Stability analysis of combined project of fish, broiler and ducks: Dynamical system in imprecise environment

Abstract

In this paper, we consider three species harvesting model and develop a solution procedure which is able to calculate the equilibrium points of the model where some biological parameters of the model are interval numbers. A parametric mathematical program is formulated to find the biological equilibrium of the model for different values of parameters. This interval-valued problem is converted into equivalent crisp model using interval operations. The main advantage of the proposed procedure is that we can present different characteristics of the model in a single framework. Analytically, the existence of steady state and stabilities are looked into. Using mathematical software, the model is illustrated and the results are obtained and presented in tabular and graphical forms. Int. J. Biomath. 2015.08. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 09/30/15. For personal use only. A. De, K. Maity & M. Maiti 1550067-2 Int. J. Biomath. 2015.08. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 09/30/15. For personal use only.Stability analysis of combined project of fish, broiler and ducks or a suitable probability distribution for the imprecise biological parameters. In the development of the above models, there are some lacunas in the formulation and solution of the models. These are:• None formulated a joint project of three related biological species such as fish, broiler and duck and studied the system. • None evaluated the three species system with uncertain biological parameters such as interval or fuzzy parameters.In this paper, we consider three species dynamic model and develop a solution procedure which is able to calculate the equilibrium points of the model where some biological parameters of the model are interval numbers. A parametric mathematical program is formulated to find the behavior of the model (biological equilibrium) for different values of parameters. The proposed procedure is more effective and interesting since we get different behavior of the model using functional form of an interval parameter for different values of parameters. We develop a new procedure to study this three species dynamic model using interval-valued technique. The main advantage of the proposed procedure is that we can present different characteristics of the model in a single framework.The paper is organized as follows. In Sec. 2, we discuss some prerequisite mathematics on interval numbers and Sec. 3 for notations. In Sec. 4, we formulate the model with crisp data. In Sec. 5, the model is extended in imprecise environment. We discuss the boundedness of the system in Sec. 6 and the local stability of the imprecise model in Sec. 7. Some particular cases have been discussed in Sec. 8. We have discussed a brief description of the results and figures of the numerical experiment in Sec. 9. This paper is concluded in Sec. 10. Prerequisite Mathematics An interval number A is represented by closed interval [a l , a u ] and defined bywhere R is the set of all real numbers and a l , a u are t...

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“…Peixoto et al [26] investigate a prey-predator model with fuzzy nature. Pal et al [27], De et al [28,29] presented optimal harvesting model with interval parameters for prey-predator system. Wang [30] presented stability analysis of prey-predator system with fuzzy impulsive control.…”

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“…Therefore researchers in last few years have given considerably high attention to construction and analysis of ecological models in fuzzy environment to capture, explain, and implement different ecological phenomena in more realistic and meaningful way. For example, Pal et al [36] studied a predator-prey model with interval biological parameters; De et al [37] presented stability analysis of combined project of fish, broiler, and ducks in imprecise environment; Zhang and Zhao [38] studied a diffusive predator-prey system with delays and interval biological parameters to obtain a sustainable optimal harvesting policy; Sharma and Samanta [39] presented a two-species com- petition model where the model parameters are taken as interval numbers. Many other notable works in this regard can be found in [40][41][42][43][44][45].…”