Inverse problem of the Holling-Tanner model and its solution

Abstract

In this paper we undertake to consider the inverse problem of parameter identification of nonlinear system of ordinary differential equations for a specific case of complete information about solution of the Holling-Tanner model for finite number of points for the finite time interval. In this model the equations are nonlinearly dependent on the unknown parameters. By means of the proposed transformation the obtained equations become linearly dependent on new parameters functionally dependent on the original ones. This simplification is achieved by the fact that the new set of parameters becomes dependent and the corresponding constraint between the parameters is nonlinear. If the conventional approach based on introduction of the Lagrange multiplier is used this circumstance will result in a nonlinear system of equations. A novel algorithm of the problem solution is proposed in which only one nonlinear equation instead of the system of six nonlinear equations has to be solved. Differentiation and integration methods of the problem solution are implemented and it is shown that the integration method produces more accurate results and uses less number of points on the given time interval.

Parameter Identification of Lorenz System with Incomplete Information: Case of One Unknown Function

“…This is the main advantage of the proposed method. In the present paper, we considered three cases of incomplete information in the Lorenz system (1). In each of the cases, we assumed that only one of three variables was unknown.…”

“…where k 1 = k 2 + H, k 2 , k 3 are constant feedback coefficients, then system (77) becomes the Lorenz system of equations (1), where x = ω ξ ; y = ω η ; z = ω ξ ;…”

“…it persists under small perturbations of the system parameters. The method of the constrained least squares minimization that is used in the special formulation of the results presented in this paper was developed by the authors and applied in several studies (see [1,3,4]). For a comprehensive exposition on the Lorenz equations, see [5].…”