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 & 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.  Google Scholar

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B. Cloez and M. Hairer, Exponential ergodicity for Markov processes with random switching, Bernoulli, 21 (2015), 505-536.  doi: 10.3150/13-BEJ577.  Google Scholar

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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.  Google Scholar

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N. H. DangN. 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.  Google Scholar

[5]

N. H. DangN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

N. Ikeda and S. Wantanabe, Stochastic Differential Equations and Diffusion Processes, Second edition, North-Holland Publishing Co. , Amsterdam; Kodansha, Ltd. , Tokyo, 1989.  Google Scholar

[11]

R. A. Khas'minskii, Stochastic Stability of Differential Equations, Springer-Verlag Berlin Heidelberg, 2012.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

F. Malrieu and T. H. Phu, Lotka-Volterra with randomly fluctuating environments: A full description, , Preprint, arXiv: 1607.04395. Google Scholar

[17]

F. Malrieu and P. A. Zitt, On the persistence regime for Lotka-Volterra in randomly fluctuating environments, Preprint, arXiv: 1601.08151. Google Scholar

[18]

X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Chichester, 1997. doi: 10.1533/9780857099402.  Google Scholar

[19]

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

[20]

X. MaoS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. V. Skorokhod, Asymptotic Methods of the Theory of Stochastic Differential Equations, A merican Mathematical Society, Providence, 1989.  Google Scholar

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.  Google Scholar

[2]

B. Cloez and M. Hairer, Exponential ergodicity for Markov processes with random switching, Bernoulli, 21 (2015), 505-536.  doi: 10.3150/13-BEJ577.  Google Scholar

[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.  Google Scholar

[4]

N. H. DangN. 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.  Google Scholar

[5]

N. H. DangN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

N. Ikeda and S. Wantanabe, Stochastic Differential Equations and Diffusion Processes, Second edition, North-Holland Publishing Co. , Amsterdam; Kodansha, Ltd. , Tokyo, 1989.  Google Scholar

[11]

R. A. Khas'minskii, Stochastic Stability of Differential Equations, Springer-Verlag Berlin Heidelberg, 2012.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

F. Malrieu and T. H. Phu, Lotka-Volterra with randomly fluctuating environments: A full description, , Preprint, arXiv: 1607.04395. Google Scholar

[17]

F. Malrieu and P. A. Zitt, On the persistence regime for Lotka-Volterra in randomly fluctuating environments, Preprint, arXiv: 1601.08151. Google Scholar

[18]

X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Chichester, 1997. doi: 10.1533/9780857099402.  Google Scholar

[19]

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

[20]

X. MaoS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. V. Skorokhod, Asymptotic Methods of the Theory of Stochastic Differential Equations, A merican Mathematical Society, Providence, 1989.  Google Scholar

Figure 1.  Trajectories of $Y(t)$ in the state 1 (blue line) and in the state 2 (red line) in Ex. 1
Figure 2.  A switching trajectory $Y(t)$ in Ex. 1.
Figure 3.  Trajectories of $Y(t)$ in the first state (blue line) and the second state (red line) respectively in Ex. 2
Figure 4.  A switching trajectory $Y(t)$ in Ex. 2
Figure 5.  Phase picture and empirical density of $\big(X(t), Y(t)\big)$ in Ex. $3.2$ in 2D and 3D settings respectively
Table 1.  Values of the coefficients in Ex. 3.1
$a_1$ $a_2$ $b_1$ $b_2$ $c_1$ $c_2$ $\sigma$ $\rho$
10.92.522.80.650.64
20.20.11430.51.54
$a_1$ $a_2$ $b_1$ $b_2$ $c_1$ $c_2$ $\sigma$ $\rho$
10.92.522.80.650.64
20.20.11430.51.54
Table 2.  Values of the coefficients in Ex. 3.2
$a_1$ $a_2$ $b_1$ $b_2$ $c_1$ $c_2$ $\sigma$ $\rho$
10.20.4519.55124
210.850.53.64.221.54
$a_1$ $a_2$ $b_1$ $b_2$ $c_1$ $c_2$ $\sigma$ $\rho$
10.20.4519.55124
210.850.53.64.221.54
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