JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2025, Vol. 60 ›› Issue (4): 72-83.doi: 10.6040/j.issn.1671-9352.0.2023.338

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Hopf bifurcation in a diffusive generalist predator-prey system with nonlocal competition and time delay

LUO Yihua, DU Yanfei*   

  1. School of Mathematics &
    Data Science, Shaanxi University of Science &
    Technology, Xian 710021, Shaanxi, China
  • Published:2025-04-08

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Abstract: The stability of positive equilibrium and the existence of Hopf bifurcation are studied by analyzing the distribution of eigenvalues. The critical time delay of Hopf bifurcation is obtained. Applying center manifold method and normal form theory, the direction of Hopf bifurcation and stability of the bifurcating periodic solution are discussed. It is shown that there are two types of bistability. The prey-free equilibrium and the positive equilibrium are both stable. The prey-free equilibrium and the periodic solution are both stable. Numerical simulations are presented to support the theoretical results.

Key words: generalist predator, nonlocal competition, time delay, Hopf bifurcation, bistability

CLC Number: 

  • O175
[1] MOFFAT C E, LALONDE R G, ENSING D J, et al. Frequency-dependent host species use by a candidate biological control insect within its native European range[J]. Biological Control, 2013, 67(3):498-508.
[2] OWEN M R, LEWIS M A. How predation can slow, stop or reverse a prey invasion[J]. Bulletin of Mathematical Biology, 2001, 63(4):655-684.
[3] FAGAN W F, LEWIS M A, NEUBERT M G, et al. Invasion theory and biological control[J]. Ecology Letters, 2002, 5(1):148-157.
[4] MAGAL C, COSNER C, RUAN S, et al. Control of invasive hosts by generalist parasitoids[J]. Mathematical Medicine and Biology, 2008, 25(1):1-20.
[5] CROWDER D W, SNYDER W E. Eating their way to the top? Mechanisms underlying the success of invasive insect generalist predators[J]. Biological Invasions, 2010, 12:2857-2876.
[6] XIANG Chuang, HUANG Jicai, RUAN Shigui, et al. Bifurcation analysis in a host-generalist parasitoid model with Holling II functional response[J]. Journal of Differential Equations, 2020, 268(8):4618-4662.
[7] ERBACH A, LUTSCHER F, SEO G. Bistability and limit cycles in generalist predator-prey dynamics[J]. Ecological Complexity, 2013, 14:48-55.
[8] CHAKRABORTY S. The influence of generalist predators in spatially extended predator-prey systems[J]. Ecological Complexity, 2015, 23:50-60.
[9] MADEC S, CASAS J, BARLES G, et al. Bistability induced by generalist natural enemies can reverse pest invasions[J]. Journal of Mathematical Biology, 2017, 75:543-575.
[10] FURTER J, GRINFELD M. Local vs. non-local interactions in population dynamics[J]. Journal of Mathematical Biology, 1989, 27:65-80.
[11] CHEN Shanshan, YU Jianshe. Stability and bifurcation on predator-prey systems with nonlocal prey competition[J]. Discrete and Continuous Dynamical Systems, 2018, 38(1):43-62.
[12] WU Shuhao, SONG Yongli. Stability and spatiotemporal dynamics in a diffusive predator-prey model with nonlocal prey competition[J]. Nonlinear Analysis: Real World Applications, 2019, 48:12-39.
[13] YANG Feng, SONG Yongli. Stability and spatiotemporal dynamics of a diffusive predator-prey system with generalist predator and nonlocal intraspecific competition[J]. Mathematics and Computers in Simulation, 2022, 194:159-168.
[14] LIU Yaqi, DUAN Daifeng, NIU Ben. Spatiotemporal dynamics in a diffusive predator-prey model with group defense and nonlocal competition[J]. Applied Mathematics Letters, 2020, 103:106175.
[15] GENG Dongxu, JIANG Weihua, LOU Yuan, et al. Spatiotemporal patterns in a diffusive predator-prey system with nonlocal intraspecific prey competition[J]. Studies in Applied Mathematics, 2022, 148(1):396-432.
[16] SHEN Zuolin, LIU Yang, WEI Junjie. Double Hopf bifurcation in nonlocal reaction-diffusion systems with spatial average kernel[J]. Discrete and Continuous Dynamical Systems B, 2023, 28(4):2424-2462.
[17] YANG Ruizhi, WANG Fatao, JIN Dan. Spatially inhomogeneous bifurcating periodic solutions induced by nonlocal competition in a predator-prey system with additional food[J]. Mathematical Methods in the Applied Sciences, 2022, 45(16):9967-9978.
[18] KUANG Y. Delay differential equations: with applications in population dynamics[M]. New York: Academic Press, 1993.
[19] LIU Zhihua, YUAN Rong. Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response[J]. Journal of Mathematical Analysis and Applications, 2004, 296(2):521-537.
[20] BERETTA E, KUANG Y. Global analyses in some delayed ratio-dependent predator-prey systems[J]. Nonlinear Analysis: Theory, Methods & Applications, 1998, 32(3):381-408.
[21] MARTIN A, RUAN S. Predator-prey models with delay and prey harvesting[J]. Journal of Mathematical Biology, 2001, 43:247-267.
[22] HASSARD B D, KAZARINOFF N D, WAN Y H. Theory and applications of Hopf bifurcation[M]. Cambridge: Cambridge University Press, 1981.
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