Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain

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 Rickerʼs birth function and Mackey–Glassʼs 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.