On evolutionary stability of carrying capacity driven dispersal in competition with regularly diffusing populations

Abstract

Two competing populations in spatially heterogeneous but temporarily constant environment are investigated: one is subject to regular movements to lower density areas (random diffusion) while the dispersal of the other is in the direction of the highest per capita available resources (carrying capacity driven diffusion). The growth of both species is subject to the same general growth law which involves Gilpin–Ayala, Gompertz and some other equations as particular cases. The growth rate, carrying capacity and dispersal rate are the same for both population types, the only difference is the dispersal strategy. The main result of the paper is that the two species cannot coexist (unless the environment is spatially homogeneous), and the carrying capacity driven diffusion strategy is evolutionarily stable in the sense that the species adopting this strategy cannot be invaded by randomly diffusing population. Moreover, once the invasive species inhabits some open nonempty domain, it would spread over any available area bringing the native species diffusing randomly to extinction. One of the important technical results used in the proofs can be interpreted in the form that the limit solution of the equation with a regular diffusion leads to lower total population fitness than the ideal free distribution.

Appendix

In this section we present auxiliary results used in the proofs of Sections 2.2 and 3. The next theorem from the monograph by Pao (1992) deals with the time-dependent solution of the system of the form

$$\begin{aligned} \begin{aligned} \frac{\partial u_i}{\partial t}-L_iu_i&=f_i(x,u_1,u_2),\;\;t>0,\;x\in \Omega \\ \frac{\partial u_i}{\partial n}&=0,\;x\in \partial \Omega \\ u_i(0,x)&=u_{i,0}(x),\;x\in \Omega , ~~i=1,2, \end{aligned} \end{aligned}$$

(7.1)

where for $i=1,2$ the operators $L_i$ defined as

$$\begin{aligned} L_iu:=\sum \limits _{i,j=1}^{n}a_{ij}(t,x)\frac{\partial ^2 u}{\partial x_i\partial x_j}+\sum \limits _{i=1}^{n}b_{i}(t,x)\frac{\partial u}{\partial x_i} \end{aligned}$$

(7.2)

are uniformly elliptic, namely, there exist positive numbers $\lambda$ and $\Lambda$ such that for every vector $\xi =(\xi _1,...,\xi _n)\in \mathbb {R}^{n}$

$$\begin{aligned} \lambda |\xi |^2\le \sum \limits _{i,j=1}^{n}a_{ij}(t,x) \xi _i\xi _j\le \Lambda |\xi |^2,\;\;(t,x)\in [0,T] \times \overline{\Omega }. \end{aligned}$$

(7.3)

We assume that the coefficients of $L$ are Hölder continuous in $[0,T)\times \Omega ,\;\forall T>0$ and the vector-function $(f_1,f_2)$ is continuously differentiable and monotone nonincreasing in $\mathbb {R}^{+}\times \mathbb {R}^{+}$.

For any $\rho =(\rho _1,\;\rho _2)\in \mathbb {R}^{2}_{+}$ define

$$\begin{aligned} \mathbf{S}_{\rho }\equiv \left\{ (u_1,u_2)\in C([0,\infty )\times \overline{\Omega });(0,0)\le (u_1,u_2)\le (\rho _1,\rho _2)\right\} \end{aligned}$$

The next result is Theorem 8.7.2 in the book of Pao (1992).

Theorem 16

Let $(f_1,f_2)$ be a quasimonotone nonincreasing Lipschitz function in $\mathbf{S}_{\rho }$ and let $(f_1,f_2)$ satisfy

$$\begin{aligned} \begin{aligned} f_1(t,x,\rho _1,0)&\le 0\le f_1 (t,x,0,\rho _2)\\ f_2(t,x,0,\rho _2)&\le 0\le f_2(t,x,\rho _1,0) \end{aligned} \end{aligned}$$

(7.4)

for any $x\in \Omega ,\;t>0$. Then for any $(u_{1,0},u_{2,0})\in \mathbf{S}_{\rho }$ there exists a unique solution of (7.1) $\mathbf{u}\equiv (u_1,u_2)$ in $\mathbf{S}_{\rho }$, and $u_i(t,x)>0$ for $x\in \Omega ,\;t>0$ when $u_{i,0}\ne 0, i=1,2$.

The next lemma (Cantrell and Cosner 2003) implies the monotonicity property for solutions of (7.1).

Lemma 5

Let $(u_i(t,x),v_i(t,x)),\;i=1,2$, be two solutions of system (7.1) such that $u_1(0,x)\ge u_2(0,x)$ and $v_1(0,x)\le v_2(0,x)$ for any $x\in \Omega$. Then $u_1(t,x)\ge u_2(t,x)$ and $v_1(t,x)\le v_2(t,x)$ for any $x\in \Omega$ and any $t>0$. Moreover, if

$$\begin{aligned} (u_1(0,x),v_1(0,x))\ne (u_2(0,x),v_2(0,x)), \end{aligned}$$

then $u_1(t,x)>u_2(t,x)$ and $v_1(t,x)<v_2(t,x)$ for any $x\in \Omega$ and any $t>0$

For the proof see Theorem 1.20 in the book by Cantrell and Cosner (2003) and remarks thereafter. The final statement of the lemma follows from the strong maximum principle for parabolic equations (Protter and Weinberger 1967) applied to the differences $u_1-u_2$ and $v_2-v_1$.

The next result by Hsu et al. (1996) classifies all possible equilibria for a monotone dynamical system. We will present a particular case of Theorem B (Hsu et al. 1996). Denote $X^{+}=C_{+}(\overline{\Omega })\times C_{+}(\overline{\Omega })$, where $C_{+}(\overline{\Omega })$ is the class of all nonnegative functions from $C(\overline{\Omega })$; $I=\left\langle 0,\tilde{u}_1\right\rangle \times \left\langle 0,\tilde{u}_2\right\rangle$, where $(\tilde{u}_1,0)$ and $(0,\tilde{u}_2)$ are the semi-trivial equilibria of (7.1). Here $\left\langle \cdot ,\cdot \right\rangle$ as in the proof of Theorem 5.

Theorem 17

Let operator $T_t$ be defined as $T_t(\mathbf{u}_0)=\mathbf{u}$, where $\mathbf{u}_0\equiv (u_{1,0},u_{2,0})$ and $\mathbf{u}\equiv (u_{1},u_{2}) = (T_t(u_{1,0}),T_t(u_{2,0}))$ is a solution of (7.1). Let the following conditions hold:

• (1) $T$ is strictly order preserving, which means that $u_1(x)\ge v_1(x)$ and $u_2(x)\le v_2(x)$ imply $T_t(u_1(x))\ge T_t(v_1(x))$ and $T_t(u_2(x))\le T_t(v_2(x))$.

• (2) $T_t(\mathbf{0})=\mathbf{0}$ for all $t\ge 0$ and $\mathbf{0}$ is a repelling equilibrium. That is there exists a neighbourhood $U$ of $\mathbf{0}$ in $X^{+}$ such that for each $(u_1,u_2)\in U,\;(u_1,u_2)\ne \mathbf{0}$, there is $t_0>0$ such that $T_{t_0}(u_1,u_2)\notin U$.

• (3) $T_t((u_{1},0))=(T_t(u_{1}),0)$ and $T_t(u_{1})\ge 0$ if $u_{1}\ge 0$. There exists $\tilde{u}_1>0$ such that $T_t((\tilde{u}_{1},0))=(\tilde{u}_{1},0)$ for any $t\ge 0$. The symmetric conditions hold for $T_t((0,u_{2}))$.

• (4) If $u_{i,0}\ge 0,\;u_{i,0}\ne 0,\;i=1,2$ then $T_t(u_{i,0})>0,\;i=1,2$. If $u_1(x)\ge v_1(x), u_2(x)\le v_2(x)$ and $u_1(x)\ne v_1(x),\;u_2(x)\ne v_2(x)$ then $T_t(u_1(x))> T_t(v_1(x))$ and $T_t(u_2(x))< T_t(v_2(x))$.

Then exactly one of the following holds:

• (a) There exists a positive coexistence equilibrium $(u_{1,s},u_{2,s})$ of (7.1).

• (b) $(u_1,u_2)\rightarrow (\tilde{u}_1,0)$ as $t\rightarrow \infty$ for every $(u_{1,0},u_{2,0})\in I$.

• (c) $(u_1,u_2)\rightarrow (0,\tilde{u}_2)$ as $t\rightarrow \infty$ for every $(u_{1,0},u_{2,0})\in I$.

Moreover, if (b) or (c) holds then for every $(u_{1,0},u_{2,0})\in X^{+}\backslash I$ and $u_{i,0}\ne 0,\;i=1,2$ either $(u_1,u_2)\rightarrow (\tilde{u}_1,0)$ or $(u_1,u_2)\rightarrow (0,\tilde{u}_2)$ as $t\rightarrow \infty$.