# American Institute of Mathematical Sciences

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

 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

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.
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.
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