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 & 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.  Google Scholar

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

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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.  Google Scholar

[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.  Google Scholar

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

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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.  Google Scholar

[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.  Google Scholar

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

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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.  Google Scholar

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

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

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

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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.  Google Scholar

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

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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