Parameter Estimation in Ecological Models: The Link Between Data and Models

This book combines ecological model building and the application of advanced statistical methods for parameter estimation. Written for model builders and statisticians in the field of ecology, it provides the statistical and numerical tools needed to estimate parameters from experimental data. These tools range from standard methods to highly advanced ones for parameter identification using ordinary differential equations. Detailed examples based on real data illustrate their use. It provides the necessary link between data and models, between simulation and statistics. The book starts with an introductory chapter which outlines the scope of problems to be treated in later chapters. Chapter 2 provides the statistical means. The first part of this chapter is mainly addressed to readers unfamiliar with statistical theory. The second part of this chapter contains some advanced techniques of parameter estimation in ordinary differential equations which might be of interest even to the experienced statistician. Chapter 3 is concerned with parameter estimation in models formulated as probability distribution functions. Since the most basic problem in many ecological studies is the estimation of species abundance, the first part of this chapter is concerned with spatial patterns and density estimation. The second part of this chapter is devoted to survival distributions. As a special feature, the influence of time dependent covariates such as temperature on the survival function is treated in the frame of the accelerated life model, which is widely used in medical statistics. In chapter 4 population models for age structured populations and for populations exhibiting both age structure and development stages are developed starting from the classical Leslie theory. In addition the book demonstrates how one reduces the number of parameters to be estimated by deriving the elements of the population projection matrix from parametric survival distributions. Chapter 5 treats models in the form of ordinary differential equations. The methods demonstrated here are advanced from the numerical point of view. As typical applications, parameter estimation for a plant growth model and a competition model is performed by use of a recently developed boundary value method.

Table of Contents:
Part 1 Introduction: parameter estimation - the link between data and model; introducing models and parameters; deterministic dynamic models; stochastic models. Part 2 Mathematical and statistical tools: probabilty distributions; discrete random variables; the binomial distribution;the hypergoemetric distribution the Poisson distribution; continuous random variables; the exponential distribution; the normal distribution; parameter estimation; maximum likelihood estimation; explicit deterministic models; general statement of the regression model; solution of the linear regression model; parameter estimation of nonlinear models; models defined by differential equations; introductory example - the classical Lotka-Volterra competition model; numerical procedures; the use of program packages; nonlinear regression with BMDP - analysis of growth data; nonlinear problems involving differential equations. Part 3 Models in the form of probability distributions: spatial processes and density estimation; the necessity of identifying types of spatial distributions; simulation of spatial patterns; mathematical models for spatial distributions; random distributions; patchy distributions; indexes of patchiness and the effect of scaling; methods of density estimation; pattern detection based on quadrat counts; estimating density by quadrat sampling; stocked quadrat surveys; distance methods; line transect methods; capture recapture methods; survival distributions and estimation of mean survival times; basic concepts; some special distributions; environmental variables as covariates; survival curves of populations. Part 4 Discrete population dynamic models: introducing discrete models - the growth of single populations; the Leslie process; introduction of age structure into population models; the model equations; parammeter estimation; survival probabilities; fertility rates; the generalized Leslie process; the model equations; parameter estimation. Part 5 Continuous ecosystem models: plant growth models; descriptive versus explanatory models; classical growth models; introduction of control functions; one dimensional models; multiexperiment regression models; whole plant models; ecosystem models; mathematical structure of continuous ecosystem models; multiexperiment problems in the case of the classical Lotka-Volterra competition model; analysis of the model; experimental design and analysis for parameter estimation. Appendices: probability and statistics; convergence of iterative schemes; evaluation of maximum likelihood estimators of the negative binomial distribution.