American Institute of Mathematical Sciences

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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

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, Harcourt/Academic Press, 2000.  Google Scholar

[2]

A. Bahar and X. Mao, Stochastic delay population dynamics, Int. J. Pure Appl. Math., 11 (2004), 377-400.  Google Scholar

[3]

J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616. doi: 10.1016/j.na.2011.06.043.  Google Scholar

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A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

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S. Cheng, Stochastic population systems, Stoch. Anal. Appl., 27 (2009), 854-874. doi: 10.1080/07362990902844348.  Google Scholar

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H. I. Freedman and S. Ruan, Uniform persistence in functional differential equations, J. Differential Equations, 115 (1995), 173-192. doi: 10.1006/jdeq.1995.1011.  Google Scholar

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T. C. Gard, Persistence in stochastic food web models, Bull. Math. Biol., 46 (1984), 357-370. doi: 10.1007/BF02462011.  Google Scholar

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T. C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal., 10 (1986), 1411-1419. doi: 10.1016/0362-546X(86)90111-2.  Google Scholar

[11]

M. Gilpin and I. Hanski, Metapopulation Dynamics: Empirical and Theoretical Investigations, Academic Press, New York, 1991. Google Scholar

[12]

B. S. Goh, Stability in models of mutualism, Amer. Natur., 113 (1979), 261-275. doi: 10.1086/283384.  Google Scholar

[13]

K. Gopalsamy, Global asymptotic stability in Volterra's population systems, J. Math. Biol., 19 (1984), 157-168. doi: 10.1007/BF00277744.  Google Scholar

[14]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.  Google Scholar

[15]

T. G. Hallam and Z. Ma, Persistence in population models with demographic fluctuations, J. Math. Biol., 24 (1986), 327-339. doi: 10.1007/BF00275641.  Google Scholar

[16]

F. B. Hanson and H. C. Tuckwell, Persistence times of populations with large random fluctuations, Theoret. Population Biol., 14 (1978), 46-61. doi: 10.1016/0040-5809(78)90003-5.  Google Scholar

[17]

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F. B. Hanson and H. C. Tuckwell, Population growth with randomly distributed jumps, J. Math. Biol., 36 (1997), 169-187. doi: 10.1007/s002850050096.  Google Scholar

[19]

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

[20]

C. Ji and D. Jiang, Persistence and non-persistence of a mutualism system with stochastic perturbation, Discrete Contin. Dyn. Syst., 32 (2012), 867-889. doi: 10.3934/dcds.2012.32.867.  Google Scholar

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Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.  Google Scholar

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Y. Kuang and H. L. Smith, Global stability for infinite delay Lotka-Valterra type systems, J. Differential Equations, 103 (1993), 221-246. doi: 10.1006/jdeq.1993.1048.  Google Scholar

[24]

R. Lande, Risks of population extinction from demographic and environmental stochasticity and random catastrophes, Amer. Natur., 142 (1993), 911-927. Google Scholar

[25]

R. Lande, Genetics and demography in biological conservation, Science, 241 (1988), 1455-1460. Google Scholar

[26]

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

[27]

M. Liu and K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations, Discrete Contin. Dyn. Syst., 33 (2013), 2495-2522. doi: 10.3934/dcds.2013.33.2495.  Google Scholar

[28]

M. Liu and K. Wang, Analysis of a stochastic autonomous mutualism model, J. Math. Anal. Appl., 402 (2013), 392-403. doi: 10.1016/j.jmaa.2012.11.043.  Google Scholar

[29]

M. Liu and K. Wang, Dynamics of a Leslie-Gower Holling-type II predator-prey system with Lévy jumps, Nonlinear Anal., 85 (2013), 204-213. doi: 10.1016/j.na.2013.02.018.  Google Scholar

[30]

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

[31]

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

[32]

R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 1974. Google Scholar

[33]

R. J. Plemmons, M-matrix characterizations. I. Nonsingular M-matrices, Linear Algebra and Appl., 18 (1977), 175-188. doi: 10.1016/0024-3795(77)90073-8.  Google Scholar

[34]

G. Poole and T. Boullion, A Survey on M-Matrices, SIAM Rev., 16 (1974), 419-427. doi: 10.1137/1016079.  Google Scholar

[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), 655-664. doi: 10.1016/j.spl.2012.11.003.  Google Scholar

show all references

References:
[1]

R. B. Ash and C. A. Doléans-Dade, Probability and Measure Theory, Second edition, Harcourt/Academic Press, 2000.  Google Scholar

[2]

A. Bahar and X. Mao, Stochastic delay population dynamics, Int. J. Pure Appl. Math., 11 (2004), 377-400.  Google Scholar

[3]

J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616. doi: 10.1016/j.na.2011.06.043.  Google Scholar

[4]

J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl., 391 (2012), 363-375. doi: 10.1016/j.jmaa.2012.02.043.  Google Scholar

[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), 279-291. doi: 10.1006/jmaa.1997.5403.  Google Scholar

[6]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

[7]

S. Cheng, Stochastic population systems, Stoch. Anal. Appl., 27 (2009), 854-874. doi: 10.1080/07362990902844348.  Google Scholar

[8]

H. I. Freedman and S. Ruan, Uniform persistence in functional differential equations, J. Differential Equations, 115 (1995), 173-192. doi: 10.1006/jdeq.1995.1011.  Google Scholar

[9]

T. C. Gard, Persistence in stochastic food web models, Bull. Math. Biol., 46 (1984), 357-370. doi: 10.1007/BF02462011.  Google Scholar

[10]

T. C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal., 10 (1986), 1411-1419. doi: 10.1016/0362-546X(86)90111-2.  Google Scholar

[11]

M. Gilpin and I. Hanski, Metapopulation Dynamics: Empirical and Theoretical Investigations, Academic Press, New York, 1991. Google Scholar

[12]

B. S. Goh, Stability in models of mutualism, Amer. Natur., 113 (1979), 261-275. doi: 10.1086/283384.  Google Scholar

[13]

K. Gopalsamy, Global asymptotic stability in Volterra's population systems, J. Math. Biol., 19 (1984), 157-168. doi: 10.1007/BF00277744.  Google Scholar

[14]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.  Google Scholar

[15]

T. G. Hallam and Z. Ma, Persistence in population models with demographic fluctuations, J. Math. Biol., 24 (1986), 327-339. doi: 10.1007/BF00275641.  Google Scholar

[16]

F. B. Hanson and H. C. Tuckwell, Persistence times of populations with large random fluctuations, Theoret. Population Biol., 14 (1978), 46-61. doi: 10.1016/0040-5809(78)90003-5.  Google Scholar

[17]

F. B. Hanson and H. C. Tuckwell, Logistic growth with random density independent disasters, Theoret. Population Biol., 19 (1981), 1-18. doi: 10.1016/0040-5809(81)90032-0.  Google Scholar

[18]

F. B. Hanson and H. C. Tuckwell, Population growth with randomly distributed jumps, J. Math. Biol., 36 (1997), 169-187. doi: 10.1007/s002850050096.  Google Scholar

[19]

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

[20]

C. Ji and D. Jiang, Persistence and non-persistence of a mutualism system with stochastic perturbation, Discrete Contin. Dyn. Syst., 32 (2012), 867-889. doi: 10.3934/dcds.2012.32.867.  Google Scholar

[21]

V. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic, Dordrecht, 1992. doi: 10.1007/978-94-015-8084-7.  Google Scholar

[22]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.  Google Scholar

[23]

Y. Kuang and H. L. Smith, Global stability for infinite delay Lotka-Valterra type systems, J. Differential Equations, 103 (1993), 221-246. doi: 10.1006/jdeq.1993.1048.  Google Scholar

[24]

R. Lande, Risks of population extinction from demographic and environmental stochasticity and random catastrophes, Amer. Natur., 142 (1993), 911-927. Google Scholar

[25]

R. Lande, Genetics and demography in biological conservation, Science, 241 (1988), 1455-1460. Google Scholar

[26]

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

[27]

M. Liu and K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations, Discrete Contin. Dyn. Syst., 33 (2013), 2495-2522. doi: 10.3934/dcds.2013.33.2495.  Google Scholar

[28]

M. Liu and K. Wang, Analysis of a stochastic autonomous mutualism model, J. Math. Anal. Appl., 402 (2013), 392-403. doi: 10.1016/j.jmaa.2012.11.043.  Google Scholar

[29]

M. Liu and K. Wang, Dynamics of a Leslie-Gower Holling-type II predator-prey system with Lévy jumps, Nonlinear Anal., 85 (2013), 204-213. doi: 10.1016/j.na.2013.02.018.  Google Scholar

[30]

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

[31]

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

[32]

R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 1974. Google Scholar

[33]

R. J. Plemmons, M-matrix characterizations. I. Nonsingular M-matrices, Linear Algebra and Appl., 18 (1977), 175-188. doi: 10.1016/0024-3795(77)90073-8.  Google Scholar

[34]

G. Poole and T. Boullion, A Survey on M-Matrices, SIAM Rev., 16 (1974), 419-427. doi: 10.1137/1016079.  Google Scholar

[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), 655-664. doi: 10.1016/j.spl.2012.11.003.  Google Scholar

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