Study On Bifurcations In Several Biological Models With Discrete And Distributed Time-delays

The diferential equations with discrete and distributed time delays are widely ap-plied in biological systems for describing biology dynamical behaviors, such as stability,periodic oscillation and chaos of systems.In this thesis, we study the dynamical behaviors of nonlinear biology models withdiscrete and distributed time-delays by using nonlinear dynamics systems, functional dif-ferential equations and other related theories and methods. We analyze the efects of timedelay on the dynamics behavior of nonlinear biology models, including the stability ofthe system, the permanence of the system, local Hopf bifurcation, global Hopf bifurca-tion, and two types of codimension two bifurcation and other related theoretical results.The corresponding biological results, which is represented by the obtained mathematicaltheoretical results from studying the biological system, are revealed.First, we investigate Hopf bifurcation and Hopf-fold bifurcation of predator-preymodel with dormancy of predators and discrete delay by using the center manifold theo-rem, the Hassard normal form theory and the Faria and Magalhaes normal form theory.The efects of time delay on the stability, persistence, periodic solution and chaos ofsystem are analyzed. The occurrence conditions of Hopf bifurcation and Hopf-fold bi-furcation are obtained. It shows that time delay in the prey-species growth can lead tothe phenomena of stability switches and the coexistence of multiple periodic orbits. Wededuce the normal form with original parameters in the model and bifurcation propertiesnear the bifurcation point. By using the Lyapunov-Razumikhin theorem and other relatedknowledge, the uniform persistence of predator-prey model with time delay is proved.Numerical simulation shows the stability as well as a large-scale existence of periodicsolution.Next, we investigate the stability and bifurcation of predator-prey model with dor-mancy of predators and impulsive perturbations. By using the theories of impulsive e-quations, small amplitude perturbation skills and the comparison technique, we get theconditions which guarantee the global asymptotical stability of the prey-eradication peri-odic solution and the permanence of the system. Further, influences of the impulsive per-turbation on the inherent oscillation are studied numerically, which shows rich dynamics,such as period-doubling bifurcation, chaos, and period-halving bifurcation. Moreover, the efects of the impulsive perturbation and hatching rate on the chaos of the system are ob-tained by numerical simulation. In addition, using the theories of the normal form and thetopological degree, we study the local and global Hopf bifurcation properties of two sun-flower equations with distributed delay. Using the Rouche theorem and the principle ofargument, we discuss the occurrence conditions of local Hopf bifurcation of the sunflowermodel with logarithmic growth and give the approximate expression of the periodic so-lution of this model. Local Hopf bifurcation properties between the linear growth modeland the logarithmic growth of the sunflower model are compared. It reveals the diferencebetween period changes of periodic solutions as well as the diference between variousform of approximate expression of the periodic solutions. From theoretical investigationand numerical simulation, we obtain a large-scale existence of periodic solution.Finally, the Hopf-pitchfork bifurcation of a two-neuron system with discrete and dis-tributed delays is investigated. Through analyzing the characteristic equation, we obtainthe occurrence conditions of Hopf-pitchfork bifurcation. Using the center manifold theo-rem and the Faria and Magalhaes normal form theory, we obtain the normal form and theirunfolding with original parameters of the system near the bifurcation point and refine thebifurcation diagram. It is revealed that when the time delay and the parameter changenear the vicinity of the critical point, some dynamical behaviors of neuron system arefound, such as stable periodic orbit, the coexistence of two stable non-trivial equilibrium,and the coexistence of a stable periodic orbit and two stable equilibrium, by theoreticalinvestigation and numerical simulation.