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.
Highlights
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
We propose a singular ecological-economic model to model such a problem
Summary
The management of renewable resources is important for the development of human and society. 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. Since a singular biological economics model with stage structure was established to model the biological systems in , some singular biological models appeared 16–19 These ideas are based on the economic theory 20 : Net Economic Revenue Total Revenue − Total Cost.
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