Study On Muti-parameter Bifurcation For A Class Of Plankton-oxygen Model

Considering that temperature fluctuations will affect the oxygen content in seawater and the dynamic behavior of plankton systems,this paper studies a plankton-oxygen dynamic model.After making appropriate assumptions to transform the threedimensional system into a two-dimensional system,various bifurcation phenomena of this model are studied.First,this paper indicates that the zero equilibrium is always stable,the boundedness of solution and the existence and uniqueness of the positive equilibrium are proved,then we discuss the distribution of roots by obtaining the linearization equations at the boundary equilibrium and the positive equilibrium.It is found that the stability of the boundary equilibrium point will change,thus system undergoes a saddle node bifurcation.And this paper obtains the sufficient conditions for Hopf bifurcation of the system at the positive equilibrium point,and the reduced equation of the system restricted to the central manifold is calculated according to central manifold theorem and normal form method,giving the bifurcation properties of the Hopf bifurcation.Secondly,it is proved that the Bogdanov-Takens bifurcation occurs at the intersection of the saddle node bifurcation line and the Hopf bifurcation line,and the normal form of the Bogdanov-Takens bifurcation is obtained by simplifying the equation,giving the complete bifurcation set near the bifurcation point and the expression of the system's homoclinic trajectory.Moreover,this paper considers the effect of diffusion on the dynamic properties of the system.By studying the distribution of the roots of the characteristic equation of the linearized system at the constant steady state solution,it is proved that the system does not have Turing instability caused by diffusion.Finally,combined with the biological significance of the model,the appropriate parameters are selected for numerical simulation.By simulating the survival of plankton,the safe parameter area for plankton survival is obtained on the parameter plane,and the asymptotic stability and periodic oscillation of plankton population density are simulated.Aiming at the three-dimensional system,the simulation work is carried out in parallel with the two-dimensional system,and it is found that the two systems have almost the same branching behavior.we prove that the boundary of the region on sustainable dynamic properties in the parameter plane is composed of the saddle node bifurcation line and the homoclinic trajectory.In this tongue-shaped area,the system is in a safe state,that is,two populations can coexist,beyond this tongue-shaped area,the oxygen in the system will be depleted,so that the population will eventually die out.