, we have that for and τ∼ 0 the function M(μ, τ) can be expanded as

where μ_τ lies in the interval or . Therefore, the following result holds.

Lemma 4.2

Suppose that and Then

Thus, as a consequence of Lemmas 4.1 and 4.2 we can prove the following result.

Proposition 4.1

For the semi-trivial steady state of Equation (1) is asymptotically stable whenever and unstable when provided that τ>0 is small enough. Moreover if and then there exists δ>0 such that for small τ we can find with such that for the steady state is unstable when and asymptotically stable when

Similarly, we can prove the following result for .

Proposition 4.2

For the semi-trivial steady state of Equation (1) is asymptotically stable whenever and unstable when provided that τ>0 is small enough. Moreover if and then there exists δ>0 such that for small τ we can find with such that for the steady state is unstable when and asymptotically stable when

4.2. Stability of positive steady states

In this subsection, we study the sign of the largest eigenvalue of Equation (48) corresponding to solutions (u, v) close to for , as given by Theorem 3.1(1) and Proposition 3.2.

Choosing appropriately the eigenfunction associated to , the principal eigenvalue of Equation (48), they satisfy ϕ→1, ψ→−1. Indeed, we have the following expansions:

where , with y¯, z¯ as in Equations (11) and (12) and the smooth functions . By Equation (23) we have that are solutions of

and after a simple expansion we have that satisfy

From now on we denote , and . Using Equation (49) and

we obtain the following formula for in terms of :

where J(μ, τ) is a smooth function and

Multiplying the first equation of Equation (57) by V ₁ and the second by U ₁ and integrating both by parts we obtain that

Similarly, if we multiply the second equation of Equation (56) by ϕ₁ and the first by ψ₁ and we integrate both by parts we get

therefore, replacing Equations (59) and (60) in Equation (58), we obtain the following expression:

where .

Proposition 4.3

We have

Proof

Using Equations (56) and (57), we can easily check that satisfies

from where we conclude that

where C ₁ is a constant. By Equation (56) we can deduce that satisfies

thus

with C ₂ a constant. Hence by Equation (61) we have that

Now, adding both equations of Equation (57) we can see that satisfies Equation (25) then with as defined in Equation (25). Analogously, by Equation (56) we have that U ₁+V ₁ satisfies Equation (24), thus where . Thus, Equation (65) can be rewritten as

where we have used the definition of G(μ) and H(μ) as stated in Equation (5). Finally, using that we obtain from the expression above that

which proves the desired result.   ▪

Observe that the above proposition gives the stability of the unique positive steady state of Equation (1), provided by Theorem 3.1, whenever H(μ)>0 and G(μ)>0. When μ* is a simple zero of H (or G) and (and ) we have, by Theorem 3.1 and Proposition 3.2, that a bifurcating branch of positive steady states arises. Since at μ* we have that , we can proceed as in the proof of Lemma 4.2 to conclude that the bifurcating positive solutions are also asymptotically stable. Then, we can state the following result.

Theorem 4.1

For every fixed μ>0, there exists a τ₀>0 such that if the unique positive steady state, if it exists, is asymptotically stable.

5. Discussions

We investigated a reaction-diffusion system that models two ecologically equivalent competing species with cross-diffusion and self-diffusion. The idea is to envision that some mutation occurs and the mutant and the resident species have the same population dynamics but differ slightly in their dispersal behaviours. We addressed the following question: Which dispersal strategy can convey some competitive advantage and thus will be selected? Our main results suggest that for homogeneous spatial environment the dispersal strategy with constant cross- and self-diffusion coefficients seem to be preferred over those with nonconstant ones. For one-dimensional homogeneous habitat, we showed that if the gradients of two cross- and self-diffusion coefficients have the same direction, the species with the smaller gradient can always invade at low densities, and, as it is strongly suggested by the stability analysis, it will win. In particular, it suggests that the species with constant cross- and self-diffusion coefficients may have competitive advantage over species with nonconstant cross- and self-diffusion coefficients, and the underlying reason may be that the environment is assumed to be spatially homogenous. However, if the two gradients have opposite directions, neither of the two dispersal strategies wins as these two species can coexist. The underlying reason for such coexistence is that each species has advection along the gradient of its own cross- and self-diffusion coefficients, i.e. the two species can move in two opposite directions that promotes spatial segregation and hence coexistence. These results also raise one interesting question: If the population dynamical terms were spatially heterogeneous, would dispersal strategies with nonconstant cross- and self-diffusion coefficients be favoured? As one of the referees commented, understanding the evolution of density-dependent dispersal with nonconstant coefficients in spatially heterogeneous environment can be interesting as certain types of spatially varying dispersal seem to be favoured in heterogeneous environments in other types of models.

Acknowledgements

Part of work was done while Y.L. was visiting Department of Ecology and Evolution at Princeton University, and he thanks Simon Levin and the department for the hospitality. S.M. wants to thank the MBI at Ohio State University for the hospitality, where part of this work was written. We sincerely thank Professor Cosner and the two anonymous referees for their helpful suggestions/comments that substantially improve the exposition of the manuscript. Y. Lou was partially supported by the National Science Foundation grant DMS–0615845. S. Martínez was partially supported by Fondecyt # 1050754, FONDAP de Matemáticas Aplicadas and Nucleus Millennium P04-069-F Information and Randomness.