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.  doi: 10.1103/PhysRevE.70.041910.  Google Scholar

[2]

G. Q. Cai and Y. K. Lin, Stochastic analysis of predator-prey type ecosystems,, Ecological Complexity, 4 (2007), 242.  doi: 10.1016/j.ecocom.2007.06.011.  Google Scholar

[3]

R. M. Capocelli and L. M. Ricciardi, A diffusion model for population growth in random environment,, Theor. Pop. Biol., 5 (1974), 28.  doi: 10.1016/0040-5809(74)90050-1.  Google Scholar

[4]

R. M. Capocelli and L. M. Ricciardi, Growth with regulation in random environment,, Kybernetik, 15 (1974), 147.  doi: 10.1007/BF00274586.  Google Scholar

[5]

M. F. Dimentberg, Lotka-Volterra system in a random environment,, Phys Rev. E, 65, 036204 (2002), 1.  doi: 10.1103/PhysRevE.65.036204.  Google Scholar

[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.  doi: 10.1017/S0308210500002304.  Google Scholar

[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.  doi: 10.1016/0040-5809(75)90014-3.  Google Scholar

[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.  doi: 10.1103/RevModPhys.43.231.  Google Scholar

[9]

A.J. Lotka, Elements of Mathematical Biology,, Dover Publications, (1958).   Google Scholar

[10]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey systems,, J. Math. Biol., 36 (1998), 389.  doi: 10.1007/s002850050105.  Google Scholar

[11]

R. M. May, Stability and Complexity in Model Ecosystems,, Princeton University Press, (1973).   Google Scholar

[12]

R. M. May, Theoretical Ecology, Principles and Applications,, Oxford University Press, (1976).   Google Scholar

[13]

E. W. Montroll, Some statistical aspects of the theory of interacting species,, in Some Mathematical Questions in Biology. III., 4 (1972), 101.   Google Scholar

[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.  doi: 10.1007/BF00334464.  Google Scholar

[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.  doi: 10.1007/BF00337078.  Google Scholar

[16]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives,, Interdisciplinary Applied Mathematics, 14 (2001).   Google Scholar

[17]

L. M. Ricciardi, Diffusion processes and related topics in biology,, Lecture Notes in Biomathematics, 14 (1977).   Google Scholar

[18]

L. M. Ricciardi, Stochastic population theory: diffusion processes,, in Mathematical Ecology (eds. T. G. Hallam and S. A. Levin), 17 (1986), 191.   Google Scholar

[19]

R. J. Swift, A Stochastic Predator-Prey Model,, Irish Math. Soc. Bulletin, 48 (2002), 57.   Google Scholar

[20]

V. Volterra, Leçon sur la Théorie Mathématique de la Lutte pour la Vie,, Les Grands Classiques Gauthier-Villars, (1931).   Google Scholar

[21]

M.C Wang and G.E. Uhlenbeck, On the theory of the Brownian motion. II,, Rev. Modern Phys., 17 (1945), 323.  doi: 10.1103/RevModPhys.17.323.  Google Scholar

[22]

A. Yagi and T.V. Ton, Dynamic of a stochastic predator-prey population,, Applied Mathematics and Computation, 218 (2011), 3100.  doi: 10.1016/j.amc.2011.08.037.  Google Scholar

[23]

A. S. Zaghrout and F. Hassan, Non-autonomous predator prey model with application,, International Mathematical Forum, 5 (2010), 3309.   Google Scholar

[24]

W. R. Zhong, Y. Z. Shao and Z. H. He, Correlated noises in a prey-predator ecosystem,, Chin. Phys. Lett., 23 (2006), 742.   Google Scholar

show all references

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.  doi: 10.1103/PhysRevE.70.041910.  Google Scholar

[2]

G. Q. Cai and Y. K. Lin, Stochastic analysis of predator-prey type ecosystems,, Ecological Complexity, 4 (2007), 242.  doi: 10.1016/j.ecocom.2007.06.011.  Google Scholar

[3]

R. M. Capocelli and L. M. Ricciardi, A diffusion model for population growth in random environment,, Theor. Pop. Biol., 5 (1974), 28.  doi: 10.1016/0040-5809(74)90050-1.  Google Scholar

[4]

R. M. Capocelli and L. M. Ricciardi, Growth with regulation in random environment,, Kybernetik, 15 (1974), 147.  doi: 10.1007/BF00274586.  Google Scholar

[5]

M. F. Dimentberg, Lotka-Volterra system in a random environment,, Phys Rev. E, 65, 036204 (2002), 1.  doi: 10.1103/PhysRevE.65.036204.  Google Scholar

[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.  doi: 10.1017/S0308210500002304.  Google Scholar

[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.  doi: 10.1016/0040-5809(75)90014-3.  Google Scholar

[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.  doi: 10.1103/RevModPhys.43.231.  Google Scholar

[9]

A.J. Lotka, Elements of Mathematical Biology,, Dover Publications, (1958).   Google Scholar

[10]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey systems,, J. Math. Biol., 36 (1998), 389.  doi: 10.1007/s002850050105.  Google Scholar

[11]

R. M. May, Stability and Complexity in Model Ecosystems,, Princeton University Press, (1973).   Google Scholar

[12]

R. M. May, Theoretical Ecology, Principles and Applications,, Oxford University Press, (1976).   Google Scholar

[13]

E. W. Montroll, Some statistical aspects of the theory of interacting species,, in Some Mathematical Questions in Biology. III., 4 (1972), 101.   Google Scholar

[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.  doi: 10.1007/BF00334464.  Google Scholar

[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.  doi: 10.1007/BF00337078.  Google Scholar

[16]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives,, Interdisciplinary Applied Mathematics, 14 (2001).   Google Scholar

[17]

L. M. Ricciardi, Diffusion processes and related topics in biology,, Lecture Notes in Biomathematics, 14 (1977).   Google Scholar

[18]

L. M. Ricciardi, Stochastic population theory: diffusion processes,, in Mathematical Ecology (eds. T. G. Hallam and S. A. Levin), 17 (1986), 191.   Google Scholar

[19]

R. J. Swift, A Stochastic Predator-Prey Model,, Irish Math. Soc. Bulletin, 48 (2002), 57.   Google Scholar

[20]

V. Volterra, Leçon sur la Théorie Mathématique de la Lutte pour la Vie,, Les Grands Classiques Gauthier-Villars, (1931).   Google Scholar

[21]

M.C Wang and G.E. Uhlenbeck, On the theory of the Brownian motion. II,, Rev. Modern Phys., 17 (1945), 323.  doi: 10.1103/RevModPhys.17.323.  Google Scholar

[22]

A. Yagi and T.V. Ton, Dynamic of a stochastic predator-prey population,, Applied Mathematics and Computation, 218 (2011), 3100.  doi: 10.1016/j.amc.2011.08.037.  Google Scholar

[23]

A. S. Zaghrout and F. Hassan, Non-autonomous predator prey model with application,, International Mathematical Forum, 5 (2010), 3309.   Google Scholar

[24]

W. R. Zhong, Y. Z. Shao and Z. H. He, Correlated noises in a prey-predator ecosystem,, Chin. Phys. Lett., 23 (2006), 742.   Google Scholar

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