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Dynamics of a predator–prey model with double Allee effects and impulse

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Abstract

In this paper, a predator–prey model with double Allee effects and impulse is studied. The existence and stability of the prey-free periodic solution are investigated. The sufficient conditions for global stability of the prey-free periodic solution are obtained. We also find a critical threshold that the predator and prey populations will coexist. The existence of the transcritical bifurcations is considered by means of the bifurcation theory when the prey population is not subject to Allee effect. Combining mathematical analysis and numerical simulations, we show that the double Allee effects and impulse greatly alter the outcome of the survival of both species.

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References

  1. Correigh, M.G.: Habitat selection reduces extinction of populations subject to Allee effects. Theor. Popul. Biol. 64, 1–10 (2003)

    Article  MATH  Google Scholar 

  2. Stephens, P.A., Sutherland, W.J.: Consequences of the Allee effect for behavior, ecology and conservation. Trends Ecol. Evol. 14, 401–405 (1999)

    Article  Google Scholar 

  3. Lin, Z.S., Li, B.L.: The maximum sustainable yield of Allee dynamic system. Ecol. Model. 154, 1–7 (2002)

    Article  Google Scholar 

  4. Berec, L., Angulo, E., Counchamp, F.: Multiple Allee effects and population management. Ecol. Model. 22, 185–191 (2006)

    Google Scholar 

  5. Wang, M.H., Kot, M.: Speeds of invasion in a model with strong or weak Allee effects. Math. Biosci. 171, 83–97 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ferdy, J.B., Molofsky, J.: Allee effect, spatial structure and species coexistance. J. Theor. Biol. 217, 413–427 (2002)

    Article  Google Scholar 

  7. Sun, G.Q., Wu, Z.Y., Wang, Z., Jin, Z.: Influence of isolation degree of spatial patterns on persistence of populations. Nonlinear Dyn. 83, 811–819 (2016)

    Article  MathSciNet  Google Scholar 

  8. Sun, G.Q., Wang, S.L., Ren, Q., Jin, Z., Wu, Y.P.: Erratum: Effects of time delay and space on herbivore dynamics: linking inducible defenses of plants to herbivore outbreak. Sci. Rep-uk. 5, 11246 (2015)

    Article  Google Scholar 

  9. Li, L., Jin, Z., Li, J.: Periodic solutions in a herbivore-plant system with time delay and spatial diffusion. Appl. Math. Model. 40, 4765–4777 (2016)

    Article  MathSciNet  Google Scholar 

  10. Sun, G.Q., Chakraborty, Amit, Liu, Q.X., Jin, Z., Anderson, Kurt E., Li, B.L.: Influence of time delay and nonlinear diffusion on herbivore outbreak. Commun. Nonlinear Sci. Numer. Simul. 19, 1507–1518 (2014)

    Article  MathSciNet  Google Scholar 

  11. Li, L., Jin, Z.: Pattern dynamics of a spatial predator–prey model with noise. Nonlinear Dyn. 67, 1737–1744 (2012)

    Article  MathSciNet  Google Scholar 

  12. Sun, G.Q., Zhang, J., Song, L.P., Jin, Z., Li, B.L.: Pattern formation of a spatial predator–prey system. Appl. Math. Comput. 218, 11151–11162 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Li, L.: Patch invasion in a spatial epidemic model. Appl. Math. Comput. 258, 342–349 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Sun, G.Q.: Pattern formation of an epidemic model with diffusion. Nonlinear Dyn. 69, 1097–1104 (2012)

    Article  MathSciNet  Google Scholar 

  15. Sun, G.Q., Zhang, Z.K.: Global stability for a sheep brucellosis model with immigration. Appl. Math. Comput. 246, 336–345 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Sun, G.Q.: Mathematical modeling of population dynamics with Allee effect. Nonlinear Dyn. 85, 1–12 (2016)

    Article  MathSciNet  Google Scholar 

  17. Boukal, D.S., Sabelis, M.W., Berec, L.: How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses. Theor. Popul. Biol. 72, 136–147 (2007)

    Article  MATH  Google Scholar 

  18. Zu, J., Mimura, M.: The impact of Allee effect on a predator–prey system with Holling type II functional response. Appl. Math. Comput. 217, 3542–3556 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Zhou, S.R., Liu, Y.F., Wang, G.: The stability of predator–prey systems subject to the Allee effects. Theor. Popul. Biol. 67, 23–31 (2005)

    Article  MATH  Google Scholar 

  20. Wang, J.F., Shi, J.P., Wei, J.J.: Predator–prey system with strong Allee effect in prey. J. Math. Biol. 62, 291–331 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Van, G.V., Hemerik, L., Boer, M.P., Kooi, B.W.: Heteroclinic orbits indicate overexploitation in predator–prey systems with a strong Allee effect. Math. Biosci. 209, 451–469 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zu, J., Mimura, M., Wakano, J.Y.: The evolution of phenotypic traits in a predator–prey system subject to Allee effect. J. Theor. Biol. 262, 528–543 (2010)

    Article  MathSciNet  Google Scholar 

  23. Zu, J.: Global qualitative analysis of a predator prey system with Allee effect on the prey species. Math. Comput. Simul. 94, 33–54 (2013)

    Article  MathSciNet  Google Scholar 

  24. Sen, M., Banerjee, M., Morozov, A.: Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect. Ecol. Complex. 11, 12–27 (2012)

    Article  Google Scholar 

  25. Aguirre, P., Gonzalez-Olivares, E., Saez, E.: Three limit cycles in a Leslie–Gower predator–prey model with additive Allee effect. SIAM. J. Appl. Math. 69, 1244–1262 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Xiao, Q.Z., Dai, B.X.: Heteroclinic bifurcation for a general predator–prey model with Allee effect and state feedback inpulsive control strategy. Math. Biosci. Eng. 12, 1065–1081 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Terry, A.J.: Predator–prey models with component Allee effect for predator reproduction. J. Math. Biol. 71, 1325–1352 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Biswas, S., Sasmal, S.K., Samanta, S., Saifuddin, M., Khan, Q.J.A., Chattopadhyay, J.: A delayed eco-epidemiological system with infected prey and predator subject to the weak Allee effect. Math. Biosci. 263, 198–208 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Terry, A.J.: Prey resurgence from mortality events in predator–prey models. Nonlinear Anal. RWA 14, 2180–2203 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, W., Zhang, Y., Liu, C.: Analysis of a discrete-time predator–prey system with Allee effect. Ecol. Complex. 8, 81–85 (2011)

  31. Feng, P., Kang, Y.: Dynamics of a modified Leslie–Gower model with double Allee effects. Nonlinear Dyn. 80, 1051–1062 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Cushing, J.M.: Periodic time-dependent predator–prey system. SIAM. J. Appl. Math. 32, 82–95 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, S., Huang, Q.D.: Bifurcation of nontrivial periodic solutions for a Beddington–DeAngelis interference model with impulsive biological control. Appl. Math. Model. 39, 1470–1479 (2015)

    Article  MathSciNet  Google Scholar 

  34. Liu, X.N., Chen, L.S.: Complex dynamics of Holling type II Lotka–Volterra predator–prey system with impulsive perturbations on the predator. Chaos Solitons Fract. 16, 311–320 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank the anonymous referees very much for their valuable comments and suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Xiangsen Liu & Binxiang Dai

  2. Department of Mathematics, North University of China, Taiyuan, 030051, Shanxi, People’s Republic of China

    Xiangsen Liu

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  1. Xiangsen Liu
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  2. Binxiang Dai
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Correspondence to Binxiang Dai.

Additional information

This work is supported by the National Natural Science Foundation of China (Nos. 11271371 and 51479215).

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Liu, X., Dai, B. Dynamics of a predator–prey model with double Allee effects and impulse. Nonlinear Dyn 88, 685–701 (2017). https://doi.org/10.1007/s11071-016-3270-7

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  • DOI: https://doi.org/10.1007/s11071-016-3270-7

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