Research On The Travelling Wave Solutions Of Reaction Diffusion Systems And Persistence Of Some Reaction Systems

The main results of our research are divided into two parts: The first oneis on the existence of travelling wave solutions of reaction-di?usion systems withspatio-temporal delays. We consider the reaction-di?usion equations with mul-tiple spatio-temporal delays and with some zero di?usion coe?cients, and dealwith the existence of travelling wave fronts when the reaction term satisfies di?er-ent monotonicity conditions, such as quasi-monotonicity condition(QM), the expo-nential quasi-monotonicity condition(QM?) and the weakened exponential quasi-monotonicity condition(QM??), etc.. By using analysis technique and fixed pointtheorem, we obtain the existence of the solutionΦ(s) of wave equations, and thenthe existence of travelling wave fronts U(t,x) =Φ(x + ct). As the applicationsof our results, we give examples, and present some conditions under which thetravelling wave fronts exists. The second part is on the persistence of di?erentialsystems with some biological background. We investigate all kinds of persistenceof nonautonomous Kolmogorov systems, such as permanence, strong persistence,average persistence, etc.. Under some conditions, we obtain the equivalence ofthe permanence and strong persistence, and the equivalence of the uniformly weakaverage persistence and weak average persistence. We also consider the periodicsystems and almost periodic systems, and prove the equivalence of permanence,strong persistence, uniformly strong average persistence, uniformly weak averagepersistence, strong average persistence and weak average persistence. As the appli-cations of these results, we study nonautonomous Lotka-Volterra systems, Holling(m,n)-type functional response systems, Beddington-DeAngelis type functional re-sponse systems and Chemostat systems, respectively. The innovative points in this dissertation are as follows:1. The reaction-di?usion systems with multiple spatio-temporal delays andwith some zero di?usion coe?cients are a couple systems of ordinary di?erentialequations and partial di?erential equations, which can be frequently found in math-ematics modeling, the existence of their travelling wave is an important researchsubject, and the results about this research area are few now. Some present re-sults are about the systems with multiple spatio-temporal delays and with positivedi?usion coe?cients(see [50]), or about the systems with finite delays and withsome zero di?usion coe?cients(see [33]). The criteria of the existence of travellingwave fronts of the systems with multiple spatio-temporal delays and with some zerodi?usion coe?cients are given by analysis technique and fixed point theorem. Ex-amples are presented to show the application of these criteria. Comparing with theconclusions in references, these results are new and extensive, and some conditionsof upper solution and lower solution are weaken.2. The persistence of the general nonautonomous Kolmogorov systems is animportant problem in mathematical biology. The necessary conditions of persis-tence of some Kolmogorov systems are discussed. The equivalence of the perma-nence and strong persistence, and the equivalence of the uniformly weak averagepersistence and weak average persistence for some Kolmogorov systems are provedunder some conditions. For periodic and almost periodic Kolmogorov systems, itis shown that the relations that permanence, strong persistence, uniformly strongaverage persistence, uniformly weak average persistence, strong average persistenceand weak average persistence are equivalent.