Dynamics of a stochastic service-resource mutualism model with Lévy noises and harvesting

Volume 10, Issue 12, pp 6205--6218 http://dx.doi.org/10.22436/jnsa.010.12.07

Publication Date: December 06, 2017 Submission Date: August 09, 2017

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2587 Downloads
• 5840 Views

Authors

Hui Wang - School of Mathematical Science, Huaiyin Normal University, Huaian 223300, P. R. China. Chenxi Du - School of Mathematical Science, Huaiyin Normal University, Huaian 223300, P. R. China. Meng Liu - School of Mathematical Science, Huaiyin Normal University, Huaian 223300, P. R. China. - School of Mathematics and Statistics, Northeast Normal University, Jilin 130024, P. R. China.

Abstract

In this paper, we propose a stochastic service-resource mutualism model with Lévy noises and harvesting. Under some assumptions, we study several dynamical properties of the model. We first obtain the thresholds between persistence and extinction for both the service species and the resource species. Then we give sharp sufficient conditions for stability in distribution of the model. Finally, we establish sufficient and necessary criteria for the existence of the optimal harvesting policy. The optimal harvesting effort and maximum of sustainable yield are also obtained. Our results reveal that the persistence, extinction, stability in distribution and optimal harvesting strategy have close relationships with the random noises.

Share and Cite

Keywords

• Service-resource mutualism model
• white noise
• Lévy jumps
• persistence
• optimal harvesting

MSC

•  92D25
•  60H10
•  60H30

References

• [1] C.-Z. Bai, Multiplicity of solutions for a class of non-local elliptic operators systems, Bull. Korean Math. Soc., 54 (2017), 715–729.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] J.-H. Bao, C.-G. Yuan , Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl., 391 (2012), 363–375.
  • View Article
  • MathSciNet
  • Google Scholar


• [3] I. Barbălat, Systémes d’équations différentielles d’oscillations non linéaires, (French) Rev. Math. Pures Appl., 4 (1959), 267–270.
  • MathSciNet
  • Google Scholar
  • MATH


• [4] M. S. Bazaraa, C. M. Shetty , Nonlinear programming , Theory and algorithms, John Wiley & Sons, New York- Chichester-Brisbane (1979)
  • Google Scholar


• [5] J. R. Beddington, R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463–465.
  • View Article
  • Google Scholar


• [6] M. Begon, J. L. Harper, C. R. Townsend, Ecology, Individuals, populations and communities, Blackwell scientific publications, Cambridge (1986)
  • Google Scholar


• [7] C. A. Braumann , Variable effort harvesting models in random environments: generalization to density-dependent noise intensities, Deterministic and stochastic modeling of biointeraction, West Lafayette, IN, (2000), Math. Biosci., 177/178 (2002), 229–245.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] E. Braverman, L. Braverman , Optimal harvesting of diffusive models in a nonhomogeneous environment , Nonlinear Anal., 71 (2009), e2173–e2181.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] E. Braverman, R. Mamdani, Continuous versus pulse harvesting for population models in constant and variable environment, J. Math. Biol., 57 (2008), 413–434.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] C. W. Clark , Mathematical bioeconomics, The optimal management of renewable resources, Second edition, With a contribution by Gordon Munro, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1990)
  • Google Scholar


• [11] G. Da Prato, J. Zabczyk, Ergodicity for infinite-dimensional systems, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge (1996)
  • View Article
  • MathSciNet
  • Google Scholar


• [12] M. Fan, K. Wang , Optimal harvesting policy for single population with periodic coefficients, Math. Biosci., 152 (1998), 165–177.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] T. C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal., 10 (1986), 1411–1419.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] T. K. Kar, Modelling and analysis of a harvested prey-predator system incorporating a prey refuge, J. Comput. Appl. Math., 185 (2006), 19–33.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] H. Kunita, Itô ’s stochastic calculus: its surprising power for applications, Stochastic Process. Appl., 120 (2010), 622–652.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] M. Liu, C.-Z. Bai, Optimal harvesting of a stochastic logistic model with time delay, J. Nonlinear Sci., 25 (2015), 277–289.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] M. Liu, C.-Z. Bai , Analysis of a stochastic tri-trophic food-chain model with harvesting, J. Math. Biol., 73 (2016), 597–625.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] M. Liu, C.-Z. Bai, Dynamics of a stochastic one-prey two-predator model with Lévy jumps, Appl. Math. Comput., 284 (2016), 308–321.
  • View Article
  • MathSciNet
  • Google Scholar


• [19] M. Liu, C.-Z. Bai, Optimal harvesting of a stochastic delay competitive model, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1493–1508.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] M. Liu, X. He, J. Yu , Dynamics of a stochastic regime-switching predator-prey model with harvesting and distributed delays, Nonlinear Anal. Hybrid Syst., 28 (2018), 87–104.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] L.-D. Liu, X.-Z. Meng, T.-H. Zhang, Optimal control strategy for an impulsive stochastic competition system with time delays and jumps, Phys. A, 477 (2017), 99–113.
  • View Article
  • MathSciNet
  • Google Scholar


• [22] M. Liu, K. Wang, Stochastic Lotka-Volterra systems with Lévy noise, J. Math. Anal. Appl., 410 (2014), 750–763.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] X.-R. Mao, C.-G. Yuan, Stochastic differential equations with Markovian switching, Imperial College Press, London (2006)
  • Google Scholar


• [24] W. M. Murdoch, C. J. Briggs, R. M. Nisbet, Consumer-resource dynamics, Princeton University Press, Princeton (2003)
  • Google Scholar


• [25] Y.-S. Wang, D. L. DeAngelis, J. N. Holland, Uni-directional consumer-resource theory characterizing transitions of interaction outcomes, Ecol. Complex., 8 (2011), 249–257.
  • View Article
  • Google Scholar


• [26] Y. Zhao, S.-L. Yuan, Optimal harvesting policy of a stochastic two-species competitive model with Lévy noise in a polluted environment, Phys. A, 477 (2017), 20–33.
  • View Article
  • MathSciNet
  • Google Scholar


• [27] Y. Zhu, M. Liu, Permanence and extinction in a stochastic service-resource mutualism model, Appl. Math. Lett., 69 (2017), 1–7.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] C. Zhu, G. Yin , On hybrid competitive Lotka-Volterra ecosystems, Nonlinear Anal., 71 (2009), e1370–e1379.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] X.-L. Zou, W.-X. Li, K. Wang, Ergodic method on optimal harvesting for a stochastic Gompertz-type diffusion process, Appl. Math. Lett., 26 (2013), 170–174.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] X.-L. Zou, K. Wang, Optimal harvesting for a logistic population dynamics driven by a Lévy process, J. Optim. Theory. Appl., 161 (2014), 969–979.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH