Statistical Analysis Of Ecological Mathematical Model Based On Differential Equation

The paper summed up several kinds of typical biomathematical models, by all-around analyzing the home and abroad developments of biostatistical and ecological mathematical theories. It outlined studies of the two fields and the deletions of their integrations. Logistic model and GM(1,1) model are definitely discussed on basis of statistical modeling methods and principles; the basic theories and principles of the seemingly unrelated linear model are applied in Logistic model groups with correlativity and the two-group Lotka-VIterra model. For the Logistic model,the GM(1,1) model, Logistic model groups with correlativity and the two-group Lotka-Vlterra model, the parameters are estimated with the two direction difference generalized weighted least squares estimate method, and the approach is used in practical problems. Mathematical ecologic in biomathematics including biostatistics and ecological species mathematical models, most of ecological species mathematical models are differential equations. Though the studies of biostatistics and ecological species mathematical models (differential equation) both have long history, complete system, abundant contents and perfect theory, the main point in the paper is how we can fuse and cross the two direction studies to advance the developments of biostatistics and ecological species mathematical model so that more practical problems can be solved. Ecological species mathematical models usually made qualitative representations and analyses only for species. Biostatistical methods and principles are used to estimate the parameters of ecological species mathematical models(differential equation), and optimize the parameters of the Logistic model and the GM(1,1)model from multi-direction and multi-angle, thereby the more practical application of ecological species differential equation model is obtained, and new development in estimating parameters came into being. The following are the main points and conclusions of this article:(1) By using multivariate statistical analytical methods and principles as a tool, Logistic function model is constructed, and the two direction difference generalized weighted least squares estimate method is put forward, and the initial predictive value is modified through weighed, and the fitting curve of the Logistic model is improved. The paper pointed out the inadequacy of the original understanding for the initial predictive value and the initial predictive value which is improved can optimize system prediction. (2) The GM(1,1) model of grey system whose parameter estimation is the main part of the GM (1,1) model is form of differential equations. The parameter estimation is processed well and optimized through modifying the weighted value and the initial predictive value in two direction difference so as to achieve the purpose of reducing the prediction error in this paper.(3) For some species, which are fitted for Logistic equation curve with correlativity, if the parameter estimation of the Logistic equation of every specie is studied respectively, inevitably lost the systematic of some species with correlativity, and thus lost the function of the system. In this paper, by using principles and methods of the seemingly unrelated linear model, interrelated modeling platform for Logistic models with correlativity is built. The parameter estimation has principles and basis of statistical estimation, then parameter estimation is optimized. In addition, the approach of issue-analyzing and problem-solving is holistic and systematic, rather than apart the system, analyze and study respectively, otherwise inevitably lost the function of the system. At last, the parameters of the Logistic equation system were estimated with the two direction difference generalized weighted least squares estimate method, examples of practical applications indicated that the results taking into account the relevance and systemic of Logistic quation system are superior to them without doing it.(4) The qualitative representation and analyses of the two species Lotka-Vlterra model, besides predator-prey model competitive model and cooperative model, have long history, complete system, abundant contents and perfect theory, so is the studies of biostatistics. On the other hand, statistical studies of the two-group Lotka-Vlterra model are missing. In this paper, the better ripe principles and methods of biostatistics are used to make statistical studies, such as parameter estimation and hypothesis test, and so on. a reference method for specific applications is obtained. Principles and methods of the seemingly unrelated linear model are used to estimate parameters of the two species Lotka-Vlterra model, and the approach is applied in two species relation of lynx and snowshoe hare. Quantitative analysis indicated predator-prey relationship of the two species, and got the environmental capital of the two species and the estimating value of stable equilibrium.(5) Based on using the sample data to build a linear statistical model and How to use different batches of samples and information to establish generalized linear model and seemingly unrelated linear models, the paper put forward to the air refreshing problem on the generalized least squares estimation of generalized linear models and seemingly unrelated linear models, present the problem taken in the fresh of the generalized least squares estimation, the problem got rid of the stale and the recursive algorithm formula of the air refreshing problem. Thus simplifying the calculation method of the generalized least squares estimation, increasing computing speed and efficiency and saving storage space of the computer. Finally the paper effectively solved the air refreshing problem of the recursive algorithm of the generalized least squares estimation of unknown parameters of generalized linear models and seemingly unrelated linear models.