Stability dynamics of a nonlocal interaction model with memory and Allee effect

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Abstract

Interaction dynamics among species yield to Allee effects, spatial memory, and nonlocal competition. When predators are not subject to the Allee effect, the coexistence equilibrium point remains locally asymptotically stable despite nonlocal competition. However, in the presence of the Allee effect, nonlocal competition leads to the destabilization of the coexistence point. Moreover, the model will undergo stability switches, Hopf bifurcation, and Turing bifurcation, and induced complex dynamics will appear. It is also found that when the memory diffusion is small, it has no effect on the stability switches; as it increases, only the nonlocal model exhibits the spatially inhomogeneous Hopf bifurcation. When it exceeds the maximum threshold, this phenomenon occurs in both models. Furthermore, when the memory diffusion coefficient exceeds the threshold, the stability range of the coexistence equilibrium will decrease in both the local and nonlocal models. Finally, numerical simulations verify the theoretical results.

Keywords

nonlocal competition / spatial memory / Allee effect / Hopf bifurcation / Turing bifurcation

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References

[1] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen., 7(1937), 355-369. https://doi.org/10.1111/j.1469-1809.1937.tb02153.x

[2] A. Kolmogoroff, I. Petrovsky, N. Piscounoff, Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem, in Dynamics of Curved Fronts(ed. P. Pelcé), Academic Press, San Diego, 1988, 105-130. https://doi.org/10.1016/B978-0-08-092523-3.50014-9

[3] 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

[4] N. Apreutesei, N. Bessonov, V. Volpert, V. Vougalter, Spatial structures and generalized travelling waves for an integro-differential equation, Discrete Contin. Dyn. Syst. B, 13(2010), 537-557. https://doi.org/10.3934/dcdsb.2010.13.537

[5] O. Aydogmus, Patterns and transitions to instability in an intraspecific competition model with nonlocal diffusion and interaction, Math. Model. Nat. Phenom., 10(2015), 17-29. https://doi.org/10.1051/mmnp/201510603

[6] M. Banerjee, V. Vougalter, V. Volpert, Doubly nonlocal reaction-diffusion equations and the emergence of species, Appl. Math. Model., 42(2017), 591-599. http://doi.org/10.1016/j.apm.2016.10.041

[7] S. M. Merchant, W. Nagata, Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition, Theor. Popul. Biol., 80(2011), 289-297. https://doi.org/10.1016/j.tpb.2011.10.001

[8] S. H. Wu, Y. L. Song, Stability and spatiotemporal dynamics in a diffusive predator-prey model with nonlocal prey competition, Nonlinear Anal. Real World Appl., 48(2019), 12-39. https://doi.org/10.1016/j.nonrwa.2019.01.004

[9] Q. Y. Shi, J. P. Shi, Y. L. Song, Effect of spatial average on the spatiotemporal pattern formation of reaction-diffusion systems, J. Dyn. Differ. Equ., 34(2022), 2123-2156. https://doi.org/10.1007/s10884-021-09995-z

[10] D. F. Duan, B. Niu, J. J. Wei, Y. Yuan, The dynamical analysis of a nonlocal predator-prey model with cannibalism, Eur. J. Appl. Math., 35(2024), 1-25. https://doi.org/10.1017/S0956792524000019

[11] X. S. Dong, B. Niu, L. Wang, Spatiotemporal patterns induced by nonlocal prey competition and prey-taxis in a diffusive Rosenzweig-MacArthur system, Nonlinear Anal. Real World Appl., 91(2026), 104561. https://doi.org/10.1016/j.nonrwa.2025.104561

[12] J. Li, Y. T. Ding, New mechanism for spatial heterogeneity pattern revealed by nonlocal competition and host-taxis in a 2D pine wilt disease model, J. Math. Biol., 92(2026), 19. https://doi.org/10.1007/s00285-025-02336-4

[13] D. X. Geng, W. H. Jiang, Y. Lou, H. B. Wang, Spatiotemporal patterns in a diffusive predator-prey system with nonlocal intraspecific prey competition, Stud. Appl. Math., 148(2022), 396-432. https://doi.org/10.1111/sapm.12444

[14] Y. H. Peng, G. Y. Zhang, Dynamics analysis of a predator-prey model with herd behavior and nonlocal prey competition, Math. Comput. Simul., 170(2020), 366-378. https://doi.org/10.1016/j.matcom.2019.11.012

[15] P. Kareiva, G. Odell, Swarms of predators exhibit “preytaxis” if individual predators use area-restricted search, Am. Nat., 130(1987), 233-270. https://doi.org/10.1086/284707

[16] S. A. Budick, M. H. Dickinson, Free-flight responses of Drosophila melanogaster to attractive odors, J. Exp. Biol., 209(2006), 3001-3017. https://doi.org/10.1242/jeb.02305

[17] I. Mori, Y. Ohshima, Molecular neurogenetics of chemotaxis and thermotaxis in the nematode Caenorhabditis elegans , BioEssays, 19(1997), 1055-1064. https://doi.org/10.1002/bies.950191204

[18] W. F. Fagan, M. A. Lewis, M. Auger-Methe, T. Avgar, S. Benhamou, G. Breed, et al., Spatial memory and animal movement, Ecol. Lett., 16(2013), 1316-1329. https://doi.org/10.1111/ele.12165

[19] C. H. Wang, S. L. Yuan, H. Wang, Spatiotemporal patterns of a diffusive prey-predator model with spatial memory and pregnancy period in an intimidatory environment, J. Math. Biol., 84(2022), 12. https://doi.org/10.1007/s00285-022-01716-4

[20] Y. H. Peng, K. Yu, Y. J. Li, Effect of spatial memory on a predator-prey model with herd behavior, Int. J. Biomath., 16(2023), 2350082. https://doi.org/10.1142/S1793524523500821

[21] D. Y. Wu, F. P. Lu, C. S. Shen, J. Gao, Impact of spatial memory on a predator-prey system with Allee effect, Int. J. Bifurcat. Chaos, 33(2023), 2350086. https://doi.org/10.1142/S0218127423500864

[22] D. Liu, W. H. Jiang, Hopf bifurcation in a memory-based diffusion predator-prey model with spatial heterogeneity, J. Differ. Equ., 397(2024), 377-403. https://doi.org/10.1016/j.jde.2024.04.015

[23] Y. L. Song, Y. H. Peng, T. H. Zhang, The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system, J. Differ. Equ., 300(2021), 597-624. https://doi.org/10.1016/j.jde.2021.08.010

[24] H. Shen, Y. L. Song, H. Wang, Bifurcations in a diffusive resource-consumer model with distributed memory, J. Differ. Equ., 347(2023), 170-211. https://doi.org/10.1016/j.jde.2022.11.044

[25] K. Manna, M. Banerjee, Dynamics of a prey-predator model with reproductive Allee effect for prey and generalist predator, Nonlinear Dyn., 112(2024), 7727-7748. https://doi.org/10.1007/s11071-024-09451-9

[26] A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific, Singapore, 1998. https://doi.org/10.1142/2284

[27] L. Berec, Impacts of foraging facilitation among predators on predator-prey dynamics, Bull. Math. Biol., 72(2010), 94-121. https://doi.org/10.1007/s11538-009-9439-1

[28] A. Kumar, K. P. Reshma, P. Shri Harine, Global dynamics of an ecological model in presence of fear and group defense in prey and Allee effect in predator, Nonlinear Dyn., 113(2025), 7483-7518. https://doi.org/10.1007/s11071-024-10706-8

[29] Y. W. Yang, D. Y. Wu, C. S. Shen, F. P. Lu, Allee effect in a diffusive predator-prey system with nonlocal prey competition, Physica A, 615(2023), 128606. https://doi.org/10.1016/j.physa.2023.128606

[30] J. LaSalle, Some extensions of Liapunov’s second method, IRE Trans. Circuit Theory, 7(1960), 520-527. https://doi.org/10.1109/TCT.1960.1086720