Dynamical Analysis And Sychronization Of Serveral Multi-Species Biological Systems

With global warming, rapid population growth and natural environmental degradation, there are many species on earth are threatened with extinction. To maintain the stability of biological populations and the diversity of biological resources has always been the concern of biological workers. In this paper, the mathematical model of the biological interaction between a number of populations is established and the dynamic behavior of the model is analyzed frorm the perspective of nonlinear dynamics. The main results are given as follows:Firstly, the stability of discrete population model with refuge is studied based on the Lyapunov stability theory, bifurcation, phase diagram, cell-to-cell mapping tools, etc. And the mainly results are given as,â… The bifurcation diagram of the active population and the population in refuge under the same bifurcation parameter set may be still different, studies show that the strange phenomena is determined by the biological significance of the model;â…¡The discrete population model enters into chaos mainly through period-doubling bifurcation in most parameters, but there may be movement from the period-1 state directly into period-4 state. And the period-1 motion may also directly to the quasi-periodic motion (Neimark-Sacker bifurcation).â…¢In order to study the coexistence of multiple attractors, ie the system with different initial conditions may eventually be attracted to different attractors, the basin of attractions are drawn.â…£When certain parameters are selected, there exists "down" phenomenon in the process of period-doubling bifurcation. And there exists "scattered" phenomenon when the system enters into chaotic state.â…¤Finally, the influence of the selected parameters on biological significance are discussed:the selection of immigration rate and emigration rate must be appropriate. The small immigration rate with large emigration rate will give rise to large population density in refuge, and then lead to the detriment of population reproduction. On the other hand, the large immigration rate with small emigration rate is not conductive to the protection of the population (ie, resulting in failure of the refuge). While loss rate can not be too larege, no matter under what circumstances.Secondly, two different type of continuous popution dynamics model are established. And the Kolmogorov theorem and the Lyapunov stability theorem are used to analysis the stability of the two different biological models. And the condition of stable equilibrium and stable limit cycle of the systems are obtained, respectively. Some interesting phenomenon called "period bubble" and "chaotic bubble" are found in the study of period-doubling bifurcation of the systems. "Chaotic bubble" refers to the system enters into chaos by period-1â†'period-2â†'period-4â†'……â†'haosâ†'……period-4â†'period-2â†'period-1 and the "period bubble" means the state of the system goes like period-1â†'period-2â†'period-1, This is the first time we found this stange phenomenon and has not yet appeared in other literatures. We also found that it takes a long time to determine the state of biological systems, which is one of the reasons why chaos is rarely detected in biological systems. In addition, we also found there are narrow parameters in parameter space can find chaos in biological systems, which is another reason why chaos is rarely detected in biological systems.Finally, we discussed the synchronization phenomenon of two different models, system without vertebrate and system with vertebrate. By the design of effective adaptive control strategy, both the synchronization of system without vertebrate itself and the synchronization of system without vertebrate and system with vertebrate can easy realize. That means the synchronization of two or more biological systems can be easy found in nature. At last, the anti-synchronization of two same model without vertebrate is realized by designing the effective control strategy.