Stochastic population models. A compartmental perspective
Mathematical models are useful in formulating and refining hypotheses about biological processes and have attracted considerable amount of research efforts [see, e.g., \textit{R.B. Banks}, Growth and diffusion phenomena. Mathematical frameworks and applications. (1994; Zbl 0788.92001); \textit{J.A. Jacquez}, Compartmental analysis in biology and medicine. (1996; Zbl 0703.92001); \textit{E. Renshaw}, Modelling biological populations in space and time. (1991; Zbl 0754.92018); \textit{D.H. Anderson}, Compartmental modeling and tracer kinetics. (1983; Zbl 0509.92001)]. This book presents a unified treatment of stochastic discrete population-growth models. The authors develop models which not only describe but also explain changes in populations. Most of the models of interest are sets of differential equations, where the underlying parameters are related to the kinetics of the process. These kinetic models could be used to analyze population data, to make statistical inferences and to predict population size. Some of these models are quite sophisticated and incorporate detailed mechanistic concepts. The monograph focuses on the development of exact and approximate solutions for analogous stochastic models based on birth-death-migration processes. The models are illustrated with numerous practical applications. The monograph consists of three parts with a total of 14 chapters. The first part presents an overview and applications to four real world problems, viz., invasion of Africanized honey bees, Muskrat spread in the Netherlands, Bioaccumulation of mercury in fish and human calcium kinetics. The second part describes stochastic birth-death models for the growth of a single population. It develops new insights for nonlinear population models by employing an effective cumulant approach. The last part employs multi-dimensional birth-death-migration models for describing biological spread through a stochastic compartmental analysis. The development of stochastic linear multicompartmental modelling is extended by introducing the practical methodology of mean residual times and non-Markovian kinetics. The authors discuss the stochastic population models and their implications rather lucidly and with a good pragmatic approach. They have done considerable work in this field and this monograph is based largely on their own work. They present with some authority the practical motivation, utility and shortcomings of each application so that user communities will find these examples and ensuing discussions useful. The reviewer enjoyed reading this book which will be immensely useful to mathematicians and statisticians interested in biological modelling. Due to the attractive topics as well as the informal and vivid style often used in the presentation and description of the challenging biological problems, they will find this book absorbing and will benefit a lot from it. It is a valuable addition to the existing line of books on stochastic population models.
scientific article; zbMATH DE number 1480162