Analysis of a Prey-Predator System with Modified Transmission Function

Transcription

1 Research Paper American Journal of Engineering Research (AJER) e-issn : p-issn : Volume-3, Issue-9, pp Open Access Analysis of a Prey-Predator System with Modified Transmission Function Neelima Daga 1, Bijendra Singh 1, Suman Jain 2, Gajendra Ujjainkar 3 1 School of Studies in Mathematics, Vikram University, Ujjain M.P.,India 2 Govt. College Kalapipal, Vikram University, Ujjain M.P.,India 3 Govt. P.G. College, Dhar, India ABSTRACT - In this paper, a predator prey model with a non-homogeneous transmission functional response is studied. It is interesting to note that the system is persistent. The purpose of this work is to offer some mathematical analysis of the dynamics of a two prey one predator system. Criteria for local stability and global stability of the non- negative equilibria are obtained. Using differential inequality, we obtain sufficient conditions that ensure the persistence of the system. KEYWORDS Transmission function, Prey-Predator interaction, Local stability and Global Stability I. INTRODUCTION Mathematical modelling is frequently an evolving process. Systematic mathematical analysis can often lead to better understanding of bio-economic models. System of differential equations has, to a certain extent, successfully described the interactions between species. There exists a huge literature documenting ecological and mathematical result from the model. Heathcoat et al. [6] proposed some epidemiological model with nonlinear incidence. Kesh et al. [3] proposed and analyzed a mathematical model of two competing prey and one predator species where the prey species follow Lotka - Volterra dynamics and predator uptake functions are ratio dependent. Some works in context of source-sink dynamics are due to Newman et al. [10]. His results show that the presence of refuge can greatly stabilize a population that otherwise would exhibit chaotic dynamics. Dubey et al. [2] analyzed a dynamic model for a single species fishery which depends partially on a logistically growing resource in a two patch environment. Ruan et al. [9] studied the global dynamics of an epidemic model with vital dynamics and nonlinear incidence rate of saturated mass action. Kar [11] considered a prey- predator fishery model and discussed the selective harvesting of fishes age or size by incorporating a time delay in the harvesting terms. Feng [14] considered a differential equation system with diffusion and time delay which models the dynamics of predator prey interactions within three biological species. Kar et al. [13] describe a prey predator model with Holling type II functional response where harvesting of each species is taken into consideration. Braza [8] considered a two predator; one prey model in which one predator interferes significantly with the other predator is analyzed. Kar and Chakraborty [12] considered a prey predator fishery model with prey dispersal in a two patch environment, one of which is a free fishing zone and other is protected zone. Sisodia et al. [1] proposed a generalized mathematical model to study the depletion of resources by two kinds of populations, one is weaker and others stronger. The dynamics of resources is governed by generalized logistic equation whereas the population of interacting species follows the logistic law. We have formulated and analyzed two species prey-predator model in which the prey dispersal in a two patch environment. Mehta et al. [4] considered prey predator model with reserved and unreserved area having modified transmission function. A model of predator-prey in homogeneous environment with Holling type-ii functionl response is introduced to Alebraheen et al. [7]. Recently Mehta et al. [5] describe the epidemic model with an asymptotically homogeneous transmission function. In this paper biological equilibria of the system are obtained and criteria for local stability and global stability of the system derived. We have investigate the model persistence with an asymptotically transmission function. w w w. a j e r. o r g Page 194

2 II. MATHEMATICAL MODEL Mathematical Model considered is based on the predator prey system WITH MODIFIED change transmission rate: III. EQUILIBRIUM ANALYSIS w w w. a j e r. o r g Page 195

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4 III. STABILITY ANALYSIS w w w. a j e r. o r g Page 197

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9 IV. CONCLUSION V. REFERENCE [1] A. Sisodia, B. Singh, B.K. Joshi, Effect of two interacting populations on resource following generalized logistic growth, Appl. Math. Sci.,5(9), 2011, [2] B. Dubey, P. Chandra, P. Sinha, A model for fishery resource with reserve area, J. Biol. Syst.,10, 2002, [3] D. Kesh., A.K. Sarkar, A.B. Roy, Persistent of Two Prey One Predator System with Ratio Dependent Predator Influence, Math. Appl.Sci., 23, 2000, [4] H. Mehta, B. Singh, N.Trivedi, R. Khandelwal, Prey predator model with reserved and unreserved area having modified transmission function, Pelagia Research Library, 3(4), 2012, [5] H. Mehta,B. Singh,N. Trivedi, An epidemic model with an asymptotically homogeneous transmission function, ARPN Journal of Science and Technology, 2013, 3,4, [6] H.W. Hethcote, P. van den Driessche, Some epidemiological model with nonlinear incidence, J. Math. Biol., 29, 1991, [7] J. Alebraheem, Persistence of predators in a two predators- one prey model with non-periodic solution, Applied Mathematical Science, 6(19), 2012, [8] P.A. Braza, A dominant predator and a prey, Math. Bio.Sci.and Engg. 5(1), 2008, [9] S. Ruan, W. Wang, Dynamical behavior of an epidemic model with nonlinear incidence rate, J. Differ. Equations, 188, 2003, [10] T.J. Newman, J. Antonovics, H. M. Wilbur, Population Dynamic with a Refuge: Fractal basins and the suppression of chaos, Theor. Popul. Biol., 62, 2002, [11] T.K. Kar, Selective Harvesting in a Prey Predator Fishery with Time Delay, Mathematical and Computer Modelling, 38, 2003, [12] T.K Kar, A. Batabyal, Persistence and Stability of a Two Prey One Predator System, International Journal of Engg. Sci. and Tech., 2(2), 2010, [13] T.K. Kar,K. Chakraborty,U.K. Pahari, A Prey Predator Model with Alternative Prey, Mathematical Model and Analysis, Canadian Applied, 18(2), 2010, [14] W. Feng, Dynamics in Three Species Predator Prey Models Unit Time Delays, Discrete Contin. Dyn. Syst. Supplement, 2007, w w w. a j e r. o r g Page 202