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

摘要

It is shown that many systems of nonlinear differential equations of interest in various fields are naturally embedded in a new family of differential equations. In this paper, we improve new and effective methods for nonautonomous systems and they produce new exact solutions to some important biological systems. The exact solution of predator and prey population for different particular cases has been derived. The numerical examples show that new exact solutions can be obtained for many biological systems such as SIR model, Lotka–Volterra model. The methods perform extremely well in terms of efficiency and simplicity to solve this historical biological models.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.