Persistence and extinction of a modified Leslie-Gower Holling-type Ⅱ predator-prey stochastic model in polluted environments with impulsive toxicant input

Yongxin Gao; Shuyuan Yao; Yongxin Gao; Shuyuan Yao

doi:10.3934/mbe.2021249

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 4: 4894-4918. doi: 10.3934/mbe.2021249

Previous Article Next Article

Theory article

Persistence and extinction of a modified Leslie-Gower Holling-type Ⅱ predator-prey stochastic model in polluted environments with impulsive toxicant input

Yongxin Gao ,
Shuyuan Yao ^,

College of Science, Civil Aviation University of China, 2898 Jinbei Road, Tianjin 300300, China

Academic Editor: Jia Li

Received: 25 March 2021 Accepted: 26 May 2021 Published: 03 June 2021

In this paper, a modified Leslie-Gower Holling-type Ⅱ two-predator one-prey stochastic model in polluted environments with impulsive toxicant input is proposed where we use an Ornstein-Uhlenbeck process to improve the stochasticity of the environment. The sharp sufficient conditions for persistence in the mean and extinction are established. The results reveal that the persistence and extinction of the species have close relationships with the toxicant and environmental stochasticity. In addition, the theoretical results are verified by numerical simulation.
- stochastic models,
- Itô's formula,
- persistence in the mean,
- extinction,
- intensity of white noise,
- toxicant effect
Citation: Yongxin Gao, Shuyuan Yao. Persistence and extinction of a modified Leslie-Gower Holling-type Ⅱ predator-prey stochastic model in polluted environments with impulsive toxicant input[J]. Mathematical Biosciences and Engineering, 2021, 18(4): 4894-4918. doi: 10.3934/mbe.2021249

Related Papers:

Abstract

In this paper, a modified Leslie-Gower Holling-type Ⅱ two-predator one-prey stochastic model in polluted environments with impulsive toxicant input is proposed where we use an Ornstein-Uhlenbeck process to improve the stochasticity of the environment. The sharp sufficient conditions for persistence in the mean and extinction are established. The results reveal that the persistence and extinction of the species have close relationships with the toxicant and environmental stochasticity. In addition, the theoretical results are verified by numerical simulation.

References

[1]	J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340. doi: 10.2307/3866
[2]	G. T. Skalski, J. F. Gilliam, Functional responses with predator interference: viable alternatives to the Holling-type Ⅱ model, Ecology, 82 (2001), 3083-3092. doi: 10.1890/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2
[3]	M. A. Aziz-Alaoui, M. D. Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Appl. Math. Lett., 16 (2003), 1069-1075. doi: 10.1016/S0893-9659(03)90096-6
[4]	A. F. Nindjin, M. A. Aziz-Alaoui, M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with time delay, Nonlinear Anal.: Real World Appl., 7 (2006), 1104-1118. doi: 10.1016/j.nonrwa.2005.10.003
[5]	X. Y. Song, Y. F. Li, Dynamic behaviors of the periodic predator-prey model with modified Leslie-Gower Holling-type Ⅱ schemes and impulsive effect, Nonlinear Anal.: Real World Appl., 9 (2008), 64-79. doi: 10.1016/j.nonrwa.2006.09.004
[6]	C. Ji, D. Jiang, 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
[7]	C. Ji, D. Jiang, N. Shi, A note on a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 377 (2011), 435-440. doi: 10.1016/j.jmaa.2010.11.008
[8]	M. Liu, K. Wang, Dynamics of a Leslie-Gower Holling-type Ⅱ predator-prey system with L$\rm\acute{e}$vy jumps, Nonlinear Anal., 85 (2013), 204-213. doi: 10.1016/j.na.2013.02.018
[9]	Y. Xu, M. Liu, Y. Yang, Analysis of a stochastic two-predators one-prey system with modified Leslie-Gower and Holling-type Ⅱ schemes, J. Appl. Anal. Comput., 7 (2017), 713-727.
[10]	T. G. Hallam, C. E. Clark, R. R. Lassiter, Effects of toxicants on populations: a qualitative approach Ⅰ. Equilibrium environmental exposure, Ecol. Model., 8 (1983), 291-304.
[11]	T. G. Hallam, C. E. Clark, G. S. Jordan, Effects of toxicant on population: a qualitative approach Ⅱ. First order kinetics, J. Math. Biol., 18 (1983), 25-37. doi: 10.1007/BF00275908
[12]	T. G. Hallam, J. Deluna, Effects of toxicant on populations: a qualitative approach Ⅲ. Environmental and food chain pathways, J. Theor. Biol., 109 (1984), 411-429. doi: 10.1016/S0022-5193(84)80090-9
[13]	B. Buonomo, A. D. Liddo, I. Sgura, A diffusive-convective model for the dynamics of population-toxicant intentions: some analytical and numerical results, Math. Biosci., 157 (1999), 37-46. doi: 10.1016/S0025-5564(98)10076-7
[14]	H. I. Freedman, J. B. Shukla, Models for the effect of toxicant in single-species and predator-prey systems, J. Math. Biol., 30 (1991), 15-30. doi: 10.1007/BF00168004
[15]	T. G. Hallam, Z. Ma, Persistence in population models with demographic fluctuations, J. Math. Biol., 24 (1986), 327-339. doi: 10.1007/BF00275641
[16]	H. P. Liu, Z. Ma, The threshold of survival for system of two species in a polluted environment, J. Math. Biol., 30 (1991), 49-51. doi: 10.1007/BF00168006
[17]	Z. Ma, T. G. Hallam, Effects of parameter fluctuations on community survival, Math. Biosci., 86 (1987), 35-49. doi: 10.1016/0025-5564(87)90062-9
[18]	J. Pan, Z. Jin, Z. Ma, Thresholds of survival for an n-dimensional volterra mutualistic system in a polluted environment, J. Math. Anal. Appl., 252 (2000), 519-531. doi: 10.1006/jmaa.2000.6853
[19]	E. L. Johnston, M. J. Keough, Field assessment of effects of timing and frequency of copper pulses on settlement of sessile marine invertebrates, Mar. Biol., 137 (2000), 1017-1029. doi: 10.1007/s002270000420
[20]	E. L. Johnston, M. J. Keough, P. Y. Qian, Maintenance of species dominance through pulse disturbances to a sessile marine invertebrate assemblage in port shelter, Mar. Ecol. Prog. Ser., 226 (2002), 103-114. doi: 10.3354/meps226103
[21]	J. Liang, S. Tang, J. J. Nieto, R. A. Cheke, Analytical methods for detecting pesticide switches with evolution of pesticide resistance, Math. Biosci., 245 (2013), 249-257. doi: 10.1016/j.mbs.2013.07.008
[22]	B. Liu, L. Chen, Y. Zhang, The effects of impulsive toxicant input on a population in a polluted environment, J. Biol. Syst., 11 (2003), 265-274. doi: 10.1142/S0218339003000907
[23]	B. Liu, L. Zhang, Dynamics of a two-species lotka-volterra competition system in a polluted environment with pulse toxicant input, Appl. Math. Comput., 214 (2009), 155-162.
[24]	X. Yang, Z. Jin, Y. Xue, Weak average persistence and extinction of a predator-prey system in a polluted environment with impulsive toxicant input, Chaos Solitons Fractals, 31 (2007), 726-735. doi: 10.1016/j.chaos.2005.10.042
[25]	R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, New Jersey, 2001.
[26]	C. Braumann, Variable effort harvesting models in random environments: generalization to density-dependent noise intensities, Math. Biosci., 177 (2002), 229-245.
[27]	B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, 4$^nd$ edition, Springer, Berlin, 1998.
[28]	M. Liu, C. Du, M. Deng, Persistence and extinction of a modified Leslie-Gower Holling-type Ⅱ stochastic predator-prey model with impulsive toxicant input in polluted environments, Nonlinear Anal. Hybrid Syst., 27 (2018), 177-190. doi: 10.1016/j.nahs.2017.08.001
[29]	Y. Zhao, S. L. Yuan, J. L. Ma, Survival and stationary distribution analysis of a stochastic competitive model of three species in a polluted environment, Bull. Math. Biol., 77 (2015), 1285-1326. doi: 10.1007/s11538-015-0086-4
[30]	Y. L. Cai, J. J. Jiao, Z. J. Gui, Y. T. Liu, Environmental variability in a stochastic epidemic model, Appl. Math. Comput., 329 (2018), 210-226.
[31]	D. Zhou, M. Liu, Z. Liu, Persistence and extinction of a stochastic predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Adv. Differ. Equations, 1 (2020), 1-15.
[32]	B. Liu, L. Chen, Y. Zhang, The effects of impulsive toxicant input on a population in a polluted environment, J. Biol. Syst., 11 (2003), 265-274. doi: 10.1142/S0218339003000907
[33]	B. Oksendal, Stochastic differential equations and diffusion processes, North Holland Press, Amsterdam, 1981.
[34]	D. Q. Jiang, N. Z. Shi, A note on non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005), 164-172. doi: 10.1016/j.jmaa.2004.08.027
[35]	M. Liu, K. Wang, 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
[36]	Higham, J. Desmond, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302
[37]	M. Liu, Dynamics of a stochastic regime-switching predator-prey model with modified Leslie-Gower Holling-type Ⅱ schemes and prey harvesting, Nonlinear Dyn., 96 (2019), 417-442. doi: 10.1007/s11071-019-04797-x
[38]	J. Bao, X. Mao, G. Yin, C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616. doi: 10.1016/j.na.2011.06.043
[39]	C. Bai, Multiplicity of solutions for a class of nonlocal elliptic operators systems, Bull. Korean. Math. Soc., 54 (2017), 715-729. doi: 10.4134/BKMS.b150489

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)