Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics

We search for traveling-wave solutions of two classes of equations: of reaction-diffusion equations t + dD dQ ∂ Q ∂ x 2 + D ( Q ) ∂ 2 Q ∂ x 2 + F ( Q ) = 0 of reaction-telegraph equations t - α ∂ 2 Q ∂ t 2 - β ∂ 2 Q ∂ x 2 - γ dF dQ ∂ Q ∂ t - F ( Q ) = 0 α, β, γ are parameters and D and F depend on the population density Q. We obtain such solutions by the modified method of simplest equation for the cases when the simplest equation is the equation of Bernoulli or the equation of Riccati. On the basis of the appropriate ansatz the PDEs are reduced to nonlinear algebraic systems of relationships among the parameters of the equations and the parameters of the solution. By means of these systems we obtain numerous solutions for PDEs belonging to the investigated classes of equations.