Relaxation Oscillations And Bifurcations In Several Singularly Perturbed Predator-Prey Systems With Weak Allee Effect

Based on geometric singular perturbation theory(GSPT),canard theory,entryexit function and slow divergence integral et al.,this thesis studies the birth and bifurcations of relaxation oscillations in several singularly perturbed predator-prey models with weak Allee effect.The thesis is divided into four chapters,which are as follows.Chapter 1 is the introduction,in which,the research background and recent advance of related researches are given,and also GSPT as well as the main works of this thesis are introduced.In Chapter 2,we study the bifurcations of canards and homoclinic orbits in a singularly perturbed modified May-Holling-Tanner predator-prey model with weak multiple Allee effect.Based on GSPT and canard theory,canard explosion is observed and the associated bifurcation curve is determined.Due to the existence of canard point,a homoclinic orbit with slow and fast segments and homoclinic to a saddle is also detected,in which,the stable and unstable manifolds of the saddle are connected under certain parameter value.By analyzing the slow divergence integral,it is proved that the cyclicity of canard cycles in this model is at most four.Finally,by calculating the entry-exit function explicitly,a unique,orbitally stable canard relaxation oscillation passing through the transcritical bifurcation point is detected.All these theoretical predictions on the births of canard explosion,canard limit cycles and homoclinic orbits are verified by numerical simulations.In Chapter 3,by using GSPT and a new entry-exit function,this chapter is concerned with the number and stability of relaxation oscillations in a non-standard singularly perturbed predator-prey model with weak Allee effect and Holling-Ⅳ functional response.We find that this model can admit at most three nested(large amplitude)relaxation oscillations.Two of them are stable while the other one is unstable.Also we find that the relative heights between the folded points and the trancritical bifurcation point of the critical curves affect the number and stability of relaxation oscillations greatly.Compared with the existing models,the weak Allee effect makes the model possess one more sustained oscillations,i.e.,the weak Allee effect promotes the stability of the predator-prey system.The theoretical predictions about the number and stability of relaxation oscillations are verified by numerical simulations.In Chapter 4,based on GSPT,canard theory and the mechanisms on the birth of mixed-mode oscillations(MMOs),we study the the existence of MMOs in a singularly perturbed tritrophic food chain system with three distinct timescales.A variety of rich and interesting dynamics,including MMOs,relaxation oscillations,canard explosion and bi-stability are observed.We find that the complex dynamics depend on the relative geometry on certain singular limit systems,specifically on the relative positions of the critical manifolds,superslow manifolds,the positive equilibrium,the fold curves and the transcritical bifurcation points.It is proved that the mechanism on the birth of locally small amplitude oscillations(SAOs)in this model can only be the delayed Hopf bifurcation.It is worth noting that the transcritical bifurcations play an important role on the global return of MMOs.Chapter 5 is the summary on this thesis.Some unsolved problems are also given in this section.