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Dynamics and control of a system of two non‐interacting preys with common predator
Author(s) -
Zaman Gul,
Saker Samir H.
Publication year - 2011
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1526
Subject(s) - mathematics , pontryagin's minimum principle , stability (learning theory) , optimal control , maximum principle , set (abstract data type) , mathematical optimization , exponential stability , control theory (sociology) , control (management) , computer science , nonlinear system , quantum mechanics , machine learning , artificial intelligence , programming language , physics
In this paper, we consider a Holling type model, which describes the interaction between two preys with a common predator. First, we give some sufficient conditions for the globally asymptotic stability and prove that local stability implies global stability. Then, we present a set of sufficient conditions for the existence of a positive periodic solution with strictly positive components. Finally, the optimal control strategy is developed to minimize the number of predator and maximize the number of preys. We also show the existence of an optimal control for the optimal control problem and derive the optimality system. The technical tool used to determine the optimal strategy is the Pontryagin Maximum Principle. Finally, the numerical simulations of global stability and the optimal problem are given as the conclusion of this paper. Copyright © 2011 John Wiley & Sons, Ltd.