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Bulletin of Mathematical Biology

, Volume 81, Issue 5, pp 1337–1351 | Cite as

Impacts of the Dispersal Delay on the Stability of the Coexistence Equilibrium of a Two-Patch Predator–Prey Model with Random Predator Dispersal

  • Ali Mai
  • Guowei Sun
  • Lin WangEmail author
Article

Abstract

In this paper, we study a predator–prey system with random predator dispersal over two habitat patches. We show that in most cases the dispersal delay does not affect the stability and instability of the coexistence equilibrium. However, if the mean time that the predator spent in one patch is much shorter than the timescale of reproduction of the prey and is larger than the double mean time of capture of prey, the dispersal delay can induce stability switches such that an otherwise unstable coexistence equilibrium can be stabilized over a finite number of stability intervals.

Keywords

Predator–prey Random dispersal Dispersal delay Stability switch 

Notes

Acknowledgements

The authors would like to thank the anonymous referee for his/her valuable comments and suggestions, which greatly helped us improve the presentation of this paper. This work is partially supported by National Natural Science Foundation of China (No. 11526183), China Scholarship Council (201608140214), Foundation of Yuncheng University (YQ-2017003), Biomathematics Laboratory of Yuncheng University (SWSX201502, SWSX201602) and by a discovery grant from NSERC.

References

  1. Cooke KL, Grossman Z (1982) Discrete delay, distributed delay and stability switches. J Math Anal Appl 86(2):592–627MathSciNetzbMATHCrossRefGoogle Scholar
  2. Edwards R, van den Driessche P, Wang L (2007) Periodicity in piecewise-linear switching networks with delay. J Math Biol 55(2):271–298MathSciNetzbMATHCrossRefGoogle Scholar
  3. El Abdllaoui A, Auger P, Kooi BW, De la Parra RB, Mchich R (2007) Effects of density-dependent migrations on stability of a two-patch predator-prey model. Math Biosci 210(1):335–354MathSciNetzbMATHCrossRefGoogle Scholar
  4. Feng W, Hinson J (2005) Stability and pattern in two-patch predator-prey population dynamics. Discrete Contin Dyn Syst Suppl 2005:268–279MathSciNetzbMATHGoogle Scholar
  5. Feng W, Rock B, Hinson J (2011) On a new model of two-patch predator prey system with migration of both species. J Appl Anal Comput 1(2):193–203MathSciNetzbMATHGoogle Scholar
  6. Freedman HI (1987) Single species migration in two habitats: persistence and extinction. Math Model 8:778–780MathSciNetCrossRefGoogle Scholar
  7. Hale JK, Verduyn Lunel SM (1993) Introduction to Functional differential equations, vol 99. Springer, BerlinzbMATHGoogle Scholar
  8. Hauzy C, Gauduchon M, Hulot FD, Loreau M (2010) Density-dependent dispersal and relative dispersal affect the stability of predator-prey metacommunities. J Theor Biol 266(3):458–469MathSciNetzbMATHCrossRefGoogle Scholar
  9. Holyoak M, Lawler SP (1996) The role of dispersal in predator-prey metapopulation dynamics. J Anim Ecol 65(5):640–652CrossRefGoogle Scholar
  10. Hsu S-B (1978) On global stability of a predator-prey system. Math Biosci 39(1–2):1–10MathSciNetzbMATHCrossRefGoogle Scholar
  11. Huffaker CB, Kennett CE (1956) Experimental studies on predation: predation and cyclamen-mite populations on strawberries in california. Hilgardia 26(4):191–222CrossRefGoogle Scholar
  12. Jansen VAA (2001) The dynamics of two diffusively coupled predator-prey populations. Theor Popul Biol 59(2):119–131zbMATHCrossRefGoogle Scholar
  13. Kang Y, Sourav KS, Komi M (2017) A two-patch prey-predator model with predator dispersal driven by the predation strength. Math Biosci Eng 14(4):843–880MathSciNetzbMATHCrossRefGoogle Scholar
  14. Klepac P, Neubert MG, van den Driessche P (2007) Dispersal delays, predator-prey stability, and the paradox of enrichment. Theor Popul Biol 71(4):436–444zbMATHCrossRefGoogle Scholar
  15. Kot M (2001) Elements of mathematical ecology. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  16. Kuang Y (1993) Delay differential equations: with applications in population dynamics, vol 191. Academic Press, New YorkzbMATHGoogle Scholar
  17. Kuang Y, Takeuchi Y (1994) Predator-prey dynamics in models of prey dispersal in two-patch environments. Math Biosci 120(1):77–98MathSciNetzbMATHCrossRefGoogle Scholar
  18. Levin SA (1974) Dispersion and population interactions. Am Nat 108(960):207–228CrossRefGoogle Scholar
  19. Liao K-L, Lou Y (2014) The effect of time delay in a two-patch model with random dispersal. Bull Math Biol 76(2):335–376MathSciNetzbMATHCrossRefGoogle Scholar
  20. Mai A, Sun G, Zhang F, Wang L (2019) The joint impacts of dispersal delay and dispersal patterns on the stability of predator-prey metacommunities. J Theor Biol 462:455–465MathSciNetzbMATHCrossRefGoogle Scholar
  21. Mchich R, Auger P, Poggiale J-C (2007) Effect of predator density dependent dispersal of prey on stability of a predator-prey system. Math Biosci 206(2):343–356MathSciNetzbMATHCrossRefGoogle Scholar
  22. Nathan R, Giuggioli L (2013) A milestone for movement ecology research. Mov Ecol 1:1–1CrossRefGoogle Scholar
  23. Neubert MG, Klepac P, van den Driessche P (2002) Stabilizing dispersal delays in predator-prey metapopulation models. Theor Popul Biol 61(3):339–347zbMATHCrossRefGoogle Scholar
  24. Pillai P, Gonzalez A, Loreau M (2011) Evolution of dispersal in a predator-prey metacommunity. Am Nat 179(2):204–216CrossRefGoogle Scholar
  25. Wall E, Guichard F, Humphries AR (2013) Synchronization in ecological systems by weak dispersal coupling with time delay. Theor Ecol 6(4):405–418CrossRefGoogle Scholar
  26. Wang W, Takeuchi Y (2009) Adaptation of prey and predators between patches. J Theor Biol 258(4):603–613MathSciNetzbMATHCrossRefGoogle Scholar
  27. Wang X, Zou X (2016) On a two-patch predator-prey model with adaptive habitancy of predators. Discrete Contin Dyn Syst Ser B 21(2):677–697MathSciNetzbMATHCrossRefGoogle Scholar
  28. Whitten KR, Garner GW (1992) Productivity and early calf survival in the porcupine caribou herd. J Wildl Manag 56(2):201CrossRefGoogle Scholar
  29. Zhang Y, Lutscher F, Guichard F (2015) The effect of predator avoidance and travel time delay on the stability of predator-prey metacommunities. Theor Ecol 8(3):273–283CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.School of Mathematics and Information TechnologyYuncheng UniversityYunchengChina
  2. 2.Department of Mathematics and StatisticsUniversity of New BrunswickFrederictonCanada

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