On the numerical study of periodic solutions to delay equations in biological models

We present the results of a numerical study of periodic solutions of to a nonlinear delay equation in connection with mathematical models having a real biological prototype. The problem is formulated as a boundary value problem for a delay equation with periodicity and transversality conditions. We propose a spline-collocation finite-difference scheme of the boundary value problem using the Hermite interpolation cubic spline of class $C^1$ with fourth-order error. For the numerical study of the system of nonlinear equations of the difference scheme, the parameter-extension method is used, which allows us to identify the possible nonuniqueness of a solution to the boundary value problem and hence the nonuniqueness of periodic solutions regardless of their stability. It is shown by examples that periodic oscillations arise for values of the parameters typical for real molecular-genetic systems of higher organisms, for which the principle of “delay” is rather easy to implement.