Variable Structure Control For A Singular Biological System With Stage Structure

When studying practical problems from the perspective of mathematics,we always build mathematical models by abstracting practical problems and then use mathematical methods to solve them.In this paper,focus on the biological systems with capture factors and invasion of alien species,a singular bioeconomic mathematical model is established to study the sustainable development and variation of populations.In the process of building the mathematical model,the more realistic population model with stage structure is considered,furthermore,the problems of modeling for singular bioeconomic system and variable structure control with sliding mode in the field of biological mathematics are also studied.First,the biological dynamical system with stage structure and its research overview is presented,the research background and status of the singular biological model and the theory of variable structure control are introduced,some basic knowledge related to the paper is given.Then,a singular biological economic model which considers a prey-predator system with time delay and stage structure is proposed by the theory of singular system in this paper.The local stability at the equilibrium points and the bifurcation behavior of the model are studied,and when the economic profit is zero and there is no time delay,singularity induced bifurcation of the biological economic model occurs at the positive equilibrium.Furthermore,a phenomenon of Hopf bifurcation of the model at the boundary equilibrium point occurs as the time delay satisfies certain conditions.In order to eliminate the complex behaviors caused by singularity induced bifurcation,the singular biological economic model is transformed into a single-input and single-output model with parameter varying within definite intervals,and then variable structure controller with siding mode based on a power reaching law is designed to stabilize the model.Numerical simulations are given to verify the effectiveness of the conclusions.Last,the problem of variable structure control for singular biological economic model with stage structure and uncertain parameters is studied.According to the T-S fuzzy theory,the uncertain T-S fuzzy singular model is established and the stability of the model at the positive equilibria is discussed.For the T-S fuzzy singular system with uncertain term,the model is stabilized based on the method of fuzzy sliding mode variable structure control,and the fuzzy integral sliding mode surface and the controller are further designed.And the numerical simulations are given to verify the effectiveness of the conclusions.