| In this dissertation, the application of the theory and methods of nonlinear dy-namical systems in population ecology and neuronal systems is discussed. We main-ly consider the following contents: first, we apply the forward Euler discrete scheme to a predator-prey system of Holling and Leslie type with a constant-yield prey har-vesting. The dynamic behavior of the discrete system is studied. Second, the stabil-ity and bifurcation analysis for a continuous predator-prey system with the nonlinear Michaelis-Menten type predator harvesting are taken into account. Third, the stability and synchronization analysis of two chaotic Rulkov maps coupled by bidirectional and symmetric chemical synapses are considered. Finally, the dynamical behaviors of a s-ingle Hindmarsh-Rose neuron model with multiple time delays are investigated. The dissertation is summarized as follows:In chapters 1 and 2, we respectively introduce the research background, research status, the development of nonlinear dynamical systems, the background of population ecology and neuronal systems and some basic concepts and facts in nonlinear dynamical systems.In chapter 3, we apply the forward Euler discrete scheme to a predator-prey system of Holling and Leslie type with a constant-yield prey harvesting. The conditions of existence for flip bifurcation and Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. Some numerical simulations are given for confirming the qualitative results. These results demonstrate that the integral step size plays a vital role to the local and global stability of the discrete-time predator-prey system with the Holling and Leslie type after the original continuous-time predator-prey system is discretized.In chapter 4, the stability and bifurcation analysis for a predator-prey system with the nonlinear Michaelis-Menten type predator harvesting are taken into account. The existence and stability of possible equilibria are investigated. The rigorous mathemati-cal proofs of the existence of codimension one bifurcations, such as saddle-node bifur-cation, transcritical bifurcation, Hopf bifurcation, and codimension 2 bifurcations such as Bogdanov-Takens bifurcation are derived. These bifurcations are ecologically impor-tant and the saddle-node bifurcation and codimension 2 Bogdanov-Takens bifurcation especially will lead to the potentially dramatic variation of the system dynamics. Exis-tence of such kinds of bifurcations indicates that the over exploitation of resource will cause extinction of the species. It can be thought as a supplement to existing literature on the dynamics of this system.In chapter 5, the stability and synchronization analysis of two chaotic Rulkov maps coupled by bidirectional and symmetric chemical synapses are considered. The syn-chronization of the coupled system and conditions of stability of the fixed point for this system are discussed. We not only consider the influence of the system parameter on the system, but also the coupling strength, especially the effect of coupling strength on the synchronization of the system. As the coupling strength increases, the coupled neurons possess rich firing patterns, such as square-wave bursting, triangle bursting and the mix-ture of these two firing patterns, and then they can achieve complete synchronization.Furthermore, the synchronized regions of the coupled system are given.In chapter 6, the dynamical behaviors of a single continuous Hindmarsh-Rose neu-ron model with multiple time delays are investigated, including the stability of equi-libria, the local Hopf bifurcation, direction and stability of the Hopf bifurcation. In order to further investigate effects of the two time delays, the bifurcation analysis of inter-spike intervals (ISIs) are given. We find that the delays have different scales when the two delay values are not equal. This phenomenon is most likely to be caused by Hindmarsh-Rose neuron model has two different time scales. |
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