Department of Mathematics and Sciences, College of Humanities and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia.
Department of Mathematics, Air University, PAF Complex E-9, Islamabad 44000, Pakistan.
Math Biosci Eng. 2023 Jan 6;20(3):5066-5093. doi: 10.3934/mbe.2023235.
This research deals with formulating a multi-species eco-epidemiological mathematical model when the interacting species compete for the same food sources and the prey species have some infection. It is assumed that infection does not spread vertically. Infectious diseases severely affect the population dynamics of prey and predator. One of the most important factors in population dynamics is the movement of species in the habitat in search of resources or protection. The ecological influences of diffusion on the population density of both species are studied. The study also deals with the analysis of the effects of diffusion on the fixed points of the proposed model. The fixed points of the model are sorted out. The Lyapunov function is constructed for the proposed model. The fixed points of the proposed model are analyzed through the use of the Lyapunov stability criterion. It is proved that coexisting fixed points remain stable under the effects of self-diffusion, whereas, in the case of cross-diffusion, Turing instability exists conditionally. Moreover, a two-stage explicit numerical scheme is constructed, and the stability of the said scheme is found by using von Neumann stability analysis. Simulations are performed by using the constructed scheme to discuss the model's phase portraits and time-series solution. Many scenarios are discussed to display the present study's significance. The impacts of the transmission parameter 𝛾 and food resource f on the population density of species are presented in plots. It is verified that the availability of common food resources greatly influences the dynamics of such models. It is shown that all three classes, i.e., the predator, susceptible prey and infected prey, can coexist in the habitat, and this coexistence has a stable nature. Hence, in the realistic scenarios of predator-prey ecology, the results of the study show the importance of food availability for the interacting species.
本研究针对同一食物来源竞争的多物种生态流行病数学模型进行了构建，同时假设被捕食物种存在某种感染，且感染不会垂直传播。传染病会严重影响猎物和捕食者的种群动态。在种群动态中，一个最重要的因素是物种在栖息地中为寻找资源或保护而进行的移动。本研究探讨了扩散对两个物种的种群密度的生态影响。该研究还涉及对扩散对所提出模型的平衡点的影响进行分析。对模型的平衡点进行了分类。为所提出的模型构建了李雅普诺夫函数。利用李雅普诺夫稳定性准则对所提出模型的平衡点进行了分析。证明了在自扩散的影响下，共存平衡点保持稳定，而在交叉扩散的情况下，条件存在图灵不稳定性。此外，构建了一个两阶段显式数值方案，并通过冯·诺依曼稳定性分析发现了该方案的稳定性。利用构建的方案进行了模拟，以讨论模型的相图和时间序列解。讨论了许多场景以展示本研究的意义。通过绘图展示了传播参数𝛾和食物资源 f 对物种种群密度的影响。验证了共同食物资源的可得性对这类模型的动力学有重大影响。结果表明，栖息地中可以共存捕食者、易感猎物和受感染猎物这三个类别，而且这种共存具有稳定的性质。因此，在实际的捕食者-猎物生态场景中，研究结果表明食物可得性对相互作用的物种的重要性。
