Renshaw, E. (2010) Stochastic Population Processes: Analysis, Approximations and Simulations. Oxford University Press.

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In the real world the vast majority of random processes have no memory. That is, the next step in their development depends only on the current state of the system and not on its previous history. Stochastic realizations are then defined in terms of the sequence of event-time pairs, whence the equations for the probability that a system is of size n at time t are straightforward to write down. So from a purely applied perspective the mathematical challenge lies in determining their solution in order to enhance understanding of the particular system under study. Now preceeding the Second World War deep and insightful theoretical developments, pioneered by A.A. Markov and A.N. Kolmogorov, enabled many pure mathematically oriented results to be constructed. Whilst these have considerable intrinsic beauty, which stimulated great interest amongst pure probabilists, applied probabilists seeking solutions to systems oriented problems gained relatively little from them. Serious development over a rapidly widening spectrum of applications only really started to grow following a series of thought-provoking and innovative papers read to the Royal Statistical Society in 1949. Although studies in this new field swiftly prospered, with new stochastic results constantly being applied to fresh areas of application, as the associated level of mathematical difficulty increased, more and more researchers started to switch their attention back onto purer probability problems. For these are more likely to lead to intellectually appealing closed-form solutions. Unfortunately, success usually depended on making unrealistic, and hence detrimental, assumptions about the way in which particular scenarios develop. In recent years, however, there has been a resurgence of interest in constructing approximate theoretical solutions and simulation analyses. The latter has prospered greatly by the availability of fast computing power on inexpensive personal computers, which enables quite complex problems to be studied within a reasonable time frame.

This text therefore has three aims: (1) to introduce a variety of stochastic processes that possess relatively simple closed-form solutions; (2) to present a range of approximation techniques which show that in more complex problems considerable behavioural information may still be extracted even when the exact probability structure is mathematically intractable to direct solution; and, (3) to illustrate how to construct basic simulation algorithms which not only yield insight into the way that particular systems develop, but which also highlight intriguing, and previously unknown, properties that may then be studied in their own right. The content is wholly based around stochastic population processes, and is driven almost entirely by the underlying Kolmogorov probability equations for population size. Thus it is not intended as a text for pure-minded mathematicians who require a deep theoretical understanding, but for researchers who want answers to real questions; there are plenty of theoretically oriented books on Markov processes already on the market. What makes this book different is that it concentrates on practical application. That is, we start with a problem, create a realistic model which encapsulates its main properties, and then construct and apply appropriate stochastic techniques in order to solve it. This is in direct contrast to `traditional' books on stochastic processes, which are generally heavy in mathematical theory and attempt to force problems into a rigid framework of closed form solutions. Because the text provides a personal tour which details the author's experiences in applying stochastic processes over a wide range of different applied arenas, the examples are slanted towards ecological and physical applications. However, only a little imagination is required in order to extend the processes discussed to problems generic to engineering and chemistry. Indeed, the book provides a rich source of ideas and reference not only for researchers in applied probability, but for also for any scientists or engineers working with random processes who are prepared to take a flexible approach to the problems they are trying to solve.

Little of the material is covered at a deep mathematical level, so the text< should be readily accessible to a wide variety of practitioners. Moreover, considerable effort has been made to ensure that it ties together as a unified whole, with connections being highlighted between all the component processes and issues wherever possible. Many outstanding problems are identified, and signposts are provided en route that point to further reading for readers who wish to acquire a deeper analytic knowledge. One of the main objectives is to use the Kolmogorov equations to expose the high degree of linkage which exists between apparently unconnected processes. For in this way, results and techniques for one may be carried over directly to the other without us having constantly to reinvent the wheel. So in this vein many of the specific processes presented can be thought of as being examples of general application. With regards to teaching, the varied nature of the material means that it is ideal for constructing both honours level undergraduate and graduate courses. Moreover, whilst many readers may wish to progress through each chapter in sequence, it is also easy to treat the book as a toolbox that one can dip into in order to select specific analytic and computational techniques.

Given that a completely definitive treatment would require a huge number of volumes, material has had to be chosen selectively. Examples have been kept sufficiently simple to make the underlying ideas easy to understand, yet sufficiently structured to convince readers that these techniques and approaches may be applied to any problem, no matter how complex. In the first half of the book basic principles and ideas are introduced through the study of single-species populations; in the second these are extended to cover a variety of different scenarios involving bivariate populations and processes that develop in both space and time. Chapter 2 explains the basic principles which underlie the concept of stochastic population dynamics by considering simple forms of model structure for which the resulting mathematical analyses are sufficiently transparent to enable useful conclusions to be drawn from them. This provides the opportunity to explain: the relationship between moments and probabilities; the role undertaken by various forms of generating function; and, a general simulation approach based on event-time pairs. Since any Markov population that either increases or decreases by one at each event can be expressed in terms of a general birth-death process, this topic is introduced at the start of Chapter 3. The examples presented demonstrate models with single and multiple equilibria, and highlight the role played by moment closure and the saddlepoint approximation in the derivation of approximate stochastic solutions. Diffusion and perturbation techniques are also introduced at this point. Attention now switches from continuous to discrete time, with random walks and Markov chains being treated in Chapters 4 and 5. Consideration of path analyses for random walks (Chapter 4) leads on to first passage and return probabilities and the Arc Sine Law. Both absorbing and reflecting barriers are considered in some depth. Since the assumption of independent steps is not universally true, a classic case being the movement of share prices driven by market sentiment, we introduce the correlated random walk. Further generalizations include the extension to Markov chains and branching processes (Chapter 5). Having dealt with processes in discrete space, and discrete and continuous time, Chapter 6 investigates Markov processes in continuous space and time. The material follows a natural progression through the Wiener process, the Fokker-Planck diffusion equation and the Ornstein-Uhlenbeck process, and tracks the relationships between them. When barriers are introduced theoretical development is not straightforward, so care needs to be
taken when constructing solutions.

Individuals often do not exist in isolation but instead co-exist with individuals from many other species, and thereby exhibit between-species interaction. Unfortunately, not all the progress made with the analysis of univariate populations carries through to the multivariate case, since the resulting stochastic equations are usually nonlinear and intractable. So developing an understanding of such processes involves greater reliance on simulation and approximation techniques. Chapter 7 examines the general bivariate process, and illustrates the basic approaches involved by first developing a simple process for which the preceding methods of solution do carry across. The univariate saddlepoint approximation is then extended to cover multivariate processes, with cumulant truncation being covered in some detail. A bivariate process of considerable practical importance involves employing total counts as a second variable, especially since this can generate intriguing effects in which the structure of the occupation probabilities depends on whether the population size is odd or even. Various examples are presented, including a family of processes that generates different probability distributions which have the same moment structure. Moreover, although complex systems often exhibit extremely rich dynamic behaviour, gaining a direct understanding of the underlying structure may not be possible if the system remains hidden and observations can only be made on external counts. We show that a surprisingly high level of analytic information can still be gained from the counts alone, and demonstrate how to employ Markov chain Monte Carlo techniques in such situations. Chapter 8 derives specific results for the bivariate logistic process in the contexts of competition, predator-prey and epidemic processes. Here minor modifications to the transition rates cause substantial changes in the structure of stochastic realizations. Earlier discussions on cumulative size and power-law processes are woven into the discussion, and strong emphasis is placed on the development of computer simulation algorithms.

Since many processes develop through both space and time, Chapter 9 introduces a spatial dimension by presuming that individuals now develop over a number of interacting sites. The level of mathematical tractability swiftly decreases as the level of spatial interaction increases, so a variety of exact and approximate approaches are introduced for constructing moments and probability distributions. The techniques covered include slightly connected processes, generating functions, integral equations and Riccati matrix representations. The removal of boundary effects eases analytic progress, with wavefront profiles and their velocities of propagation producing interesting results. For their construction not only ties in with the earlier saddlepoint work, but for non-exponentially bounded contact distributions the velocities explode. Moreover, moving from single to multi-type systems generates Turing processes in which inherently unstable stochastic systems can be made stable by injecting spatial structure. Whilst the concept of a `stochastic dynamic' enables highly volatile systems which exhibit frequent local extinctions to persist indefinitely. Further insight is gained by replacing Markov processes by approximating Markov chains, thereby giving rise to stochastic cellular automata; two simple examples highlight the spread and control of forest fires and foot-and-mouth disease. Although the exact event-time pairs algorithm is easy to construct, and hence is ideal for simulating fairly simple dynamical systems, it is typically too computationally expensive to be of use in more complex situations. A suite of alternative approximate strategies are therefore presented based on time increments, tau-leaping, Langevin-leaping and stochastic differential equations.

Finally, in Chapter 10 we consider two spatial-temporal extensions. First, large-scale spatial interaction allows the general contact distribution to possess a power-law structure. Spectral results are derived which intertwine with those obtained via a parallel fractional integration route. The solution to the corresponding inverse problem is particularly useful since it enables us to construct processes which possess any observed spectral structure; examples show the generation of sea waves and optical caustic surfaces. Second, employing marked point processes allows us to replace the discrete (usually lattice) site-structure by a continuous space domain. To enable fast compute times, and to draw parallels with Gibbs processes, analysis proceeds within a deterministic space-time framework with stochasticity being induced through random arrivals and departures; interactive death occurs when the mark size decays to zero. As the resulting pattern structures depend heavily on the chosen growth and spatial interaction functions, a wide variety of different pattern types may be produced, many of which have direct practical application to packed systems in physics, forestry, geology, chemistry and ecology. Specific issues covered include convergence, the generation of fractal structure, and the construction of densely packed systems for particles moving under interaction pressure.

The potential for extending many of the single- and multi-species model structures considered in this text is virtually without limit. Not only may we have different types of species, or marks, involving competition, attraction, predator-prey, infection, chemical reaction, etc., but severe edge-effect problems may arise when these processes are placed in a finite bounded domain. The aim is therefore to ensure that readers are sufficiently enthused to explore both theory and application in this ever-widening arena of applied stochastic processes, which, for far too long, has been ignored in favour of developing appealing results in pure probability theory that often have scant practical relevance to the real world. The development of genuinely applied stochastic temporal and spatial-temporal analyses generates an exciting field of study which provides great mathematical, statistical and computational challenges.