Spreading Speeds And Traveling Waves Of Reaction-Diffusion Equations With Non-Local Delays

As is well known, the time delay and diffusion are inevitable in many evolutionary processes. In recent years, many researchers began to study the combined effects of both time delay and spatial diffusion on the dynamic behaviors of differential equations. The study leads to a new class of infinite dimensional dynamical equations:reaction-diffusion systems with nonlocal delays, where the nonlinearities involving a spatial averaging the whole of the infinite spatial domain and the whole of the previous times. In contrast to (delayed) reaction-diffusion equations, these kinds of systems are more closer to reality. On the other hand, the time delay and spatial nonlocality also lead to many mathematical difficulty and complicated dynamics. Therefore, it is interesting and valuable both in theory and practice to study such kinds of the systems. This thesis is mainly concerned with the spreading speeds and traveling waves of reaction-diffusion equations with nonlocal delay. The main results in this thesis are as follows:●For reaction-diffusion equations with nonlocal delay and crossing-monostability, the existence of oscillatory waves is established based on the construction of two associated auxiliary reaction-diffusion equations with quasi-monotonicity and using the Schauder's fixed point theorem and Hopf bifurcation theorem. Furthermore, by rewriting the equa-tion as an integral equation and using the theory on nontrivial solutions of a convolution equation, we show that the non-monotone traveling waves are unique up to translation and also obtain the the exact asymptotic behavior of the profile asξ→-∞. Finally, we employ the the technical weighted energy method to prove the asymptotic stability of all traveling waves, including even the slower waves whose speeds are close to the minimal speed. As applications of the main results, we investigate the traveling waves of two reaction-diffusion models with nonlocal delay for the crossing-monostable nonlinearity, respectively.●For a class of nonlocal reaction-diffusion equations with delay, the exponential sta-bility of traveling fronts is proved by means of the (technical) weighted energy method. Furthermore, we consider monostable reaction-advection-diffusion equations with nonlocal delay. The existence and comparison theorem of solutions of the corresponding Cauchy problem in a weighted Sobolev space are established for the systems on R, by appealing to the theory of semigroup and abstract functional differential equations. The exponen-tial stability of traveling wave fronts is then proved by the comparison principle and the (technical) weighted energy method. When applied to some biological and epidemiological models, we obtain many meaningful results in practice.●For monostable reaction-diffusion equations with spatio-temporal delay(and/or in- finite delay), we consider the spreading speeds and traveling fronts. The existence of spreading speeds is established firstly by using the comparison method and finite time delay approximation. The existence of traveling fronts is then proved using the monotone iterations together with super-and subsolutions techniques and a limiting argument. Fur-thermore, the minimal wave speed and nonexistence of traveling fronts are also obtained. The result implies that the spreading speed is coincident with the minimal wave speed even for the monostable reaction-diffusion equations with spatio-temporal delay(and/or infinite delay), this kind of results have both theoretical and applied significance.●For a class of bistable reaction-diffusion systems whose reaction terms do not satisfy the so-called monotone condition, by proposing a weak monotone condition for the reaction terms, which is called interval monotone condition, the global asymptotic stability and uniqueness (up to translation) of traveling fronts are proved using the elementary super-and sub-solution comparison and the convergence results of monotone semiflows. Furthermore, we consider a bistable reaction-diffusion system with nonlocal delay. By introducing an undelayed reaction-diffusion system with more variables, the existence and uniqueness of traveling waves are established for the system with nonlocal delay. The result implies that the bistable wave is persistent even for large delay.●For non-monotone integral equations, the existence of traveling waves is established firstly via the construction of two auxiliary monotone integral equations and using the Schauder's fixed point theorem. Then we show that the traveling waves are unique up to translation by using the theory on nontrivial solutions of a convolution equation. The exact asymptotic behavior of the profile asξ→-∞and the minimal wave speed are also obtained. As an application for the main results, we carefully study the traveling waves for a nonlocal epidemic model with non-monotone "force of infection" by transforming the model to an integral system.