Global stability of single-species diffusion volterra models with continuous time delays

In this paper we consider the stability property of single-species patches connected by diffusion with a within-patch dynamics of Volterra type and with continuous time delays. We prove that this system can only have two kinds of equilibria: the positive and the trivial one. By the assumption that the delay kernels are convex combinations of suitable non-negative and normalized functions, the linear chain trick gives an expanded system of O.D.E. with the same stability properties as the original integro-differential system. Homotopy function techniques provide sufficient conditions for the existence of the positive equilibrium and for its global stability. We also prove the local stability of any positive equilibrium and the local instability both of positive and trivial equilibria. The biological meanings of the results obtained are compared with known results from the literature. This work was performed under the auspices of G.N.F.M., C.N.R. (Italy) and within the activity of the Evolution Equations and Applications group, M.P.I. (Italy). I thank the Department of Applied Mathematics, Shizuoka University, Japan, which enabled me to visit Urbino.