Global stability of single-species diffusion volterra models with continuous time delays |
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Authors: | E Beretta Y Takeuchi |
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Affiliation: | (1) Istituto di Biomatematica, Università di Urbino, I-61029 Urbino, Italy;(2) Department of Applied Mathematics, Faculty of Engineering, Shizuoka University, 432 Hamamatsu, Japan |
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Abstract: | 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. |
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