A Class Of Predator-prey Reaction-diffusion Models With Age Structure

The tight connections between chemostat models and real life make it a hot topic.With the development of study,scholars have improved and promoted the well-stirred chemostat by introducing delay,inhibitors,periodic input and output.The study of well-stirred chemostat models is becoming more and more completed.In recent years,the study on unstirred chemostat models has received more and more attention.In contrast to the competitive unstirred chemostat models,the study on the unstirred chemostats with prey and predator is more different.This paper deals with a predator-prey reaction-diffusion model with age structure.The introduction of age structure leads to the failure of the conservation law in this model,and the more equations of the model increases the complexity of the study.All of these have caused some difficulties in studying the properties of the model solution.The dynamic behavior of the model was studied by applying the classical maximum principle,upper and lower solution methods,uniform persistence theory,and bifurcation theory.The main results are as follows:Firstly,the existence of the steady-state solution of the single-species model is discussed.With the help of the maximum principle,the method of upper and lower solutions,and some analytical techniques,the boundedness and global existence of the model solution are obtained.Secondly,the stability of the semi-trivial solution is obtained by means of the linearized eigenvalue problems.The sufficient conditions for uniform persistence of the model solution are obtained by means of the uniform persistence theory and the maximum principle.The bifurcation theory is applied to investigate the branch of the positive steady-state solution generated from the semi-trivial solution by taking the mortality m and the diffusion coefficient d as bifurcation parameters.With the help of perturbation theory,the stability of the local bifurcation solution is studied.By using the global bifurcation theory,we extend the local bifurcation to the global bifurcation,and the sufficient conditions for the existence of coexistence solutions are given.Finally,by numerical simulations,we further investigate the effects of main parameters on the dynamical behavior of the model.Numerical simulations show that when the mortality rate m and diffusion coefficient d change,we will observe the predator-prey coexistence or prey survival.Moreover,numerical results also show that there may exist Hopf bifurcation,which verifies and supplements our theoretical results.In summary,the results indicate that coexistence between species exists under certain conditions,and the death rate m and the diffusion coefficient d have an effect on the coexistence of species.