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Global stability of single-species diffusion Volterra models with continuous time delays. (English) Zbl 0627.92021

Consider an ecological system composed of multiple heterogeneous patches connected by discrete diffusion, and each patch is assumed to be occupied by a single species whose evolution equation for the ${i}^{th}$ patch is:

$\left(1\right)\phantom{\rule{1.em}{0ex}}{\stackrel{˙}{x}}_{i}={x}_{i}\left({e}_{i}-{a}_{i}{x}_{i}+{\gamma }_{i}{\int }_{-\infty }^{t}{F}_{i}\left(t-\tau \right){x}_{i}\left(\tau \right)d\tau \right)+\sum _{\mu =1}^{n}{D}_{i\mu }\left({x}_{\mu }-{x}_{i}\right),\phantom{\rule{1.em}{0ex}}i\in N,$

where $N=\left\{1,···,n\right\}$, n is the number of patches and ${x}_{i}$ is the population density in the ${i}^{th}$ patch. (1) may be thought as a generalization of the Volterra integral-differential equation to the n-patch case in which ${a}_{i},{e}_{i}\in {ℝ}^{+}$; ${\gamma }_{i}\in ℝ$ for all $i\in N$, where ${e}_{i}$, $i\in N$, are the intrinsic growth rates, and ${a}_{i}$, $i\in N$, represent the intraspecific relationships.

By introducing the supplementary functions ${x}_{i}^{\left(j\right)}$, $j=1,···,{k}_{i}$, $i\in N$, (1) is transformed into the expanded system of O.D.E.:

$\left(2\right)\phantom{\rule{1.em}{0ex}}{\stackrel{˙}{x}}_{i}={x}_{i}\left({e}_{i}-{a}_{i}{x}_{i}+{\gamma }_{i}\sum _{j=1}^{{k}_{i}}{C}_{i}^{\left(j\right)}{x}_{i}^{\left(j\right)}\right)+\sum _{\mu =1}^{n}{D}_{i\mu }\left({x}_{\mu }-{x}_{i}\right),$
${\stackrel{˙}{x}}_{i}^{\left(j\right)}={\alpha }_{i}{x}_{i}^{\left(j-1\right)}-{\alpha }_{i}{x}_{i}^{\left(j\right)},\phantom{\rule{1.em}{0ex}}j=1,···,{k}_{i},\phantom{\rule{1.em}{0ex}}{x}_{i}^{\left(0\right)}={x}_{i}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}i\in N·$

The dynamical behavior of (2) implies the same kind of dynamical behavior of (1). By applying homotopy function techniques [see e.g. C. B. Garcia and W. I. Zangwill, Pathways to solutions, fixed points, and equilibria (1981; Zbl 0512.90070)] the authors give sufficient conditions for the existence of a positive equilibrium and for its global and local stability. The biological meanings of the results are considered and compared with some known results.

Reviewer: Li Bingxi

##### MSC:
 92D25 Population dynamics (general) 34D20 Stability of ODE 45J05 Integro-ordinary differential equations 92D40 Ecology