On an Analytic Solution of a Class of Strictly non-Volterra Quadratic Stochastic Operators with Continuous Time

Abstract We study a continuous analog of strictly non-Volterra quadratic stochastic operators with parameters on a two-dimensional simplex. The explicit form of the fixed point and its type are found. For particular parameter values, an analytic solution of the Cauchy problem is obtained and a comparative analysis with the numerical solution of the system is conducted. We prove that the trajectories of the system on the invariant manifold converge to the equilibrium position exponentially rapidly. As an example, an analytic solution of the Cauchy problem for the linearized system in a neighborhood of the fixed point is presented, which demonstrates the vanishing on the invariant manifold of the coefficient of the exponential term with a positive eigenvalue in the exponent. An analytic solution of the system for general parameters is constructed as well. The solutions are represented in the form of special power series whose coefficients are determined from recurrence relations. It is shown that the representation of the solution is uniquely determined by the set of free coefficients. The uniform convergence of the constructed series on an infinite interval starting from a certain time is proved. The efficiency of this method is demonstrated by comparison with the numerical solution.

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