A fractional prey-predator model with functional carrying capacity

In our ecosystem, predation is the primary interaction between different species. The relationship between prey and predators plays a crucial role in species survival and biodiversity conservation. To understand the long-term behavior of populations, researchers study the dynamics of predator-prey models. As a result, various models have been proposed to incorporate different biological phenomena. Typically, prey growth is modeled using a logistic equation that includes a growth rate and carrying capacity. In population biology, carrying capacity is defined as the maximum load an environment can support, and it is often treated as a constant value. However, due to environmental changes, carrying capacity can vary over time, and a species may even influence its own intrinsic carrying capacity [1]. Some models have been developed and analyzed to address this, often using delay differential equations [2,3]. In this work, we use fractional differential equations to introduce a general memory effect into the system. Additionally, we consider a model where prey’s growth is subject to the Allee effect, a phenomenon that describes a positive correlation between average individual fitness and population size [4]. So, at low population densities, the presence of conspecifics can increase the per capita growth rate. We will conduct stability and bifurcation analyses and provide numerical simulations to support our findings.