Population dynamic study of two prey one predator system with disease in first prey using fuzzy impulsive control

The prey-predator model provides a mathematical framework for understanding the population dynamics of interacting species, highlighting the delicate balance between predator and prey populations in ecological systems. The four-species predator-prey model extends the Lotka-Volterra framework to explore the dynamics of ecosystems with multiple interacting species. It provides a theoretical foundation for understanding how the populations of multiple prey and predator species influence each other over time. Apart from the traditional methods like direct approach for solving the non-linear system of equations, recent Fuzzy method approaches have been developed. The solution of non-linear systems using classical methods is not easy due to its non-linearity, analytical complexity, chaotic behavior, etc. and the T-S method is very much effective to analyze the non-linear models. In this study, we considered an eco-epidemic model with two populations of prey and one population of predator, with the only infectious disease infecting the first prey population. The four-dimensional Lotka-Volterra predator-prey system’s model stability has been examined using the Takagi-Sugeno (T-S) impulsive control model and the Fuzzy impulsive control model. Following the formulation of the model, the global stability and the Fuzzy solution are carried out through numerical simulations and graphical representations with appropriate discussion for a better understanding the dynamics of our proposed model. The Takagi-Sugeno method has diverse applications in modeling, control, pattern recognition, and decision-making in systems where uncertainty and non-linearity play a significant role. Its ability to combine fuzzy logic with traditional mathematical models provides a powerful tool for addressing complex real-world problems. The impulse control approach, what is considered within the foundation of fuzzy systems established on T-S model, is found to be suitable for extremely complex and non-linear systems with impulse effects.