Department of Mathematics, NIT Patna, Bihar 800005, India.
MS2 Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo N2L3C5, Canada.
Chaos. 2023 Oct 1;33(10). doi: 10.1063/5.0167466.
Pattern formation is a central process that helps to understand the individuals' organizations according to different environmental conditions. This paper investigates a nonlocal spatiotemporal behavior of a prey-predator model with the Allee effect in the prey population and hunting cooperation in the predator population. The nonlocal interaction is considered in the intra-specific prey competition, and we find the analytical conditions for Turing and Hopf bifurcations for local and nonlocal models and the spatial-Hopf bifurcation for the nonlocal model. Different comparisons have been made between the local and nonlocal models through extensive numerical investigation to study the impact of nonlocal interaction. In particular, a legitimate range of nonlocal interaction coefficients causes the occurrence of spatial-Hopf bifurcation, which is the emergence of periodic patterns in both time and space from homogeneous periodic solutions. With an increase in the range of nonlocal interaction, the whole Turing pattern suppresses after a certain threshold, and no pure Turing pattern exists for such cases. Specifically, at low diffusion rates for the predators, nonlocal interaction in the prey population leads to the extinction of predators. As the diffusion rate of predators increases, impulsive wave solutions emerge in both prey and predator populations in a one-dimensional spatial domain. This study also includes the effect of nonlocal interaction on the invasion of populations in a two-dimensional spatial domain, and the nonlocal model produces a patchy structure behind the invasion where the local model predicts only the homogeneous structure for such cases.
模式形成是一种有助于根据不同环境条件理解个体组织的核心过程。本文研究了具有种群中被捕食者正相互作用和捕食者中捕食合作的被捕食者-捕食者模型的非局部时空行为。在种内被捕食者竞争中考虑了非局部相互作用，我们发现了局部和非局部模型的 Turing 和 Hopf 分岔的分析条件，以及非局部模型的空间-Hopf 分岔。通过广泛的数值研究对局部和非局部模型进行了不同的比较，以研究非局部相互作用的影响。特别是，在一定的非局部相互作用系数范围内，会导致空间-Hopf 分岔的发生，即从均匀周期解中出现时空周期模式。随着非局部相互作用范围的增加，在一定阈值之后，整个 Turing 模式会被抑制，并且在这种情况下不存在纯 Turing 模式。具体而言，在捕食者的低扩散率下，被捕食者种群中的非局部相互作用会导致捕食者的灭绝。随着捕食者扩散率的增加，在一维空间域中，捕食者和被捕食者种群中都会出现脉冲波解。本研究还包括非局部相互作用对二维空间域中种群入侵的影响，并且非局部模型在这种情况下会产生斑块结构，而局部模型仅预测均匀结构。
