Dynamic Behavior Of A Plankton Diffusion Model With Allelopathy

In this paper,the dynamic behavior of a plankton diffusion model with allelopathy is studied.It mainly consists of three parts.Firstly,the stability of the non-negative equilibria of the ordinary differential model,the existence and stability of Hopf bifurcation at the positive equilibrium are discussed.Secondly,for the reaction-diffusion model,the detailed properties of the branches around the positive equilibrium are analyzed,including the conditions of the Turing bifurcation and the situation of the Turing spot diagram,the direction and stability of the periodic solutions of the Hopf bifurcation.Especially,by selecting two parameters,the condition for the existence of Turing-Hopf bifurcation is given.By using the central manifold theory and the normal method,the normal form of the Turing-Hopf bifurcation is calculated.The results of theoretical analysis are illustrated by numerical simulation,the simulation graphs of constant steady-state,non-normal steady-state,spatially homogeneous periodic solutions and spatially inhomogeneous periodic solutions are obtained.The results show that allelopathy plays a key role in determining the stability and branching properties of the model,which is fundamentally different from the case without allelopathy.Finally,we discuss the non-existence and existence of positive nonconstant solutions.