Study On Spatial Dynamics Of A Competition Model For Populations With Saturation Effect

Population dynamics is one of the most mature and widely used branches of mathematics biology.In ecology,it mainly studies the relationship among different species,and the interaction between species and environment.Now,population competition model and predator-prey model are two kinds of most popular models used in the study of population dynamics.Because of the universality and importance of competitive relationship,the dynamic behavior of competition model has been and will continue to be an important research field of ecosystem for a long time.Based on the classical Lotka-Volterra competition model,we establish several types of two species competition model with saturation effect.We mainly discuss the dynamic behavior of these models separately,and study the effects of spatial diffusion,time delay on the dynamic behavior of the models.The outline of this paper is as follows:In the first Chapter,we briefly introduce the research background,significance and research status of our study,and than we summarize the main research results of this paper.In the second Chapter,based on the classical two-dimensional Lotka-Volterra competition model,we establish a competition model with saturation effect,and analyze the stability of the equilibriums of the model.Firstly,we obtain the existence conditions of the positive equilibrium of the model and the conditions of its local stability by using the geometric theory of isocline.The global stability of the positive equilibrium is proved by constructing Dulac function.Finally,the conditions of stability of the boundary equilibrium are demonstrated by using the eigenvalue theory.In the third Chapter,based on the two-dimensional competition model of Chapter 2,considered the influence of a spatial factor,a model of reaction-diffusion competition model with a saturated effect is established.We discuss the spatial dynamics behavior of the model under the influence of cross-diffusion.Firstly,for selecting control parameters,the conditions for generating the Turing pattern near the positive equilibrium are obtained by linear stability analysis.Then,we deduce the amplitude equations around the Turing bifurcation point by using the standard multiple scale analysis.Finally,a series of numerical simulations are delivered to explain our theoretical analysis.It is shown that self-diffusion does not change the stability of the positive equilibrium of model,but under the effect of cross-diffusion,the model produces complex patch structure.In the fourth Chapter,considering two factors of time delay and diffusion,we consider a delayed diffusive competition model with saturation effect.Firstly,using the linear stability theory,the condition of local stability of the positive equilibrium for the model without time delay are obtained.Then,the condition for spatial Hopf bifurcation of model with time-delay is obtained.It is shown that time delay can destabilize the positive equilibrium and induce spatially periodic solutions.Then,by calculating the normal form on the center manifold,the formulas determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are obtained.Finally,in order to support our theoretical results,we present some numerical simulations.In the fifth Chapter,we summarize the research content of our paper and look forward to further investigation.