Generalized system of trial equation methods and their applications to biological systems

The Lotka-Volterra nonlinear differential equations for two competing species, namely X and Y, contain six independent parameters. Their general analytic solutions, valid for arbitrary values of the parameters, are at present unknown. However, when two or more of these parameters are interrelated, it is possible to obtain the exact solutions in the X, Y phase plane, and six cases of solvability are given in this paper. The dependence of the solutions on the parameters and the initial conditions can thus be readily investigated.