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This chapter analyzes the different ways to describe brain behaviour with the goal to provide a basis for an informed discussion of the nature of decisions and actions that humans perform in their lives. The chapter is organized as follows. Section 2 outlines a number of concepts exhibiting how many subtle details and distinctions lie behind the broad notions of determinacy and stochasticity. These details are necessary for a discussion, in Section 3, of particular aspects relevant for the characterization of brain states and their dynamics. The descriptions of brain behaviour currently provided by neuroscience depend on the level and context of the descriptions. There is no clear-cut evidence for ultimately determinate or ultimately stochastic brain behaviour. As a consequence, there is no solid neurobiological basis to argue either in favour of or against any fundamental determination or openness of human decisions and actions.

This epilogue presents some concluding thoughts from the author. Increased stochasticity, beginning at the level of the genome, may work its way through the system and eventually translate into aging as a pattern of cells and tissues acting increasingly out of tune. It is now possible to explore a wealth of ever-more sophisticated tools in both cell and molecular biology and the computational sciences to systematically study these cascades of stochastic variation in the various model systems for aging that are now available. This is likely to finally unlock the door to a full understanding of how we grow old.

The effective management of natural resources use requires a mechanistic understanding of the system, not just correlations between variables of the kind discussed in Chapter 4. Understanding may simply be in the form of a conceptual model, but is much more powerful when formalized as a mathematical model. This chapter introduces methods for building a model of the system that can be used to predict future sustainability with or without management interventions. The emphasis is on the simulation of biological and bioeconomic dynamics, for which step-by-step worked examples are given. These examples start with conceptual models, then show how to formalize these as mathematical equations, build these into computer code; test model sensitivity, validity, and alternative structures; and finally, explore future scenarios. Methods for modelling stochasticity and human behaviour are also introduced, as well as the use of Bayesian methods for understanding dynamic systems and exploring management interventions.

This closing chapter first recapitulates the main notions and information seen in this book, then discusses the main problems related to the definition and to the demonstration of Allee effects; for example with regards to demographic stochasticity or to confounding variables. It also considers interesting ideas that are slightly off the mainstream of Allee effect reflections, such as ecosystem shifts or similar processes in some other scientific disciplines (such as mathematics, social sciences, linguistics). Finally, it considers the likely future of studies on this concept and proposes some avenues for new research, including finding Allee effects in new species, looking for new model taxa, for new types of Allee effects, and considering the broader perspective of community and ecosystem level consequences of this process.

Ecological hypercycles are auto-catalytic processes by which different organisms (or guilds) improve each others environments. For example, autotrophs producing material of use to decomposers and the decomposers releasing nutrients which are reused by the autotrophs. Such an auto-catalytic process clearly has a positive Gaian effect, and is one of several reasons for expecting life to cover quickly most of a planet's surface once it has evolved. Environmental stochasticity makes it unlikely that ecologies restricted to a small area of a planet will survive for a geological period of time. However, once widespread, life may have a long drawn out end on a dying planet, with some cells surviving for at least millions of years after the biogeochemical cycles returned to being mere geochemical cycles.

Establishment is the stage that has been the focus of most invasion studies in birds. However, the research on this topic has not always been congruent in its findings, and considerable disagreement exists over the causes of successful establishment. One point on which most writers agree is that the numbers of individuals contributing to an introduction event (i.e., propagule pressure) is an important contributory factor. This chapter presents reasons why this would be expected, which relate to the problems of survival that face small populations of any organism. Evidence for a relationship between propagule pressure and establishment success is assessed. The chapter concludes by considering some of the other associations of propagule pressure, especially those which may affect the conclusions of studies of establishment that do not account for it.

Many organisms adjust their thermal sensitivities in response to environmental conditions. This form of phenotypic plasticity, referred to as acclimation, involves potentially reversible changes in the quantity and quality of cellular structures. Although these adjustments impose energetic costs, a more serious cost results from the time lag required to acclimate following changes in temperature. Optimality models, based on the costs imposed by time lags, can be used to predict the acclimation of performance curves in fluctuating environments. Contrary to current theory, acclimation of the thermal optimum rarely occurs in laboratory experiments. Furthermore, a genotype's capacity for acclimation rarely correlates with the magnitude or predictability of thermal heterogeneity in its natural environment. Even with their shortcomings, current optimality models constitute a major advance over verbal models, such as the beneficial acclimation hypothesis.

Martingales form the mathematical counterpart of betting strategies. This chapter studies computable randomness, where the tests are computable martingales, and separates it from both Martin–Löf randomness and Schnorr randomness. The chapter shows that each high degree contains a properly computably random (Schnorr random) set. It varies computable martingales, discussing stochasticity and non-monotonic betting strategies.

The central premise of the neuroeconomic endeavor is that the iterative process of reductively linking neuroscience, psychology, and economics through theoretical modifications to each discipline will maximize predictive power. This chapter examines further neurobiological, psychological, and economic constraints on the choice mechanism to test that premise. First, it examines in greater detail the relationship between expected subjective value and expected utility, focusing on the interrelationship between neuronal and behavioral stochasticity as revealed by existing psychological models. Second, it looks at the precise nature of cortical representation in the nervous system. Theories of cortical representation anchored to normative models of efficient coding identify constraints all neural representations must acknowledge. These constraints predict a specific class of choice behaviors that violate traditional Soft-REU, behaviors that have already been observed but not yet linked to the structure of the choice mechanism. This suggests that a version of Hard-REU that incorporates these constraints has significantly greater predictive power at both the neural and behavioral levels than a model more closely aligned with traditional Soft-REU. These are the final issues that need to be engaged.

This chapter defines and formulates demographic and environmental stochasticity, and illustrates statistical methods for estimating them from field data. Population fluctuations in most species are produced by demographic and environmental stochasticity, rather than by internally driven cycles or chaos. Demographic stochasticity results from chance independent events of individual mortality and reproduction, causing random fluctuations in population growth rate, primarily in small populations. Environmental stochasticity results from temporal fluctuations in mortality and reproductive rates of all individuals in a population in the same or similar fashion, causing population growth rate to fluctuate randomly in populations of all sizes. In populations much larger than the ratio of the demographic variance to the environmental variance, demographic stochasticity can be neglected. The demographic variance can be estimated from data on individual survival and reproduction, and using this, the environmental variance can then be estimated from population time series. Under density-independent growth in a random environment, the eventual rate of increase or decrease of population size is given by the long-run growth rate, the mean rate of increase of log population size, which is reduced by stochasticity. Density-dependent population growth in a stochastic environment produces temporal autocorrelation in population size, even in the absence of temporal autocorrelation in the environment. Time series for terrestrial populations of birds and mammals show little evidence of temporal environmental autocorrelation.

This chapter reviews recent analytical models of sustainable harvesting of fluctuating populations without age structure that incorporate the risk of population collapse or extinction. Using diffusion theory it compares three classical harvesting strategies and one new strategy in terms of their harvest statistics and the mean time to population collapse or extinction. It uses simple analytical models to derive general principles and then apply these principles to more realistic, age-structured models of particular species to derive by simulation the optimal harvesting strategies.

Most of conservation biology deals with one species at a time. However, increasing attention is being devoted to the conservation and restoration of landscapes, communities and ecosystems, and how their dynamics are altered by human activities. One reason there is relatively little discussion of community dynamics in conservation is the absence of any guiding theory. This chapter develops a general model of community dynamics that allows species to differ in their basic ecology and responses to environmental fluctuations. It investigates the extent to which species in a community differ consistently in abundance because of differences in their ecology, and explores how environmental stochasticity influences community composition in space and time. This theory is applied to analyze the dynamics of a tropical butterfly community sampled intensively through a limited space and time, emphasizing the advantages of studying taxa with short generations to facilitate sampling on temporal scales longer than the generation times of species in the community.

This chapter defends a stochastic dynamical interpretation of evolution under which drift does not emerge as an evolutionary cause, unlike mutation and selection. Rather, whether drift occurs in a population model depends on its constitutive assumptions, namely, whether the population size is finite. The same amount of selection makes quantitatively and qualitatively different predictions in finite and infinite population models: this is all that there is to drift. This argument is illustrated through the explicit solution of a haploid model in which differences in vital parameters lead to drift in the presence of selection. In the absence of these differences, the model reduces to a neutral model. In the infinite population limit, the standard results without drift also obtain. The stochastic dynamical interpretation is contrasted with the views that evolution is a theory of forces and the statistical interpretation of evolution.

This chapter outlines four future research questions where the size- and trait-based theory can be applied: stochasticity, behavioral ecology, coupling to primary production, and thermal ecology and climate change. The chapter first argues that differences in growth can be modeled with the size-based framework by introducing stochasticity into the feeding interaction. Next, the chapter contends that the behavioral response to food and predation risk has potentially big implications for community dynamics because it changes a key element in the model—namely, the interaction between individuals. On the matter of production, the chapter demonstrates that changing the carrying capacity or the productivity of the resource changes the food environment for the fish and that changes in the primary–secondary production would also have an impact on the carrying capacity of the stock-recruitment relation. Finally, the chapter looks at how increasing temperatures affect fish populations and communities on at least two time scales: on the short term is the direct physiological response to a temperature increase in terms of increasing metabolic demands. On the longer time scale is the ecological response where some species in a community will be replaced by other, better adapted, species.

Although we must understand the mechanical nuts and bolts of inheritance, evolution occurs through changes in the population frequencies of traits, trait values, and genes in time and space, values that depend at least as much on the dynamics of populations as on the mechanics of inheritance. This chapter demonstrates the crucial roles of ecology in evolutionary change. There are two major types of dynamics that are essential to model evolution. The chapter develops the principles of strategy dynamics to examine the processes responsible for the success and failure of some traits, and trait values, over others. Different traits and their values represent competing strategies to be tested by adaptive evolution. The success of each strategy depends on the spatial and temporal dynamics of populations, and their respective influences on those dynamics. The chapter merges strategy and population dynamics to evaluate the evolutionary stability of competing strategies.

This chapter focuses on the study of dispersal in butterfly populations, in particular the Glanville fritillary butterfly. Many of the processes that influence, and are influenced by, dispersal in the Glanville fritillary are likely to be equally significant in other species of insects and invertebrates. Butterflies are small bodied and ectothermic; they are strongly influenced by the prevailing environmental, and especially thermal, conditions, which increase the impact of environmental stochasticity on their population dynamics. Butterflies also often exhibit high host plant and habitat specificities. In many landscapes, the host plants occur in habitats that have a highly fragmented distribution, and hence the respective butterfly species have spatially structured populations, often in the form of a network of local populations called a metapopulation.

Climate variation strongly influences fluctuations in size of avian populations. In this chapter, we show that it is difficult to predict how the abundance of birds will respond to climate change. A major reason for this is that most available time series of fluctuations in population size are in a statistical sense short, thus often resulting in large uncertainties in parameter estimates. We therefore argue that reliable population predictions must be based on models that capture how climate change will affect vital rates as well as including other processes (e.g. density-dependences) known to affect the population dynamics of the species in question. Our survey of examples of such forecast studies show that reliable predictions necessarily contain a high level of uncertainty. A major reason for this is that avian population dynamics are strongly influenced by environmental stochasticity, which is for most species, irrespective of their life history, the most important driver of fluctuations in population size. Credible population predictions must therefore assess the effects of such uncertainties as well as biases in population estimates.

This chapter addresses the problem of controllability for stochastic complex electromagnetic media. The starting point is the stochastic integrodifferential equations that model the evolution of the fields, with a control procedure to be selected so that the system is driven to a desired final state. The controllability problem for stochastic media is more complicated than that for deterministic media and includes subtleties that must be addressed to reach a satisfactory answer. Hence, this chapter sets the model and then discusses the subtle issues introduced by the stochasticity in the controllability problem. It then proposes two different approaches towards controllability: an approach using PDEs with random coefficients and an approach using backward stochastic evolution equations (BSEEs). Afterward, the chapter lists several comments concerning boundary controllability and optimal control.