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Survival analysis of stochastic three-species food chain model with white noise and general Lévy jumps

  • College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2018.08.004
  • Received Date: 05 June 2017
  • Accepted Date: 18 July 2017
  • Rev Recd Date: 18 July 2017
  • Publish Date: 31 August 2018
    Abstract

  • Abstract

    The survival of a stochastic three species food chain model with white noise and general Lévy jumps was analyzed, which includes existence and uniqueness of global positive solution as well as permanence in the mean of the highest species.Situations where some species are extinct while others are persistent in the mean were also clarified.The results show that Lévy jumps can obviously change the survival of population, which can make the persistent population become extinct, or vice versa.

    Abstract

    The survival of a stochastic three species food chain model with white noise and general Lévy jumps was analyzed, which includes existence and uniqueness of global positive solution as well as permanence in the mean of the highest species.Situations where some species are extinct while others are persistent in the mean were also clarified.The results show that Lévy jumps can obviously change the survival of population, which can make the persistent population become extinct, or vice versa.
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    • References(20)
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    HSU S B, HWANG T W, KUANG Y. A ratio-dependent food chain model and its applications to biological control[J]. Math Biosci, 2003, 181: 55-83.
    [2]
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    LIU M, BAI C. Analysis of a stochastic tri-trophic food-chain model with harvesting[J]. J Math Biol, 2016, 73: 597-625.
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    LI M, GAO H, WANG B. Analysis of a non-autonomous mutualism model driven by Lévy jumps[J]. Discrete & Continuous Dynamical Systems - B, 2016, 21(4): 1189-1202.
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    LIU M, WANG K. Stochastic Lotka-Volterra systems with Lévy noise[J]. J Math Anal Appl, 2014, 410: 750-763.
    [9]
    LIU M, BAI C. Dynamics of a stochastic one-prey two-predator model with Lévy jumps[J]. Appl Math Comput, 2016, 284: 308-321.
    [10]
    LIU Q, JIANG D, SHI N. Stochastic mutualism model with Lévy jumps[J]. Comm Nonl Sci Num Simul, 2016, 43: 78-90.
    [11]
    ZHOU Y, YUAN S, ZHAO D. Threshold behavior of a stochastic SIS model with Lévy jumps[J]. Appl Math Comput, 2016, 275: 255-267.
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    GE Q, JI G, XU J, et al. Extinction and persistence of a stochastic nonlinear SIS epidemic model with jumps[J]. Physica A: Statistical Mechanics and Its Applications, 2016, 462: 1120-1127.
    [13]
    CHEN C, KANG Y. Dynamics of a stochastic multi-strain SIS epidemic model driven by Lévy noise[J]. Comm Nonl Sci Num Simul, 2017, 42: 379-395.
    [14]
    MAO W, MAO X. On the asymptotic stability and numerical analysis of solutions to nonlinear stochastic differential equations with jumps[J]. J Comput Appl Math, 2016, 301: 1-15.
    [15]
    ZOU X,WANG K. Numerical simulations and modeling for stochastic biological systems with jumps[J]. Comm Nonl Sci Nume Simul, 2014, 19: 1557-1568.
    [16]
    ZHANG X, WANG K. Stability analysis of a stochastic Gilpin-Ayala model driven by Lévy noise[J]. Comm Nonl Sci Nume Simul, 2014, 19(5): 1391-1399.
    [17]
    WU R, ZOU X, WANG K. Asymptotic properties of stochastic hybrid Gilpin-Ayala system with jumps[J]. Appl Math Comput, 2014, 249: 53-66.
    [18]
    JIANG D, SHI N, LI X. Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation[J]. J Math Anal Appl, 2008, 340: 588-597.
    [19]
    LIPTSER R S. A strong law of large numbers for local martingales[J]. Stoch Inter J Prob Stoch Proc, 1980, 3: 217-228.
    [20]
    MAO X, MARION G, RENSHAW E. Environmental Brownian noise suppresses explosions in population dynamics[J]. Stoch Proc Appl, 2002, 97: 95-110.)
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    [1]
    HSU S B, HWANG T W, KUANG Y. A ratio-dependent food chain model and its applications to biological control[J]. Math Biosci, 2003, 181: 55-83.
    [2]
    李海红.随机种群模型的渐进行为[D].长春：吉林大学, 2014.
    [3]
    SUN Y, SAKER S H. Positive periodic solutions of discrete three-level food-chain model of Holling type Ⅱ[J]. Appl Math Comput, 2006, 180: 353-365.
    [4]
    GOH B S. Global stability in many-species systems[J] The American Naturalist, 1977, 111(977): 135-143.
    [5]
    LIU M, BAI C. Analysis of a stochastic tri-trophic food-chain model with harvesting[J]. J Math Biol, 2016, 73: 597-625.
    [6]
    BAO J, YUAN C. Stochastic population dynamics driven by Lévy noise[J]. J Math Anal Appl, 2012, 391: 363-375.
    [7]
    LI M, GAO H, WANG B. Analysis of a non-autonomous mutualism model driven by Lévy jumps[J]. Discrete & Continuous Dynamical Systems - B, 2016, 21(4): 1189-1202.
    [8]
    LIU M, WANG K. Stochastic Lotka-Volterra systems with Lévy noise[J]. J Math Anal Appl, 2014, 410: 750-763.
    [9]
    LIU M, BAI C. Dynamics of a stochastic one-prey two-predator model with Lévy jumps[J]. Appl Math Comput, 2016, 284: 308-321.
    [10]
    LIU Q, JIANG D, SHI N. Stochastic mutualism model with Lévy jumps[J]. Comm Nonl Sci Num Simul, 2016, 43: 78-90.
    [11]
    ZHOU Y, YUAN S, ZHAO D. Threshold behavior of a stochastic SIS model with Lévy jumps[J]. Appl Math Comput, 2016, 275: 255-267.
    [12]
    GE Q, JI G, XU J, et al. Extinction and persistence of a stochastic nonlinear SIS epidemic model with jumps[J]. Physica A: Statistical Mechanics and Its Applications, 2016, 462: 1120-1127.
    [13]
    CHEN C, KANG Y. Dynamics of a stochastic multi-strain SIS epidemic model driven by Lévy noise[J]. Comm Nonl Sci Num Simul, 2017, 42: 379-395.
    [14]
    MAO W, MAO X. On the asymptotic stability and numerical analysis of solutions to nonlinear stochastic differential equations with jumps[J]. J Comput Appl Math, 2016, 301: 1-15.
    [15]
    ZOU X,WANG K. Numerical simulations and modeling for stochastic biological systems with jumps[J]. Comm Nonl Sci Nume Simul, 2014, 19: 1557-1568.
    [16]
    ZHANG X, WANG K. Stability analysis of a stochastic Gilpin-Ayala model driven by Lévy noise[J]. Comm Nonl Sci Nume Simul, 2014, 19(5): 1391-1399.
    [17]
    WU R, ZOU X, WANG K. Asymptotic properties of stochastic hybrid Gilpin-Ayala system with jumps[J]. Appl Math Comput, 2014, 249: 53-66.
    [18]
    JIANG D, SHI N, LI X. Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation[J]. J Math Anal Appl, 2008, 340: 588-597.
    [19]
    LIPTSER R S. A strong law of large numbers for local martingales[J]. Stoch Inter J Prob Stoch Proc, 1980, 3: 217-228.
    [20]
    MAO X, MARION G, RENSHAW E. Environmental Brownian noise suppresses explosions in population dynamics[J]. Stoch Proc Appl, 2002, 97: 95-110.)
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