The Analysis and Control for Singular Ecological-Economic Model with Harvesting and Migration
The Analysis and Control for Singular Ecological-Economic Model with Harvesting and Migration
Qingling Zhang,1 Hong Niu,1 Lichun Zhao,2 and Fenglan Bai3
1Institute of Systems Science, Northeastern University, Liaoning, Shenyang 110819, China
2Department of Mathematics, Anshan Normal University, Liaoning, Anshan 114007, China
3School of Science, Dalian Jiaotong University, Liaoning, Dalian 116028, China
Received 12 March 2012; Accepted 11 April 2012
Academic Editor: Zhiwei Gao
Copyright © 2012 Qingling Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
To keep the resources renewable, a singular ecological-economic model is proposed for the populations with harvesting and migration. The local stability and the dynamic behavior of the model are studied. Singular induced bifurcation appears when economic interest is zero, which is different from the ordinary differential models. In order to apply variable structure control to eliminate these complex behaviors, the singular model is transformed into a single-input and single-output model with parameter varying within definite intervals. And then, a variable structure controller is designed to make the model stable. Finally, an inshore-offshore fishery model is given to illustrate the proposed method, and some numerical simulations are shown to demonstrate the control results.
Journal of Applied Mathematics
Volume 2012 (2012), Article ID 973869, 17 pages
doi:10.1155/2012/973869
Research Article
The Analysis and Control for Singular Ecological-Economic Model with Harvesting and Migration
Qingling Zhang,1 Hong Niu,1 Lichun Zhao,2 and Fenglan Bai3
1Institute of Systems Science, Northeastern University, Liaoning, Shenyang 110819, China
2Department of Mathematics, Anshan Normal University, Liaoning, Anshan 114007, China
3School of Science, Dalian Jiaotong University, Liaoning, Dalian 116028, China
Received 12 March 2012; Accepted 11 April 2012
Academic Editor: Zhiwei Gao
Copyright © 2012 Qingling Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
To keep the resources renewable, a singular ecological-economic model is proposed for the populations with harvesting and migration. The local stability and the dynamic behavior of the model are studied. Singular induced bifurcation appears when economic interest is zero, which is different from the ordinary differential models. In order to apply variable structure control to eliminate these complex behaviors, the singular model is transformed into a single-input and single-output model with parameter varying within definite intervals. And then, a variable structure controller is designed to make the model stable. Finally, an inshore-offshore fishery model is given to illustrate the proposed method, and some numerical simulations are shown to demonstrate the control results.
1. Introduction
The management of renewable resources is important for the development of human and society. In exploiting the biological resources, both the economic profit and the environmental effects should be taken into account, which initiates a new research area: biomathematics. Interactions of mathematics and biology promote the development of the biosciences greatly in a certain extent. Since most of biological theories evolve rapidly, it is necessary to develop some useful mathematical models to describe the consequences of these biological models.
Singular model as a branch of modern control theory can describe a class of practical models more accurately. Compared with the ordinary differential models, singular models exhibit more complicated dynamics, such as the impulse phenomenon. They have more applications in power systems, aerospace engineering, chemical processes, social economic systems, biological systems, network analysis, and so forth. With the help of the singular models for the power systems and bifurcation theory, complex dynamical behaviors of the power systems have been extensively studied, which reveal the instability mechanism of power systems [1–3]. Applications of singular models are also found in neural networks [4], fault diagnosis [5, 6], robotics [7, 8] and epidemic [9–11], economics [12, 13], and chemistry [14]. As far as the singular system theory is concerned, there are a few research results in biology. Since a singular biological economics model with stage structure was established to model the biological systems in [15], some singular biological models appeared [16–19]. These ideas are based on the economic theory [20]: This formula presents some solid preliminary on singular biological systems.
In biology, many math (...truncated)