Global Dynamics Of A Non-Local Delayed Feedback Differential Equation In Half Unbounded Domain

As is known to all, the living environment of the species is not only the entire domain or an bounded domain. Sometimes, it may be an half unbounded domain. So, it is of great significance to investigate the species living in half unbounded domain.In this paper, we first derive the delayed and diffusive equation for a single species population with two age classes period living in a spatially half unbounded environment. We show that if the mature death and diffusion rates are age inde-pendent, the total mature population is governed by a reaction-diffusion equation with time delay and non-local effect. But there are also some species whose imma-ture individuals diffuse, but their mature individuals do not, such as Barnacles.We establish the corresponding differential equation according to the living habits of the special species. Then adopting the compact open topology, we describe asymp-totic properties of the nonlocal delayed effect and establish some a prior 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 modal with Ricker’s birth function and Mackey-Glass’s hematopoiesis function, we obtain threshold results for the global dynamics of these two models. Then we explain why our results on the global attractivity of the positive equilib-rium in C+\{0} with respect to the usual supremum is not valid while it is valid in C+\{0} with respect to compact topology, and we identify a subset of C+\{0} in which the positive equilibrium remains attractive with respect to the supremum norm.The paper consists of the following four chapters:In the first chapter, we describe the background and significance of the differ-ential equation with time delay and no-local effect, and illustrate the problems we further study in this paper.In the second chapter, firstly, we present some basic notations and concepts. Then, we derive a delayed reaction-diffusion equation with spatial non-locality on half unbounded domain.In the third chapter, considering that the immature population of some species move while the mature can’t, we establish the corresponding differential equation according to the living habits of the special species, and then investigate the global dynamics of the equation.The last chapter we apply the main result to the model with Ricker’s birth function and Mackey-Glass’s hematopoiesis function, and then obtain the threshold results for the global dynamics of these two models.