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Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics. (English) Zbl 1222.35201
Summary: We search for traveling-wave solutions of two classes of equations:
(I) Class of reaction-diffusion equations $\frac{\partial Q}{\partial t}+\frac{dD}{dQ} \left(\frac{\partial Q}{\partial x}\right)^2+D(Q) \frac{\partial^2Q}{\partial x^2}+F(Q)=0;$
(II) Class of reaction-telegraph equations $\frac{\partial Q}{\partial t}-\alpha\frac{\partial^2Q}{\partial t^2}- \beta\frac{\partial^2Q}{\partial x^2}-\gamma\frac{dF}{dQ} \frac{\partial Q}{\partial t}-F(Q)=0.$ 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 Bernoulli equation or the Riccati equation. On the basis of an 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.

##### MSC:
 35Q92 PDEs in connection with biology, chemistry and other natural sciences 35C07 Traveling wave solutions 92D25 Population dynamics (general) 92D40 Ecology 35K57 Reaction-diffusion equations
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