Complexity Analysis And Control For Some Kinds Of Biological Dynamical Systems

Biological dynamical system is a typical nonlinear complex system. Although some research results about its dynamics analysis and control have been obtained, there still exist many problems to be dealt with. In this dissertation, by using quali-tative theory of ordinary differential equation, bifurcation, chaos and chaotic control theory of normal system, bifurcation and control theory of differential-algebraic sys-tem, dynamics including bifurcation, chaos, impulse and the corresponding control strategies are investigated for some kinds of population models and singular biolog-ical economic systems. Main contents of this dissertation are as follows.(1) Considering the effect of harvesting on the biological population system, bifurcations of a two dimensional continuous population model with constant rate harvesting are studied. Research results show that compared with the model with no harvesting, this model has more equilibria and the dynamical behaviors near the equilibria become more complex. Choosing the constant rate harvesting as the bifur-cation parameter, the existence of the saddle-node bifurcation, the Hopf bifurcation and the limit cycle are proved. Moreover, the existence of the codimension two Bogdanov-Takens bifurcation is also verified. From the point of view of biology, the above results mean that the rate of harvesting should lie in the suitable range such that the prey and predator can coexist. Overharvesting will result in the extinction of the populations and the collapse of the biological system. When the parameters lie in the bifurcation sets, periodic vibration and even more complex dynamics may occur.(2) In order to consider the economic interest of harvesting in a biological popu-lation system, singular system theory is used to model a singular biological economic system. The model is governed by a differential-algebraic equation. Then the effect of the change of the economic interest on the singular biological economic system is considered. Local dynamics near each equilibrium are studied systematically. Re- search results show that when the economic profit increases from negative to positive and passes through zero, this system undergoes the singularity induced bifurcation near the positive equilibrium. Moreover, the positive equilibrium becomes unstable when the economic profit is positive. Biologically, singularity induced bifurcation means impulse, i.e., rapid expansion of the population which may exceed the carry-ing capacity of the environment and lead to the unbalance of the ecosystem. And the unstable state in the case of positive economic interest is obviously unfavor-able for the sustainable exploitation of the valuable biological resource. Then, using the singular system control methods, state feedback controllers are designed in the following two steps:First, eliminate the unexpected singularity induced bifurcation which may result in the impulse and the collapse of the system as the economic profit increases through zero. Then, stabilize the positive equilibrium of the differential-algebraic system when the economic profit is positive. These results tell people suitable policy management of harvesting can make the population develop contin-uously. Furthermore, timely adjustment can keep the economic interest stay in an ideal level which can satisfy people’s needs for the biological resource.(3) The dynamical behaviors of a discrete-time generalized Lotka-Volterra sys-tem are investigated. In this system, nonlinear density growth rate is considered. It is shown that the system undergoes the flip bifurcation and the Hopf bifurcation near the unique positive fixed point. Chaos induced by the regular nondegenerate snap-back repeller is also studied. Numerical simulations are presented to exhibit new complex dynamics, such as attracting and non-attracting chaotic sets and periodic orbits in different chaotic regions. Especially, we can see when the prey is in chaotic dynamic, the predator may tend to extinction. Impulsive hybrid control method proposed firstly in Internet TCP-RED congestion control system and small-world networks is adopted in biological population system to control the period-doubling bifurcation and chaos such that the populations can grow orderly. For the resource manager, this is a new idea. In practice, if the adjustment of system parameters (e.g. competition coefficient) and state (population density) feedback are combined in a proper weighting coefficient, the undesirable period-doubling route to chaos can be delayed or eliminated completely.(4) The dynamical behaviors of a discrete-time generalized Lotka-Volterra sys- tem are investigated. It is shown that the system undergoes the flip bifurcation and the Hopf bifurcation near the unique positive fixed point. Numerical simula-tions are presented to illustrate more complex dynamics. Periodic feedback control is proposed to control the period-doubling bifurcation and chaos. Then whether one can achieve the same objective with a control input with less dimensionality is considered. Scalar periodic feedback control is then proposed. The applications of periodic feedback and scalar periodic feedback in biological population system are realized firstly.