Department of Mathematics and Statistics, IIT Kanpur, Kanpur 208016, India.
Math Biosci. 2019 Apr;310:96-107. doi: 10.1016/j.mbs.2019.01.011. Epub 2019 Feb 5.
Non-local interaction describes the effects of mobility of a population species in their spatial locations. Non-local interaction is incorporated into a prey-predator model by introducing an integral term with appropriate kernel function. The kernel function determines the nature and range of the accessibility of the resources. We consider a nonlocal spatio-temporal prey-predator model with additive weak Allee effect in prey growth and density-dependent predator mortality. The dynamics of the model is examined in presence of a parabolic and a triangular kernel functions. Comparisons are made between the resulting dynamics of the nonlocal model with these kernels for the same range of nonlocal interaction. A linear stability analysis is performed for both the kernels to derive Turing and spatial-Hopf bifurcation conditions. In general, Turing bifurcation curves are different for the two kernels and the same is true for spatial-Hopf bifurcation curves. This results in different dynamics for the two kernels for some parameter values. However, for a fixed range of nonlocal interaction, the dynamics of the model with these two kernels are almost similar for most of the parameter values explored. Increase in the range of nonlocal interaction up to a certain value stabilizes the system dynamics, and then, it destabilizes. Extensive numerical simulations are performed to illustrate the system dynamics.
非局部相互作用描述了种群在其空间位置上的流动性对种群的影响。通过引入具有适当核函数的积分项，将非局部相互作用纳入到捕食者-被捕食者模型中。核函数决定了资源的可及性的性质和范围。我们考虑了一个具有附加弱阿利效应的非局部时空调食者-被捕食者模型，其中猎物的生长和密度依赖的捕食者死亡率是依赖于密度的。在存在抛物线和三角形核函数的情况下，对模型的动力学进行了检验。对于相同的非局部相互作用范围，比较了这两种核函数对非局部模型的动力学的影响。对两种核函数都进行了线性稳定性分析，以推导出 Turing 和空间 Hopf 分岔条件。一般来说，对于这两种核函数，Turing 分岔曲线是不同的，空间 Hopf 分岔曲线也是如此。这导致了对于两种核函数来说，在某些参数值下会有不同的动力学。然而，对于固定的非局部相互作用范围，对于大多数探索的参数值来说，这两个核函数的模型动力学几乎是相似的。非局部相互作用范围的增加会稳定系统的动力学，然后又会使其不稳定。进行了广泛的数值模拟来演示系统的动力学。
