Legendre wavelet method based solution of fractional order prey–predator model in type-2 fuzzy environment

Abstract

Researchers have recently examined several fractional-order prey–predator population models in a crisp or type-1 fuzzy environment. But incompleteness in the membership may also be present. In this regard, type-2 fuzzy numbers are employed in this inquiry to address the system’s imprecision since they are more realistic than type-1 fuzzy numbers in light of the uncertain membership grades. To numerically estimate the solution, the Legendre wavelet method (LWM) is used in conjunction with the r2-cut of the r1-plane form of the type-2 fuzzy parameters. To test the applicability of the present method, three cases are examined. The present solutions are quantitatively compared with the RK4 solutions in the crisp case. Approximations of the type-2 fuzzy prey and predator populations are also included. In addition, two- and three-dimensional plots are depicted to examine the behaviour of prey and predator populations in crisp and type-2 fuzzy cases. To assess the LWM’s efficacy, comparison graphs of the acquired solutions and the results of the RK4 approach are also provided. Numerical validation against the RK4 method in crisp cases demonstrates strong agreement, while revealing contrasting population behaviours in first two scenarios, whereas in the third case, both populations reduce with time. Further, the behaviour of the fuzzy solution has also been shown graphically using the footprint of uncertainty (FOU).