Bifurcation And Control In The Discrete Biological Model And Bio-economic Model

Since the natural balance is closely correlated to the existence and development ofhuman being, the kinetic analysis of the ecosystem attracts more and more attention. Amongthe different kinds of biodynamical system, the prey-predator model is a typical one forresearch，whose stability, bifurcation mechanism and control principle play an important rolein the ecological mathematics. This paper is to investigate its dynamical properties from bothdiscrete and continuous respective.When it comes to the discrete prey-predator model, the paper classifies differentcategories of the multidimensional parameter space, and then investigates the existence andstability of the interior fixed point of this model. By using the central manifold theorem andbifurcation theories, the mechanism of Flip bifurcation and Neimark-Sacker bifurcation arestudied. In order to eliminate the bifurcations and stabilize those interior fixed point of thismodel, a feedback controller is designed. Finally, numerical simulations by Matlab areimplemented, which present the bifurcation maps and phrase maps. Not only the resultsillustrate the validity of the bifurcation analysis and effectiveness of the feedback controller,but they also reveal that the discrete models have more complex and abundant dynamicalphenomena than the continuous ones, such as period-doubling orbits, quasi-periodic orbitsand the chaotic sets.When it comes to the continuous prey-predator model, a food chain that includes threepopulations is investigated. After incorporating a harvesting effort for the top predator, adifferential-algebraic model is constructed. By using the theories of singular system andsingularity induced bifurcation, the unstable mechanism around the inner fixed point when the economical benefits are nonnegative is studied. In other words, when the economic interest iszero, there exists a singularity induced bifurcation around the equilibrium; while when theinterest is positive, the model is unstable around its equilibrium. Further study shows that ifthe benefits are restricted in a proper range, the instability of the fixed point can be eliminatedby a feedback controller. Finally, numerical simulations by Matlab are presented to illustratethe effectiveness of the proposed controller. Meanwhile, the ecological explanations about therelated theoretical fruits are also given for reference.