*(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:

where $N=\{1,\xb7\xb7\xb7,n\}$, 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 ${a}_{i},{e}_{i}\in {\mathbb{R}}^{+}$; ${\gamma}_{i}\in \mathbb{R}$ 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 ${x}_{i}^{\left(j\right)}$, $j=1,\xb7\xb7\xb7,{k}_{i}$, $i\in N$, (1) is transformed into the expanded system of O.D.E.:

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.

##### MSC:

92D25 | Population dynamics (general) |

34D20 | Stability of ODE |

45J05 | Integro-ordinary differential equations |

92D40 | Ecology |