Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Optimal harvesting of a competitive n-species stochastic model with delayed diffusions

1 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, PR China
2 Department of Mathematics, Swinburne University of Technology, Hawthorn, VIC 3122 Australia

Special Issues: Non-smooth biological dynamical systems and applications

In this study, we propose an n-species stochastic model which considers the influences of the competitions and delayed diffusions among populations on dynamics of species. We then investigate the stochastic dynamics of the model, such as the persistence in mean of the species, and the asymptotic stability in distribution of the model. Then, by using the Hessian matrix and theory of optimal harvesting, we investigate the optimal harvesting problem, obtaining the optimal harvesting effort and the maximum of expectation of sustainable yield (ESY). Finally, we numerically discuss some examples to illustrate our theoretical findings, and conclude our study by a brief discussion.
  Figure/Table
  Supplementary
  Article Metrics

References

1. Z. Lu and Y. Takeuchi, Global asymptotic behavior in single-species discrete diffusion systems, J. Math. Biol., 32 (1993), 67–77.

2. E. Beretta and Y. Takeuchi, Global stability of single-species diffusion Volterra models with continuous time delays, Bull. Math. Biol. 49 (1987), 431–448.

3. D. Li, J. Cui and G. Song, Permanence and extinction for a single-species system with jump diffusion, J. Math. Anal. Appl., 430 (2015), 438–464.

4. L. J. Allen, Persistence and extinction in single-species reaction-diffusion models, Bull. Math. Biol. 45.2 (1983), 209–227.

5. W. Wang and T. Zhang, Caspase-1-Mediated Pyroptosis of the Predominance for Driving CD$_{4}$$^+$ T Cells Death: A Nonlocal Spatial Mathematical Model, B. Math. Biol., 80 (2018), 540--582.

6. Y. Cai andW.Wang, Fish-hook bifurcation branch in a spatial heterogeneous epidemic model with cross-diffusion, Nonlinear Anal. Real World Appl., 30 (2016), 99–125.

7. T. Zhang, X. Liu, X. Meng and T. Zhang, Spatio-temporal dynamics near the steady state of a planktonic system, Comput. Math. Appl., 75 (2018), 4490–4504.

8. T. Zhang, T. Zhang, X. Meng, Stability analysis of a chemostat model with maintenance energy, Appl. Math. Lett., 68 (2017), 1–7.

9. J. Zhou, C. Sang, X. Li, M. Fang and Z. Wang, H1 consensus for nonlinear stochastic multi-agent systems with time delay, Appl. Math. Comput., 325 (2018), 41–58.

10. K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Kluwer Academic, Dorecht. 1992.

11. Y. Kuang, Delay differential equations with applications in population dynamics, Academic Press, 1993.

12. X. Meng, F. Li and S. Gao, Global analysis and numerical simulations of a novel stochastic ecoepidemiological model with time delay, Appl. Math. Comput., 339 (2018), 701–726.

13. T. Zhang, W. Ma and X. Meng, Global dynamics of a delayed chemostat model with harvest by impulsive flocculant input, Adv. Difference Equ., 2017 (2017), 115.

14. G. Liu, X. Wang, X. Meng and S. Gao, Extinction and persistence in mean of a novel delay impulsive stochastic infected predator-prey system with jumps, Complexity, 2017 (2017), 15.

15. Y. Tan, S. Tang, J. Yang and Z. Liu, Robust stability analysis of impulsive complex-valued neural networks with time delays and parameter uncertainties, J. Inequal. Appl., 2017 (2017), 215.

16. T. Zhang and H. Zang, Delay-induced Turing instability in reaction-diffusion equations, Phys. Rev. E, 90 (2014) 052908.

17. R. M. May, Stability and Complexity in Model Ecosystems, Princeton Univ. Press, NewYork, 2001.

18. X. Yu, S. Yuan and T. Zhang, Survival and ergodicity of a stochastic phytoplankton-zooplankton model with toxin producing phytoplankton in an impulsive polluted environment, Appl. Math. Comput., 347 (2019), 249-264.

19. H. Qi, X. Leng, X. Meng and T. Zhang, Periodic Solution and Ergodic Stationary Distribution of SEIS Dynamical Systems with Active and Latent Patients, Qual. Theory Dyn. Syst., 18 (2019).

20. S. Zhang, X. Meng, T. Feng and T. Zhang, Dynamics analysis and numerical simulations of a stochastic non-autonomous predator-prey system with impulsive effects, Nonlinear Anal. Hybrid Syst., 26 (2017), 19–37.

21. M. Liu, H. Qiu and K. Wang, A remark on a stochastic predator-prey system with time delays, Appl. Math. Lett., 26 (2013), 318-323.

22. X. Meng and L. Zhang, Evolutionary dynamics in a Lotka-Volterra competition model with impulsive periodic disturbance, Math. Method. Appl. Sci., 39 (2016), 177–188.

23. L. Zhu and H. Hu, A stochastic SIR epidemic model with density dependent birth rate, Adv. Difference Equ., 2015 (2015), 1.

24. X. Meng, S. Zhao, T. Feng and T. Zhang, Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl., 433 (2016), 227–242.

25. M. Liu and C. Bai, A remark on a stochastic logistic model with diffusion, Appl. Math. Lett., 228 (2014), 141–146.

26. F. Bian, W. Zhao, Y. Song and R. Yue, Dynamical analysis of a class of prey-predator model with Beddington-Deangelis functional response, stochastic perturbation, and impulsive toxicant input, Complexity, 2017 (2017) Article ID 3742197.

27. W. Wang, Y. Cai, J. Li and Z. Gui, Periodic behavior in a FIV model with seasonality as well as environment fluctuations, J. Franklin Inst., 354 (2017), 7410–7428.

28. X. Song and L. Chen, Optimal harvesting and stability for a two-species competitive system with stage structure, Math. Biosci., 170 (2001), 173–186.

29. J. Liang, S. Tang and R. A. Cheke, Pure Bt-crop and mixed seed sowing strategies for optimal economic profit in the face of pest resistance to pesticides and Bt-corn, Appl. Math. Comput., 283 (2016), 6–21.

30. S. Sharma and G. P. Samanta, Optimal harvesting of a two species competition model with imprecise biological parameters, Nonlinear Dynam., 77 (2014), 1101–1119.

31. J. Xia, Z. Liu and R. Yuan, The effects of harvesting and time delay on predator-prey systems with Holling type II functional response, SIAM J. Appl. Math., 70 (2009), 1178–1200.

32. W. Li, K. Wang, H. Su, Optimal harvesting policy for stochastic logistic population model, Appl. Math. Comput., 218 (2011), 157–162.

33. X. Zou,W. Li and K.Wang, Ergodic method on optimal harvesting for a stochastic Gompertz-type diffusion process, Appl. Math. Lett., 26 (2013), 170–174.

34. Q. Song, R. H. Stockbridge and C. Zhu, On optimal harvesting problems in random environments, J. Soc. Ind. Appl. Math., 49 (2011), 859–889.

35. G. Zeng, F. Wangand J. J. Nieto, Complexity of a delayed predator-prey model with impulsive harvesting and Holling type II functional response, Adv. Complex Syst., 11 (2008), 77–97.

36. X. Zou, W. Li and K. Wang, Ergodic method on optimal harvesting for a stochastic Gom-pertztype diffusion process, Appl. Math. Lett., 26 (2013), 170–174.

37. C. Ji, D. Jiang and N. Shi, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482–498.

38. J. Bao and C. Yuan, Comparison theorem for stochastic differential delay equations with jumps, Acta Appl. Math., 116 (2011), 119.

39. D. Jiang and N. Shi, A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303 (2005), 164–172.

40. D. Jiang, C. Ji, X. Li and D. O'Regan, Analysis of autonomous Lotka-Volterra competition systems with random perturbation, J. Math. Anal. Appl., 390 (2012), 582–595.

41. I. Barbalat, Systems d'equations differentielles d'osci d'oscillations nonlineaires, Rev. Roumaine Math. Pures Appl. 1959.

42. X. Leng, T. Feng and X. Meng, Stochastic inequalities and applications to dynamics analysis of a novel SIVS epidemic model with jumps, J. Inequel. Appl., 2017 (2017), 138.

43. J. Prato and J. Zabczyk, Ergodicity for infinite dimensional systems, Cambridge Univ. Press, Cambridge, (1996), 229.

44. Y. Zhao, S. Yuan and I. Barbalat, Systems dequations differentielles doscillations nonlineaires, Phys. A., 477 (2017), 20–33.

45. L. Liu and X. Meng, Optimal harvesting control and dynamics of two species stochastic model with delays, Adv. Differ Equat., 2017 (2017), 1–18.

46. H. Qi, L. Liu and X. Meng, Dynamics of a non-autonomous stochastic SIS epidemic model with double epidemic hypothesis, Complexity, 2017 (2017), 14.

47. D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525–546.

48. N. Bruti-Liberati and E. Platen, Monte Carlo simulation for stochastic differential equations, Encyclopedia of Quantitative Finance, 10 (2010), 23–37.

49. M. Liu and C. Bai, Optimal harvesting of a stochastic delay competitive model, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 1493–1508.

50. H. Ma and Y. Jia, Stability Analysis For Stochastic Differential EquationsWith Infinite Markovian Switchings, J. Math. Anal. Appl., 435 (2016), 593–605.

51. M. Liu, X. He and J. Yu, Dynamics of a stochastic regime-switching predator-prey model with harvesting and distributed delays, Nonlinear Anal. Hybrid Syst., 28 (2018), 87–104.

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved