Turing instability and bifurcation in a predator-prey model with nonlocal fear and nonlinear cross-diffusion

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Abstract

This paper has studied a predator-prey model that incorporates a nonlocal fear effect and nonlinear cross-diffusion under homogeneous Neumann boundary conditions. We have derived necessary and sufficient conditions for Turing instability in the presence of both nonlocal fear and nonlinear cross-diffusion by means of linear stability analysis. Moreover, we have investigated steady-state bifurcations induced by the nonlocal fear effect using the Lyapunov-Schmidt reduction.

Keywords

predator-prey model / nonlinear cross-diffusion / nonlocal fear effect / Turing instability / steady-state bifurcation

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References

[1] L. Y. Zanette, A. F. White, M. C. Allen, M. Clinchy, Perceived predation risk reduces the number of offspring songbirds produce per year, Science, 334 (2011), 1398-1401. https://doi.org/10.1126/science.1210908

[2] X. Wang, L. Y. Zanette, X. Zou, Modelling the fear effect in predator-prey interactions, J. Math. Biol., 73 (2016), 1179-1204. https://doi.org/10.1007/s00285-016-0989-1

[3] X. Wang, X. Zou, Modeling the fear effect in predator-prey interactions with adaptive avoidance of predators, Bull. Math. Biol., 79 (2017), 1325-1359. https://doi.org/10.1007/s11538-017-0287-0

[4] P. Panday, N. Pal, S. Samanta, J. Chattopadhyay, Stability and bifurcation analysis of a three-species food chain model with fear, Int. J. Bifurcation Chaos, 28 (2018), 1850009. https://doi.org/10.1142/S0218127418500098

[5] J. Wang, Y. Cai, S. Fu, W. Wang, The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge, Chaos, 29 (2019), 083109. https://doi.org/10.1063/1.5111121

[6] C. Wang, H. Wang, S. Yuan, Complex bifurcations and noise-induced transitions: A predation model with fear effect in prey and crowding effect in predators, Discrete Contin. Dyn. Syst. B, 28 (2023), 3837-3867. https://doi.org/10.3934/dcdsb.2022243

[7] Q. Liu, D. Jiang, Influence of the fear factor on the dynamics of a stochastic predator-prey model, Appl. Math. Lett., 112 (2021), 106756. https://doi.org/10.1016/j.aml.2020.106756

[8] S. Chen, Z. Liu, J. Shi, Nonexistence of nonconstant positive steady states of a diffusive predatorprey model with fear effect, J. Nonlinear Modell. Anal., 1 (2019), 47-56. https://doi.org/10.12150/jnma.2019.47

[9] D. Duan, B. Niu, J. Wei, Hopf-Hopf bifurcation and chaotic attractors in a delayed diffusive predator-prey model with fear effect, Chaos Solitons Fractals, 123 (2019), 206-216. https://doi.org/10.1016/j.chaos.2019.04.012

[10] W. Li, L. Zhang, J. Cao, A note on Turing-Hopf bifurcation in a diffusive Leslie-Gower model with weak Allee effect on prey and fear effect on predator, Appl. Math. Lett., 172 (2026), 109741. https://doi.org/10.1016/j.aml.2025.109741

[11] R. Han, L. N. Guin, B. Dai, Cross-diffusion-driven pattern formation and selection in a modified Leslie-Gower predator-prey model with fear effect, J. Biol. Syst., 28 (2020), 27-64. https://doi.org/10.1142/s0218339020500023

[12] Z. Bi, S. Liu, M. Ouyang, X. Wu, Pattern dynamics analysis of spatial fractional predator- prey system with fear factor and refuge, Nonlinear Dyn., 111 (2023), 10653-10676. https://doi.org/10.1007/s11071-023-08353-6

[13] B. Dai, G. Sun, Turing-Hopf bifurcation of a delayed diffusive predator-prey system with chemotaxis and fear effect, Appl. Math. Lett., 111 (2021), 106644. https://doi.org/10.1016/j.aml.2020.106644

[14] Y. Lv, Turing-Hopf bifurcation analysis and normal form in delayed diffusive predator- prey system with taxis and fear effect, J. Appl. Math. Comput., 70 (2024), 5721-5761. https://doi.org/10.1007/s12190-024-02183-4

[15] J. Furter, M. Grinfeld, Local vs. non-local interactions in population dynamics, J. Math. Biol., 27 (1989), 65-80. https://doi.org/10.1007/bf00276081

[16] X. Dong, B. Niu, On a diffusive predator-prey model with nonlocal fear effect, Appl. Math. Lett., 132 (2022), 108156. https://doi.org/10.1016/j.aml.2022.108156

[17] X. Sun, Dynamics of a diffusive predator-prey model with nonlocal fear effect, Chaos Solitons Fractals, 177 (2023), 114221. https://doi.org/10.1016/j.chaos.2023.114221

[18] H. Yuan, J. Wu, Y. Jia, H. Nie, Coexistence states of a predator-prey model with cross-diffusion, Nonlinear Anal. Real World Appl., 41 (2018), 179-203. https://doi.org/10.1016/j.nonrwa.2017.10.009

[19] W. Li, L. Yang, J. Cao, Threshold dynamics of a degenerated diffusive incubation perio host-Cpathogen model with saturation incidence rate, Appl. Math. Lett., 160 (2025), 109312. https://doi.org/10.1016/j.aml.2024.109312

[20] W. Li, L. Zhang, J. Cao, F. Xu, Z. Cai, Finite time attractivity and exponentially stable of a multi- stage epidemic system with discontinuous incidence, Qual. Theory Dyn. Syst., 24 (2025), 199. https://doi.org/10.1007/s12346-025-01358-z

[21] A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B Biol. Sci., 237 (1952), 37-72. https://doi.org/10.1098/rstb.1952.0012

[22] L. N. Guin, Existence of spatial patterns in a predator-prey model with self-and cross-diffusion, Appl. Math. Comput., 226 (2014), 320-335. https://doi.org/10.1016/j.amc.2013.10.005

[23] Z. Wen, S. Fu, Turing instability for a competitor-competitor-mutualist model with nonlinear cross-diffusion effects, Chaos Solitons Fractals, 91 (2016), 379-385. https://doi.org/10.1016/j.chaos.2016.06.019

[24] B. Liu, R. Wu, L. Chen, Patterns induced by super cross-diffusion in a predator- prey system with Michaelis-Menten type harvesting, Math. Biosci., 298 (2018), 71-79. https://doi.org/10.1016/j.mbs.2018.02.002

[25] C. Liu, S. Guo, Steady states of Lotka-Volterra competition models with nonlinear cross-diffusion, J. Differ. Equations, 292 (2021), 247-286. https://doi.org/10.1016/j.jde.2021.05.014

[26] M. Fu, P. Liu, Q. Shi, Turing bifurcation in activator-inhibitor (depletion) models with cross-diffusion and nonlocal terms, Stud. Appl. Math., 153 (2024), e12749. https://doi.org/10.1111/sapm.12749

[27] Y. Song, H. Wang, J. Wang, Cognitive consumer-resource spatiotemporal dynamics with nonlocal perception, J. Nonlinear Sci., 34 (2024), 19. https://doi.org/10.1007/s00332-023-09996-w

[28] J. Cao, H. Sun, P. Hao, P. Wang, Bifurcation and Turing instability for a predator-prey model with nonlinear reaction cross-diffusion, Appl. Math. Modell., 89 (2021), 1663-1677. https://doi.org/10.1016/j.apm.2020.08.030