Formation Of Spatial Patterns Of Two Interacting Populations In River Ecosystems

This article aims to study the formation of spatial patterns of two interacting population in river ecosystems.Based on a traditional predator-prey model,a predator-prey model with advection terms and diffusion terms are established by considering the influence of river environments on populations.The reaction-diffusion-advection models with one-dimensional bounded domains and one-dimensional unbounded domains are considered,respectively.In Chapter 1,we briefly introduce the research background and related existing research,the main work of this paper and the required mathematical preliminaries.In Chapter 2,we establish a reaction-diffusion-advection model in one dimensional bounded domain.First,we obtain the necessary conditions for the model to produce Turing instability,and then we analyze the effects of advection terms on the Turing instability of space homogeneous steady-state solutions.In the case of Turing instability of spatially homogeneous steady-state solution,to take into account the influence of advection terms,we let the convection coefficients gradually increase from zero to higher levels.When the advection coefficient is greater than a critical value.We find that the spatially homogeneous steady-state solution will be stabilized again.In Chapter 3,we establish a reaction-diffusion-advection model with a one-dimensional unbounded domain.First,we analyze and obtain the necessary conditions for the occurrence of Turing instability of a spatial homogeneous steady-state solution.Then,we consider the effects of the advection terms on the stability of the spatial homogeneous vi steady-state solutions.The results show that in the case of Turing instability,with the effects of the advection terms,the spatial homogeneous steady-state solutions are still unstable;When the spatial homogeneous steady-state solution is stable,with the influence of advection terms,it can be found that when the coefficient of advection terms is greater than a certain critical value,advection will make the spatial homogeneous steady-state solution unstable.Moreover,the larger the difference between the two advection terms in the system,the easier the spatial homogeneous steady-state solution becomes unstable.In Chapter 4,we study two types of predator-prey spatiotemporal models.Firstly,we analyze the spatial non-homogeneous Hopf bifurcation of the constant steady-state solution in a predator-prey model with a harvesting rate and a diffusion term.We also obtain the conditions under which the homogeneous steady-state solution of the modelundergoes Turing instability,We numerically analyze the effects of convection terms on the occurrence of Turing instability using Matlab.Then,we study a predator-prey spatiotemporal model with competition terms and provide the conditions for the occurrence of Turing instability for spatial homogeneous steady-state solutions.We numerically analyze the effects of convection terms on the occurrence of Turing instability for spatial homogeneous steady-state solutions using Matlab.In Chapter 5,we briefly summarize the main conclusions of this theses.We discuss the shortcomings of the current research and further research directions.