# American Institute of Mathematical Sciences

November  2017, 22(9): 3483-3498. doi: 10.3934/dcdsb.2017176

## Dynamic behavior of a stochastic predator-prey system under regime switching

 1 Faculty of Mathematics, Mechanics, and Informatics, University of Science-VNU, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam 2 Department of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam 3 Faculty of Basic Sciences, Ho Chi Minh University of Transport, 2 D3, Ho Chi Minh, Vietnam

* Corresponding author: Nguyen Thanh Dieu

Received  July 2016 Revised  May 2017 Published  July 2017

Fund Project: This work was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) n0 101.03-2017.23.

In this paper we deal with regime switching predator-prey models perturbed by white noise. We give a threshold by which we know whenever a switching predator-prey system is eventually extinct or permanent. We also give some numerical solutions to illustrate that under the regime switching, the permanence or extinction of the switching system may be very different from the dynamics in each fixed state.

Citation: Nguyen Huu Du, Nguyen Thanh Dieu, Tran Dinh Tuong. Dynamic behavior of a stochastic predator-prey system under regime switching. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3483-3498. doi: 10.3934/dcdsb.2017176
##### References:
 [1] M. Benaïm and C. Lobry, Lotka-Volterra with randomly fluctuating environments or "how switching between beneficial environments can make survival harder'', Ann. Appl. Probab., 26 (2016), 3754-3785.  doi: 10.1214/16-AAP1192. [2] B. Cloez and M. Hairer, Exponential ergodicity for Markov processes with random switching, Bernoulli, 21 (2015), 505-536.  doi: 10.3150/13-BEJ577. [3] N. H. Dang, A note on sufficient conditions for asymptotic stability in distribution of stochastic differential equations with Markovian switching, Nonlinear Analysis, 95 (2014), 625-631.  doi: 10.1016/j.na.2013.09.030. [4] N. H. Dang, N. H. Du and T. V. Ton, Asymptotic behavior of predator-prey systems perturbed by white noise, Acta Appl. Math., 115 (2011), 351-370.  doi: 10.1007/s10440-011-9628-4. [5] N. H. Dang, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise, J. Differential Equations, 257 (2014), 2078-2101.  doi: 10.1016/j.jde.2014.05.029. [6] A. d'Onofrio, Bounded Noises in Physics, Biology, and Engineering, Springer Science+Business Media New York, 2013. doi: 10.1007/978-1-4614-7385-5. [7] N. H. Du and N. H. Dang, Asymptotic behavior of Kolmogorov systems with predator-prey type in random environment, Commun. Pure Appl. Anal., 13 (2014), 2693-2712.  doi: 10.3934/cpaa.2014.13.2693. [8] N. H. Du and N. H. Dang, Dynamics of Kolmogorov systems of competitive type under the telegraph noise, J. Differential Equations, 250 (2011), 386-409.  doi: 10.1016/j.jde.2010.08.023. [9] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [10] N. Ikeda and S. Wantanabe, Stochastic Differential Equations and Diffusion Processes, Second edition, North-Holland Publishing Co. , Amsterdam; Kodansha, Ltd. , Tokyo, 1989. [11] R. A. Khas'minskii, Stochastic Stability of Differential Equations, Springer-Verlag Berlin Heidelberg, 2012. [12] X. Li and X. Mao, 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. [13] M. Liu and K. Wang, The threshold between permanence and extinction for a stochastic logistic model with regime switching, J. Appl. Math. Comput., 43 (2013), 329-349.  doi: 10.1007/s12190-013-0666-0. [14] M. Liu and K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations, Discrete Contin. Dyn. Syst., 33 (2013), 2495-2522.  doi: 10.3934/dcds.2013.33.2495. [15] Q. Luo and X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69-84.  doi: 10.1016/j.jmaa.2006.12.032. [16] F. Malrieu and T. H. Phu, Lotka-Volterra with randomly fluctuating environments: A full description, , Preprint, arXiv: 1607.04395. [17] F. Malrieu and P. A. Zitt, On the persistence regime for Lotka-Volterra in randomly fluctuating environments, Preprint, arXiv: 1601.08151. [18] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Chichester, 1997. doi: 10.1533/9780857099402. [19] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. doi: 10.1142/p473. [20] X. Mao, S. Sabanis and E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156.  doi: 10.1016/S0022-247X(03)00539-0. [21] G. Maruyama and H. Tanaka, Ergodic property of N-dimentional recurrent Markov processes, Mem. Fac. Sci. Kyushu. Ser. A, 13 (1959), 157-172.  doi: 10.2206/kyushumfs.13.157. [22] S. P. Meyn and R. L. Tweedie, Stability of Markovian processes Ⅲ: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Prob., 25 (1993), 518-548.  doi: 10.2307/1427522. [23] D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Equations, 262 (2017), 1192-1225.  doi: 10.1016/j.jde.2016.10.005. [24] D. H. Nguyen, G. Yin and C. Zhu, Certain properties related to well posedness of switching diffusions Stochastic Process. Appl. , (2017). doi: 10.1016/j.spa.2017.02.004. [25] M. Pinsky and R. Pinsky, Transience recurrence and central limit theorem behavior for diffusions in random temporal environments, Ann. Probab., 21 (1993), 433-452.  doi: 10.1214/aop/1176989410. [26] R. Rudnicki, Long-time behaviour of a stochastic prey-predator model, Stochastic Process. Appl., 108 (2003), 93-107.  doi: 10.1016/S0304-4149(03)00090-5. [27] A. V. Skorokhod, Asymptotic Methods of the Theory of Stochastic Differential Equations, A merican Mathematical Society, Providence, 1989.

show all references

##### References:
 [1] M. Benaïm and C. Lobry, Lotka-Volterra with randomly fluctuating environments or "how switching between beneficial environments can make survival harder'', Ann. Appl. Probab., 26 (2016), 3754-3785.  doi: 10.1214/16-AAP1192. [2] B. Cloez and M. Hairer, Exponential ergodicity for Markov processes with random switching, Bernoulli, 21 (2015), 505-536.  doi: 10.3150/13-BEJ577. [3] N. H. Dang, A note on sufficient conditions for asymptotic stability in distribution of stochastic differential equations with Markovian switching, Nonlinear Analysis, 95 (2014), 625-631.  doi: 10.1016/j.na.2013.09.030. [4] N. H. Dang, N. H. Du and T. V. Ton, Asymptotic behavior of predator-prey systems perturbed by white noise, Acta Appl. Math., 115 (2011), 351-370.  doi: 10.1007/s10440-011-9628-4. [5] N. H. Dang, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise, J. Differential Equations, 257 (2014), 2078-2101.  doi: 10.1016/j.jde.2014.05.029. [6] A. d'Onofrio, Bounded Noises in Physics, Biology, and Engineering, Springer Science+Business Media New York, 2013. doi: 10.1007/978-1-4614-7385-5. [7] N. H. Du and N. H. Dang, Asymptotic behavior of Kolmogorov systems with predator-prey type in random environment, Commun. Pure Appl. Anal., 13 (2014), 2693-2712.  doi: 10.3934/cpaa.2014.13.2693. [8] N. H. Du and N. H. Dang, Dynamics of Kolmogorov systems of competitive type under the telegraph noise, J. Differential Equations, 250 (2011), 386-409.  doi: 10.1016/j.jde.2010.08.023. [9] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [10] N. Ikeda and S. Wantanabe, Stochastic Differential Equations and Diffusion Processes, Second edition, North-Holland Publishing Co. , Amsterdam; Kodansha, Ltd. , Tokyo, 1989. [11] R. A. Khas'minskii, Stochastic Stability of Differential Equations, Springer-Verlag Berlin Heidelberg, 2012. [12] X. Li and X. Mao, 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. [13] M. Liu and K. Wang, The threshold between permanence and extinction for a stochastic logistic model with regime switching, J. Appl. Math. Comput., 43 (2013), 329-349.  doi: 10.1007/s12190-013-0666-0. [14] M. Liu and K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations, Discrete Contin. Dyn. Syst., 33 (2013), 2495-2522.  doi: 10.3934/dcds.2013.33.2495. [15] Q. Luo and X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69-84.  doi: 10.1016/j.jmaa.2006.12.032. [16] F. Malrieu and T. H. Phu, Lotka-Volterra with randomly fluctuating environments: A full description, , Preprint, arXiv: 1607.04395. [17] F. Malrieu and P. A. Zitt, On the persistence regime for Lotka-Volterra in randomly fluctuating environments, Preprint, arXiv: 1601.08151. [18] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Chichester, 1997. doi: 10.1533/9780857099402. [19] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. doi: 10.1142/p473. [20] X. Mao, S. Sabanis and E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156.  doi: 10.1016/S0022-247X(03)00539-0. [21] G. Maruyama and H. Tanaka, Ergodic property of N-dimentional recurrent Markov processes, Mem. Fac. Sci. Kyushu. Ser. A, 13 (1959), 157-172.  doi: 10.2206/kyushumfs.13.157. [22] S. P. Meyn and R. L. Tweedie, Stability of Markovian processes Ⅲ: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Prob., 25 (1993), 518-548.  doi: 10.2307/1427522. [23] D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Equations, 262 (2017), 1192-1225.  doi: 10.1016/j.jde.2016.10.005. [24] D. H. Nguyen, G. Yin and C. Zhu, Certain properties related to well posedness of switching diffusions Stochastic Process. Appl. , (2017). doi: 10.1016/j.spa.2017.02.004. [25] M. Pinsky and R. Pinsky, Transience recurrence and central limit theorem behavior for diffusions in random temporal environments, Ann. Probab., 21 (1993), 433-452.  doi: 10.1214/aop/1176989410. [26] R. Rudnicki, Long-time behaviour of a stochastic prey-predator model, Stochastic Process. Appl., 108 (2003), 93-107.  doi: 10.1016/S0304-4149(03)00090-5. [27] A. V. Skorokhod, Asymptotic Methods of the Theory of Stochastic Differential Equations, A merican Mathematical Society, Providence, 1989.
Trajectories of $Y(t)$ in the state 1 (blue line) and in the state 2 (red line) in Ex. 1
A switching trajectory $Y(t)$ in Ex. 1.
Trajectories of $Y(t)$ in the first state (blue line) and the second state (red line) respectively in Ex. 2
A switching trajectory $Y(t)$ in Ex. 2
Phase picture and empirical density of $\big(X(t), Y(t)\big)$ in Ex. $3.2$ in 2D and 3D settings respectively
Values of the coefficients in Ex. 3.1
 $a_1$ $a_2$ $b_1$ $b_2$ $c_1$ $c_2$ $\sigma$ $\rho$ 1 0.9 2.5 2 2.8 0.6 5 0.6 4 2 0.2 0.1 1 4 3 0.5 1.5 4
 $a_1$ $a_2$ $b_1$ $b_2$ $c_1$ $c_2$ $\sigma$ $\rho$ 1 0.9 2.5 2 2.8 0.6 5 0.6 4 2 0.2 0.1 1 4 3 0.5 1.5 4
Values of the coefficients in Ex. 3.2
 $a_1$ $a_2$ $b_1$ $b_2$ $c_1$ $c_2$ $\sigma$ $\rho$ 1 0.2 0.45 1 9.5 5 1 2 4 2 1 0.85 0.5 3.6 4.2 2 1.5 4
 $a_1$ $a_2$ $b_1$ $b_2$ $c_1$ $c_2$ $\sigma$ $\rho$ 1 0.2 0.45 1 9.5 5 1 2 4 2 1 0.85 0.5 3.6 4.2 2 1.5 4
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