Research On Nonlinear Dynamical Problems In Some Predator-prey Ecosystems

The author concentrate on the consideration of predator-prey systems with Holling type ? functional response,Beddington-De Angelis functional responses and Allee-like effect on predator.Chapter 1 mainly presents the Bazykin's predator-prey system with the Holling type ? functional response and interspecific density-restricted effect on predator,and deals with interior equilibria in special cases and their corresponding Bogdanov-Takens bifurcations.Firstly,it derives a multiple focus of multiplicity one,cusps of codimension 2 and a degenerate Bogdanov-Takens singularity(focus or center)of codimension 3 in this system under some critical conditions,respectively.Secondly,the distinction of two types of codimension 2 cusps is discussed,and it points out that the threshold of these two types exhibits a cusp which is a special case of the mentioned degenerate Bogdanov-Takens singularity of codimension 3.Finally,numerical simulations show that the system undergoes two types of Bogdanov-Takens bifurcations of codimension 2 and a degenerate focus type Bogdanov-Takens bifurcation of codimension 3.According to the Bazykin's predator-prey system,chapter 2 firstly proposes a homogeneous reaction-diffusion predator-prey system with Holling type ? functional response subject to Neumann boundary conditions.Several new sufficient conditions are analytically established to ensure that this system has globally asymptotically stable interior equilibria.Based on the phenomenon of Turing instability,the occurrence of Hopf bifurcation incorporating an example of numerical simulations is presented.Secondly,an useful algorithm for determining direction of Hopf bifurcation and stability of bifurcating periodic solutions is respectively derived.At last,for the corresponding steady state system,prior estimate and non-existence of non-constant positive solutions are deduced with the help of the maximum principle,the Harnack inequality and energy integration.Chapter 3 constructs a predator-prey system with Beddington-De Angelis functional response and Allee-like effect on predator.Complex dynamical behaviors were studied by qualitative analysis and numerical simulations,and theoretical derivations give sufficient conditions to guarantee the occurrence of transcritical,saddle-node,pitchfork and non-degenerate Hopf bifurcations.Numerical simulations verifies its effective and feasibility.In short,these works provide a theoretical basis for complexity problems in more predatorprey ecosystems.