On the Extended Simple Equations Method (SEsM) and its Application to the Time--Fractional Diffusive Predator--Prey System incorporating an Allee Effect

In this paper, we extend the Simple Equations Method (SEsM) and adapt it to obtain exact solutions of systems of fractional nonlinear partial differential equations (FNPDEs). The novelty in the extended SEsM algorithm is that, in addition to introducing more simple equations in the construction of the solutions of the studied FNPDEs, we assume that the selected simple equations have different independent variables (i.e. different coordinates moving with the wave). As a consequence, nonlinear waves propagating with different wave velocities will be observed. A generalization of the SEsM algorithm is provided, including suggestions for possible solution constructions for the studied equations, potential traveling wave transformations, and types of simple equations that can be used. Several scenarios from the proposed generalized SEsM have been applied to the time-fractional predator--prey model under the Allee effect. Based on this, new analytical solutions have been derived. Numerical simulations of some of these solutions are presented, adequately capturing the expected diverse wave dynamics of predator--prey interactions.