Fractal-fractional mathematical model of four species comprising of prey-predation

This work is devoted to the consideration of the four species N-1, N-2, N-3 and N-4 of syn-ecosymbiosis mathematical model under fractal fractional (FF) Caputo derivative. This problem has four compartmental species in which two compartments are predator and prey, while the other two are related with each other and also with predator and prey. The relation of the two species has been described by the concept of commensalism and mutualism. The Considered system has been investigated for qualitative analysis in sense of fractal fractional Caputo derivative by using some well-known theorem of fixed point theory. Also, some results for stability are developed by Ulam-Hyers (UH) concept of stability. The proposed system has been analyzed for a numerical solution using fractal-fractional Adams-Bashforth iterative techniques. The numerical procedure has been simulated to show the validity of the model under fractal dimension and fractional-order derivatives. The numerical techniques have been done for four sets of different data. All The sets of data provide comparable results in different arbitrary order and show that how to survive different species in an ecological system.