Dynamical Analysis Of A Diffusive Predator-prey Model

Predator-prey systems are important components of the ecological system and they can explain some related phenomena of the nature. They also have been investigated by many researchers in the view of dynamical systems. Various models, such as ODE systems, DDE systems and PDE systems have been built to understand predator-prey interaction. This thesis mainly considers a diffusive predator-prey model and investigates the effect of the conversion rate on the dynamics of the model. The main contents are as follows.For the case of small conversion rate, we first investigate the global existence of solutions and the global attractivity of the constant equilibria through upper-lower solution method. Then based on the stability theories of reaction-diffusion equations, we analyze the eigenvalue equations associated with the constant equilibria and prove the locally stability of the constant equilibria. Finally, we obtain the globally stability of the constant equilibria.For the case of large conversion rate, we first give the priori estimates for positive steady state solutions through the maximum principle and Harnack inequality. Then through Schauder theory and pL theory, we obtain the limit profile of the positive solutions. This, combined with the Implicit Function theorem, implies the non-existence of non-constant positive steady state solutions.Finally, in order to show theoretical results of general models are universal, we apply the previously obtained results to some concrete predator-prey models and give some ranges of their parameters where complex pattern formation cannot occur.