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.

[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.

[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.

[4]

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

[5]

M. F. Dimentberg, Lotka-Volterra system in a random environment,, Phys Rev. E, 65, 036204 (2002), 1. 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. 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. 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. doi: 10.1103/RevModPhys.43.231.

[9]

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

[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.

[11]

R. M. May, Stability and Complexity in Model Ecosystems,, Princeton University Press, (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., 4 (1972), 101.

[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.

[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.

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[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.

[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.

[23]

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

[24]

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

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.

[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.

[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.

[4]

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

[5]

M. F. Dimentberg, Lotka-Volterra system in a random environment,, Phys Rev. E, 65, 036204 (2002), 1. 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. 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. 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. doi: 10.1103/RevModPhys.43.231.

[9]

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

[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.

[11]

R. M. May, Stability and Complexity in Model Ecosystems,, Princeton University Press, (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., 4 (1972), 101.

[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.

[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.

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[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.

[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.

[23]

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

[24]

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

[1]

Nguyen Huu Du, Nguyen Thanh Dieu, Tran Dinh Tuong. Dynamic behavior of a stochastic predator-prey system under regime switching. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3483-3498. doi: 10.3934/dcdsb.2017176

[2]

Inkyung Ahn, Wonlyul Ko, Kimun Ryu. Asymptotic behavior of a ratio-dependent predator-prey system with disease in the prey. Conference Publications, 2013, 2013 (special) : 11-19. doi: 10.3934/proc.2013.2013.11

[3]

Ovide Arino, Manuel Delgado, Mónica Molina-Becerra. Asymptotic behavior of disease-free equilibriums of an age-structured predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 501-515. doi: 10.3934/dcdsb.2004.4.501

[4]

E. J. Avila–Vales, T. Montañez–May. Asymptotic behavior in a general diffusive three-species predator-prey model. Communications on Pure & Applied Analysis, 2002, 1 (2) : 253-267. doi: 10.3934/cpaa.2002.1.253

[5]

Nguyen Huu Du, Nguyen Hai Dang. Asymptotic behavior of Kolmogorov systems with predator-prey type in random environment. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2693-2712. doi: 10.3934/cpaa.2014.13.2693

[6]

Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129

[7]

Canan Çelik. Dynamical behavior of a ratio dependent predator-prey system with distributed delay. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 719-738. doi: 10.3934/dcdsb.2011.16.719

[8]

Xinjian Wang, Guo Lin. Asymptotic spreading for a time-periodic predator-prey system. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2983-2999. doi: 10.3934/cpaa.2019133

[9]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507

[10]

Gianni Gilioli, Sara Pasquali, Fabrizio Ruggeri. Nonlinear functional response parameter estimation in a stochastic predator-prey model. Mathematical Biosciences & Engineering, 2012, 9 (1) : 75-96. doi: 10.3934/mbe.2012.9.75

[11]

Guanqi Liu, Yuwen Wang. Stochastic spatiotemporal diffusive predator-prey systems. Communications on Pure & Applied Analysis, 2018, 17 (1) : 67-84. doi: 10.3934/cpaa.2018005

[12]

S. Nakaoka, Y. Saito, Y. Takeuchi. Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system. Mathematical Biosciences & Engineering, 2006, 3 (1) : 173-187. doi: 10.3934/mbe.2006.3.173

[13]

Jing-An Cui, Xinyu Song. Permanence of predator-prey system with stage structure. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 547-554. doi: 10.3934/dcdsb.2004.4.547

[14]

Dongmei Xiao, Kate Fang Zhang. Multiple bifurcations of a predator-prey system. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 417-433. doi: 10.3934/dcdsb.2007.8.417

[15]

Laura Martín-Fernández, Gianni Gilioli, Ettore Lanzarone, Joaquín Míguez, Sara Pasquali, Fabrizio Ruggeri, Diego P. Ruiz. A Rao-Blackwellized particle filter for joint parameter estimation and biomass tracking in a stochastic predator-prey system. Mathematical Biosciences & Engineering, 2014, 11 (3) : 573-597. doi: 10.3934/mbe.2014.11.573

[16]

Hongxiao Hu, Liguang Xu, Kai Wang. A comparison of deterministic and stochastic predator-prey models with disease in the predator. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2837-2863. doi: 10.3934/dcdsb.2018289

[17]

Peng Feng. On a diffusive predator-prey model with nonlinear harvesting. Mathematical Biosciences & Engineering, 2014, 11 (4) : 807-821. doi: 10.3934/mbe.2014.11.807

[18]

Ronald E. Mickens. Analysis of a new class of predator-prey model. Conference Publications, 2001, 2001 (Special) : 265-269. doi: 10.3934/proc.2001.2001.265

[19]

Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124

[20]

Yun Kang, Sourav Kumar Sasmal, Amiya Ranjan Bhowmick, Joydev Chattopadhyay. Dynamics of a predator-prey system with prey subject to Allee effects and disease. Mathematical Biosciences & Engineering, 2014, 11 (4) : 877-918. doi: 10.3934/mbe.2014.11.877

2017 Impact Factor: 1.23

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (0)

[Back to Top]