Smooth Models of Biological Populations

We propose a method for constructing models to express the size of biological populations based on time series. At the first stage we construct a smoothed-out collection of empirical data reflecting the common features of an actual time series by using an optimizing spline, which is a piecewise polynomial function at the minimal distance from the empirical data in the sense of the least-squares method. Then we construct a system of ODEs which has the minimal least-squares distance from the derivative of the optimizing spline on certain finer lattice. We take the solution to the Cauchy problem for this system as the forecast by the model. We apply the method to concrete time series, estimate the error of the forecast and study its dependence on the parameters of the method. In addition, we apply the method to an artificial time series containing random perturbations. We study the dependence of error in the forecast on the size of random perturbation.