November  2017, 22(9): 3499-3528. doi: 10.3934/dcdsb.2017177

On stochastic multi-group Lotka-Volterra ecosystems with regime switching

School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China

* Corresponding author: Xiaoyue Li

Received  August 2016 Revised  May 2017 Published  July 2017

Fund Project: The second author is s supported by Natural Science Foundation of Jilin Province (No. 20170101044JC), the Education Department of Jilin Province (No.JJKH20170904KJ) and by the National Natural Science Foundation of China (11171056,11571065,11671072).

Focusing on stochastic dynamics involving continuous states as well as discrete events, this paper investigates dynamical behaviors of stochastic multi-group Lotka-Volterra model with regime switching. The contributions of the paper lie on: (a) giving the sufficient conditions of stochastic permanence for generic stochastic multi-group Lotka-Volterra model, which are much weaker than the existing results in the literature; (b) obtaining the stochastic strong permanence and ergodic property for the mutualistic systems; (c) establishing the almost surely asymptotic estimate of solutions. These can specify some realistic recurring phenomena and reveal the fact that regime switching can suppress the impermanence. A couple of examples and numerical simulations are given to illustrate our results.

Citation: Rui Wang, Xiaoyue Li, Denis S. Mukama. On stochastic multi-group Lotka-Volterra ecosystems with regime switching. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3499-3528. doi: 10.3934/dcdsb.2017177
References:
[1]

J. Bao and J. Shao, Permanence and extinction of regime-switching predator-prey models, SIAM J. Math. Anal., 48 (2016), 725-739.  doi: 10.1137/15M1024512.

[2]

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.

[3]

N. H. DuR. KonK. Sato and Y. Takeuchi, Dynamical behavior of Lotka-Volterra competition systems: Non-autonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math., 170 (2004), 399-422.  doi: 10.1016/j.cam.2004.02.001.

[4]

A. Friedman, Stochastic Differential Equations and Applications, Dover Publications, Inc. , Mineola, NY, 2006.

[5]

X. He and K. Gopalsamy, Persistence, attractivity, and delay in facultative mutualism, J. Math. Anal. Appl., 215 (1997), 154-173.  doi: 10.1006/jmaa.1997.5632.

[6]

Y. HuF. Wu and C. Huang, Stochastic Lotka-Volterra models with multiple delays, J. Math. Anal. Appl., 375 (2011), 42-57.  doi: 10.1016/j.jmaa.2010.08.017.

[7]

G. E. Hutchinson, The Paradox of the plankton, Amer. Nat., 95 (1961), 137-145.  doi: 10.1086/282171.

[8]

A. M. Il'inR. Z. Khasminskii and G. Yin, Asymptotic expansions of solutions of integro-differential equations for transition densities of singularly perturbed switching diffusions: Rapid switching, J. Math. Anal. Appl., 238 (1999), 516-539.  doi: 10.1006/jmaa.1998.6532.

[9]

R. Khasminskii, Stochastic Stability of Differential Equations, 2nd edition, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0.

[10]

S. D. LawleyJ. C. Mattingly and M. C. Reed, Sensitivity to switching rates in stochastically switched ODEs, Commun. Math. Sci., 12 (2014), 1343-1352.  doi: 10.4310/CMS.2014.v12.n7.a9.

[11]

X. LiD. Jiang and X. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448.  doi: 10.1016/j.cam.2009.06.021.

[12]

X. Li and G. Yin, Logistic models with regime switching: Permanence and ergodicity, J. Math. Anal. Appl., 441 (2016), 593-611.  doi: 10.1016/j.jmaa.2016.04.016.

[13]

R. Liptser, A strong law of large numbers for local martingale, Stochastics, 3 (1980), 217-228.  doi: 10.1080/17442508008833146.

[14]

H. LiuX. Li and Q. Yang, The ergodic property and positive recurrence of a multi-group Lotka-Volterra mutualistic system with regime switching, Systems Control Lett., 62 (2013), 805-810.  doi: 10.1016/j.sysconle.2013.06.002.

[15]

M. Liu, Global asymptotic stability of stochastic Lotka-Volterra systems with infinite delays, IMA J. Appl. Math., 80 (2015), 1431-1453.  doi: 10.1093/imamat/hxv002.

[16]

A. J. Lotka, Elements of Physical Biology, William and Wilkins, Baltimore, 1925.

[17]

Q. Luo and X. Mao, tochastic population dynamics under regime switching Ⅱ, J. Math. Anal. Appl., 355 (2009), 577-593.  doi: 10.1016/j.jmaa.2009.02.010.

[18]

X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Horwwood Publishing, Chichester, 2008. doi: 10.1533/9780857099402.

[19]

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

[20]

X. Mao, Stationary distribution of stochastic population systems, Systems Control Lett., 60 (2011), 398-405.  doi: 10.1016/j.sysconle.2011.02.013.

[21]

D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Equations, 267 (2017), 1192-1225.  doi: 10.1016/j.jde.2016.10.005.

[22]

M. Slatkin, The dynamics of a population in a Markovian environment, Ecology, 59 (1978), 249-256.  doi: 10.2307/1936370.

[23]

Y. TakeuchiN. H. DuN. T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957.  doi: 10.1016/j.jmaa.2005.11.009.

[24]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. Acad. Lincei., 2 (1926), 31-113. 

[25]

F. Wu and Y. Xu, Stochastic Lotka-Volterra population dynamics with infinite delay, SIAM J. Appl. Math., 70 (2009), 641-657.  doi: 10.1137/080719194.

[26]

G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications: A Two-Time-Scale Approach, 2nd edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4346-9.

[27]

G. Yin and C. Zhu, Hybrid Switching Diffusions Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.

[28]

C. Zhu and G. Yin, On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Anal., 71 (2009), e1370-e1379.  doi: 10.1016/j.na.2009.01.166.

[29]

C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments, J. Math. Anal. Appl., 357 (2009), 154-170.  doi: 10.1016/j.jmaa.2009.03.066.

show all references

References:
[1]

J. Bao and J. Shao, Permanence and extinction of regime-switching predator-prey models, SIAM J. Math. Anal., 48 (2016), 725-739.  doi: 10.1137/15M1024512.

[2]

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.

[3]

N. H. DuR. KonK. Sato and Y. Takeuchi, Dynamical behavior of Lotka-Volterra competition systems: Non-autonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math., 170 (2004), 399-422.  doi: 10.1016/j.cam.2004.02.001.

[4]

A. Friedman, Stochastic Differential Equations and Applications, Dover Publications, Inc. , Mineola, NY, 2006.

[5]

X. He and K. Gopalsamy, Persistence, attractivity, and delay in facultative mutualism, J. Math. Anal. Appl., 215 (1997), 154-173.  doi: 10.1006/jmaa.1997.5632.

[6]

Y. HuF. Wu and C. Huang, Stochastic Lotka-Volterra models with multiple delays, J. Math. Anal. Appl., 375 (2011), 42-57.  doi: 10.1016/j.jmaa.2010.08.017.

[7]

G. E. Hutchinson, The Paradox of the plankton, Amer. Nat., 95 (1961), 137-145.  doi: 10.1086/282171.

[8]

A. M. Il'inR. Z. Khasminskii and G. Yin, Asymptotic expansions of solutions of integro-differential equations for transition densities of singularly perturbed switching diffusions: Rapid switching, J. Math. Anal. Appl., 238 (1999), 516-539.  doi: 10.1006/jmaa.1998.6532.

[9]

R. Khasminskii, Stochastic Stability of Differential Equations, 2nd edition, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0.

[10]

S. D. LawleyJ. C. Mattingly and M. C. Reed, Sensitivity to switching rates in stochastically switched ODEs, Commun. Math. Sci., 12 (2014), 1343-1352.  doi: 10.4310/CMS.2014.v12.n7.a9.

[11]

X. LiD. Jiang and X. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448.  doi: 10.1016/j.cam.2009.06.021.

[12]

X. Li and G. Yin, Logistic models with regime switching: Permanence and ergodicity, J. Math. Anal. Appl., 441 (2016), 593-611.  doi: 10.1016/j.jmaa.2016.04.016.

[13]

R. Liptser, A strong law of large numbers for local martingale, Stochastics, 3 (1980), 217-228.  doi: 10.1080/17442508008833146.

[14]

H. LiuX. Li and Q. Yang, The ergodic property and positive recurrence of a multi-group Lotka-Volterra mutualistic system with regime switching, Systems Control Lett., 62 (2013), 805-810.  doi: 10.1016/j.sysconle.2013.06.002.

[15]

M. Liu, Global asymptotic stability of stochastic Lotka-Volterra systems with infinite delays, IMA J. Appl. Math., 80 (2015), 1431-1453.  doi: 10.1093/imamat/hxv002.

[16]

A. J. Lotka, Elements of Physical Biology, William and Wilkins, Baltimore, 1925.

[17]

Q. Luo and X. Mao, tochastic population dynamics under regime switching Ⅱ, J. Math. Anal. Appl., 355 (2009), 577-593.  doi: 10.1016/j.jmaa.2009.02.010.

[18]

X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Horwwood Publishing, Chichester, 2008. doi: 10.1533/9780857099402.

[19]

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

[20]

X. Mao, Stationary distribution of stochastic population systems, Systems Control Lett., 60 (2011), 398-405.  doi: 10.1016/j.sysconle.2011.02.013.

[21]

D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Equations, 267 (2017), 1192-1225.  doi: 10.1016/j.jde.2016.10.005.

[22]

M. Slatkin, The dynamics of a population in a Markovian environment, Ecology, 59 (1978), 249-256.  doi: 10.2307/1936370.

[23]

Y. TakeuchiN. H. DuN. T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957.  doi: 10.1016/j.jmaa.2005.11.009.

[24]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. Acad. Lincei., 2 (1926), 31-113. 

[25]

F. Wu and Y. Xu, Stochastic Lotka-Volterra population dynamics with infinite delay, SIAM J. Appl. Math., 70 (2009), 641-657.  doi: 10.1137/080719194.

[26]

G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications: A Two-Time-Scale Approach, 2nd edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4346-9.

[27]

G. Yin and C. Zhu, Hybrid Switching Diffusions Properties and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4419-1105-6.

[28]

C. Zhu and G. Yin, On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Anal., 71 (2009), e1370-e1379.  doi: 10.1016/j.na.2009.01.166.

[29]

C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments, J. Math. Anal. Appl., 357 (2009), 154-170.  doi: 10.1016/j.jmaa.2009.03.066.

Figure 1.  Sample paths of $|x(t)|$ of (4) (left) and (5) (right)
Figure 2.  A sample path of $|x(t)|$ of the switching system
Figure 3.  Sample paths of $|x(t)|$ of state-$1$ (left) and state-$2$ (right) in Example 7.1
Figure 4.  Case 1. A sample path of $|x(t)|$ of the switching system in Example 7.1
Figure 5.  Case 2. A sample path of $|x(t)|$ of the switching system in Example 7.1
Figure 6.  A sample path of $ x_1(t) $ and $ x_2(t) $ of state-$1$, state-$2$ and state-$3$ in Example 7.2
Figure 7.  Stationary distribution and scatter plot of a sample path of state-$1$ in Example 7.2
Figure 8.  Stationary distribution and scatter plot of a sample path of state-$3$ in Example 7.2
Figure 9.  Case 1. A sample path of $ x_1(t) $ and $x_2(t)$ of the switching system in Example 7.2
Figure 10.  Case 1. Stationary distribution and scatter plot of a sample path of the switching system in Example 7.2
Figure 11.  Case 1. A sample path in time average of the switching system in Example 7.2
Figure 12.  Case2. A sample path of $x_1(t)$ and $x_2(t)$ of the switching system in Example 7.2
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