Study On The Estimation Of Parameters Of The 2-species Predator-prey Model

The research of mathematical biology model has got the extensive application through the development of a century. At the same time, estimation of the ordinary Differential equations (ODE) has appeared. Presently, many methods for estimating parameters from noisy data are computationally intensive and often fit poorly with the real data. Consequently, study for a method with high precision becomes more and more important.Lately, Ramsay etc. have given an estimation method of parameters in the 2-species predator-prey model by treating the spine functions as the basis system. However, they didn't make most use of qualitative property of the model such as the periodicity. In this paper, we introduce a new Generalized Profiling Method (GPM for short) to study the estimation of parameters in the 2-species predator-prey model. First, we identify the order of the model by treating the trigonometric function as the basis system and using the estimation method of the Hidden Periodic Model (HPM). Next, in order to estimate the structural parameters, we introduce the local and smoothing parameters and adopt different optimization criterion to estimate different parameter respectively. Then, we study the variances of the estimation value of these three distinct parameters by applying the Delta method. Moreover, we show the asymptotic behavior of our parameter estimate as smoothing parameterλtends to be infinite. Finally, combined with real data, numeric simulation has been done and shows that the method is truly feasible.This paper is composed of four parts. In the first chapter, we introduce the historical background of the problems which will be investigated, the main results and some definitions and lemmas which are required in this paper. In the second chapter, we discuss the estimation method of each parameter in the 2-species predator-prey model in detail. And in the third chapter, we give the estimation of variance of the estimation value of each parameter by using the Delta method and show that the asymptotic property that the estimation value tend to the real value as X tends to be infinite. At last, we produce a group of observation data by using the real value of the parameter. According to the observation data we use the method introduced in this paper to estimate the parameters, which shows the results are better than those given in pertinent references. So the method in this paper is feasible.