Qualitative Analysis And Bifurcation Study Of Two Difference Biomathematical Models

This paper focuses on the stability and the bifurcation problems of two difference biomathematical models near the fixed point.Mainly applying the center manifold reduction theory,the normal form theory and the bifurcation theory,this paper presents the stability of two models at the fixed points,the codimension-1 flip bifurcation and the codimension-2 generalized flip bifurcation.The study is separated into two parts:(1)A two-dimensional difference system derived from the grass litter decomposition model is studied.Firstly,the appropriate parameters are chosen so that the fixed point is non-hyperbolic and one of the corresponding eigenvalues is-1.Then the twodimensional difference system is transformed into a one-dimensional one by the center manifold reduction theorem and Mathematica software.Finally,we show that the system undergoes a flip bifurcation or a generalized flip bifurcation near the fixed point using the normal form theory and the bifurcation theory.(2)A discrete cooperative hunting predator-prey model is studied near the degenerate fixed point,in which the eigenvalues are ±1.Takens' s Theorem is applied to convert the difference system into the corresponding ordinary differential system so that its fixed point corresponds to the high-order degenerate equilibrium point of the ordinary differential system.With the blow-up technique and the differential homeomorphism,the stability near the equilibrium point of the ordinary differential system is obtained,which derives the orbital structure of the corresponding discrete system near the degenerate fixed point.Then the phase portraits of the ordinary differential system and the difference system are simulated by Maple and Matlab software respectively.Finally,the validity and the correctness of the theoretical results are verified by numerical simulations.The obtained dynamical properties of the two models can theoretically explain some phenomena observed in the real world.