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Impact of the Fear Effect on the Stability and Bifurcation of a Leslie–Gower Predator–Prey Model

Xiaoqin Wang

School of Science and Arts, Shaanxi University of Science and Technology, Xi’an 710021, P. R. China

E-mail Address: wangxiaoqin@sust.edu.cn

Corresponding authors.

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Yiping Tan

School of Science and Arts, Shaanxi University of Science and Technology, Xi’an 710021, P. R. China

School of Mathematics and Statistics, Huaiyin Normal University, Huaian 223300, P. R. China

E-mail Address: yptan16@163.com

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Yongli Cai

School of Mathematics and Statistics, Huaiyin Normal University, Huaian 223300, P. R. China

E-mail Address: caiyongli@hytc.edu.cn

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, and

Weiming Wang

http://orcid.org/0000-0002-7562-8260

School of Mathematics and Statistics, Huaiyin Normal University, Huaian 223300, P. R. China

E-mail Address: wangwm_math@hytc.edu.cn

Corresponding authors.

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https://doi.org/10.1142/S0218127420502107Cited by:20 (Source: Crossref)

Abstract

In this paper, we investigate analytically and numerically the dynamics of a modified Leslie–Gower predator–prey model which is characterized by the reduction of prey growth rate due to the anti-predator behavior. We prove the existence and local/global stability of equilibria of the model, and verify the existence of Hopf bifurcation. In addition, we focus on the influence of the fear effect on the population dynamics of the model and find that the fear effect can not only reduce the population density of both predator and prey, but also destabilize the coexistence equilibrium, which are beneficial to the occurrence of limit-cycle-induced oscillation, or prevent the occurrence of limit cycle oscillation and increase the stability of the system.

Keywords:

References

Abbey-Lee, R. N., Mathot, K. J. & Dingemanse, N. J. [2016] “ Behavioral and morphological responses to perceived predation risk: A field experiment in passerines,” Behav. Ecol. 27, 857–864. Crossref, Web of Science, Google Scholar
Abrams, P. & Ginzburg, L. [2000] “ The nature of predation: Prey dependent, ratio dependent or neither,” Trends Ecol. Evol. 15, 337–341. Crossref, Web of Science, Google Scholar
Arditi, R. & Ginzburg, L. [1989] “ Coupling in predator–prey dynamics: Ratio-dependence,” J. Theor. Biol. 139, 311–326. Crossref, Web of Science, Google Scholar
Aziz-Alaoui, M. A. & Okiye, M. D. [2003] “ Boundedness and global stability for a predator–prey model with modified Leslie–Gower and Holling-type II schemes,” Appl. Math. Lett. 16, 1069–1075. Crossref, Web of Science, Google Scholar
Biktashev, V. N., Brindley, J., Holden, A. V. & Tsyganov, M. A. [2004] “ Pursuit-evasion predator–prey waves in two spatial dimensions,” Chaos 14, 988–994. Crossref, Web of Science, Google Scholar
Burton, T. A. [2005] Volterra Integral and Differential Equations (Elsevier). Google Scholar
Cai, Y., Zhao, C., Wang, W. & Wang, J. [2015] “ Dynamics of a Leslie–Gower predator–prey model with additive Allee effect,” Appl. Math. Model. 39, 2092–2106. Crossref, Web of Science, Google Scholar
Cai, Y., Gui, Z., Zhang, X., Shi, H. & Wang, W. [2018] “ Bifurcations and pattern formation in a predator–prey model,” Int. J. Bifurcation and Chaos 28, 1850140-1–17. Link, Web of Science, Google Scholar
Chen, F., Chen, L. & Xie, X. [2009] “ On a Leslie–Gower predator–prey model incorporating a prey refuge,” Nonlin. Anal.: Real World Appl. 10, 2905–2908. Crossref, Web of Science, Google Scholar
Hua, F., Sieving, K. E., Fletcher, R. J. & Wright, C. A. [2014] “ Increased perception of predation risk to adults and offspring alters avian reproductive strategy and performance,” Behav. Ecol. 25, 509–519. Crossref, Web of Science, Google Scholar
Leslie, P. H. [1948] “ Some further notes on the use of matrices in population mathematics,” Biometrika 35, 213–245. Crossref, Web of Science, Google Scholar
Leslie, P. H. [1958] “ A stochastic model for studying the properties of certain biological systems by numerical methods,” Biometrika 45, 16–31. Crossref, Web of Science, Google Scholar
Pijush, P., Nikhil, P., Sudip, S. & Joydev, C. [2018] “ Stability and bifurcation analysis of a three-species food chain model with fear,” Int. J. Bifurcation and Chaos 28, 1850009-1–20. Web of Science, Google Scholar
Pijush, P., Nikhil, P., Sudip, S. & Joydev, C. [2019] “ A three species food chain model with fear induced trophic cascade,” Int. J. Appl. Comput. Math. 5, 100-1–26. Google Scholar
Qiao, T., Cai, Y., Fu, S. & Wang, W. [2019] “ Stability and Hopf bifurcation in a predator–prey model with the cost of anti-predator behaviors,” Int. J. Bifurcation and Chaos 29, 1950185-1–10. Link, Web of Science, Google Scholar
Saheb, P., Nikhil, P., Sudip, S. & Joydev, C. [2019] “ Stability and bifurcation analysis of a three-species food chain model with fear,” Math. Biosci. Eng. 16, 5146–5179. Google Scholar
Sasmal, S. [2018] “ Population dynamics with multiple allee effects induced by fear factors induced by fear factors — A mathematical study on prey–predator,” Appl. Math. Model. 64, 1–14. Crossref, Web of Science, Google Scholar
Sasmal, S. K. & Takeuchi, Y. [2020] “ Dynamics of a predator–prey system with fear and group defense,” J. Math. Anal. Appl. 481, 123471. Crossref, Web of Science, Google Scholar
Suraci, J. P., Clinchy, M., Dill, L. M., Roberts, D. & Zanette, L. Y. [2016] “ Fear of large carnivores causes a trophic cascade,” Nat. Commun. 7, 10698. Crossref, Web of Science, Google Scholar
Tsyganov, M. A., Brindley, J., Holden, A. V. & Biktashev, V. N. [2003] “ Quasisoliton interaction of pursuit–evasion waves in a predator–prey system,” Phys. Rev. Lett. 91, 218102. Crossref, Web of Science, Google Scholar
Upadhyay, R. & Mishra, S. [2018] “ Population dynamic consequences of fearful prey in a spatiotemporal predator–prey system,” Math. Biosci. Eng. 16, 338–372. Crossref, Web of Science, Google Scholar
Volkeltigoe, M. [1992] “ The orgins and evolution of predator–prey theory,” Ecology 73, 1530–1535. Crossref, Web of Science, Google Scholar
Wang, W., Lin, Y., Zhang, L., Rao, F. & Tan, Y. [2011] “ Complex patterns in a predator–prey model with self and cross-diffusion,” Commun. Nonlin. Sci. Numer. Simulat. 16, 2006–2015. Crossref, Web of Science, Google Scholar
Wang, X., Zanette, L. & Zou, X. [2016] “ Modeling the fear effect in predator–prey interactions,” J. Math. Biol. 73, 1179–1204. Crossref, Web of Science, Google Scholar
Wang, X. & Zou, X. [2017] “ Modeling the fear effect in predator–prey interactions with adaptive avoidance of predators,” Bull. Math. Biol. 79, 1325–1359. Crossref, Web of Science, Google Scholar
Wang, J., Cai, Y., Fu, S. & Wang, W. [2019] “ The effect of the fear factor on the dynamics of a predator–prey model incorporating the prey refuge,” Chaos 29, 083109. Crossref, Web of Science, Google Scholar
Zanette, L. Y., White, A. F., Allen, M. C. & Michael, C. [2011] “ Perceived predation risk reduces the number of offspring songbirds produce per year,” Science 334, 1398–1401. Crossref, Web of Science, Google Scholar
Zhang, H., Cai, Y., Fu, S. & Wang, W. [2019] “ Impact of fear effect in a prey–predator model incorporating a prey refuge,” Appl. Math. Comput. 356, 328–337. Crossref, Web of Science, Google Scholar