Figure 1.
Sample paths of $|x(t)|$ of (4) (left) and (5) (right)
-
Previous Article
A preconditioned fast Hermite finite element method for space-fractional diffusion equations
- DCDS-B Home
- This Issue
-
Next Article
Dynamic behavior of a stochastic predator-prey system under regime switching
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 |
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.
Keywords:
Stochastic Lotka-Volterra model,
stochastic Lyapunov analysis,
asymptotic properties,
stochastic permanence,
ergodicity.
Mathematics Subject Classification:
Primary:60H10;Secondary:92B05.
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. Du, R. Kon, K. 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. Hu, F. 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'in, R. 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. Lawley, J. 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. Li, D. 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. Liu, X. 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. Takeuchi, N. H. Du, N. 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. Du, R. Kon, K. 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. Hu, F. 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'in, R. 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. Lawley, J. 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. Li, D. 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. Liu, X. 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. Takeuchi, N. H. Du, N. 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 2.
A sample path of $|x(t)|$ of the switching system
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 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
| [1] |
Michel Benaïm, Antoine Bourquin, Dang H. Nguyen. Stochastic persistence in degenerate stochastic Lotka-Volterra food chains. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022023 |
| [2] |
Hongfu Yang, Xiaoyue Li, George Yin. Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3743-3766. doi: 10.3934/dcdsb.2016119 |
| [3] |
Fuke Wu, Yangzi Hu. Stochastic Lotka-Volterra system with unbounded distributed delay. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 275-288. doi: 10.3934/dcdsb.2010.14.275 |
| [4] |
Dejun Fan, Xiaoyu Yi, Ling Xia, Jingliang Lv. Dynamical behaviors of stochastic type K monotone Lotka-Volterra systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2901-2922. doi: 10.3934/dcdsb.2018291 |
| [5] |
Xiaoling Zou, Ke Wang. Optimal harvesting for a stochastic N-dimensional competitive Lotka-Volterra model with jumps. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 683-701. doi: 10.3934/dcdsb.2015.20.683 |
| [6] |
Suqing Lin, Zhengyi Lu. Permanence for two-species Lotka-Volterra systems with delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 137-144. doi: 10.3934/mbe.2006.3.137 |
| [7] |
Guichen Lu, Zhengyi Lu. Permanence for two-species Lotka-Volterra cooperative systems with delays. Mathematical Biosciences & Engineering, 2008, 5 (3) : 477-484. doi: 10.3934/mbe.2008.5.477 |
| [8] |
Xingwang Yu, Sanling Yuan. Asymptotic properties of a stochastic chemostat model with two distributed delays and nonlinear perturbation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2373-2390. doi: 10.3934/dcdsb.2020014 |
| [9] |
Xiao He, Sining Zheng. Protection zone in a modified Lotka-Volterra model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2027-2038. doi: 10.3934/dcdsb.2015.20.2027 |
| [10] |
Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201 |
| [11] |
Dan Li, Jing'an Cui, Yan Zhang. Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2069-2088. doi: 10.3934/dcdsb.2015.20.2069 |
| [12] |
Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521 |
| [13] |
Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953 |
| [14] |
Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197 |
| [15] |
Qi Wang, Chunyi Gai, Jingda Yan. Qualitative analysis of a Lotka-Volterra competition system with advection. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1239-1284. doi: 10.3934/dcds.2015.35.1239 |
| [16] |
Lin Niu, Yi Wang, Xizhuang Xie. Carrying simplex in the Lotka-Volterra competition model with seasonal succession with applications. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2161-2172. doi: 10.3934/dcdsb.2021014 |
| [17] |
Yueding Yuan, Yang Wang, Xingfu Zou. Spatial dynamics of a Lotka-Volterra model with a shifting habitat. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5633-5671. doi: 10.3934/dcdsb.2019076 |
| [18] |
Yukio Kan-On. Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 147-162. doi: 10.3934/dcds.2002.8.147 |
| [19] |
Li Ma, Shangjiang Guo. Bifurcation and stability of a two-species diffusive Lotka-Volterra model. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1205-1232. doi: 10.3934/cpaa.2020056 |
| [20] |
Wentao Meng, Yuanxi Yue, Manjun Ma. The minimal wave speed of the Lotka-Volterra competition model with seasonal succession. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021265 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]