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  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 & Continuous Dynamical Systems - B, 2018, 23 (8) : 3275-3296. doi: 10.3934/dcdsb.2018244
References:
[1]

M. V. BaalenV. KřivanP. 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. Google Scholar

[2]

T. CaraballoR. 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. Google Scholar

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T. CaraballoR. 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. Google Scholar

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T. CaraballoR. Colucci and X. Han, Predation with indirect effects in fluctuating environments, Nonlinear Dynamics, 84 (2016), 115-126. doi: 10.1007/s11071-015-2238-3. Google Scholar

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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. Google Scholar

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Available from: https://www.conservationnw.org/what-we-do/predators-and-prey/carnivores-predators-and-their-prey (accessed 10 October 2017).Google Scholar

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N. H. DuN. 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. Google Scholar

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Functional and numerical response, Available from: https://www.ma.utexas.edu/users/davis/375/popecol/lec10/funcresp.html.Google Scholar

[9]

P. GhoshP. 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. Google Scholar

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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. Google Scholar

[11]

T. F. HansenN. C. StensethH. 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. Google Scholar

[12]

D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546. doi: 10.1137/S0036144500378302. Google Scholar

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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. Google Scholar

[14]

Intraspecific Competition: Example and Definition, Chapter 18 / Lesson 30, Available from: https://study.com/academy/lesson/intraspecific-competition-example-definition-quiz.html.Google Scholar

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C. JiD. 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. Google Scholar

[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.Google Scholar

[17]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. Google Scholar

[18]

Lesser-known examples of intraspecific competition that exist, Available from: https://www.buzzle.com/articles/examples-of-intraspecific-competition.html.Google Scholar

[19]

J. P. Lasalle and R. J. Rath, Eventual stability, Proc. of Sec. of the I.F.A.C, (1963), 556-560. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

M. LiuK. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

A. Lotka, Elements of physical biology, Williams and Wilkins, Baltimore, 1925.Google Scholar

[27]

A. M. Lyapunov, The General Problem of the Stability of Motion, Kharkov Mathematical Society, Kharkov, 1892. Google Scholar

[28]

X. Y. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402. Google Scholar

[29]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial college press, 2006. doi: 10.1142/p473. Google Scholar

[30]

R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, NJ, 2001.Google Scholar

[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. Google Scholar

[32]

W. W. MurdochS. Avery and E. B. Michael, Switching in predatory fish, Ecology, 56 (1975), 1094-1105. doi: 10.2307/1936149. Google Scholar

[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. Google Scholar

[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).Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

What are examples of intraspecific competition?, Available from: https://www.quora.com/What-are-examples-of-intraspecific-competition.Google Scholar

[39]

R. WuX. 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. Google Scholar

[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. Google Scholar

show all references

References:
[1]

M. V. BaalenV. KřivanP. 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. Google Scholar

[2]

T. CaraballoR. 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. Google Scholar

[3]

T. CaraballoR. 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. Google Scholar

[4]

T. CaraballoR. Colucci and X. Han, Predation with indirect effects in fluctuating environments, Nonlinear Dynamics, 84 (2016), 115-126. doi: 10.1007/s11071-015-2238-3. Google Scholar

[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. Google Scholar

[6]

Available from: https://www.conservationnw.org/what-we-do/predators-and-prey/carnivores-predators-and-their-prey (accessed 10 October 2017).Google Scholar

[7]

N. H. DuN. 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. Google Scholar

[8]

Functional and numerical response, Available from: https://www.ma.utexas.edu/users/davis/375/popecol/lec10/funcresp.html.Google Scholar

[9]

P. GhoshP. 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. Google Scholar

[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. Google Scholar

[11]

T. F. HansenN. C. StensethH. 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. Google Scholar

[12]

D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546. doi: 10.1137/S0036144500378302. Google Scholar

[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. Google Scholar

[14]

Intraspecific Competition: Example and Definition, Chapter 18 / Lesson 30, Available from: https://study.com/academy/lesson/intraspecific-competition-example-definition-quiz.html.Google Scholar

[15]

C. JiD. 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. Google Scholar

[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.Google Scholar

[17]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. Google Scholar

[18]

Lesser-known examples of intraspecific competition that exist, Available from: https://www.buzzle.com/articles/examples-of-intraspecific-competition.html.Google Scholar

[19]

J. P. Lasalle and R. J. Rath, Eventual stability, Proc. of Sec. of the I.F.A.C, (1963), 556-560. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

M. LiuK. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

A. Lotka, Elements of physical biology, Williams and Wilkins, Baltimore, 1925.Google Scholar

[27]

A. M. Lyapunov, The General Problem of the Stability of Motion, Kharkov Mathematical Society, Kharkov, 1892. Google Scholar

[28]

X. Y. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402. Google Scholar

[29]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial college press, 2006. doi: 10.1142/p473. Google Scholar

[30]

R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, NJ, 2001.Google Scholar

[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. Google Scholar

[32]

W. W. MurdochS. Avery and E. B. Michael, Switching in predatory fish, Ecology, 56 (1975), 1094-1105. doi: 10.2307/1936149. Google Scholar

[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. Google Scholar

[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).Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

What are examples of intraspecific competition?, Available from: https://www.quora.com/What-are-examples-of-intraspecific-competition.Google Scholar

[39]

R. WuX. 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. Google Scholar

[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. Google Scholar

Figure 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
Figure 2.  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
Figure 3.  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
Figure 4.  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
Figure 5.  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
Figure 6.  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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