Existence and Uniqueness of Asymptotic Periodic Solution in the Cyclic Four Species Predator- Prey Model

Document Type : Original Paper

Authors

Department of Mathematics, University of Neyshabur, Neyshabur, Iran

Abstract

In the past decades, in the area of mathematical ecology, the dynamical properties occurring in the predator-prey models have been studied. Moreover, the stability and boundedness of the solution for population model such as cyclic, delayed and etc. have been studied. In the present paper, a nonlinear cyclic predator-prey system with sigmoidal type functional response is analyzed. Indeed, a model of four species predator-prey system has been investigated and the sufficient conditions for stability and boundedness of the solutions of predator-prey system have been presented. For this purpose, the differential inequality theory is employed and finally, by constructing a suitable Lyapanov function the existence and uniqueness of asymptotically periodic solution which is globally asymptotically stable are proved.

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Main Subjects


[1] Rahmani Doust, M.H. and Gholizade, S. (2014). An analysis of the modified Lotka-Volterra predator-prey equations, Gen. Math. Notes, 25, 2, 1-5.
[2] Rahmani Doust, M.H. and Gholizade, S. (2014). The lotka- Volterra predator-prey equations, Caspian Journal of Mathematical Sciences, 3(1), 227-231.
[3] Rahmani Doust, M. H. (2015). The efficiency of harvest factor; Lotka-Volterra predator-prey model, Caspian Journal of Mathematical Sciences, 4(1), 51-59.
[4] Li Y. K., and Ye, Y. (2013). Multiple positive almost periodic solutions to an impulsive non-autonomous Lotka-Volterra predator-prey system with harvesting terms, common. Non-linear sci. Number. Simul. 18, 3190-3201.
 [5] XU, C. and Zhang, Q. (2014) Permanence and asymptotically periodic solution for a cyclic predator-prey model with sigmoidal type functional response, Wseas transactions on Systems, 13, 668-678.
 [6] Yang, Y. and Chen, W.C. (2006). Uniformly strong persistence of a nonlinear asymptotically periodic multispecies competition predator-prey system with general functional response, Appl. Math. Comput. 183, 423-426.
 [7] Yuan, R., (1992). Existence of almost periodic solutions of functional differential equations of neutral type, J. Math. Anal. Appl. 165, 524-538.
[8] Murray, J.D. (2002). Mathematical biology (Vol. 1: An Introduction). Springer, New York.
[9] Murray, J.D. (2003). Mathematical biology (Vol. 2: Spatial models and biomedical applications). Springer, New York.
[10] Cornin, J. (2008). Ordinary differential equations. Third Edition, Chapman & Hall/CRC.New York.
[11] Perco, L. (2001). Differential equations and dynamical systems. Springer, New York.
 [12] Miller, R.K. and Michel A.N. (1982). Ordinary differential equations, Academic press, New York.
[13] رحمانی دوست، محمدحسین، (1392). معادلات دیفرانسیل و اکولوژی جلد اول، انتشارات نوروزی، گرگان. دانشگاه نیشابور.
[14] Montes F. de Oca, and Vivas, M. (2006). Extinction in a two dimensional Lotka-Volterra system with infinite delay. Nonlinear Anal. Real World Appl. 7, 1042-1047.