American Institute of Mathematical Sciences

2014, 11(2): 167-188. doi: 10.3934/mbe.2014.11.167

A non-autonomous stochastic predator-prey model

 1 Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli Federico II, Via Cintia, 80126 Napoli, Italy, Italy, Italy 2 Dipartimento di Studi e Ricerche Aziendali, (Management & Information Technology), Università di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy

Received  September 2012 Revised  January 2013 Published  October 2013

The aim of this paper is to consider a non-autonomous predator-prey-like system, with a Gompertz growth law for the prey. By introducing random variations in both prey birth and predator death rates, a stochastic model for the predator-prey-like system in a random environment is proposed and investigated. The corresponding Fokker-Planck equation is solved to obtain the joint probability density for the prey and predator populations and the marginal probability densities. The asymptotic behavior of the predator-prey stochastic model is also analyzed.
Citation: Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Amelia G. Nobile. A non-autonomous stochastic predator-prey model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 167-188. doi: 10.3934/mbe.2014.11.167
References:
 [1] G. Q. Cai and Y. K. Lin, Stochastic analysis of the Lotka-Volterra model for ecosystems, Phys Rev. E, 70, 041910 (2004) 1-7. doi: 10.1103/PhysRevE.70.041910. [2] G. Q. Cai and Y. K. Lin, Stochastic analysis of predator-prey type ecosystems, Ecological Complexity, 4 (2007), 242-249. doi: 10.1016/j.ecocom.2007.06.011. [3] R. M. Capocelli and L. M. Ricciardi, A diffusion model for population growth in random environment, Theor. Pop. Biol., 5 (1974), 28-41. doi: 10.1016/0040-5809(74)90050-1. [4] R. M. Capocelli and L. M. Ricciardi, Growth with regulation in random environment, Kybernetik, 15 (1974), 147-157. doi: 10.1007/BF00274586. [5] M. F. Dimentberg, Lotka-Volterra system in a random environment, Phys Rev. E, 65, 036204 (2002), 1-7. doi: 10.1103/PhysRevE.65.036204. [6] M. Fan, Q. Wang and X. Zou, Dynamics of a non-autonomous ratio-dependent predator-prey system, Proceedings of the Royal Society of Edinburgh, 133A (2003), 97-118. doi: 10.1017/S0308210500002304. [7] M. W. Feldman and J. Roughgarden, A population's stationary distribution and chance of extinction in a stochastic environment with remarks on the theory of species packing, Theor. Popul. Biol., 7 (1975), 197-207. doi: 10.1016/0040-5809(75)90014-3. [8] N.S. Goel, S.C. Maitra and E.W. Montroll, On the Volterra and other nonlinear models of interacting populations, Reviews of Modern Physics, 43, Part 1 (1971), 231-276. doi: 10.1103/RevModPhys.43.231. [9] A.J. Lotka, Elements of Mathematical Biology, Dover Publications, Inc., New York, 1958. [10] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey systems, J. Math. Biol., 36 (1998), 389-406. doi: 10.1007/s002850050105. [11] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 1973. [12] R. M. May, Theoretical Ecology, Principles and Applications, Oxford University Press, 1976. [13] E. W. Montroll, Some statistical aspects of the theory of interacting species, in Some Mathematical Questions in Biology. III., Lectures on Mathematics in the Life Sciences, 4, The American Mathematical Society, Providence, Rhode Island, (1972), 101-143. [14] A. G. Nobile and L. M. Ricciardi, Growth with regulation in fluctuating environments. I. Alternative logistic-like diffusion models, Biol. Cybern., 49 (1984), 179-188. doi: 10.1007/BF00334464. [15] A. G. Nobile and L. M. Ricciardi, Growth with regulation in fluctuating environments. II. Intrinsic lower bounds to population size, Biol. Cybern., 50 (1984), 285-299. doi: 10.1007/BF00337078. [16] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Interdisciplinary Applied Mathematics, 14, Mathematical Biology, Springer, 2001. [17] L. M. Ricciardi, Diffusion processes and related topics in biology, Lecture Notes in Biomathematics, 14, Berlin, Heidelberg, New York, Springer, 1977. [18] L. M. Ricciardi, Stochastic population theory: diffusion processes, in Mathematical Ecology (eds. T. G. Hallam and S. A. Levin), (Miramare Trieste, 1982), Biomathematics, 17, Springer Verlag, Berlin, (1986), 191-238. [19] R. J. Swift, A Stochastic Predator-Prey Model, Irish Math. Soc. Bulletin, 48 (2002), 57-63. [20] V. Volterra, Leçon sur la Théorie Mathématique de la Lutte pour la Vie, Les Grands Classiques Gauthier-Villars, Paris, 1931. [21] M.C Wang and G.E. Uhlenbeck, On the theory of the Brownian motion. II, Rev. Modern Phys., 17 (1945), 323-342. doi: 10.1103/RevModPhys.17.323. [22] A. Yagi and T.V. Ton, Dynamic of a stochastic predator-prey population, Applied Mathematics and Computation, 218 (2011), 3100-3109. doi: 10.1016/j.amc.2011.08.037. [23] A. S. Zaghrout and F. Hassan, Non-autonomous predator prey model with application, International Mathematical Forum, 5 (2010), 3309-3322. [24] W. R. Zhong, Y. Z. Shao and Z. H. He, Correlated noises in a prey-predator ecosystem, Chin. Phys. Lett., 23 (2006), 742-745.

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References:
 [1] G. Q. Cai and Y. K. Lin, Stochastic analysis of the Lotka-Volterra model for ecosystems, Phys Rev. E, 70, 041910 (2004) 1-7. doi: 10.1103/PhysRevE.70.041910. [2] G. Q. Cai and Y. K. Lin, Stochastic analysis of predator-prey type ecosystems, Ecological Complexity, 4 (2007), 242-249. doi: 10.1016/j.ecocom.2007.06.011. [3] R. M. Capocelli and L. M. Ricciardi, A diffusion model for population growth in random environment, Theor. Pop. Biol., 5 (1974), 28-41. doi: 10.1016/0040-5809(74)90050-1. [4] R. M. Capocelli and L. M. Ricciardi, Growth with regulation in random environment, Kybernetik, 15 (1974), 147-157. doi: 10.1007/BF00274586. [5] M. F. Dimentberg, Lotka-Volterra system in a random environment, Phys Rev. E, 65, 036204 (2002), 1-7. doi: 10.1103/PhysRevE.65.036204. [6] M. Fan, Q. Wang and X. Zou, Dynamics of a non-autonomous ratio-dependent predator-prey system, Proceedings of the Royal Society of Edinburgh, 133A (2003), 97-118. doi: 10.1017/S0308210500002304. [7] M. W. Feldman and J. Roughgarden, A population's stationary distribution and chance of extinction in a stochastic environment with remarks on the theory of species packing, Theor. Popul. Biol., 7 (1975), 197-207. doi: 10.1016/0040-5809(75)90014-3. [8] N.S. Goel, S.C. Maitra and E.W. Montroll, On the Volterra and other nonlinear models of interacting populations, Reviews of Modern Physics, 43, Part 1 (1971), 231-276. doi: 10.1103/RevModPhys.43.231. [9] A.J. Lotka, Elements of Mathematical Biology, Dover Publications, Inc., New York, 1958. [10] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey systems, J. Math. Biol., 36 (1998), 389-406. doi: 10.1007/s002850050105. [11] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 1973. [12] R. M. May, Theoretical Ecology, Principles and Applications, Oxford University Press, 1976. [13] E. W. Montroll, Some statistical aspects of the theory of interacting species, in Some Mathematical Questions in Biology. III., Lectures on Mathematics in the Life Sciences, 4, The American Mathematical Society, Providence, Rhode Island, (1972), 101-143. [14] A. G. Nobile and L. M. Ricciardi, Growth with regulation in fluctuating environments. I. Alternative logistic-like diffusion models, Biol. Cybern., 49 (1984), 179-188. doi: 10.1007/BF00334464. [15] A. G. Nobile and L. M. Ricciardi, Growth with regulation in fluctuating environments. II. Intrinsic lower bounds to population size, Biol. Cybern., 50 (1984), 285-299. doi: 10.1007/BF00337078. [16] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Interdisciplinary Applied Mathematics, 14, Mathematical Biology, Springer, 2001. [17] L. M. Ricciardi, Diffusion processes and related topics in biology, Lecture Notes in Biomathematics, 14, Berlin, Heidelberg, New York, Springer, 1977. [18] L. M. Ricciardi, Stochastic population theory: diffusion processes, in Mathematical Ecology (eds. T. G. Hallam and S. A. Levin), (Miramare Trieste, 1982), Biomathematics, 17, Springer Verlag, Berlin, (1986), 191-238. [19] R. J. Swift, A Stochastic Predator-Prey Model, Irish Math. Soc. Bulletin, 48 (2002), 57-63. [20] V. Volterra, Leçon sur la Théorie Mathématique de la Lutte pour la Vie, Les Grands Classiques Gauthier-Villars, Paris, 1931. [21] M.C Wang and G.E. Uhlenbeck, On the theory of the Brownian motion. II, Rev. Modern Phys., 17 (1945), 323-342. doi: 10.1103/RevModPhys.17.323. [22] A. Yagi and T.V. Ton, Dynamic of a stochastic predator-prey population, Applied Mathematics and Computation, 218 (2011), 3100-3109. doi: 10.1016/j.amc.2011.08.037. [23] A. S. Zaghrout and F. Hassan, Non-autonomous predator prey model with application, International Mathematical Forum, 5 (2010), 3309-3322. [24] W. R. Zhong, Y. Z. Shao and Z. H. He, Correlated noises in a prey-predator ecosystem, Chin. Phys. Lett., 23 (2006), 742-745.
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