Hopf bifurcation in a diffusive predator-prey model with a square-root singularity
DOI:
https://doi.org/10.12775/TMNA.2021.024Keywords
Hopf bifurcation, reaction-diffusion system, predator-prey model, square-root singularityAbstract
In this paper, stability and Hopf bifurcation in a diffusive predator-prey system are discussed considering prey species' group behavior. The interaction term is of Holling type II with the prey density X under the square root. The local behavior is first discussed for the corresponding homogeneous system. Then, the diffusive system's linear stability is discussed around a homogeneous equilibrium state followed by bifurcations in the infinite-dimensional system. By the linear stability analysis, we see that a Hopf bifurcation occurs in the homogeneous system. By choosing a proper bifurcation parameter, we prove that a Hopf bifurcation occurs in the nonhomogeneous system. Furthermore, the direction of the Hopf bifurcation is obtained.References
V. Ajraldi, M. Pittavino and E. Venturino, Modeling Herd behavior in population systems, Nonlinear Anal. Real World Appl. 12 (2011), 2319–2338.
V. Ajraldi and E. Venturino, Mimicking spatial effects in predator-prey models with group defense, Proceedings of the 2009 International Conference on Computational and Mathematical Methods in Science and Engineering 1 (2009), 57–67.
I. Boudjema and S. Djilali, Turing–Hopf bifurcation in Gauss-type model with crossdiffusion and its application, Nonlinear Stud. 25 (2018), 665–687.
P.A. Braza, Predator-prey dynamics with square root functional responses, Nonlinear Anal. Real World. Appl. 13 (2012), 1837–43.
S. Djilali, Herd behavior in a predator-prey model with spatial diffusion: bifurcation analysis and Turing instability, J. Appl. Math. Comput. 58 (2018), 125–149.
S. Djilali, Impact of prey herd shape on the predator-prey interaction, Chaos Solitons Fractals 120 (2019), 139–148.
M. Haragus and G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, Universitext, Springer, London, Dordrecht, Heidelberg, New York, 2011.
A.J. Lotka, Relation between birth rates and death rates, Adv. Science 26 (1907), 21–22.
A.J. Lotka, Elements of Mathematical Biology, Dover, New York, 1956.
Y. Song and X. 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.
Y. Song, T. Yin and H. Shu, Dynamics of a ratio-dependent stage-structured predatorprey model with delay, Math. Methods Appl. Sci. 40 (2017), 6451–6467.
X. Tang and Y. Song, Bifurcation analysis and Turing instability in a diffusive predatorprey model with herd behavior and hyperbolic mortality, Chaos Solitons Fractals 81 (2015), 303–314.
E. Venturino, A minimal model for ecoepidemics with group defense, J. Biol. Syst. 19 (2011), 763–785.
E. Venturino and S. Petrovskiı̆, Spatiotemporal behavior of a prey-predator system with a group defense for prey, Ecol. Compl. 14 (2013), 37–47.
V. Volterra, Sui tentutive di applicazione delle mathematiche alle seienze biologiche e sociali, Ann. Radioelectr. Univ. Romandes 23 (1901), 436–458.
V. Volterra, La Concorrenza Vitale Tra le Specie Nellámbiente Marino, Soc. Nouv. Delı́mpr. du Loiret, 1931.
X.P. Yan, Stability and Hopf bifurcation for a delayed prey-predator system with diffusion effects, Appl. Math. Comp. 192 (2007), 552–566.
S. Yuan, C. Xu and T. Zhang, Spatial dynamics in a predator-prey model with herd behavior, Chaos 23 (2013), 033102.
X.C. Zhang, G.Q. Sun and Z. Jin, Spatial dynamics in a predator-prey model with Beddington–DeAngelis functional response, Phys. Rev. E 85 (2012), 0219241–02192414.
W.J. Zuo and J.J. Wei, Stability and bifurcation in a ratio-dependent Holling III system with diffusion and delay, Nonlinear Anal. Model. Control 19 (2014), 132–153.
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Copyright (c) 2022 Rasoul Asheghi
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