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Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain
Authors:Taishan Yi
Institution:a College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, PR China
b Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B7
Abstract:In this paper, we study the global dynamics of a class of differential equations with temporal delay and spatial non-locality in an unbounded domain. Adopting the compact open topology, we describe the delicate asymptotic properties of the nonlocal delayed effect and establish some a priori estimate for nontrivial solutions which enables us to show the permanence of the equation. Combining these results with a dynamical systems approach, we determine the global dynamics of the equation under appropriate conditions. Applying the main results to the model with Ricker?s birth function and Mackey-Glass?s hematopoiesis function, we obtain threshold results for the global dynamics of these two models. We explain why our results on the global attractivity of the positive equilibrium in C+?{0} under the compact open topology becomes invalid in C+?{0} with respect to the usual supremum norm, and we identify a subset of C+?{0} in which the positive equilibrium remains attractive with respect to the supremum norm.
Keywords:34D23  39A30
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