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**Modelling biological populations in space and time.**
*(English)*
Zbl 0754.92018

Cambridge Studies in Mathematical Biology. 11. Cambridge: Cambridge University Press. xvii, 403 p. (1991).

The majority of mathematical models that appear in the vast literature on population dynamics and multi-species interactions utilize autonomous ordinary differential (or difference) equations which describe the rates of change of a population level statistic such as total population numbers or biomass. This is no doubt true because of the powerful tools that are available for the analysis of autonomous systems of ODE’s, both qualitative and quantitative. While valuable insights have been gained from these kinds of models (e.g. the concepts of logistic growth and carrying capacity, competitive exclusion and ecological niche, and predator-prey oscillations), in order to make further progress it is important to remember the simplifying assumptions upon which they are based. One way to think about these assumptions is that they reflect various kinds of homogeneities and uniformities that greatly simplify the mathematical equations that are utilized in the model. These include homogeneities in time, in space, and between individuals within the population, amongst others. Furthermore, these models are deterministic and there is the fundamental question about the validity of their predictions, particularly in a natural setting, given the often extreme stochasticity in natural populations and their environments.

The author’s primary goal in this book is to promote the use of both deterministic and stochastic models in conjunction with one another. While the relative tractability of simple deterministic models makes them valuable guides to the discovery of some basic principles and patterns in a highly complex biological world, the author argues that they should never be used in isolation from their stochastic counterparts, at least not without considerable risk. Since the exact relationship between a deterministic and a stochastic model is usually not known or is itself an intractable mathematical problem, this means that one must rely on stochastic simulations of the model. With the current availability of inexpensive, yet powerful computers, such model simulations can be done in sufficient numbers to make them a valuable analytical tool.

What the reader will find in the main portion of this book is a stochastic treatment of the most familiar deterministic differential equations in population dynamics, including the simple exponential and logistic (Pearl-Verhulst) equations, the Lotka-Volterra predator-prey and two species competition equations, Hutchinson’s delayed logistic, as well as some others perhaps less familiar model equations. The general format is to provide an analysis of the deterministic model and its asymptotic dynamics and then compare the results with simulation of a stochastic version of the model. In some cases considerable mathematical analysis of the stochastic model is also given, with interest focusing on the probability of extinction, the mean time to extinction, and other probabilistic events. In each study the author compares significant predictions of the deterministic model with the behavior of the stochastic model and draws conclusions about the robustness of the prediction when subjected to stochasticities in the birth and death processes.

Also treated are (compartmental) spatial models and, for single species, models in fluctuating environments (deterministic or stochastic). There is also a chapter on models of epidemics.

Readers well versed in theoretical population dynamics can profit from the author’s unifying treatment of deterministic and stochastic models. The author provides for those with less mathematical backgrounds by indicating sections of the book that contain technicalities that can be skipped, if desired. The book is well written and very readable. Both readers of mathematical and of biological interests should find the book of interest.

The author’s primary goal in this book is to promote the use of both deterministic and stochastic models in conjunction with one another. While the relative tractability of simple deterministic models makes them valuable guides to the discovery of some basic principles and patterns in a highly complex biological world, the author argues that they should never be used in isolation from their stochastic counterparts, at least not without considerable risk. Since the exact relationship between a deterministic and a stochastic model is usually not known or is itself an intractable mathematical problem, this means that one must rely on stochastic simulations of the model. With the current availability of inexpensive, yet powerful computers, such model simulations can be done in sufficient numbers to make them a valuable analytical tool.

What the reader will find in the main portion of this book is a stochastic treatment of the most familiar deterministic differential equations in population dynamics, including the simple exponential and logistic (Pearl-Verhulst) equations, the Lotka-Volterra predator-prey and two species competition equations, Hutchinson’s delayed logistic, as well as some others perhaps less familiar model equations. The general format is to provide an analysis of the deterministic model and its asymptotic dynamics and then compare the results with simulation of a stochastic version of the model. In some cases considerable mathematical analysis of the stochastic model is also given, with interest focusing on the probability of extinction, the mean time to extinction, and other probabilistic events. In each study the author compares significant predictions of the deterministic model with the behavior of the stochastic model and draws conclusions about the robustness of the prediction when subjected to stochasticities in the birth and death processes.

Also treated are (compartmental) spatial models and, for single species, models in fluctuating environments (deterministic or stochastic). There is also a chapter on models of epidemics.

Readers well versed in theoretical population dynamics can profit from the author’s unifying treatment of deterministic and stochastic models. The author provides for those with less mathematical backgrounds by indicating sections of the book that contain technicalities that can be skipped, if desired. The book is well written and very readable. Both readers of mathematical and of biological interests should find the book of interest.

Reviewer: J.M.Cushing (Tucson)

### MSC:

92D25 | Population dynamics (general) |

92D30 | Epidemiology |

92D40 | Ecology |

92-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to biology |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

34D99 | Stability theory for ordinary differential equations |