T. K. Kar and Kunal Chakraborty. 2010. Bioeconomic modelling of a prey predator system using differential algebraic equations. International Journal of Engineering, Science and Technology . 2(1): 13-34
Article Abstract: We propose a biological economic model based on prey-predator dynamics where the prey species are continuously harvested and predation is considered with type II functional response. The dynamic behavior of the proposed biological economic prey- predator model is discussed. Continuous type gestational delay of predators is incorporated and its effect on the dynamical behavior of the model system is analyzed. Through considering delay as a bifurcation parameter, the occurrence of Hopf bifurcation of the proposed model system with positive economic profit is shown in the neighborhood of the co-existing equilibrium point. Finally, some numerical simulations are given to verify the analytical results and the system is analyzed through graphical illustrations.
This paper is a tour de force of multiple notions and would serve to point out the use in context of many concepts such as Hopf bifurcation, economic application, fisheries, and functional responses.
From the Conclusion of the paper we read:
The paper analyzes the dynamical behavior of a prey predator model using differential-algebraic systems theory. In general, delay differential equations exhibit much more complicated dynamics than ordinary
differential equations thus we have studied the effects of continuous time-delay on the dynamics of prey predator system. It is found that singularity induced bifurcation takes place when net economic revenue of the fishery is considered to be positive. In consequence to the aforesaid bifurcation, an impulsive phenomenon occurs and the system becomes unstable. The most important realistic feature of the paper is the state feedback controller which is designed to stabilize the model system when positive economic rent is taken into consideration. Numerical simulations are used to show that state feedback controller can be designed to resume the stability of a model system in case of positive economic profit. In the second part of the paper we have discussed the behavior of the model system with positive economic profit, here we have divided our discussion in two parts with and without time delay. In case of without time delay it is observed that though the model system is stable but it is possible to get a critical value of total economic profit so that the model system becomes unstable when total economic profit passes through the critical value and the model system enters into Hopf type small amplitude periodic solution. It is noted that continuous time delay also plays an important role to the dynamics of the model system. It is evident from the obtained results that the time delay can cause a stable equilibrium to become unstable and even a simple Hopf bifurcation occurs when the time delay passes through its critical value.
The entire study of the paper is mainly based on the deterministic framework. On the other hand it will be more realistic if it is possible to consider the model system in the stochastic environment due to some ecological fluctuations and other factors. Thus, a future research problem would be considered in stochastic environment. Again, to achieve the commercial purpose of the fishery it is also possible to determine optimal harvesting strategies using game theory.
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