On several conjectures from evolution of dispersal

Abstract

We address several conjectures raised in Cantrell et al. [Evolution of dispersal and ideal free distribution, Math. Biosci. Eng. 7 (2010), pp. 17–36 [9]] concerning the dynamics of a diffusion–advection–competition model for two competing species. A conditional dispersal strategy, which results in the ideal free distribution of a single population at equilibrium, was found in Cantrell et al. [9]. It was shown in [9] that this special dispersal strategy is a local evolutionarily stable strategy (ESS) when the random diffusion rates of the two species are equal, and here we show that it is a global ESS for arbitrary random diffusion rates. The conditions in [9] for the coexistence of two species are substantially improved. Finally, we show that this special dispersal strategy is not globally convergent stable for certain resource functions, in contrast with the result from [9], which roughly says that this dispersal strategy is globally convergent stable for any monotone resource function.

Keywords:

• evolution of dispersal
• ideal free distribution
• evolutionarily stable strategy
• reaction–diffusion–advection

AMS Classification :

• 35K57
• 92D25

1. Introduction

Within the broad scope of theoretical ecology, the notion of dispersal is indispensable in determining the distribution, dynamics, and persistence of a species within its habitat. More specifically, one can ask how the spread and movement of a population evolves over time. Recent studies have identified several mechanisms which play significant roles in this evolution 18, one of which is temporal and spatial variability in the environment. Hastings 23 focused on spatial variation in the environment, utilizing a reaction–diffusion model to study its effect on the evolution of passive dispersal (see also 19 30). Following Hastings’ work, Belgacem and Cosner 1 added an advection term to the well-known logistic reaction–diffusion model, realizing that in a spatially variable environment, a population may move towards regions that are more favourable. The endeavour to understand the evolution of this combination of passive and biased dispersal, via a reaction–diffusion–advection model in a spatially inhomogeneous environment, prompted the work of Cosner and Lou 13, Cantrell et al. 4 6 8 9, Chen and Lou 10, Chen et al. 11, Hambrook and Lou 22, Bezuglyy and Lou 2, Lam 34 35, and Lam and Ni 36.

Our paper emerges from the above context with the aim of addressing several conjectures raised in Cantrell et al. 9 concerning the dynamics of the two species diffusion–advection–competition model

where u(x, t) and v(x, t) represent the densities of two competing species, μ and ν are their random diffusion coefficients, , and m(x) is the intrinsic growth rate of both species. Throughout this paper we will always assume that m>0 in ¯Ω, where Ω is a bounded domain in ℝ ^N with smooth boundary , n is the outward unit normal vector on , and the boundary condition in Equation (1) says that there is no flux across the boundary.

To motivate our discussion, we first consider the dynamics of Equation (1) when , i.e. only species u is present. For such a situation, Equation (1) is reduced to the single species reaction–diffusion–advection model

It is known that if m>0 in Ω, then Equation (2) has a unique positive steady state, denoted by u*, which is globally asymptotically stable among non-negative non-trivial initial data, and u* solves

Integrating the equation of u* and applying the divergence theorem, we have

Since u*>0 in Ω, either m−u* changes sign in Ω or in Ω. If we regard m−u* as the fitness of the species u at equilibrium, m−u* changes sign means that there is some mismatch between the species density and its resource distribution. On the other hand, means that the population matches the environmental quality perfectly and the fitness of the population is the same everywhere in the habitat. Furthermore, if , then in Ω, i.e. zero net movement of individuals. Hence, if the scenario occurs, then the species’ equilibrium density with dispersal present is the same as that with dispersal absent. Such rather peculiar spatial distribution of species is usually referred to as the ideal free distribution 5.

A natural question is: for what kind of functions P(x) can it happen that ? It was found 9 that if and only if is equal to some constant. Since for any constant C, we may simply restrict our discussion to the case instead of . It is known that population models with nonlinear diffusion can support or approximate ideal free dispersal strategies (see 7 12). It is quite interesting that even biased movement of a species along its resource gradient alone can also produce ideal free distributions of populations at equilibrium.

A steady state of Equation (1) with both components being positive is called a coexistence state; is a semi-trivial steady state if one component is positive and the other is the zero function. It is known that if m>0, Equation (1) has exactly two semi-trivial steady states, denoted as (u*, 0) and (0, v*), where u* is the unique positive solution of Equation (3) and v* can be defined similarly.

The first result of Cantrell et al. 9 can be stated as follows.

Theorem 1.1

[9, Theorem 1] Suppose that μ=ν, and m>0 in ¯Ω.

	(a) Suppose that where . If R is non-constant, then (0, v*) is unstable and (u*, 0) is globally asymptotically stable for .
	(b) Suppose that P(x)−ln m is non-constant. Then there exists some such that for (u*, 0) is unstable for .

An important idea in adaptive dynamics 15 17 16 20 is the idea of evolutionarily stable strategies (ESS). A strategy is said to be evolutionarily stable if a population using it cannot be invaded by any small population using a different strategy. Part (a) of Theorem 1.1 shows that P=ln m is a local ESS and part (b) shows that no other strategy can be a local ESS. It was conjectured in 9 that P=ln m is a global ESS, i.e. part (a) of Theorem 1.1 holds for any . Our first result is to answer this conjecture positively.

Theorem 1.2

Given any μ, ν>0. Suppose that P(x)=ln m, and Q(x)−ln m is not a constant function. Then, the semi-trivial steady state (u*, 0) is globally asymptotically stable.

Remark 1.1

This theorem proves the conjecture that P(x)=ln m is a global ESS. In fact, we also allow μ, ν to be arbitrary here. The condition on Q(x) is also necessary: if Q(m)−ln m is also a constant function, Equation (1) has a continuum family of positive steady states, all of them are of the form , where s∈(0, 1).

Remark 1.2

Note that u*≡ m when . It is easy to see that the semi-trivial steady state (m, 0) is neutrally stable. Thus, even the local asymptotic stability of (m, 0) is non-trivial and is of independent interest.

The second main result in 9 concerns the coexistence of two competing species and it can be stated as the following.

Theorem 1.3

[9, Theorem 2, part (b)] Suppose that μ=ν, m>0. We further assume that Ω=(0, 1) and R _x ≠0 in [0, 1]. If then both (u*, 0) and (0, v*) are unstable, and system (1) has at least one stable positive steady state.

Theorem 1.3 implies that the two species can coexist provided that their dispersal strategies lie on two ‘opposite sides’ of the optimal strategy ln m. Our second result is to sharpen Theorem 1.3 as follows.

Theorem 1.4

Suppose that and is non-constant. If then both (u*, 0) and (0, v*) are unstable, and system (1) has at least one stable positive steady state.

The third main result of 9 concerns whether ln m is a convergent stable strategy (CSS) of system (1). A strategy is convergent stable if selection favours strategies that are closer to it over strategies that are further away. More precisely, the following result is established in 9.

Theorem 1.5

[9, Theorem 2, part (a)] Suppose that μ=ν, Ω=(0, 1), and R _x ≠0 in [0, 1]. If or then (u*, 0) is unstable and (0, v*) is stable. Moreover, given any η>0, there exists κ>0 such that if either (i) and or (ii) and then (0, v*) is globally asymptotically stable.

It is natural to inquire whether the assumption R _x non-vanishing in Ω is essential for Theorem 1.5 to hold. A little surprisingly, it turns out to be possible to construct non-monotone functions R(x) such that for and , both (u*, 0) and (0, v*) are unstable for suitably chosen positive constants . To this end, we first give a description of such non-monotone functions R(x). Given any function m>0 in ¯Ω, we assume that R(x) satisfies the following.

(A) There exists some such that x ₀ is a local maximum of R(x) and

It is not difficult to see that for any positive function m (even if m is a positive constant), there exist functions which satisfy assumption (A). If we perturb R slightly, we may further assume that all critical points of R are non-degenerate. Clearly, any function R(x) which satisfies assumption (A) will have at least two local maxima and thus cannot be monotone. To see this, let x* be any global maximum point of R(x), then we have

In other words, for any local maximum point x ₀ of R satisfying assumption (A), x ₀ cannot be a global maximum point of R.

Our main goal is to show that under assumption (A), (u*, 0) is unstable for suitably chosen parameters . The key ingredient is to find α>0 and μ>0 such that , i.e. the species u at equilibrium undermatches its resource at some local maximum point of R. Once this is done, we can choose β sufficiently large, i.e. the species v has a strong tendency to concentrate near the local maxima of R, such that small populations of v can invade in a neighbourhood of x ₀ since the effective growth rate for v is m(x)−u(x), which is positive for x close to x ₀. The precise statement of our result is as follows.

Theorem 1.6

Suppose that R(x) satisfies assumption (A) and all critical points of R are non-degenerate. Assume that and in Equation (1). Then there exists some α₀>0 such that for every we can find some μ₀>0 such that if μ>μ₀, then given any ν>0, both (u*, 0) and (0, v*) are unstable for sufficiently large β>0. Furthermore, system (1) has at least a stable positive steady state.

Note that ν can be arbitrarily chosen, so we allow μ=ν in our constructions. This immediately gives a counterexample to some conjecture raised in part (c), Remark 1.1 of 9.

Theorem 1.6 suggests that there is some such that the strategy may be a local ESS and/or CSS. At first look this appears to contradict part (b) of Theorem 1.1, which says that no other strategy can be a local ESS except P=ln m. Actually they are consistent with each other since part (b) of Theorem 1.1 allows R to vary arbitrarily, while here we are fixing m and R and only allow the parameters to vary. In other words, if we only consider the evolution of a single trait , then system (1) may have other local ESS and/or CSS besides P=ln m. Hence, while the global ESS exists and is unique, this global ESS may not be a global CSS, and there may exist multiple local ESS and/or CSS for system (1) if we only allow one single trait to evolve.

This paper is organized as follows. In Section 2, we give some preliminary results on monotone dynamical systems and criteria on the local stability of semi-trivial steady states. Sections 3–5 are devoted to proofs of Theorems 1.2, 1.4, and 1.6, respectively. Some discussions of the results are given in Section 6.

2. Preliminary results

In this section, we summarize some statements regarding solutions of system (1) and the stability of its steady states, which will be useful in later sections. By the maximum principle for cooperative systems 40 and the standard theory for parabolic equations 25, if the initial conditions of Equation (1) are non-negative and not identically zero, system (1) has a unique positive smooth solution which exists for all time and it defines a smooth dynamical system on 3 26 41. The stability of steady states of Equation (1) is understood with respect to the topology of . The following result is a consequence of the maximum principle and the structure of Equation (1) (see [9, Theorem 3]).

Theorem 2.1

The system (1) is a strongly monotone dynamical system, i.e.

	(a)  and for all x∈Ω and
	(b)  implies and for all and t>0.

The following result is a consequence of Theorem 2.1 and the monotone dynamical system theory 26 41.

Theorem 2.2

If system (1) has no coexistence state, then one of the semi-trivial steady states is unstable and the other one is globally asymptotically stable 29; if both semi-trivial steady states are unstable, then Equation(1) has at least one stable coexistence state 14 37.

The following result concerns the linear stability of semi-trivial steady states of Equation (1) (see, e.g. [11, Lemma 5.5]).

Lemma 2.1

The steady state (u*, 0) is linearly stable/unstable if and only if the following eigenvalue problem, for has a positive/negative eigenvalue:

The criterion for the linearized stability of the semi-trivial steady state (0, v*) is analogous.

3. Proof of Theorem 2

In this section, we focus on the case when P(x)=ln m, i.e. the following model:

Theorem 3.1

Given any μ, ν>0. Suppose that Q(x)−ln m is not a constant function. Then, the semi-trivial steady state (u*, 0) is globally asymptotically stable.

Proof

Step 1. We show that Equation (4) has no positive steady states. To this end, we argue by contradiction. Suppose that u, v are positive steady states of Equation (4), i.e. they satisfy

Set w=u/m. Then w satisfies

Since w>0, dividing the equation of w by w and integrating in Ω, we have

Integrating the equations of u and v, we have

and

respectively.

Adding up Equations (7) and (8), we have

Subtracting Equation (9) from Equation (6), we obtain

which implies that and w=s for some positive constant s>0; i.e. u/m=s for some constant s. Since u>0 and v>0, from m−u−v=0, we see that s∈(0, 1) and v=(1−s)m. Substituting into the equation of v and dividing the result by (1−s), we see that

By the maximum principle 40, Q(x)−ln m must be equal to some constant, which contradicts our assumption. This proves that Equation (4) has no positive steady states.

Step 2. We show that (0, v*) is unstable. By Lemma 2.1, it suffices to show the smallest eigenvalue, denoted by λ₁, of the linear eigenvalue problem

satisfies λ₁<0. Let ϕ₁ denote the positive eigenfunction of λ₁ uniquely determined by . Set . Then the previous equation can be written as

Dividing the equation of ψ by ψ and integrating the result in Ω, we have

Integrating the equation of v*, we have

Subtracting Equation (12) from Equation (11), we find that

Hence, λ₁<0 as long as . To this end, we argue by contradiction and suppose that v*≡ m. Then by the equation of v*, we see that Equation (10) holds, which implies that Q(x)−ln m is constant and we reach a contradiction. Hence, and thus λ₁<0.

Step 3. We show that the semi-trivial steady state (u*, 0) is globally asymptotically stable. This follows from Theorem 2.2, system (4) has no positive steady state (Step 1), and the semi-trivial steady state (0, v*) is unstable (Step 2).   ▪

4. Proof of Theorem 4

In this section, we generalize previous results in Cantrell et al. 9 on the coexistence of two competing species. In particular, we focus on the case where and , i.e. we consider

Theorem 4.1

Suppose that and is non-constant. Then, both semi-trivial steady states (u*, 0) and (0, v*) are unstable, and system (13) has at least one stable positive steady state.

Proof

Step 1. We show that (0, v*) is unstable. Let λ₁ denotes the smallest eigenvalue of the following linear problem

and let ϕ₁ denote the unique positive eigenfunction of λ₁ which satisfies . Set . Then, ψ satisfies

Dividing the equation of ψ by ψ and integrating in Ω, we have

Recall that v* satisfies

Set . Then w satisfies

Multiplying the equation of w by w ^l and integrating in Ω, we have

where l>0 is to be chosen later.

By Equations (16) and (18) we have

Choose

By our assumption , we have l>0. Hence,

where the equality holds if and only if ψ and w are both equal to constants. Since l>0, we see that

in Ω, where the equality holds if and only if m=v*. Therefore, , and λ₁=0 if and only if . To complete the proof, it suffices to rule out the possibility m≡ v*. To this end, we see that if m≡ v*, v* satisfies

By the maximum principle 40 we see that is equal to some constant. This together with m≡ v* implies that e^{β R} must be equal to some constant. Since β≠0, we see that R must be equal to some constant, which contradicts our assumption. Hence, λ₁<0, which together with Lemma 2.1 implies that (0, v*) is unstable.

Step 2. Similarly, by symmetry we see that if , (u*, 0) is unstable. Since the system (13) is a strongly monotone dynamical system, by Theorem 2.2 we see that system (13) has at least a stable positive steady state.   ▪

5. Proof of Theorem 1.6

This section is devoted to the case when both α and β are positive. It is shown in Cantrell et al. 9 that if Ω is an interval and R _x >0 in ¯Ω, , then (u*, 0) is stable and (0, v*) is unstable. A natural question is whether the monotonicity of R(x) is essential. In this section, we will construct non-monotone functions R(x) such that for and , both (u*, 0) and (0, v*) are unstable for suitably chosen positive constants .

Lemma 5.1

Let x ₀ be a local maximum of R which satisfies assumption (A). There exists some α₀>0 such that for every

Proof

For sufficiently small α,

  ▪

Lemma 5.2

Let x ₀ be a local maximum of R which satisfies assumption (A). Then there exists some μ₀ such that if μ>μ₀, .

Proof

Set . Then w satisfies

By the maximum principle 40, w and u* are both uniformly bounded for all μ≥1. By L ^p theory for second-order elliptic operators (see 21), for any p>1, is uniformly bounded for all μ≥1. By the Sobolev embedding theorem, is uniformly bounded for some τ∈(0, 1). Passing to a subsequence if necessary, w converges to some function . Multiplying the equation of w by w and integrating the result in Ω, we have

By letting , we see that w̄ satisfies

i.e. w̄ is a constant. To determine w̄, by integrating the equation of u* in Ω we find

If w̄=0, i.e. in , then in as . Since m>0 in Ω, $m - u^*>0$ in Ω for large μ. This implies that in Ω for large μ, which contradicts Equation (20). Hence w̄ must be a positive constant. This together with Equation (20) implies that

Since w̄ is uniquely determined, the convergence of w to w̄ is independent of the subsequence. Hence,

uniformly in ¯Ω as . In particular, this, together with Lemma 5.1, implies that for every , for sufficiently large μ.   ▪

Lemma 5.3

Suppose that R satisfies assumption (A) and all critical points of R are non-degenerate. Then for μ>μ₀, ν>0, the semi-trivial steady state (u*, 0) is unstable for sufficiently large β>0.

Proof

By Lemma 2.1, it suffices to show the least eigenvalue, denoted by λ₁, of the eigenvalue problem

satisfies λ₁<0. Set . Then ψ satisfies

By the variational characterization, λ₁ is determined by

Hence, to show λ₁<0, we need to find ψ such that

Let denote the maximum of m. It suffices to find ψ such that

To establish Equation (21), consider another linear eigenvalue problem

Let λ* denote the principal eigenvalue of Equation (22). Rewrite Equation (22) as

Hence, by Theorem 1 of 10 we have

where ℛ denotes the set of local maxima of R. Note that

where the last inequality follows from Lemma 5.2, provided that and μ>μ₀. This implies that λ*<0. Let ψ*>0 denote an eigenfunction of λ*. Note that λ* can be characterized as

which is attained by ψ*. It then follows from λ*<0 and Equation (24) that Equation (21) holds for ψ=ψ*. This shows that λ₁<0.   ▪

The proof of the following result is identical to that of Theorem 3.5 in Cantrell et al. 6, so we omit the details.

Lemma 5.4

Suppose that the set of critical points of R(x) has Lebesgue measure zero and v* is given by Equation (17). Then v*→0 in L ²(Ω) as .

Lemma 5.5

Suppose that the set of critical points of R(x) has measure zero. Given any μ>0, ν>0, and α>0. If β is sufficiently large, then (0, v*) is unstable.

Proof

By Lemma 2.1 it suffices to show the principal eigenvalue, denoted by λ₁, of the eigenvalue problem

is negative. Let ϕ₁ be the positive eigenfunction of λ₁ uniquely determined by . Set . Then ψ>0 satisfies

Dividing the above equation by ψ and integrating the result in Ω, we have

where the last inequality follows from Lemma 5.4, provided that β is sufficiently large.   ▪

Proof of Theorem 1.6

It follows from Lemmas 5.3 and 5.5 and Theorem 2.2.   ▪

6. Discussion

In this paper, we addressed several conjectures raised in Cantrell et al. 9 concerning the dynamics of some diffusion–advection–competition model for two competing species. Both species are assumed to have the same population dynamics but different dispersal strategies: they both disperse by random diffusion and advection along certain gradients, but possibly do so with different rates and/or gradients. A conditional dispersal strategy, which results in the ideal free distribution of a single population at equilibrium, was found in 9. It was shown in 9 that this special dispersal strategy is a local ESS when random diffusion rates of two species are equal, and we show that it is actually a global ESS for arbitrary random diffusion rates. The conditions in 9 for the coexistence of two species are also substantially improved. Finally, we construct some examples to show that this special strategy may not be a globally CSS for certain resource functions with two or more local maxima, in strong contrast with the result from 9, which roughly says that this dispersal strategy is always a globally CSS for any monotone resource function. Our results seem to suggest that for resource functions with two or more local maxima, there may exist some other local ESS and/or CSS, besides the obvious candidate – the special conditional dispersal strategy found in 9. The biological intuition behind this is that if resource functions have two or more local maxima, the resident species at equilibrium may undermatch its resource at some local maximum of the resource, which makes it vulnerable to invasion by other species near such local maxima.

Some ideas from this work might be useful in studying the evolutionary stability of dispersal strategies in reaction–diffusion models 19 24 42 43, patch models 5 27 28 33 38 39, non-local dispersal models 31 32 44, or metapopulation models 23 45. These findings will be reported in some forthcoming paper(s).

We conjecture that the special dispersal strategy P=ln m is a globally CSS when the function R has a unique local maximum (and thus it must be the global maximum). For such functions R, the construction of the counterexample in Theorem 1.6 breaks down since one always has for any global maximum x ₀; i.e. the population at equilibrium always overmatches its resource at the global maximum of R. To see this, following the proofs of Theorem 1.3 in 7 or Lemma 5.2 in 11, if α≥0, we have the following inequality:

for every . In particular, for any global maximum x ₀ of R. We further refer to 34 35 36 for recent important development on the qualitative profiles of u* and also steady-state solutions of two species competition model with one sufficiently large advection coefficient.

Acknowledgements

The authors thank the reviewer and the editor for their comments which helped improve the content of the manuscript. This research was partially supported by the NSF grant DMS-1021179.