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 patch is:
where , n is the number of patches and is the population density in the patch. (1) may be thought as a generalization of the Volterra integral-differential equation to the n-patch case in which ; for all , where , , are the intrinsic growth rates, and , , represent the intraspecific relationships.
By introducing the supplementary functions , , , (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.