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

Youwei Yang; Ranchao Wu; Qigang Deng; Youwei Yang; Ranchao Wu; Qigang Deng

doi:10.3934/nhm.2026042

Networks and Heterogeneous Media

2026, Volume 21, Issue 3: 1017-1040. doi: 10.3934/nhm.2026042

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

School of Mathematical Sciences, Anhui University, Hefei 230601, China

Received: 14 April 2026 Revised: 06 May 2026 Accepted: 08 May 2026 Published: 18 May 2026

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.
- nonlocal competition,
- spatial memory,
- Allee effect,
- Hopf bifurcation,
- Turing bifurcation
Citation: Youwei Yang, Ranchao Wu, Qigang Deng. Stability dynamics of a nonlocal interaction model with memory and Allee effect[J]. Networks and Heterogeneous Media, 2026, 21(3): 1017-1040. doi: 10.3934/nhm.2026042

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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.

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