Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan.
Department of Mathematics, University of Narowal, Narowal, Pakistan.
Sci Rep. 2023 Feb 3;13(1):1990. doi: 10.1038/s41598-023-28324-6.
In this article, the ratio-dependent prey-predator system perturbed with time noise is numerically investigated. It relates to the population densities of the prey and predator in an ecological system. The initial prey-predator models only depend on the time and a couple of the differential equations. We are considering a model where the prey-predator interaction is influenced by both space and time and the need for a coupled nonlinear partial differential equation with the effect of the random behavior of the environment. The existence of the solutions is guaranteed by using Schauder's fixed point theorem. The computation of the underlying model is carried out by two schemes. The proposed stochastic forward Euler scheme is conditionally stable and consistent with the system of the equations. The proposed stochastic non-standard finite difference scheme is unconditionally stable and consistent with the system of the equations. The graphical behavior of a test problem for different values of the parameters is shown which depicts the efficacy of the schemes. Our numerical results will help the researchers to consider the effect of the noise on the prey-predator model.
本文通过数值研究，探讨了时变噪声干扰下的比例依赖型捕食者-猎物系统。该系统与生态系统中猎物和捕食者的种群密度有关。初始的捕食者-猎物模型仅依赖于时间和几个微分方程。我们考虑的模型中，捕食者-猎物的相互作用受到空间和时间的影响，需要一个耦合的非线性偏微分方程来描述环境随机行为的影响。通过使用 Schauder 不动点定理来保证解的存在性。通过两种方案对基础模型进行计算。提出的随机前向 Euler 格式是条件稳定的，与方程组一致。提出的随机非标准有限差分格式无条件稳定，与方程组一致。针对不同参数值的测试问题的图形行为表明了这些方案的有效性。我们的数值结果将帮助研究人员考虑噪声对捕食者-猎物模型的影响。
