Exact and approximate solutions to the quasilinear parabolic system “predator-prey” with zero fronts

In this paper, we consider the second-order quasilinear parabolic system known in population biology as the predator-prey model and examine exact and approximate solutions with two zero fronts on which at least one of two unknown functions vanish; both these functions are positive between the fronts. We search for exact solutions in the form of polynomials in powers of the spatial variable with the coefficients depending on time. To construct approximate solutions, we propose a numerical algorithm, which is a combination of the collocation method based on the expansion of the right-hand sides by the radial basis functions and the finite-difference approximation of the derivatives in time. The algorithm is verified by model examples; the results obtained are consistent with the exact solutions found.