Simple Summary
This paper presents a new mathematical model that explores how predators, prey, and diseases interact in environments that change over time and space. The model combines several important ecological and epidemiological factors, including the movement of species across landscapes, the hunting behavior of predators (described by the Beddington–DeAngelis functional response), and the spread of diseases among prey populations. This comprehensive approach makes the model more realistic and useful for understanding complex natural systems. Key theoretical findings include proving that the model always produces biologically meaningful solutions for any set of initial conditions, ensuring its reliability over long periods. The study also identifies conditions under which species can coexist despite ongoing disease threats, helping to explain how biodiversity is maintained. Additionally, the model shows that, under certain time-varying conditions, there can be stable, repeating patterns in species populations, reflecting natural rhythms like seasons. The stability of these patterns is further confirmed, providing a solid basis for predicting long-term ecological outcomes. These theoretical insights offer practical tools for addressing real-world challenges, such as predicting when diseases might invade new areas, evaluating how diseases persist in different habitats, and designing strategies to manage wildlife and protect ecosystems. A numerical example is included to demonstrate the model’s accuracy and usefulness in real-world scenarios, highlighting its potential to guide policy and promote sustainable ecosystem management.
Abstract
Based on existing models, this paper incorporates some key ecological factors, thereby obtaining a class of eco-epidemiological models that can more objectively reflect natural phenomena. This model simultaneously integrates disease dynamics within the prey population and the Beddington–DeAngelis functional response, thus achieving an organic combination of ecological dynamics, epidemic transmission, and spatial movement under time-varying environmental conditions. The proposed framework significantly enhances ecological realism by simultaneously accounting for spatial dispersal, predator–prey interactions, disease transmission within prey species, and seasonal or temporal variations, providing a comprehensive mathematical tool for analyzing complex eco-epidemiological systems. The theoretical results obtained from this study can be summarized as follows: Firstly, the existence and uniqueness of globally positive solutions for any positive initial data are rigorously established, ensuring the well-posedness and biological feasibility of the model over extended temporal scales. Secondly, analytically tractable sufficient conditions for uniform population persistence are derived, which elucidate the mechanisms of species coexistence and biodiversity preservation even under sustained epidemiological pressure. Thirdly, by employing innovative applications of differential inequalities and fixed point theory, the existence and uniqueness of a positive spatially homogeneous periodic solution in the presence of time-periodic coefficients are conclusively demonstrated, capturing essential rhythmicities inherent in natural systems. Fourthly, through a sophisticated combination of the upper and lower solution method for parabolic partial differential equations and Lyapunov stability theory, the global asymptotic stability of this periodic solution is rigorously established, offering a powerful analytical guarantee for long-term predictive modeling. Beyond theoretical contributions, these research findings provide actionable insights and quantitative analytical tools to tackle pressing ecological and public health challenges. They facilitate the prediction of thresholds for maintaining ecosystem stability using real-world data, enable the analysis and assessment of disease persistence in spatially structured environments, and offer robust theoretical support for the planning and design of wildlife management and conservation strategies. The derived criteria support evidence-based decision-making in areas such as controlling zoonotic disease outbreaks, maintaining ecosystem stability, and mitigating anthropogenic impacts on ecological communities. A representative numerical case study has been integrated into the analysis to verify all of the theoretical findings. In doing so, it effectively highlights the model’s substantial theoretical value in informing policy-making and advancing sustainable ecosystem management practices.