Modeling And Control For Singular Bio-economic System With Invasion Of Alien Species

Modeling and control for singular bio-economic system with invasion of alien species is the research on the field of biological mathematics. Biological mathematics is a new interdisciplinary subject between biology and mathematics. Biological mathematics achieved the first brilliant in the early 20 th Century, and further blended with other subjects in the middle of the 20 th Century. In particular, the further development of mathematics and computer technology has made a breakthrough in the research of complex systems such as human and cell. From the view of mathematics and physics, this young discipline combines the problem of dynamics and thinking method of mathematics to study and solve biological problems. It is applied to the economic production of the fishery and life practice, and it is a new chapter for the exploration of ecological mystery.Based on the theory of generalized bio economic system and the law of alien species invasion, this paper takes the breeding fishery as the background and constructs the model for carbon sinks fisheries of multi-trophic levels, the model for a single-species fish population logistic model with the invasion of alien species and the model for H∞ control of T-S fuzzy fish population with the invasion of alien species. According to the theory of singular systems, interval function theory and control theory, the stability has been analyzed on each equilibrium point of the model. The paper further studies on the optimal harvesting for carbon sinks fisheries of multi-trophic levels with interval biological parameters and the bifurcation and control of a single-species fish population logistic model with the invasion of alien species. At last, the problem of the H∞ control of fish population logistic model with the invasion of alien species is studied via T-S fuzzy control approach in this paper. The main works of this paper are summarized as follows:(1) Backgrounds of the research work in the paper are introduced. Firstly, The Lotka-Volterra model and Logistic model of biological mathematics are given, and the singular biological economic system and its research status are introduced. Secondly, it summarizes the development and research status of China’s fishery economy and the properties of exotic species(taking Spartina anglica as an example), the introduction of the reasons, the domestic and foreign research situation. Finally, the main research contents of this paper are stated.(2) The paper presents the study of optimal harvesting for carbon sinks fisheries of multi-trophic levels with interval biological parameters. Due to the lack of precise numerical information of the biological parameters, we consider the model with imprecise data as form of an interval in nature. This paper studies on the model of kelp-abalone-sea cucumber under impreciseness and introduces parametric functional form of an interval. We identify the equilibrium points of the model and discuss their stabilities. The existence of bionomic equilibrium of the model is discussed. We study the optimal harvest policy and obtain the solution in the interior equilibrium using Pontryagin’s maximum principle. Numerical examples are presented to support the proposed theory.(3) The singular biological dynamic system with the invasion of alien species and the purification capacity is studied. Considering the capture behavior of fish population and the purification behavior of alien species anglica Spartina, the singular biological economic dynamical system with the invasion species is constructed. In the absence of economic interest of harvesting, namely the phenomenon of biological economic equilibrium, equilibria of the system are obtained. Firstly, the differential-algebraic system and related lemmas are established, and then using singular system theory and bifurcation theory, the existence of trans-critical bifurcation and singularity induced bifurcation of the system are studied. The linear controller is designed by the singular system control method to eliminate the singularity induced bifurcation and stabilize the system.(4) Due to the biological properties of the invasion species Spartina anglica, the T -S fuzzy control theory and the H∞ control method are used to design the state feedback controller and maintain the continuous stability of the system. For investigating the problem step by step, the bio-economic system with added input disturbance has been considered in the later section. The modified differential transform method provides analytical solution of invasion model with periodic disturbance in convergent series form. Furthermore, the application of H∞ controller according to actual events demonstrates the effectiveness of the method used in this paper. Our work generates a profound impact on better developing the relationship of alien species and fishery.