Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion

Previous Article
Dynamics of stochastic fractional Boussinesq equations
DCDS-B Home
This Issue
Next Article
The spreading fronts in a mutualistic model with advection

September 2015, 20(7): 2069-2088. doi: 10.3934/dcdsb.2015.20.2069

Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion

Dan Li ^1, , Jing'an Cui ^2, and Yan Zhang ^2,

1.	Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China
2.	School of Science, Beijing University of Civil Engineering and Architecture, Beijing, 100044, China, China

Received May 2014 Revised February 2015 Published July 2015

Abstract
Related Papers

Using stochastic differential equations with Lévy jumps, this paper studies the effect of environmental stochasticity and random catastrophes on the permanence of Lotka-Volterra facultative systems. Under certain simple assumptions, we establish the sufficient conditions for weak permanence in the mean and extinction of the non-autonomous system, respectively. In particular, a necessary and sufficient condition for permanence and extinction of autonomous system with jump-diffusion are obtained. We generalize some former results under weaker assumptions. Finally, we discuss the biological implications of the main results.

Keywords: Lévy jump, random catastrophe., permanence, environmental stochasticity, Stochastic differential equation, facultative system.

Mathematics Subject Classification: 60H10, 60J75, 92D25, 62P1.

Citation: Dan Li, Jing'an Cui, Yan Zhang. Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2069-2088. doi: 10.3934/dcdsb.2015.20.2069

References:

[1]	R. B. Ash and C. A. Doléans-Dade, Probability and Measure Theory,, Second edition, (2000).
[2]	A. Bahar and X. Mao, Stochastic delay population dynamics,, Int. J. Pure Appl. Math., 11 (2004), 377.
[3]	J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps,, Nonlinear Anal., 74 (2011), 6601. doi: 10.1016/j.na.2011.06.043.
[4]	J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise,, J. Math. Anal. Appl., 391 (2012), 363. doi: 10.1016/j.jmaa.2012.02.043.
[5]	H. Bereketoglu and I. Győri, Global asymptotic stability in a nonautonomous Lotka-Volterra type system with infinite delay,, J. Math. Anal. Appl., 210 (1997), 279. doi: 10.1006/jmaa.1997.5403.
[6]	A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, SIAM, (1994). doi: 10.1137/1.9781611971262.
[7]	S. Cheng, Stochastic population systems,, Stoch. Anal. Appl., 27 (2009), 854. doi: 10.1080/07362990902844348.
[8]	H. I. Freedman and S. Ruan, Uniform persistence in functional differential equations,, J. Differential Equations, 115 (1995), 173. doi: 10.1006/jdeq.1995.1011.
[9]	T. C. Gard, Persistence in stochastic food web models,, Bull. Math. Biol., 46 (1984), 357. doi: 10.1007/BF02462011.
[10]	T. C. Gard, Stability for multispecies population models in random environments,, Nonlinear Anal., 10 (1986), 1411. doi: 10.1016/0362-546X(86)90111-2.
[11]	M. Gilpin and I. Hanski, Metapopulation Dynamics: Empirical and Theoretical Investigations,, Academic Press, (1991).
[12]	B. S. Goh, Stability in models of mutualism,, Amer. Natur., 113 (1979), 261. doi: 10.1086/283384.
[13]	K. Gopalsamy, Global asymptotic stability in Volterra's population systems,, J. Math. Biol., 19 (1984), 157. doi: 10.1007/BF00277744.
[14]	K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics,, Kluwer Academic, (1992). doi: 10.1007/978-94-015-7920-9.
[15]	T. G. Hallam and Z. Ma, Persistence in population models with demographic fluctuations,, J. Math. Biol., 24 (1986), 327. doi: 10.1007/BF00275641.
[16]	F. B. Hanson and H. C. Tuckwell, Persistence times of populations with large random fluctuations,, Theoret. Population Biol., 14 (1978), 46. doi: 10.1016/0040-5809(78)90003-5.
[17]	F. B. Hanson and H. C. Tuckwell, Logistic growth with random density independent disasters,, Theoret. Population Biol., 19 (1981), 1. doi: 10.1016/0040-5809(81)90032-0.
[18]	F. B. Hanson and H. C. Tuckwell, Population growth with randomly distributed jumps,, J. Math. Biol., 36 (1997), 169. doi: 10.1007/s002850050096.
[19]	X. He and K. Gopalsamy, Persistence, attractivity, and delay in facultative mutualism,, J. Math. Anal. Appl., 215 (1997), 154. doi: 10.1006/jmaa.1997.5632.
[20]	C. Ji and D. Jiang, Persistence and non-persistence of a mutualism system with stochastic perturbation,, Discrete Contin. Dyn. Syst., 32 (2012), 867. doi: 10.3934/dcds.2012.32.867.
[21]	V. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations,, Kluwer Academic, (1992). doi: 10.1007/978-94-015-8084-7.
[22]	Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic Press, (1993).
[23]	Y. Kuang and H. L. Smith, Global stability for infinite delay Lotka-Valterra type systems,, J. Differential Equations, 103 (1993), 221. doi: 10.1006/jdeq.1993.1048.
[24]	R. Lande, Risks of population extinction from demographic and environmental stochasticity and random catastrophes,, Amer. Natur., 142 (1993), 911.
[25]	R. Lande, Genetics and demography in biological conservation,, Science, 241 (1988), 1455.
[26]	R. Sh. Lipster, A strong law of large numbers for local martingales,, Stochastics, 3 (1980), 217. doi: 10.1080/17442508008833146.
[27]	M. Liu and K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations,, Discrete Contin. Dyn. Syst., 33 (2013), 2495. doi: 10.3934/dcds.2013.33.2495.
[28]	M. Liu and K. Wang, Analysis of a stochastic autonomous mutualism model,, J. Math. Anal. Appl., 402 (2013), 392. doi: 10.1016/j.jmaa.2012.11.043.
[29]	M. Liu and K. Wang, Dynamics of a Leslie-Gower Holling-type II predator-prey system with Lévy jumps,, Nonlinear Anal., 85 (2013), 204. doi: 10.1016/j.na.2013.02.018.
[30]	X. Mao, Stochastic Differential Equations and Applications,, Second edition, (2008). doi: 10.1533/9780857099402.
[31]	X. Mao, Stationary distribution of stochastic population systems,, Systems Control Lett., 60 (2011), 398. doi: 10.1016/j.sysconle.2011.02.013.
[32]	R. M. May, Stability and Complexity in Model Ecosystems,, Princeton University Press, (1974).
[33]	R. J. Plemmons, M-matrix characterizations. I. Nonsingular M-matrices,, Linear Algebra and Appl., 18 (1977), 175. doi: 10.1016/0024-3795(77)90073-8.
[34]	G. Poole and T. Boullion, A Survey on M-Matrices,, SIAM Rev., 16 (1974), 419. doi: 10.1137/1016079.
[35]	J. Tong, Z. Zhang and J. Bao, The stationary distribution of the facultative population model with a degenerate noise,, Statist. Probab. Lett., 83 (2013), 655. doi: 10.1016/j.spl.2012.11.003.

show all references

References:

[1]	R. B. Ash and C. A. Doléans-Dade, Probability and Measure Theory,, Second edition, (2000).
[2]	A. Bahar and X. Mao, Stochastic delay population dynamics,, Int. J. Pure Appl. Math., 11 (2004), 377.
[3]	J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps,, Nonlinear Anal., 74 (2011), 6601. doi: 10.1016/j.na.2011.06.043.
[4]	J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise,, J. Math. Anal. Appl., 391 (2012), 363. doi: 10.1016/j.jmaa.2012.02.043.
[5]	H. Bereketoglu and I. Győri, Global asymptotic stability in a nonautonomous Lotka-Volterra type system with infinite delay,, J. Math. Anal. Appl., 210 (1997), 279. doi: 10.1006/jmaa.1997.5403.
[6]	A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, SIAM, (1994). doi: 10.1137/1.9781611971262.
[7]	S. Cheng, Stochastic population systems,, Stoch. Anal. Appl., 27 (2009), 854. doi: 10.1080/07362990902844348.
[8]	H. I. Freedman and S. Ruan, Uniform persistence in functional differential equations,, J. Differential Equations, 115 (1995), 173. doi: 10.1006/jdeq.1995.1011.
[9]	T. C. Gard, Persistence in stochastic food web models,, Bull. Math. Biol., 46 (1984), 357. doi: 10.1007/BF02462011.
[10]	T. C. Gard, Stability for multispecies population models in random environments,, Nonlinear Anal., 10 (1986), 1411. doi: 10.1016/0362-546X(86)90111-2.
[11]	M. Gilpin and I. Hanski, Metapopulation Dynamics: Empirical and Theoretical Investigations,, Academic Press, (1991).
[12]	B. S. Goh, Stability in models of mutualism,, Amer. Natur., 113 (1979), 261. doi: 10.1086/283384.
[13]	K. Gopalsamy, Global asymptotic stability in Volterra's population systems,, J. Math. Biol., 19 (1984), 157. doi: 10.1007/BF00277744.
[14]	K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics,, Kluwer Academic, (1992). doi: 10.1007/978-94-015-7920-9.
[15]	T. G. Hallam and Z. Ma, Persistence in population models with demographic fluctuations,, J. Math. Biol., 24 (1986), 327. doi: 10.1007/BF00275641.
[16]	F. B. Hanson and H. C. Tuckwell, Persistence times of populations with large random fluctuations,, Theoret. Population Biol., 14 (1978), 46. doi: 10.1016/0040-5809(78)90003-5.
[17]	F. B. Hanson and H. C. Tuckwell, Logistic growth with random density independent disasters,, Theoret. Population Biol., 19 (1981), 1. doi: 10.1016/0040-5809(81)90032-0.
[18]	F. B. Hanson and H. C. Tuckwell, Population growth with randomly distributed jumps,, J. Math. Biol., 36 (1997), 169. doi: 10.1007/s002850050096.
[19]	X. He and K. Gopalsamy, Persistence, attractivity, and delay in facultative mutualism,, J. Math. Anal. Appl., 215 (1997), 154. doi: 10.1006/jmaa.1997.5632.
[20]	C. Ji and D. Jiang, Persistence and non-persistence of a mutualism system with stochastic perturbation,, Discrete Contin. Dyn. Syst., 32 (2012), 867. doi: 10.3934/dcds.2012.32.867.
[21]	V. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations,, Kluwer Academic, (1992). doi: 10.1007/978-94-015-8084-7.
[22]	Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic Press, (1993).
[23]	Y. Kuang and H. L. Smith, Global stability for infinite delay Lotka-Valterra type systems,, J. Differential Equations, 103 (1993), 221. doi: 10.1006/jdeq.1993.1048.
[24]	R. Lande, Risks of population extinction from demographic and environmental stochasticity and random catastrophes,, Amer. Natur., 142 (1993), 911.
[25]	R. Lande, Genetics and demography in biological conservation,, Science, 241 (1988), 1455.
[26]	R. Sh. Lipster, A strong law of large numbers for local martingales,, Stochastics, 3 (1980), 217. doi: 10.1080/17442508008833146.
[27]	M. Liu and K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations,, Discrete Contin. Dyn. Syst., 33 (2013), 2495. doi: 10.3934/dcds.2013.33.2495.
[28]	M. Liu and K. Wang, Analysis of a stochastic autonomous mutualism model,, J. Math. Anal. Appl., 402 (2013), 392. doi: 10.1016/j.jmaa.2012.11.043.
[29]	M. Liu and K. Wang, Dynamics of a Leslie-Gower Holling-type II predator-prey system with Lévy jumps,, Nonlinear Anal., 85 (2013), 204. doi: 10.1016/j.na.2013.02.018.
[30]	X. Mao, Stochastic Differential Equations and Applications,, Second edition, (2008). doi: 10.1533/9780857099402.
[31]	X. Mao, Stationary distribution of stochastic population systems,, Systems Control Lett., 60 (2011), 398. doi: 10.1016/j.sysconle.2011.02.013.
[32]	R. M. May, Stability and Complexity in Model Ecosystems,, Princeton University Press, (1974).
[33]	R. J. Plemmons, M-matrix characterizations. I. Nonsingular M-matrices,, Linear Algebra and Appl., 18 (1977), 175. doi: 10.1016/0024-3795(77)90073-8.
[34]	G. Poole and T. Boullion, A Survey on M-Matrices,, SIAM Rev., 16 (1974), 419. doi: 10.1137/1016079.
[35]	J. Tong, Z. Zhang and J. Bao, The stationary distribution of the facultative population model with a degenerate noise,, Statist. Probab. Lett., 83 (2013), 655. doi: 10.1016/j.spl.2012.11.003.

[1]	Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221
[2]	Hongjun Gao, Fei Liang. On the stochastic beam equation driven by a Non-Gaussian Lévy process. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1027-1045. doi: 10.3934/dcdsb.2014.19.1027
[3]	Jiahui Zhu, Zdzisław Brzeźniak. Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3269-3299. doi: 10.3934/dcdsb.2016097
[4]	Graeme D. Chalmers, Desmond J. Higham. Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 47-64. doi: 10.3934/dcdsb.2008.9.47
[5]	Badr-eddine Berrhazi, Mohamed El Fatini, Tomás Caraballo, Roger Pettersson. A stochastic SIRI epidemic model with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2415-2431. doi: 10.3934/dcdsb.2018057
[6]	Yong-Kum Cho. On the Boltzmann equation with the symmetric stable Lévy process. Kinetic & Related Models, 2015, 8 (1) : 53-77. doi: 10.3934/krm.2015.8.53
[7]	Ziheng Chen, Siqing Gan, Xiaojie Wang. Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-33. doi: 10.3934/dcdsb.2019154
[8]	Justin Cyr, Phuong Nguyen, Roger Temam. Stochastic one layer shallow water equations with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-54. doi: 10.3934/dcdsb.2018331
[9]	Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1
[10]	Martin Redmann, Melina A. Freitag. Balanced model order reduction for linear random dynamical systems driven by Lévy noise. Journal of Computational Dynamics, 2018, 5 (1&2) : 33-59. doi: 10.3934/jcd.2018002
[11]	Markus Riedle, Jianliang Zhai. Large deviations for stochastic heat equations with memory driven by Lévy-type noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1983-2005. doi: 10.3934/dcds.2018080
[12]	Kumarasamy Sakthivel, Sivaguru S. Sritharan. Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise. Evolution Equations & Control Theory, 2012, 1 (2) : 355-392. doi: 10.3934/eect.2012.1.355
[13]	Xueqin Li, Chao Tang, Tianmin Huang. Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3309-3345. doi: 10.3934/dcdsb.2018282
[14]	Min Niu, Bin Xie. Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 2989-3009. doi: 10.3934/dcdsb.2018296
[15]	Xu Chen, Jianping Wan. Integro-differential equations for foreign currency option prices in exponential Lévy models. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 529-537. doi: 10.3934/dcdsb.2007.8.529
[16]	Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757
[17]	Hwa-Sung Kim, Bara Kim, Jerim Kim. Catastrophe equity put options under stochastic volatility and catastrophe-dependent jumps. Journal of Industrial & Management Optimization, 2014, 10 (1) : 41-55. doi: 10.3934/jimo.2014.10.41
[18]	Lena-Susanne Hartmann, Ilya Pavlyukevich. Advection-diffusion equation on a half-line with boundary Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 637-655. doi: 10.3934/dcdsb.2018200
[19]	Manil T. Mohan, Sivaguru S. Sritharan. $\mathbb{L}^p-$solutions of the stochastic Navier-Stokes equations subject to Lévy noise with $\mathbb{L}^m(\mathbb{R}^m)$ initial data. Evolution Equations & Control Theory, 2017, 6 (3) : 409-425. doi: 10.3934/eect.2017021
[20]	Dan Stanescu, Benito Chen-Charpentier. Random coefficient differential equation models for Monod kinetics. Conference Publications, 2009, 2009 (Special) : 719-728. doi: 10.3934/proc.2009.2009.719

2018 Impact Factor: 1.008

American Institute of Mathematical Sciences