Bifurcation Analysis Of Some Ecological Models

In this thesis, we firstly study the global dynamics of an ordinary differential equation model for one biomass (shrubs or trees) and one resource (water) By carrying out qualitative analysis, we present a detailed partition of the pa-rameter space and find rich dynamic including a "bubble", or a "heart", or a "lotus", or a "pepper". Furthermore, we find that the system has bistability of two equilibria or one equilibrium and one limit cycle, or even tristability of two equilibria and one limit cycle. This is perhaps a minimal model to have the properties of backward transcritical bifurcation, saddle-node bifurcations for equilibria and limit cycles, Hopf bifurcations, limit cycle bubble, homoclinic bi-furcation, Bogdanov-Takens bifurcation, all in one simple model. In the model, oscillatory vegetation change may be an indicator of catastrophic shift. Our model apparently can also be considered as an epidemic model. This work pro-vides a fundamental framework for alternative stable states or stable cycles in native biomass density, also provides a theoretical foundation for the existence of bistability in natural water-biomass system.Then, we consider the global bifurcation of the following Lotka-Volterra predator-prey-taxis model By Crandall-Rabinowitz Bifurcation Theorem and Shi-Wang Bifurcation Theo-rem, we establish the result of the bifurcation of nonconstant solutions from the positive constant solution. Furthermore, we show that the nonconstant solutions are locally stable in some cases. Our study indicates that the predator-prey sys-tem undergoes a homogeneous stable state if the prey does not have group defense ability, but a heterogeneous stable state exists if the prey has the group defense ability. Biologically, the foraging behavior of predators may transform heteroge-neous environments into homogeneous environments, while the retreating behav-ior of predators may transform homogeneous environments into heterogeneous environments which induces richer dynamical behaviors.Next, we study the Turing bifurcation of the following water-biomass model which captures the "infiltration feedback" of biomass By using Turing’s idea, we obtain the existence of Turing pattern and its numer-ical simulations in 1-dimension space by finite difference methods. Our results imply that the diffusion may lead to the pattern formation, while the rainfall rate and the infiltration feedback between the water and plant may affect the final s-tate of vegetation evolution. If considering the root suction capability of water, our study shows that a vegetation cover comprised of biomass with stronger suc-tion ability is more likely to exhibit pattern formation than a vegetation cover comprised of biomass with weaker suction ability.Finally, we study the Turing bifurcation of the following water-biomass model By Turing’s idea, we present a detailed parameter space for Turing instability. By finite difference methods, we give the numerical simulations in 2-dimension space which in turn help to study the effects of the competition between plants and roots-induced positive feedback on Turing bifurcation and explain the diversity of dryland vegetation. Our results show that with the increase of the intensity of competition, the possibility of generating pattern formation is reduced, but as the positive feedback effect of root is increased, the possibility of generating pattern formation becomes larger. Thus, under the influence of the positive feedback effect and the negative feedback effect of plant competition, the positive feedback effect of root can promote the formation of pattern, but the competition among plants has the opposite effect.