# American Institute of Mathematical Sciences

October  2018, 23(8): 3275-3296. doi: 10.3934/dcdsb.2018244

## Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition

 1 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal 711103, India 2 Department of Mathematics, Vivekananda College, Thakurpukur, Kolkata-700063, India

* Corresponding author: +91 9831022438. E-mail address: prithadas02@math.iiests.ac.in (Pritha Das)

Received  December 2017 Revised  February 2018 Published  October 2018 Early access  August 2018

The objective of this article is to study the significance of dynamical properties of non-autonomous deterministic as well as stochastic prey-predator model with Holling type-Ⅲ functional response. Firstly, uniform persistence of the deterministic model has been demonstrated. Secondly, stochastic non-autonomous prey-predator system with Holling type-Ⅲ functional response is proposed. The existence of a global positive solution has been derived. Sufficient conditions for non-persistence in mean, weakly persistence in mean, extinction have been derived. Moreover the sufficient conditions for permanence of the system have been established. The analytical results are verified by numerical simulation.

Citation: Sampurna Sengupta, Pritha Das, Debasis Mukherjee. Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3275-3296. doi: 10.3934/dcdsb.2018244
##### References:
 [1] M. V. Baalen, V. Křivan, P. C. J. van Rijn and M. W. Sabelis, Alternative food, switching predators and the persistence of predator prey systems, Ameri. Naturalist, 157 (2001), 512-524. [2] T. Caraballo, R. Colucci and X. Han, Non-autonomous dynamics of a semi-Kolmogorov population model with periodic forcing, Nonlinear Analysis: Real World Applications, 31 (2016), 661-680.  doi: 10.1016/j.nonrwa.2016.03.007. [3] T. Caraballo, R. Colucci and X. Han, Semi-Kolmogorov models for predation with indirect effects in random environments, Discrete and Continuous Dynamical Systems - Series B, 21 (2016), 2129-2143.  doi: 10.3934/dcdsb.2016040. [4] T. Caraballo, R. Colucci and X. Han, Predation with indirect effects in fluctuating environments, Nonlinear Dynamics, 84 (2016), 115-126.  doi: 10.1007/s11071-015-2238-3. [5] F. Chen and J. Shi, On a delayed non-autonomous ratio dependent predator prey model with Holling type functional response and diffusion, Appl. Math. and Compu., 192 (2007), 358-369.  doi: 10.1016/j.amc.2007.03.012. [6] Available from: https://www.conservationnw.org/what-we-do/predators-and-prey/carnivores-predators-and-their-prey (accessed 10 October 2017). [7] N. H. Du, N. T. Dieu and T. D. Tuong, Dynamic behavior of a stochastic predator-prey system under regime switching, Discrete and Continuous Dynamical Systems - Series B (DCDS-B), 22 (2017), 3483-3498.  doi: 10.3934/dcdsb.2017176. [8] Functional and numerical response, Available from: https://www.ma.utexas.edu/users/davis/375/popecol/lec10/funcresp.html. [9] P. Ghosh, P. Das and D. Mukherjee, Persistence and Stability of a Seasonally Perturbed Three Species Stochastic Model of Salmonoid Aquaculture, Diff. Eqn and Dyn. Sys, (2016), 1-17.  doi: 10.1007/s12591-016-0283-0. [10] P. Ghosh, P. Das and D. Mukherjee, Chaos to order: Effect Of random predation in a holling type iv tri-trophic food chain system with closure terms, Int. J. Biomath., 9 (2016), 1650073, 18pp. doi: 10.1142/S179352451650073X. [11] T. F. Hansen, N. C. Stenseth, H. Henttonen and J. Tast, Interspecific and intraspecific competition as causes of direct and delayed density dependence in a fluctuating vole population, Proc. Natl. Acad. Sci. USA, 96 (1999), 986-991.  doi: 10.1073/pnas.96.3.986. [12] D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [13] C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine-sawfly, Canad. Entomologist., 91 (1959), 293-320.  doi: 10.4039/Ent91293-5. [14] Intraspecific Competition: Example and Definition, Chapter 18 / Lesson 30, Available from: https://study.com/academy/lesson/intraspecific-competition-example-definition-quiz.html. [15] C. Ji, D. Jiang and N. Shi, Analysis of a predator-prey model with modified Leslie-Gower and Holling Type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482-498.  doi: 10.1016/j.jmaa.2009.05.039. [16] P. A. Khaiter and M. G. Erechtchoukova, The notion of stability in mathematics, biology, ecology and environmental sustainability, 18th World IMACS/MODSIM Congress, Cairns, Australia, 2009, 13–17. [17] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. [18] Lesser-known examples of intraspecific competition that exist, Available from: https://www.buzzle.com/articles/examples-of-intraspecific-competition.html. [19] J. P. Lasalle and R. J. Rath, Eventual stability, Proc. of Sec. of the I.F.A.C, (1963), 556-560. [20] M. Liu and M. Fan, Permanence of stochastic Lotka-Volterra systems, Journal of Nonlinear Science, 27 (2017), 425-452.  doi: 10.1007/s00332-016-9337-2. [21] M. Liu and K. Wang, Persistence, extinction and global asymptotical stability of a non-autonomous predator-prey model with random perturbation, Appl. Math. Model., 36 (2012), 5344-5353.  doi: 10.1016/j.apm.2011.12.057. [22] M. Liu and K. Wang, Global stability of a nonlinear stochastic predator-prey system with Beddington-DeAngelis functional response, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 1114-1121.  doi: 10.1016/j.cnsns.2010.06.015. [23] M. Liu, K. Wang and Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol., 73 (2011), 1969-2012.  doi: 10.1007/s11538-010-9569-5. [24] S. Li and X. Zhang, Dynamics of a stochastic non-autonomous predator-prey system with Beddington-DeAngelis functional response, Adv. Diff. Eqns., 2014 (2014), 19pp. doi: 10.1186/1687-1847-2014-314. [25] X. Y. Li, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete Contin. Dyn. Syst., 24 (2009), 523-545.  doi: 10.3934/dcds.2009.24.523. [26] A. Lotka, Elements of physical biology, Williams and Wilkins, Baltimore, 1925. [27] A. M. Lyapunov, The General Problem of the Stability of Motion, Kharkov Mathematical Society, Kharkov, 1892. [28] X. Y. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402. [29] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial college press, 2006. doi: 10.1142/p473. [30] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, NJ, 2001. [31] P. S. Mondal and M. Banerjee, Stochastic persistence and stationary distribution in a Holling-Tanner type prey-predator model, Physica A: Stat. Mech. and Appl., 391 (2012), 1216-1233.  doi: 10.1016/j.physa.2011.10.019. [32] W. W. Murdoch, S. Avery and E. B. Michael, Switching in predatory fish, Ecology, 56 (1975), 1094-1105.  doi: 10.2307/1936149. [33] S. J. Schreiber, Persistence for stochastic difference equations: A mini-review, J. Diff. Eqns and Appl., 18 (2012), 1381-1403.  doi: 10.1080/10236198.2011.628662. [34] Shut up: how noise pollution is affecting 10 animals, Available from: http://webecoist.momtastic.com/2010/02/28/please-shut-up-10-animals-affected-by-noise-pollution/ (accessed 8 October 2017). [35] T. Takahashi and S. Matsuura, Laboratory studies on molting and growth of the shore crab, Hemigrapsus sanguineus de Haan, Parasitized by a Rhizocephalan Barnacle, Biol. Bull., 186 (1994), 300-308.  doi: 10.2307/1542276. [36] V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, J. Cons. Int. Explor., 3 (1928), 3-51.  doi: 10.1093/icesjms/3.1.3. [37] L. L. Wang and W. T. Li, Existence and global stability of positive periodic solutions of a predator-prey system with delays, Appl. Math. and Compu., 146 (2003), 167-185.  doi: 10.1016/S0096-3003(02)00534-9. [38] What are examples of intraspecific competition?, Available from: https://www.quora.com/What-are-examples-of-intraspecific-competition. [39] R. Wu, X. Zou and K. Wang, Asymptotic behavior of a stochastic non-autonomous predator-prey model with impulsive perturbations, Common. Nonlinear. Sci. Number. Simulat., 20 (2015), 965-974.  doi: 10.1016/j.cnsns.2014.06.023. [40] Y. Zhang, S. Chen and S. Gao, Analysis of a non-autonomous stochastic predator-prey model with Crowley-Martin functional response, Adv. Diff. Eqns, 2016 (2016), Paper No. 264, 28 pp. doi: 10.1186/s13662-016-0993-1.

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##### References:
 [1] M. V. Baalen, V. Křivan, P. C. J. van Rijn and M. W. Sabelis, Alternative food, switching predators and the persistence of predator prey systems, Ameri. Naturalist, 157 (2001), 512-524. [2] T. Caraballo, R. Colucci and X. Han, Non-autonomous dynamics of a semi-Kolmogorov population model with periodic forcing, Nonlinear Analysis: Real World Applications, 31 (2016), 661-680.  doi: 10.1016/j.nonrwa.2016.03.007. [3] T. Caraballo, R. Colucci and X. Han, Semi-Kolmogorov models for predation with indirect effects in random environments, Discrete and Continuous Dynamical Systems - Series B, 21 (2016), 2129-2143.  doi: 10.3934/dcdsb.2016040. [4] T. Caraballo, R. Colucci and X. Han, Predation with indirect effects in fluctuating environments, Nonlinear Dynamics, 84 (2016), 115-126.  doi: 10.1007/s11071-015-2238-3. [5] F. Chen and J. Shi, On a delayed non-autonomous ratio dependent predator prey model with Holling type functional response and diffusion, Appl. Math. and Compu., 192 (2007), 358-369.  doi: 10.1016/j.amc.2007.03.012. [6] Available from: https://www.conservationnw.org/what-we-do/predators-and-prey/carnivores-predators-and-their-prey (accessed 10 October 2017). [7] N. H. Du, N. T. Dieu and T. D. Tuong, Dynamic behavior of a stochastic predator-prey system under regime switching, Discrete and Continuous Dynamical Systems - Series B (DCDS-B), 22 (2017), 3483-3498.  doi: 10.3934/dcdsb.2017176. [8] Functional and numerical response, Available from: https://www.ma.utexas.edu/users/davis/375/popecol/lec10/funcresp.html. [9] P. Ghosh, P. Das and D. Mukherjee, Persistence and Stability of a Seasonally Perturbed Three Species Stochastic Model of Salmonoid Aquaculture, Diff. Eqn and Dyn. Sys, (2016), 1-17.  doi: 10.1007/s12591-016-0283-0. [10] P. Ghosh, P. Das and D. Mukherjee, Chaos to order: Effect Of random predation in a holling type iv tri-trophic food chain system with closure terms, Int. J. Biomath., 9 (2016), 1650073, 18pp. doi: 10.1142/S179352451650073X. [11] T. F. Hansen, N. C. Stenseth, H. Henttonen and J. Tast, Interspecific and intraspecific competition as causes of direct and delayed density dependence in a fluctuating vole population, Proc. Natl. Acad. Sci. USA, 96 (1999), 986-991.  doi: 10.1073/pnas.96.3.986. [12] D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [13] C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine-sawfly, Canad. Entomologist., 91 (1959), 293-320.  doi: 10.4039/Ent91293-5. [14] Intraspecific Competition: Example and Definition, Chapter 18 / Lesson 30, Available from: https://study.com/academy/lesson/intraspecific-competition-example-definition-quiz.html. [15] C. Ji, D. Jiang and N. Shi, Analysis of a predator-prey model with modified Leslie-Gower and Holling Type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482-498.  doi: 10.1016/j.jmaa.2009.05.039. [16] P. A. Khaiter and M. G. Erechtchoukova, The notion of stability in mathematics, biology, ecology and environmental sustainability, 18th World IMACS/MODSIM Congress, Cairns, Australia, 2009, 13–17. [17] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. [18] Lesser-known examples of intraspecific competition that exist, Available from: https://www.buzzle.com/articles/examples-of-intraspecific-competition.html. [19] J. P. Lasalle and R. J. Rath, Eventual stability, Proc. of Sec. of the I.F.A.C, (1963), 556-560. [20] M. Liu and M. Fan, Permanence of stochastic Lotka-Volterra systems, Journal of Nonlinear Science, 27 (2017), 425-452.  doi: 10.1007/s00332-016-9337-2. [21] M. Liu and K. Wang, Persistence, extinction and global asymptotical stability of a non-autonomous predator-prey model with random perturbation, Appl. Math. Model., 36 (2012), 5344-5353.  doi: 10.1016/j.apm.2011.12.057. [22] M. Liu and K. Wang, Global stability of a nonlinear stochastic predator-prey system with Beddington-DeAngelis functional response, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 1114-1121.  doi: 10.1016/j.cnsns.2010.06.015. [23] M. Liu, K. Wang and Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol., 73 (2011), 1969-2012.  doi: 10.1007/s11538-010-9569-5. [24] S. Li and X. Zhang, Dynamics of a stochastic non-autonomous predator-prey system with Beddington-DeAngelis functional response, Adv. Diff. Eqns., 2014 (2014), 19pp. doi: 10.1186/1687-1847-2014-314. [25] X. Y. Li, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete Contin. Dyn. Syst., 24 (2009), 523-545.  doi: 10.3934/dcds.2009.24.523. [26] A. Lotka, Elements of physical biology, Williams and Wilkins, Baltimore, 1925. [27] A. M. Lyapunov, The General Problem of the Stability of Motion, Kharkov Mathematical Society, Kharkov, 1892. [28] X. Y. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402. [29] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial college press, 2006. doi: 10.1142/p473. [30] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, NJ, 2001. [31] P. S. Mondal and M. Banerjee, Stochastic persistence and stationary distribution in a Holling-Tanner type prey-predator model, Physica A: Stat. Mech. and Appl., 391 (2012), 1216-1233.  doi: 10.1016/j.physa.2011.10.019. [32] W. W. Murdoch, S. Avery and E. B. Michael, Switching in predatory fish, Ecology, 56 (1975), 1094-1105.  doi: 10.2307/1936149. [33] S. J. Schreiber, Persistence for stochastic difference equations: A mini-review, J. Diff. Eqns and Appl., 18 (2012), 1381-1403.  doi: 10.1080/10236198.2011.628662. [34] Shut up: how noise pollution is affecting 10 animals, Available from: http://webecoist.momtastic.com/2010/02/28/please-shut-up-10-animals-affected-by-noise-pollution/ (accessed 8 October 2017). [35] T. Takahashi and S. Matsuura, Laboratory studies on molting and growth of the shore crab, Hemigrapsus sanguineus de Haan, Parasitized by a Rhizocephalan Barnacle, Biol. Bull., 186 (1994), 300-308.  doi: 10.2307/1542276. [36] V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, J. Cons. Int. Explor., 3 (1928), 3-51.  doi: 10.1093/icesjms/3.1.3. [37] L. L. Wang and W. T. Li, Existence and global stability of positive periodic solutions of a predator-prey system with delays, Appl. Math. and Compu., 146 (2003), 167-185.  doi: 10.1016/S0096-3003(02)00534-9. [38] What are examples of intraspecific competition?, Available from: https://www.quora.com/What-are-examples-of-intraspecific-competition. [39] R. Wu, X. Zou and K. Wang, Asymptotic behavior of a stochastic non-autonomous predator-prey model with impulsive perturbations, Common. Nonlinear. Sci. Number. Simulat., 20 (2015), 965-974.  doi: 10.1016/j.cnsns.2014.06.023. [40] Y. Zhang, S. Chen and S. Gao, Analysis of a non-autonomous stochastic predator-prey model with Crowley-Martin functional response, Adv. Diff. Eqns, 2016 (2016), Paper No. 264, 28 pp. doi: 10.1186/s13662-016-0993-1.
Numerical simulation for the deterministic system (1) with initial condition (0.2, 0.3) by $a_1(t) = 0.1+0.01 \sin t, \ a_2(t) = 0.02+0.01\sin t$ shows the stable behavior of prey and predator
Numerical simulation for the deterministic system (1) with initial condition (0.2, 0.3) by $a_1(t) = 2+0.1 \sin t,\ a_2(t) = 1+0.1\sin t$ shows the unstable behavior of prey and predator
Numerical simulation for the deterministic system (1) with (0.2, 0.3) by $r(t) = 5 + 2.5 \sin t,~b(t) = 0.22+0.02\sin t,~c(t) = 0.01+0.005\sin t,~d(t)$ $= 0.2+0.01\sin t,~a_1(t) = 0.1+0.1\sin t,~a_2(t) = 1+0.1\sin t$ shows that system is persistent
Numerical simulation for the system (5) with $\frac{\sigma_1^2}{2} = \frac{\sigma_2^2}{2} = 0.21+0.02\sin t$ shows that both prey and predator population goes to extinction
Numerical simulation for the system (5) with $\frac{\sigma_1^2}{2} = 0.19+0.02\sin t,~\frac{\sigma_2^2}{2} = 0.09+0.02\sin t$ shows weakly persistence in the mean of prey and extinction of predator
Numerical simulation for the system (5) with $r(t) = 2.2+0.01 \sin t , \ \sigma_1 = \sigma_2 = 0.02+0.01\sin t$ shows permanence of both prey and predator
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