Extinction and ergodic stationary distribution of a Markovian-switching prey-predator model with additional food for predator

Xiaoxia Guo^1, 2 ; Dehan Ruan³

¹ School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, PR China.
² Department of Mathematics and Statistics, Concordia University, Montreal-H3G2W1, Canada.
³ Guangzhou International Institute of Finance and Guangzhou University, Guangzhou, Guangdong 510405, PR China.

Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 46.

Résumé

In this work we have studied a stochastic predator-prey model where the prey grows logistically in the absence of predator. All parameters but carrying capacity have been perturbed with telephone noise. The prey’s growth rate and the predator’s death rate have also been perturbed with white noises. Both of these noises have been proved extremely useful to model rapidly fluctuating phenomena Dimentberg (1988). The conditions under which extinction of predator and prey populations occur have been established. We also give sufficient conditions for positive recurrence and the existence of an ergodic stationary distribution of the positive solution, red which in stochastic predator-prey systems means that the predator and prey populations can be persistent, that is to say, the predator and prey populations can be sustain a quantity that is neither too much nor too little. In our analysis, it is found that the environmental noise plays an important role in extinction as well as coexistence of prey and predator populations. It is shown in numerical simulation that larger white noise intensity will lead to the extinction of the population, while telephone noise may delay or reduce the risk of species extinction.

Détail

DOI : 10.1051/mmnp/2019055

Bibliographie

[1] P. Aguirre, E. González-Olivares, S. Torres Stochastic predator-prey model with Allee effect on prey Nonlinear Anal. Real World Appl 2013 768 779

[2] W. Allee, Animal aggregations: A study in general sociology, University of Chicago Press, Chicago (1931).

[3] J. Bao, J. Shao Permanence and extinction of regime-switching predator-prey models J. Math. Anal 2015 725 739

[4] Y. Cai, X. Mao A stochastic prey-predator model with time-dependent delays Appl. Math. Model 2018 357 371

[5] J. Chattopadhyay, O. Arino A predator-prey model with disease in the prey Nonlinear Anal 1999 747 766

[6] T. Chowdhury, S. Chakraborty, J. Chattopadhyay Migratory effect of middle predator in a tri-trophic food chain model Math. Methods Appl. Sci 2010 1699 1711

[7] N. Dang, N. Du, T. Ton Asymptotic behavior of predator-prey systems perturbed by white noise Acta Appl. Math 2011 351 370

[8] A. Das, G. Samanta Stochastic prey-predator model with additional food for predator Physica A 2018 121 141

[9] P. DeBach, Biological Control by Natural Enemies, Cambridge University Press, UK, (1974).

[10] S. Ghorai, S. Poria Impacts of additional food on diffusion induced instabilities in a predator-prey system with mutually interfering predator Chaos Solitons Fractals 2017 68 78

[11] E. González-Olivares, R. Ramos-Jiliberto Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability Ecol. Model 2003 135 146

[12] Q. Han, D. Jiang, C. Ji Analysis of a delayed stochastic predator-prey model in a polluted environment Appl. Math. Model 2014 3067 3080

[13] R.Z. Hasminskii, Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Maryland (1980).

[14] D. Higham An algorithmic introduction to numerical simulation of stochastic differential equations SIAM. Rev 2001 525 546

[15] C. Holling Some characteristics of simple types of predation and parasitism Mem. Entomol. Soc. Canada 1965 1 60

[16] D. Kumar, S. Chakrabarty A predator-prey model with additional food supply to predators: dynamics and applications J. Comp. Appl. Math 2018 763 784

[17] A. Lahrouz, A. Settati, P.S. Mandal Dynamics of a switching diffusion modified Leslie Gower Predator-prey system with Beddington-DeAngelis functional response Nonlinear Dynam 2016 853 870

[18] M. Liu, X. He, J. Yu Dynamics of a stochastic regime-switching predator-prey model with harvesting and distributed delays Nonlinear Anal. Hybrid 2018 87 104

[19] R. Liu, G. Liu Analysis of a stochastic predator-prey population model with Allee effect and jumps J. Inequal. Appl 2019 1 16

[20] A. Lotka Analytical note on certain rhythmic relations in organic systems Proc. Natl. Acad. Sci 1920 410 415

[21] X. Mao, Stochastic Differential Equations and Applications. Horwood Publishing, Chichester (1997).

[22] R. May Stability in randomly fluctuating versus deterministic environments Am. Nat 1973 621 650

[23] J. Margaritopoulos, J. Tsitsipis, D. Perdikis Biological characteristics of the mirids Macrolophus costalis and Macroplophus pygmaeus preying on the tobacco form of Myzus persicae (Hemiptera: Aphididae) B. Entomol. Res 2003 39

[24] B. Sahoo, S. Poria Effects of additional food in a delayed predator-prey model Math. Biosc 2015 62 73

[25] P. Srinivasu, B. Prasad Role of quantity of additional food to predators as a control in predator-prey systems with relevance to pest management and biological conservation Bull. Math. Biol 2011 2249 2276

[26] P. Srinivasu, D. Vamsi, V. Ananth Additional food supplements as a tool for biological conservation of predator-prey systems involving type III functional response: A qualitative and quantitative investigation J. Theor. Biol 2018 303 318

[27] V. Volterra Variazioni e fluttuazioni del numero d’individui in specie d’animali conviventi Mem. Acad. Lincei 1989 31 113

[28] G. Yin, C. Zhu Asymptotic properties of hybrid diffusion systems SIAM J. Control Optim 2007 1155 1179

[29] G. Yin, C. Zhu, Hybrid Switching Diffusions Properties and Applications. Springer-Verlag, New York (2009).

[30] L. Zu, D. Jiang, D. O’Regand, T. Hayat Ergodic property of a Lotka-Volterra predator-prey model with white noise higher order perturbation under regime switching Appl. Math. Comput 2018 93 102