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: (I.) Class of reaction-diffusion equations partial derivative Q/partial derivative t + dD/dQ(partial derivative Q/partial derivative x)(2) + D(Q)partial derivative(2)Q/partial derivative x(2) + F(Q) = 0 (II.) Class of reaction-telegraph equations partial derivative Q/partial derivative t - alpha partial derivative(2)Q/partial derivative t(2) -beta partial derivative(2)Q/partial derivative x(2) - gamma dF/dQ partial derivative Q/partial derivative t - F(Q) = 0 Above alpha, beta, gamma 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. (C) 2009 Elsevier B.V. All rights reserved.