Insight into Caputo Fractional-Order Extension of Lotka–Volterra Model with Emphasis on Immigration Effect

For over a hundred years, the model of prey–predator has been handled extensively. It proved that Lotka–Volterra (LV) models are not stable from all previous studies. In these models, there is divergent extinction one species or cyclic oscillation. There no solution for asymptotic stability models. this paper, Caputo fractional-order prey-two predators’ with an immigration effect investigated. Then, adding small term to prey populations studied. The extension both inclusion second predator and use derivatives. Although LV systems have extensively investigated, studies on our proposed nonlinear modifications. Grünwald–Letnikov method was applied find numerical solutions validate findings. Several different cases were studied reach confirmed conclusions. variation derivative in study. From results, terms positive stability. Hence, it can be concluded stabilize natural model. An optimal control strategy shown at end paper.