Fractional modelling of gradual incorporation of infected prey into the predator-prey system with consideration of seasonality

Prey-predator models are essential for understanding the dynamics of ecological systems, where predators consume prey and population fluctuations are influenced by factors such as birth rates, death rates, and resource availability. The Lotka-Volterra model is a classical example, where prey populations grow exponentially in the absence of predators, and predator populations increase with the availability of prey. However, real-world ecosystems become more complex, particularly when diseases affect prey populations. Infected prey may exhibit altered behaviours or reduced fitness, making them more vulnerable to predation, thereby impacting both prey and predator populations. To model these interactions more realistically, disease dynamics will be incorporated to investigate the effects of infections on species behaviour, survival, and ecosystem stability. The gradual introduction of infected prey into the population will be modelled using an advanced framework of partial differential equations, which includes fractional derivatives to capture memory effects and non-local interactions. Furthermore, the model will incorporate almost periodic functions to account for seasonality, effectively reflecting the cyclical nature of environmental fluctuations.