Evolution of anisotropic diffusion in two-dimensional heterogeneous environments

Abstract

We consider a system of two competing populations in two-dimensional heterogeneous environments. The populations are assumed to move horizontally and vertically with different probabilities, but are otherwise identical. We regard these probabilities as dispersal strategies. We show that the evolutionarily stable strategies are to move in one direction only. Our results predict that it is more beneficial for the species to choose the direction with smaller variation in the resource distribution. This finding seems to be in agreement with the classical results of Hastings (1983) and Dockery et al. (1998) for the evolution of slow dispersal, i.e. random diffusion is selected against in spatially heterogeneous environments. These conclusions also suggest that broader dispersal strategies should be considered regarding the movement in heterogeneous habitats.

Appendix

1.1 Some remarks on solutions to a wave equation

In the proofs of Lemmas 4.3 and 4.4, the following result, which seems to be of self interest, plays an important role in eliminating the degeneracy of the function F and in establishing the strict concavity of the function $\Lambda (p,q)$ with respect to q.

Lemma 8.1

Let W in $C^2(\Omega )\cap C^1(\overline{\Omega })$ be a solution to the system

$$\begin{aligned} \left\{ \begin{aligned}&W_{xx}-W_{yy}=0\text { in }\Omega ,\\&W_x\nu _x=W_y\nu _y=0\text { on }\partial \Omega . \end{aligned} \right. \end{aligned}$$

(8.1)

Then, the function W is constant.

Proof

By the strict convexity assumption on the domain, the components $\nu _x$ and $\nu _y$ of the outward normal vector $\nu$ are non-zero on the boundary $\partial \Omega$, except possibly over a set of measure zero. Hence, $W_x$ and $W_y$ both vanish almost everywhere on $\partial \Omega$. Since W belongs to $C^1(\overline{\Omega })$, the gradient $\nabla W$ vanishes on $\partial \Omega$.

Set $\eta =x+y$, $\zeta =x-y$ and $Z(\eta ,\zeta ):=W(x,y)$. The function Z then satisfies

$$\begin{aligned} Z_{\eta \zeta }=0\text { in }\Omega '\quad \text { and }\quad {(Z_\eta ,Z_\zeta )=(0,0)}\text { on }\partial \Omega ', \end{aligned}$$

where $\Omega '$ is the image of $\Omega$ under the map $(x,y)\mapsto (\eta ,\zeta )$. It follows from the first relation that $Z(\eta ,\zeta )=f(\eta )+g(\zeta )$ for some functions f and g, and the second one then implies that both f and g have to be constant functions. As a consequence, Z is a constant function, and so is W. $\square$

It is possible to construct domains such that problem (8.1) admits non-constant solutions, if we allow Lipschitz domains with flat parts on their boundaries.

Example 1

Consider $\Omega =(0, 1)\times (0, 1)$ and let f be an even and 2-periodic function in $\mathbb {R}$. Set $W(x,y)=f(x+y)+f(x-y)$, which then clearly satisfies problem (8.1), and is a positive non-constant function if f is taken positive and non-constant.

On the other hand, the type of domain given in the above example seems to be non-generic, as illustrated by the following result.

Lemma 8.2

Suppose that $\Omega =(0, L_1)\times (0, L_2)$ for some positive numbers $L_1$ and $L_2$. If $L_1/L_2$ is not a rational number, then problem (8.1) has only constant solutions.

Proof

For any W satisfying problem (8.1), we have $W(x,y)=f(x+y)+f(x-y)$ for some scalar function f and then $W_x=f'(x+y)+f'(x-y)$. Since $W_x(0, y)=0$, we have $f'(y)=-f'(-y)$, i.e. $f'$ is an odd function. Since $W_x(L_1, y)=0$, we have $f'(y+L_1)=-f'(L_1-y)=f'(y-L_1)$, i.e. $f'$ is $2L_1$-periodic.

Similarly, one has $W_y=f'(x+y)-f'(x-y)$. Note that $W_y(x,0)=0$ automatically holds. By $W_y(x,L_2)=0$, we have $f'(x+L_2)=f'(x-L_2)$, that is $f'$ is also $2L_2$-periodic. Hence, if $L_1/L_2$ is not rational, then $f'$ must be a constant function. Since $f'$ is an odd function, then $f'=0$, that is W is a constant function. $\square$

1.2 A remark about a possible degeneracy induced by the domain $\Omega$

Throughout the paper, we have assumed that $\Omega$ is a strictly convex domain. We now comment on this point, showing with a very basic example that a domain with flat parts on its boundary may lead to a degeneracy of the function F.

Consider $\Omega =(0, 1)\times (0, 1)$ and let f and D be given as in Example 1. Setting

$$\begin{aligned} a:=-(\overline{D}+\underline{D})\frac{W_{xx}}{W}+W, \end{aligned}$$

it is easy to check that, for each $\theta$ in [0, 1], W also solves (2.3), that is $N_\theta \equiv W$ for each $\theta$ in [0, 1]. Since the function W is non-constant, the free growth rate a is also non-constant. Furthermore, for each value of p and q in [0, 1], problem (2.5) has a continuum of positive steady states of the form $(U, V)=(sW, (1-s)W)$, with s in (0, 1). Moreover, in this case, $F\equiv 0$ in [0, 1], even though the function a is not constant. Indeed, since f is even, we have $W(x,y)=W(y,x)$, which implies that $F\equiv 0$ in [0, 1].