Exact Nonclassical Symmetry Solutions of Lotka-Volterra Type Population Systems

Abstract

New classes of conditionally integrable systems of nonlinear reaction-diffusion equations are introduced. They are obtained by extending a well known nonclassical symmetry of a scalar partial differential equation to a vector equation. New exact solutions of nonlinear predator-prey systems, related to the diffusive Lotka-Volterra system, are constructed. An infinite dimensional class of exact solutions is made available. Unlike in the standard Lotka-Volterra system, in the absence of predators, the prey population has a finite carrying capacity, as in the Fisher equation.

Figures (8)

• Figure 1: $t=0$, $x=x_1, \ y=x_2$
• Figure 2: $t=\frac{\pi}{4},$$x=x_1, \ y=x_2$
• Figure 3: $t=\frac{\pi}{2}$, $x=x_1, \ y=x_2$
• Figure 4: $t=\pi$, $x=x_1, \ y=x_2$
• Figure 5: At $t=7\pi/3$, contours of prey population $\theta_2$=0.9, 0.924, 1.01, 1.02 ,1.04 with flow line of predator Species 1 from t=0 to 30, initial value $(x,y)=(1,-0.2)$.