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This chapter discusses adaptive diversification due to predator–prey interactions. It has long been recognized that consumption, that is, predation, can not only exert strong selection pressure on the consumer, but also on the consumed species. However, predation has traditionally received much less attention than competition as a cause for the origin and maintenance of diversity. By using adaptive dynamics theory as well as individual-based models, the chapter then illustrates that adaptive diversification in prey species due to frequency-dependent predator–prey interactions is a theoretically plausible scenario. It also describes conditions for diversification due to predator–prey interactions in classical Lotka–Volterra models, which requires analysis of coevolutionary dynamics between two interacting species, and hence of adaptive dynamics in two-dimensional phenotype spaces.

This chapter introduces the major themes of the book. It focuses on the use of models to study John Reber’s plan to dam up the San Francisco Bay and the anomalous fishery statistics discovered by Vito Volterra after World War I.

Ecological character displacement is phenotypic evolution wrought or maintained by resource competition between species. By resource competition, I mean the negative impact of one species (or individual) on another arising from depletion of shared resources. Character evolution driven by other mutually harmful interactions, such as intraguild predation or behavioral interference, is not included in the current definition of character displacement but perhaps should be in future. For the purposes of this chapter, however, character displacement is synonymous with the coevolution of resource competitors. The idea that competition between species has a significant impact on character evolution has a lively history. Prior to about 20 years ago, competition was seen as one of the major factors responsible for the evolution of species differences, particularly in traits affecting resource exploitation (e.g., body size, beak shape). The idea is seen again and again in the early literature not because it was rigorously established but because it so readily accounted for observed patterns of species differences in nature. Support for the idea began to slip soon afterward, however, as alternative hypotheses were developed and as it became clear that the quality of most of the available evidence was poor. More recently, the idea has become respectable again as evidence from several systems has become more solid. My goal in this chapter is to present an overview of some of this evidence and how it has affected our understanding of the process. I begin with a brief historical sketch of character displacement and the expectations from theory. I then present a few of the highlights emerging from observational studies of patterns suggesting character displacement, their limitations, and their implications. I follow with an overview of recent experimental work that complements studies of pattern but goes beyond them by testing novel predictions of character displacement hypotheses. I end with suggestions about where the most significant future discoveries lie. The history of ideas on competition and character divergence begins with Darwin (1859), who regarded interspecific competition for resources as a fundamental and ubiquitous agent of divergent natural selection: “Natural selection . . . leads to divergence of character, for more living beings can be supported on the same area the more they diverge in structure, habitats, and constitution”.

This introductory chapter discusses the development of the study of predator-prey interactions. It outlines some of the popular population models applied in the study of population dynamics and predation theory, starting with the Lotka-Volterra equation which is considered to be the standard predator-prey model of ecology. It then briefly looks into the Leslie matrix which supports the Lotka-Volterra equation. The chapter also briefly explains the authors' argument of viewing predator density or interference as a fundamental feature governing predator-prey interactions, an opinion that differs from most of the other proponents of population models.

Many of the most remarkable achievements of chemical science involve either synthesis (the design and construction of molecules) or analysis (the identification and structural characterization of molecules). We have organized our discussion of oscillating reactions along similar lines. In the previous chapter, we described how chemists have learned to build chemical oscillators. Now, we will consider how to dissect an oscillatory reaction into its component parts—the question of mechanism. A persuasive argument can be made that it was progress in unraveling the mechanism of the prototype BZ reaction in the 1970s that gave the study of chemical oscillators the scientific respectability that had been denied it since the discovery of the earliest oscillating reactions. The formulation by Field, Körös, and Noyes (Field et al., 1972) of a set of chemically and thermodynamically plausible elementary steps consistent with the observed “exotic” behavior of an acidic solution of bromate and cerium ions and malonic acid was a major breakthrough. Numerical integration (Edelson et al., 1975) of the differential equations corresponding to the FKN mechanism demonstrated beyond a doubt that chemical oscillations in a real system were consistent with, and could be explained by, the same physicochemical principles that govern "normal" chemical reactions. No special rules, no dust particles, and no vitalism need be invoked to generate oscillations in chemical reactions. All we need is an appropriate set of uni- and bimolecular steps with mass action kinetics to produce a sufficiently nonlinear set of rate equations. Just as the study of molecular structure has benefited from new experimental and theoretical developments, mechanistic studies of complex chemical reactions, including oscillating reactions, have advanced because of new techniques. Just as any structural method has its limitations (e.g., x-ray diffraction cannot achieve a resolution that is better than the wavelength of the x-rays employed), mechanistic studies, too, have their limitations. The development of a mechanism, however, has an even more fundamental and more frustrating limitation, sometimes referred to as the fundamental dogma of chemical kinetics. It is not possible to prove that a reaction mechanism is correct. We can only disprove mechanisms.

After having discussed how life might have originated in the early Earth (chapter 6), chapter 7 looks at how living organisms evolve by natural selection. It does that from a quantitative perspective, appropriate in the context of Artificial Chemistries. It starts with an introduction to evolutionary dynamics, the mathematical modelling of evolutionary processes. Basic concepts in evolutionary dynamics such as replication, death, selection, fitness landscapes, resource limitations, neutrality, drift and mutations are briefly explained. The classical Lotka-Volterra system is illustrated as a chemistry involving these basic concepts. An early artificial chemistry called random catalytic reaction network is then discussed, which links to the final part of the chapter, where various artificial chemistries that model evolutionary processes are reviewed, together with a brief overview of some algorithms in this area.

This chapter introduces the concept of the consumer-resource link, the idea that each species in a community consumes resources and is itself consumed by other species. The consumer–resource link is the fundamental building block from which more-complex food chains and food webs are constructed. The chapter continues by exploring what is arguably the simplest consumer–resource interaction—one predator species feeding on one species of prey. Important topics discussed in the context of predator–prey interactions are the predator’s functional response, the Lotka–Volterra predator–prey model, the Rosenzweig–MacArthur predator–prey model, and the suppression-stability trade-off. Isocline analysis is introduced as a method for visualizing the outcome of species interactions at steady-state or equilibrium. Herbivory and parasitism are briefly discussed within the context of general predator–prey models.

Interspecific competition is a major factor influencing the structure of communities. This chapter examines the principles of interspecific completion, defined as a reduction in the population growth rate of one species due to presence of one (or more) other species due to their shared use of limiting resources or active interference. The chapter begins with a presentation of the classic Lotka–Volterra competition model, but quickly moves on to more recent consumer–resource competition models. Conditions leading to competitive exclusion and species coexistence are discussed, as are empirical tests of the predictions of resource competition theory. In general, coexistence requires that each species has a greater negative effect on its own population growth rate than on the population growth rate of another species. Shared predation also can result in species having negative effects on each other’s population growth rate, a condition known as “apparent competition”.

This chapter deals with two-dimensional differential equations and examines whether they are capable of a richer set of behaviours compared with their one-dimensional counterparts. It begins by considering a model of two interacting populations, known as the Lotka-Volterra model or the Lotka-Volterra equations, a standard example of interacting populations in mathematical ecology. The example involves two populations of different creatures, perhaps rabbits and foxes, a two-dimensional system in the sense that both animal populations are both unknown functions. The chapter also illustrates how to adapt Euler's method to solve coupled equations, summarises the solutions to one-dimensional differential equations with a phase line, and discusses phase space and phase portraits. It concludes by describing the van der Pol equation and two types of stable, attracting behavior: a fixed point and a limit cycle.

This chapter focuses on the role of protozoa (purely heterotrophic protists) and other protists in grazing on other microbes. Heterotrophic nanoflagellates, 3–5 microns long, are the most important grazers of bacteria and small phytoplankton in aquatic environments. In soils, flagellates are also important, followed by naked amoebae, testate amoebae, and ciliates. Many of these protists feed on their prey by phagocytosis, in which the prey particle is engulfed into a food vacuole into which digestive enzymes are released. This mechanism of grazing explains many factors affecting grazing rates, such as prey numbers, size, and composition. Ingestion rates increase with prey numbers before reaching a maximum, similar to the Michaelis-Menten equation describing uptake as a function of substrate concentration. Protists generally eat prey that are about 10-fold smaller than the equivalent spherical diameter of the protistan predator. In addition to flagellates, ciliates and dinoflagellates are often important predators in the microbial world, and are critical links between microbial food chains and larger organisms. Many protists, especially in aquatic habitats, are capable of photosynthesis. In some cases, the predator benefits from photosynthesis carried out by engulfed, but undigested, photosynthetic prey or its chloroplasts.

Contemporary scientific practice employs at least three major categories of models: concrete models, mathematical models, and computational models. This chapter describes an example of each type in detail: The San Francisco Bay model (concrete), the Lotka–Volterra Model (mathematical), and Schelling’s model of segregation (computational).

This chapter, which discusses the similarities and differences between the classical and contemporary approaches to the niche, begins with comparisons between consumer–resource and Lotka–Volterra models. It then describes a revised niche framework that seeks to clarify and resurrect the niche concept in a more quantifiable and biologically meaningful way than the previously loosely associated ideas of the niche. The niche framework is used to illuminate a series of issues from conventional niche theory.

Protists are involved in many ecological roles in natural environments, including primary production, herbivory and carnivory, and parasitism. Microbial ecologists have been interested in these single-cell eukaryotes since Antonie van Leeuwenhoek saw them in his stool and scum from his teeth. This chapter focuses on the role of protozoa (purely heterotrophic protists) and other protists in grazing on other microbes. Heterotrophic nanoflagellates, 3–5 microns long, are the most important grazers of bacteria and small phytoplankton in aquatic environments. In soils, flagellates are also important, followed by naked amoebae, testate amoebae, and ciliates. Many of these protists feed on their prey by phagocytosis, in which the prey particle is engulfed into a food vacuole into which digestive enzymes are released. This mechanism of grazing explains many factors affecting grazing rates, such as prey numbers, size, and composition. Ingestion rates increase with prey numbers before reaching a maximum, similar to the Michaelis–Menten equation describing uptake as a function of substrate concentration. Protists generally eat prey that are about ten-fold smaller than they are. In addition to flagellates, ciliates and dinoflagellates are often important predators in the microbial world and are critical links between microbial food chains and larger organisms Many protists are capable of photosynthesis. In some cases, the predator benefits from photosynthesis carried out by engulfed, but undigested photosynthetic prey or its chloroplasts. Although much can be learnt from the morphology of large protists, small protists (<10 μ‎m) often cannot be distinguished by morphology, and as seen several times in this book, many of the most abundant and presumably important protists are difficult to cultivate, necessitating the use of cultivation-independent methods analogous to those developed for prokaryotes. Instead of the 16S rRNA gene used for bacteria and archaea, the 18S rRNA gene is key for protists. Studies of this gene have uncovered high diversity in natural protist communities and, along with sequences of other genes, have upended models of eukaryote evolution. These studies indicate that the eukaryotic Tree of Life consists almost entirely of protists, with higher plants, fungi, and animals as mere branches.

Lotka and Volterra were among the first to attempt to mathematize the dynamics of interacting populations. While their work had a profound influence on ecology, leading to many of the results that were covered in the preceding chapters, their approach is difficult to generalize to the case of many interacting species. When the number of species in a community is sufficiently large, there is little hope of obtaining analytical results by carefully studying the system of dynamical equations describing their interactions. Here, we introduce an approach based on the theory of random matrices that exploits the very large number of species to derive cogent mathematical results. We review basic concepts in random matrix theory by illustrating their applications to the study of multispecies systems. We introduce tools that can be used to yield new insights into community ecology and conclude with a list of open problems.

This chapter considers the theoretical equilibrium consequences of ratio-dependent and prey-dependent views and compares them with evidence from nature. It provides evidence, mostly from lakes and marine systems, regarding the responses of food chains to enrichment. It discusses a number of theoretical observations such as the trophic cascade pattern. It also looks into how food webs based on donor control, a case of ratio dependence, are shown to be much more stable than those based on the Lotka-Volterra model, and explains the persistence of complex food webs in nature.

This chapter examines the controversy surrounding the topic of ratio-dependence model of predator-prey interactions. It considers the practice of many ecologists in using the Lotka-Volterra model and other similar prey-dependent models when building complex food web models, either for theoretical purposes or for applied studies. It looks into the issue of concealing evidence of ratio dependence in collected works, and highlights the continuing acceptance of prey-dependent views with the paradox of enrichment and the cascading response of trophic chains. It tackles some of the arguments raised against ratio-dependency such as the theory's lack of a mechanical basis, and discusses the agreement of a “middle opinion” that predator dependence has to be included in the theory of predator-prey interactions.

Chapter 3 introduces the use of mathematical models and modeling practices in contemporary biological and cognitive sciences. The familiar Lotka–Volterra model of predator–prey relations is used to explain these practices and show how they promote the extensions of predicates, including psychological predicates, into new and often unexpected domains. It presents two models of cognitive capacities that were developed to explain human behavioral data: Ratcliff’s drift-diffusion model of decision-making and Sutton and Barto’s temporal difference model of reinforcement learning. These are now used for fruit flies and neural populations. It also discusses contemporary and ongoing attempts to revise psychological concepts in response to empirical discovery.

The need to solve analytically intractable models has led to the rise of a new kind of science, computational science, of which computer simulations are a special case. It is noted that the development of novel mathematical techniques often drives scientific progress and that even relatively simple models require numerical treatments. A working definition of a computer simulation is given and the relation of simulations to numerical methods is explored. Examples where computational methods are unavoidable are provided. Some epistemological consequences for philosophy of science are suggested and the need to take into account what is possible in practice is emphasized.