Export file:


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


  • Citation Only
  • Citation and Abstract

Periodic solution of a stage-structured predator-prey model incorporating prey refuge

1 Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
2 School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

Special Issues: Advances in Mathematical Modelling and Analysis of Bioprocesses

In this paper, we consider a delayed stage-structured predator-prey model incorporating prey refuge with Holling type II functional response. It is assumed that prey can live in two different regions. One is the prey refuge and the other is the predatory region. Moreover, in real world application, we should consider the stage-structured model. It is assumed that the prey in the predatory region can divided by two stages: Mature predators and immature predators, and the immature predators have no ability to attack prey. Based on Mawhin’s coincidence degree and novel estimation techniques for a priori bounds of unknown solutions to Lu = λNu, some sufficient conditions for the existence of periodic solution is obtained. Finally, an example demonstrate the validity of our main results.
  Article Metrics


1. A. Lotka, Elements of Physical Biology, USA: Williams Wilkins Co., Balitmore,1925.

2. V. Volterra, Variazioni e fluttuazioni del numero dindividui in specie animali conviventi, Mem. Acad Lincei Roma, 2 (1926), 31-113.

3. G. F. Gause, N. P. Smaragdova, A. A. Witt, Further studies of interaction between predators and prey, J. Anim. Ecol., 5 (1936), 1-18.

4. G. F. Gause, The Struggle for Existence, USA: Williams Wilkins Co., Balitmore, 1934.

5. S. Magalhães, P. C. J. V. Rijn, M. Montserrat, A. Pallini, M. W. Sabelis, Population dynamics of thrips prey and their mite predators in a refuge, Oecologia, 150 (2007), 557-568.

6. J. Ghosh, B. Sahoo, S. Poria, Prey-predator dynamics with prey refuge providing additional food to predator, Chaos Soliton. Fract., 96 (2017), 110-119.

7. B. Sahoo, S. Poria, Effects of additional food in a delayed predator-prey model, Math. Biosci., 261 (2015), 62-73.

8. B. Sahoo, S. Poria, Dynamics of predator-prey system with fading memory, Appl. Math. Comput., 347 (2019), 319-333.

9. U. Ufuktepe, B. Kulahcioglu, O. Akman, Stability analysis of a prey refuge predator-prey model with Allee effects, J. Biosciences, 44 (2019), 85.

10. Y. Xie, J. Lu, Z. Wang, Stability analysis of a fractional-order diffused prey-predator model with prey refuges, Physica A., 526 (2019), 120773.

11. C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Canada, 45 (1965), 1-60.

12. Q. Y. Bie, Q. R. Wang, Z. A. Yao, Cross-diffusion induced instability and pattern formation for a Holling type-II predator-prey model, Appl. Math. Comput., 247 (2014), 1-12.

13. L. Chen, F. Chen, L. Chen, Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge, Nonlinear. Anal-Real., 11 (2010), 246-252.

14. Z. J. Du, X. Chen, Z. S. Feng, Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms, Discrete. Contin. Dyn. Syst., 7 (2014), 1203-1214.

15. J. J. Jiao, L. S. Chen, S. H. Cai, A delayed stage-structured Holling II predator-prey model with mutual interference and impulsive perturbations on predator, Chaos Soliton. Fract., 40 (2009), 1946-1955.

16. W. Ko, K. Ryu, Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge, J. Differ. Equations, 231 (2006), 534-550.

17. V. Krivan, J. Eisner, The effect of the Holling type II functional response on apparent competition, Theor. Popul. Biol., 70 (2006), 421-430.

18. V. Krivan, On the Gause predator prey model with a refuge: A fresh look at the history, J. Theor. Biol., 274 (2011), 67-73.

19. Q. Liu, D. Q.Jiang, H. Tasawar, A. Ahmed, Dynamics of a stochastic predator-prey model with stage structure for predator and holling type II functional response, J. Nonlinear Sci., 28 (2018), 1151-1187.

20. S. P. Li, W. N. Zhang, Bifurcations of a discrete prey-predator model with Holling type II functional response, Discrete Cont. Dyn-B., 14 (2010), 159-176.

21. H. Molla, S. R. Md, S. Sahabuddin, Dynamics of a predator-prey model with holling type II functional response incorporating a prey refuge depending on both the species, Int. J. Nonlin. Sci. Num., 20 (2019), 1-16.

22. J. Song, Y. Xia, Y. Bai, Y. Cai, D. O'Regan, A non-autonomous Leslie-Gower model with Holling type IV functional response and harvesting complexity, Adv. Differ. Equ-Ny., 2019 (2019), 1-12.

23. D. Ye, M. Fan, W. P. Zhang, Periodic solutions of density dependent predator-prey systems with Holling Type 2 functional response and infinite delays, J. Appl. Math. Mec., 85 (2005), 213-221.

24. S. W. Zhang, L. S. Chen, A Holling II functional response food chain model with impulsive perturbations, Chaos Soliton. Fract., 24 (2005), 1269-1278.

25. J. Zhou, C. L. Mu, Coexistence states of a Holling type-II predator-prey system, J. Math. Anal. Appl., 369 (2010), 555-563.

26. S. Jana, M. Chakraborty, K. Chakraborty, T. K. Kar, Global stability and bifurcation of time delayed prey-predator system incorporating prey refuge. Math. Comput. Simulat., 85 (2012), 57-77.

27. W. G. Aiello, H. I. Freedman, J. Wu, Analysis of a model representing stage-structured population growth with state-dependent timedelay, SIAM J. Appl. Math., 52 (1992), 885-889.

28. F. Brauer, Z. Ma, Stability of stage-structured population models, J. Math. Anal. Appl., 126 (1987), 301-315.

29. H. I. Freedman, J. Wu, Persistence and global asymptotic stability of single species dispersal models with stage-structure, Q. Appl. Math., 49 (1991), 351-371.

30. W. Wang, L. Chen, A predator-prey system with stage-structure for predator, Comput. Math. Appl., 33 (1997), 83-91.

31. W. Wang, G. Mulone, F. Salemi, V. Salone, Permanence and stability of a stage-structured predator prey model, J. Math. Anal. Appl., 262 (2001), 499-528.

32. Y. Chen, Multiple periodic solution of delayed predator-prey systems with type IV functional responses, Nonlinear. Anal-Hybri., 5 (2004), 45-53.

33. M. Fan, Q. Wang, X. F. Zou, Dynamics of a nonautonomous ratio-dependent predator-prey system, P. Roy. Soc. Lond. A. Math., 133 (2003), 97-118.

34. M. Fan, P. J. Y. Wong, R. P. Agarwal, Periodicity and stability in periodic n-species Lotka-Volterra competition system with feedback controls and deviating arguments, Acta. Math. Sin., 19 (2003), 801-822.

35. R. Gaines, J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Mathematics, Springer, Berlin, 1977.

36. H. Zheng, L. Guo, Y. Z. Bai, Y. H. Xia, Periodic solutions of a non-autonomous predator-prey system with migrating prey and disease infection: Via Mawhin's coincidence degree theory, J. Fix. Point Theory A., 21 (2019), 21-37.

37. Y. H. Xia, Y. Shen, An nonautonomous predator-prey model with refuge effect, J. Xuzhou Inst. Tech., 34 (2019), 1-7.

38. F. Chen, On a periodic multi-species ecological model, Appl. Math. Comput., 171 (2005), 492-510.

39. F. Chen, Positive periodic solutions of neutral Lotka-Volterra system with feedback control, Appl. Math. Comput., 162 (2005), 1279-1302.

40. F. Chen, F. Lin, X. Chen, Sufficient conditions for the existence positive periodic solutions of a class of neutral delay models with feedback control, Appl. Math. Comput., 158 (2004), 45-68.

41. L. Chen. Mathematical Models and Methods in Ecology, Science Press, Beijing Chinese, 1998.

42. X. Chen, Z. J. Du, Existence of positive periodic solutions for a neutral delay predator-prey model with Hassell-Varley type functional response and impulse, Qual. Theor. Dyn. Syst., 17 (2018), 67-80.

43. Z. J. Du, Z. S. Feng, Periodic solutions of a neutral impulsive predator-prey model with Beddington-DeAngelis functional response with delays, J. Comput. Appl. Math., 258 (2014), 87-98.

44. S. Gao, L. Chen, Z. Teng, Hopf bifurcation and global stability for a delayed predator-prey system with stage structure for predator, Appl. Math. Comput., 202 (2008), 721-729.

45. S. Kant, V. Kumar, Stability analysis of predator-prey system with migrating prey and disease infection in both species. Appl. Math. Model., 42 (2017), 509-539.

46. Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, San Diego, 1993.

47. S. Liu, L. Chen, Z. Liu, Extinction and permanence in nonautonomous competitive system with stage structure, J. Math. Anal. Appl., 274 (2002), 667-684.

48. S. Lu, W. Ge, Existence of positive periodic solutions for neutral population model with multiple delays, Appl. Math. Comput., 153 (2004), 885-902.

49. X. Z. Meng, S. N. Zhao, T. Feng, T. H. Zhang, Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl., 433 (2016), 227-242.

50. J. Song, M. Hu, Y. Z. Bai, Y. Xia, Dynamic analysis of a non-autonomous ratio-dependent predator-prey model with additional food. J. Comput. Anal. Appl., 8 (2018), 1893-1909.

51. Y. L. Song, H. P. Jiang, Q. X. Liu, Y. Yuan, Spatiotemporal dynamics of the diffusive Mussel-Algae model near Turing-Hopf bifurcation, SIAM J. Appl. Dyn. Syst., 16 (2017), 2030-2062.

52. Y. L. Song, X. S. Tang, Stability, steady-state bifurcations and turing patterns in a predator-prey model with herd behavior and prey-taxis, Stud. Appl. Math., 139 (2017), 371-404.

53. Y. L. Song, S. H. Wu, H. Wang, Spatiotemporal dynamics in the single population modelwith memory-based diffusion and nonlocal effect, J. Differ. Equations. 267 (2019), 6316-6351.

54. J. J. Wei, M. Y. Li, Hopf bifurcation analysis in a delayed nicholson blowflies equation, Nonlinear Anal-Theor., 60 (2005), 1351-1367.

55. Z. Wei, Y. H. Xia, T. Zhang, Stability and bifurcation analysis of a amensalism model with weak Allee effect, Qual. Theor. Dyn. Syst., 2020.

56. R. Xu, Z. Ma, Stability and Hopf bifurcation in a ratio-dependent predator prey system with stage structure, Chaos Soliton. Fract., 38 (2008), 669-684.

57. J. Y. Xu, T. H. Zhang, K. Y. Song, A stochastic model of bacterial infection associated with neutrophils, Appl. Math. Comput., 373 (2020), 125025.

58. F. Xu, C. Ross, K. Vlastimil, Evolution of mobility in predator-prey systems, Discrete Cont. DynB., 19 (2014), 3397-3432.

59. F. Xu, M. Connell, An investigation of the combined effect of an annual mass gathering event and seasonal infectiousness on disease outbreak, Math. Biosci., 312 (2019), 50-58.

60. J. Y. Yang, Z. Jin, F. Xu, Threshold dynamics of an age-space structured SIR model on heterogeneous environment, Appl. Math. Lett., 96 (2019), 69-74.

61. F. Q. Yi, J. J. Wei, J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predater-prey system, J. Differ. Equations, 246 (2009), 1944-1977.

62. F. Q. Yi, J. J. Wei, J. P. Shi, Diffusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonlinear Anal-Real., 9 (2008), 1038-1051.

63. T. H. Zhang, T. Q. Zhang, X. Z. Meng, Stability analysis of a chemostat model with maintenance energy, Appl. Math. Lett., 68 (2017), 1-7.

64. T. H. Zhang, Z. W. Geem, Review of harmony search with respect to algorithm structure, Swarm Evol. Comput., 48 (2019), 31-43.

65. X. G. Zhang, C. H. Shan, Z. Jin, H. P. Zhu, Complex dynamics of epidemic models on adaptive networks, J. Differ. Equations, 266 (2019), 803-832.

© 2020 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