Application of the caputo fractional domain in stage structured predator-prey dynamics

One of the most exciting fields of applied mathematics in recent years has been the study of biological species behavior. The significance of protecting endangered species in nature has led academics from a variety of disciplines to focus on this problem. The development of fractional order derivatives and new ideas in numerical analysis have made it possible for the researchers to save more details about the development of the behaviour of the dynamical systems through time and to use those details for future traits. Sea lions are the prey and sharks are the predators. The prey population is divided into two categories: immature and mature prey. In this model, suppose that the shark (predator) feeds on both mature and immature sea lions, followed by a Crowley-Martin type functional response (CMFR), and that there would be intra-species competition among sea lions as well as sharks. Caputo fractional derivative is used to highlight the link between mature sea lions, juvenile sea lions, and sharks. In this study, we examine the given solution's positivity, boundedness, existence, and uniqueness and shown permanence of the system is highly dependent on the predation rate, conversion rate, death rate and competition rate. Sharks and sea lions are two key components of the marine environment, specifically decided to focus on these two population groups for research. As shark feeds on both adult and juvenile sea lion, they are coupled with predator-prey interaction to produce a complex combined effect as regulators of their population sizes. Caputo fractional derivative is concluded to glean further knowledge about the dynamics of this interaction. The predator feeds not just on mature prey but also on immature prey because they are unable to defend themselves or hide. On prey-predator dynamics, the effect of fractional derivative as well as the influences of various parameters, including the predation rate of mature sea lions, the rate at which mature sea lions turn into sharks, the rate at which immature sea lions die naturally, the rate at which immature sea lions are preyed upon, and the rate at which mature sea lions compete with one another, have been investigated numerically. The number of sea lions and sharks are directly correlated. One of the most significant findings indicate that the predator population is under control, even while the rate of predation is on the rise since not all predators survive for a variety of reasons, including diseases, competition for food, and its own predators. With the aid of a new fractional operator and projected technique, the analysis of the current dynamical system can assist us in comprehending a variety of models in predator-prey ecosystems that are home to fascinating phenomena.