Existence Of Travelling Waves With Strong Generic Kernel In A Vector-Disease Model

From the 1970s until now, there have been enormous developments in the theory of travelling wave solutions of non-local interaction differential equations. It was found that travelling waves can be well used in the fields of physics, chemistry and biology, so the existence, uniqueness and stability of traveling wave solutions have been widely studied. Due to time delays which usually exist in nature, there have been a lot of works devoted to the studies of delayed reaction-diffusion equations from the dynamical systems and semigroups point of view. And, it has become recognized that the delayed reaction-diffusion equations can accurately describe the spatial-temporal patterns of the objects of study.In this paper, travelling wave solutions of a vector-disease model incorporating time-delays and diffusion have been studied. When the spatial variable is one dimensional and strong generic delay kernel assumes some special forms, travelling wave equations have been transformed into a finite-dimensional system (6) of ordinary differential equations by using the linear chain trick, then the geometric singular perturbation theory have been applied to prove the existence of heteroclinic orbits, which are travelling wave solutions for the vector-disease model.In the chapter 3, we have studied the travelling wave solutions of the delayed reaction-diffusion equation with the strong generic delay kernel. It reflects a natural connection between the existence of traveling wave solutions for the reaction-diffusion equation with distributed delays and the existence of traveling wave solutions for the corresponding undelayed reaction-diffusion equation. The invariant manifold theory, geometric singular perturbation theory and implicit function theory and dynamical system theory have been used in this chapter.