Permanence and extinction in logistic and Lotka-Volterra systems with diffusion.(English)Zbl 0985.34061

The authors study the effect of diffusion on the permanence and extinction of endangered species that live in changing patch environment. Two models are considered. The first one is a logistic system with diffusion $\dot{x}_{i}=x_{i}[b_{i}(t)-a_{i}(t)x_{i}]+ \sum\limits_{j=1}^{n}D_{ij}(t)(x_{j}-\alpha_{ij}(t)x_{i}), \;\;i=1,\ldots,n,\tag{1}$ where $$x_{i}$$, $$i=1,\ldots,n$$, denotes the species $$x$$ in the patch $$i$$, $$b_{i}(t)$$ belongs to the set $$C$$ of all bounded continuous functions on $$\mathbb{R}$$, $$a_{i}(t)$$, $$D_{ij}(t)$$ belong to the set $$C_{+}\subset C$$ of continuous functions bounded below by a positive constant, $$D_{ii}(t)\equiv 0$$. $$b_{i}(t)$$ is the intrinsic growth rate for species $$x$$ in patch $$i$$, $$a_{i}(t)$$ represents the self-inhibition coefficient, $$D_{ij}(t)$$ is the diffusion coefficient of species $$x$$ from patch $$j$$ to patch $$i$$. The parameter $$\alpha_{ij}(t)\in C$$ corresponds to the boundary conditions of the continuous diffusion case.
The system of differential equations is said to be permanent if there exists a compact set $$K$$ in the interior of $${\mathbb{R}^{n}_{+}=\{x\in\mathbb{R}^{n}:x_{i}\geq 0, i=1,\ldots,n\}}$$, such that all solutions starting in the interior of $$\mathbb{R}^{n}_{+}$$ ultimately enter $$K$$. Denote by $A_{L}(f)=\lim\limits_{r\to\infty}\inf\limits_{t-s\geq r} \frac{1}{t-s}\int_{s}^{t}f(\tau) d\tau$ the lower average of $${f\in C}$$.
The following results concerning system (1) are obtained: There exists $${M>0}$$ such that the region $${D=\{0<x_{i}\leq M,i=1,\ldots,n\}}$$ is positively invariant with respect to (1). Moreover, any solution to (1) with $${x(0)>0}$$ is also ultimately bounded below away from zero provided that one of the following conditions is satisfied.
(H.1) There exists $${i_{0}\in\{1,\ldots,n\}}$$ such that $${A_{L}(\theta)>0}$$, with $$\theta(t)=b_{i_{0}}(t)-\sum_{j=1}^{n}D_{i_{0}j}(t) \alpha_{i_{0}j}(t)$$.
(H.2) $${A_{L}(\phi)>0}$$, with $$\phi(t)=\min_{1\leq i\leq n} \{b_{i}-\sum _{j=1}^{n} D_{ij}(t)\alpha_{ij}(t)+ \sum _{j=1}^{n}D_{ji}(t) \}.$$
It means that under these hypotheses system (1) is permanent.
If all coefficients in (1) are periodic with a common period $${\omega}$$, then at least one positive $$\omega$$-periodic solution exists. The uniqueness and stability of the periodic solution is also discussed.
In section 4, ehe authors consider the Lotka-Volterra diffusion model \begin{aligned} \dot{x}_{i}&=x_{i}[b_{i}(t)-a_{i}(t)x_{i}-c_{i}(t) y_{i}]+ \sum_{j=1}^{n}D_{ij}(t)(x_{j}- \alpha_{ij}(t)x_{i}),\\ \dot{y}_{i}&=y_{i}[-d_{i}(t)+e_{i}(t)x_{i}- f_{i}(t)y_{i}]+ \sum_{j=1}^{n}\lambda_{ij}(t)(y_{j}- \beta_{ij}(t)y_{i}),\end{aligned} \qquad i=1,\ldots,n, \tag{2} where $$y_{i}$$ is the density of predator species $$y$$ in patch $$i$$ and $$d_{i}(t)$$, $$e_{i}(t)$$, $$c_{i}(t)$$ and $$\beta_{ij}(t)$$ are all nonnegative and bounded continuous functions. Furthermore, $$f_{i}(t), \lambda_{ij}(t) \in C_{+}$$, $$i\neq j$$ and $$\lambda_{ii}(t)\equiv 0$$.
Under hypotheses analogous to conditions (H.1) and (H.2) given above, the permanence of predator and prey species in model (2) is proved.
The effect of a competitor $$y$$ and diffusion on the survival of the native endangered species $$x$$ is studied in section 5.
The biological meaning of the main results is discussed in the final section. The authors conclude that the diffusion rates play an important role in the determination of the permanence and extinction of single and multiple endangered species that live in weak patchy environments. Some illustrative examples are given.

MSC:

 34K13 Periodic solutions to functional-differential equations 92D25 Population dynamics (general) 34K05 General theory of functional-differential equations 34K20 Stability theory of functional-differential equations 34K25 Asymptotic theory of functional-differential equations 34K10 Boundary value problems for functional-differential equations
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