Two Types Of Population Model The Existence Of Travelling Wave Solutions

In this paper, three diffusion-reaction models are established, which are cholera model, nutrient-bacteria model and influenza model with treatment. By shooting method and Schauder’s fixed point theorem, the existence and non-existence of traveling wave solutions are proved and the minimal wave speed is obtained. These results provide a theoretical basis for the control of diseases and bacteria.The first chapter devoted into the introduction on the background of models, mathematical methods and the comparison of different methods.In Chapter2, a cholera model with contaminants diffusive is established. Firstly, we study the case with man-man and man-environment transmissions considered and mortality due to illness ignored. The existence of traveling wave solutions are proved by changing the original system into its limit one and, furthermore, using shooting method. At the same time, the minimal wave speed is obtained. In the second case, the influence of illness death on cholera is considered with natural birth and death pro-cess ignored. By constant variation method, original system is transformed reaction-diffusion system with distributed delay. The existence of traveling wave solutions are get by constructing upper and lower solutions and using Schauder’s fixed point the-orem. To prove the non-existence of traveling wave solutions by two-sided Laplace transform, it is necessary to show the exponential decay of traveling wave solutions and a new method is proposed for this aim.In Chapter3, based on Mimura’s nutrient-bacteria model, s simple model is pro-posed. Similar to the second chapter, original system is transformed reaction-diffusion one with distributed delay. The formula for minimal wave speed is deduced by com-paring two high-order polynomials with a linear function. An auxiliary system is con-structed and the existence of a sequence of traveling wave solutions for the auxiliary system is proved by Schauder’s fixed point theorem. The limit of the sequence of trav-eling wave solutions for the auxiliary system is shown to be the traveling wave solution of original system. The non-existence of traveling wave solutions is proved be defining a negative one-sided Laplace transform.In Chapter4, a diffusive influenza model with treatment is established. Firstly, we construct an auxiliary system and prove the existence of traveling wave solutions by the methods similar to that of Chapter3. However, it is more technical since the dimension of this system can not be reduced.The innovation consists of two aspects:interpretation of the real world and the methods for traveling wave solutions. The results of this paper reveal the internal mechanism of the spread of infectious diseases and the spread of bacteria and provide a theoretical basis for the control of cholera, influenza and bacteria. The innovation on mathematical methods are as follows.1. The first innovation is the analysis of linearization. The formula for minimal wave speed is deduced by comparing two high-order polynomials with a linear function. This method can be applied to most of cubic polynomials2. The definition of "negative one-sided Laplace transform" is the second innovation which is motivated by Wang and Wu(2010) and makes proofs simpler.3. The third innovation lies in the introduction of an auxiliary system and the bound-edness of upper and lower solutions, which is different from that of Wang and Wu (2010). It is difficult to get the minimal wave speed by constructing bounded upper and lower solutions for non-cooperative systems.4. The fourth innovation lies in the new method to prove the exponential decay for traveling wave solutions. The method used by Wang and Wu (2010) is applicable for one equation and not for a system. However, our method can be applied to general systems.