Dynamics of a stochastic population model with Allee effect and jumps

Rong Liu¹ ; Guirong Liu²

¹ School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, Shanxi 030006, PR China.
² School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, PR China.

Mathematical modelling of natural phenomena, Tome 17 (2022), article no. 1

Résumé

This paper is concerned with a stochastic population model with Allee effect and jumps. First, we show the global existence of almost surely positive solution to the model. Next, exponential extinction and persistence in mean are discussed. Then, we investigated the global attractivity and stability in distribution. At last, some numerical results are given. The results show that if attack rate a is in the intermediate range or very large, the population will go extinct. Under the premise that attack rate a is less than growth rate r, if the noise intensity or jump is relatively large, the population will become extinct; on the contrary, the population will be persistent in mean. The results in this paper generalize and improve the previous related results.

Détail

DOI : 10.1051/mmnp/2022002

Bibliographie

[1] W. Allee, Animal Aggregations. A study in General Sociology. University of Chicago Press, Chicago (1931).

[2] J. Bao, X. Mao, G. Yin, C. Yuan Competitive Lotka-Volterra population dynamics with jumps Nonlinear Anal. 2011 6601 6616

[3] I. Barbalat Systems d’equations differentisl d’oscillations nonlinearies Rev. Roum. Math. Pures Appl. 1959 267 270

[4] S. Biswas, Md. Saifuddin, S. Sasmal, S. Samanta, N. Pal, F. Ababneh, J. Chattopadhyay A delayed prey-predator system with prey subject to the strong Allee effect and disease Nonlinear Dyn. 2016 1569 1594

[5] D. Boukal, L. Berec Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters J. Theor. Biol. 2002 375 394

[6] J. Cushing Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations J. Biol. Dynam. 2014 57 73

[7] J. Cushing The evolutionary dynamics of a population model with a strong Allee effect Math. Biosci. Eng. 2015 643 660

[8] M. Jovanović, M. Krstić Extinction in stochastic predator-prey population model with Allee effect on prey Discret. Contin. Dyn. Syst. Ser. B 2017 2651 2667

[9] M. Jovanović, M. Krstić The influence of time-dependent delay on behavior of stochastic population model with the Allee effect Appl. Math. Model. 2015 733 746

[10] Y. Kang, O. Udiani Dynamics of a single species evolutionary model with Allee effects J. Math. Anal. Appl. 2014 492 515

[11] M. Krstić, M. Jovanović On stochastic population model with the Allee effect Math. Comput. Model. 2010 370 379

[12] X. Li, A. Alison, D. Jiang, X. Mao Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regimes witching J. Math. Anal. Appl. 2011 11 28

[13] X. Li, D. Jiang, X. Mao Population dynamical behavior of Lotka-Volterra system under regime switching J. Comput. Appl. Math. 2009 427 448

[14] D. Li, Y. Yang Impact of time delay on population model with Allee effect Commun. Nonlinear Sci. Numer. Simulat. 2019 282 293

[15] R. Lipster A strong law of large numbers for local martingales Stochastics 1980 217 228

[16] Q. Liu, Q. Chen Asymptotic behavior of a stochastic non-autonomous predator-prey system with jumps Appl. Math. Comput. 2015 418 428

[17] Q. Liu, D. Jiang, T. Hayat, A. Alsaedi Dynamical behavior of a hybrid switching SIS epidemic model with vaccination and Lévy jumps Stoch. Anal. Appl. 2019 388 411

[18] M. Liu, K. Wang Dynamics of a Leslie-Gower Holling-type II predator-prey system with Lévy jumps Nonlinear Anal. 2013 204 213

[19] X. Mao, Stochsatic Differential Equations and Applications. Horwood Publishing Limited, Chichester (2007).

[20] S. Peng, X. Zhu Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations Stoch. Proc. Appl. 2006 370 380

[21] J. Ripa, P. Lunddberg Noise colour and the risk of population extinctions P. Roy. Soc. B Biol. Sci. 1996 1751 1753

[22] S. Sun, Y. Sun, G. Zhang, X. Liu Dynamical behavior of a stochastic two-species Monod competition chemostat model Appl. Math. Comput. 2017 153 170

[23] L. Tan, W. Jin, Y. Suo Stability in distribution of neutral stochastic functional differential equations Stat. Probabil. Lett. 2015 27 36

[24] R. Wu, K. Wang Population dynamical behaviors of stochastic logistic system with jumps Turk. J. Math. 2014 935 948

[25] Q. Yang, D. Jiang A note on asymptotic behaviors of stochastic population model with Allee effect Appl. Math. Model. 2011 4611 4619

[26] X. Yu, S. Yuan, T. Zhang Persistence and ergodicity of a stochastic single species model with Allee effect under regime switching Commun. Nonlinear Sci. Numer. Simulat. 2018 359 374

[27] Y. Zhao, S. Yuan Stability in distribution of a stochastic hybrid competitive Lotka-Volterra model with Lévy jumps Chaos Soliton. Fract. 2016 98 109

[28] Y. Zhao, S. Yuan, Q. Zhang The effect of Lévy noise onthe survival of a stochastic competitive model in an impulsive polluted environment Appl. Math. Model. 2016 7583 7600

[29] Q. Zhang, D. Jiang, Y. Zhao, D. O’Regan Asymptotic behavior of a stochastic population model with Allee effect by Lévy jumps Nonlinear Anal. Hybri. 2017 1 12

[30] B. Zhang, H. Wang, G. Lv Exponential extinction of a stochastic predator-prey model with Allee effect Physica A 2018 192 204

[31] S. Zhang, T. Zhang, S. Yuan Dynamics of a stochastic predator-prey model with habitat complexity and prey aggregation Ecol. Complexity 2021 100889

[32] Q. Zhu Asymptotic stability in the pth moment for stochastic differential equations with Lévy noise J. Math. Anal. Appl. 2014 126 142

[33] X. Zou, K. Wang Numerical simulations and modeling for stochastic biological systems with jumps Commun. Nonlin. Sci. Numer. Simul. 2014 1557 1568

[34] J. Zu, M. Mimura The impact of Allee effect on a predator-prey system with Holling type II functional response Appl. Math. Comput. 2010 3542 3556