Analysis of Evolutionary Processes: The Adaptive Dynamics Approach and Its Applications

Analysis of Evolutionary Processes: The Adaptive Dynamics Approach and Its Applications

Fabio Dercole
Sergio Rinaldi
Copyright Date: 2008
Pages: 352
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  • Book Info
    Analysis of Evolutionary Processes: The Adaptive Dynamics Approach and Its Applications
    Book Description:

    Quantitative approaches to evolutionary biology traditionally consider evolutionary change in isolation from an important pressure in natural selection: the demography of coevolving populations. InAnalysis of Evolutionary Processes, Fabio Dercole and Sergio Rinaldi have written the first comprehensive book on Adaptive Dynamics (AD), a quantitative modeling approach that explicitly links evolutionary changes to demographic ones. The book shows how the so-called AD canonical equation can answer questions of paramount interest in biology, engineering, and the social sciences, especially economics.

    After introducing the basics of evolutionary processes and classifying available modeling approaches, Dercole and Rinaldi give a detailed presentation of the derivation of the AD canonical equation, an ordinary differential equation that focuses on evolutionary processes driven by rare and small innovations. The authors then look at important features of evolutionary dynamics as viewed through the lens of AD. They present their discovery of the first chaotic evolutionary attractor, which calls into question the common view that coevolution produces exquisitely harmonious adaptations between species. And, opening up potential new lines of research by providing the first application of AD to economics, they show how AD can explain the emergence of technological variety.

    Analysis of Evolutionary Processeswill interest anyone looking for a self-contained treatment of AD for self-study or teaching, including graduate students and researchers in mathematical and theoretical biology, applied mathematics, and theoretical economics.

    eISBN: 978-1-4008-2834-0
    Subjects: Mathematics, Biological Sciences

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-x)
  3. Preface
    (pp. xi-xviii)
  4. Chapter One Introduction to Evolutionary Processes
    (pp. 1-42)

    In this chapter we introduce the basic elements and the empirical evidence of evolutionary processes. Since the groundbreaking workThe Origin of Speciesby Charles Darwin (1859), a great deal of effort has been dedicated to the subject (see, e.g., Fisher, 1930; Haldane, 1932; Dobzanski, 1937; Mayr, 1942, 1963, 1982; Wright, 1969; Dawkins, 1976, 1982, 1986; Cavalli-Sforza and Feldman, 1981; Maynard Smith, 1989, 1993; Maynard Smith and Szathmary, 1995, just to mention a few masterpieces). Our discussion on the origin of evolutionary theory is mainly taken from the introduction by Ernst Mayr (2001) to the seventeenth printing of Darwin’s famous...

  5. Chapter Two Modeling Approaches
    (pp. 43-73)

    In this chapter we survey the quantitative approaches proposed in the literature for modeling evolutionary dynamics. The evolution of biological as well as social and economic systems is determined by so many interacting factors that a detailed description is practically impossible. Any mathematical model takes into account some mechanisms (hopefully the relevant ones) and sacrifices the others. The trade-off between realism and mathematical tractability has produced, starting with Charles Darwin and Gregor Mendel, a rich variety of different studies, focused on different aspects of the evolutionary process. A sharp classification of these studies into well-confined modeling approaches is problematic because...

  6. Chapter Three The Canonical Equation of Adaptive Dynamics
    (pp. 74-118)

    In this chapter we derive the canonical equation of Adaptive Dynamics (AD). As anticipated in Chapter 2, the AD approach focuses on the long-term evolutionary dynamics of continuous (quantitative) adaptive traits and bypasses genetic details by using asexual demographic models. This is justified under contrasting demographic and evolutionary timescales, i.e., if mutations are rare events at birth (see discussion of condition 1 in Section 2.7). We consider physiologically unstructured populations coexisting at equilibrium in spatially homogeneous environments and characterized by traits that mutate independently (conditions a, b, and c″ of Section 2.8). If mutations are rare, the resident populations are...

  7. Chapter Four Evolutionary Branching and the Origin of Diversity
    (pp. 119-137)

    In this chapter we show how continuous marginal innovations subject to severe competition may give rise to increasing diversity in evolving systems. The analysis is performed by pointing out that the AD canonical equations describing the evolution of a family of systems with increasing number of adaptive traits always lead to a branching point. The application that has motivated this study comes from economics, where the emergence of technological variety arising from market interaction and technological innovation has been ascertained. Existing products in the market compete with innovative ones, resulting in a slow and continuous evolution of the underlying technological...

  8. Chapter Five Multiple Attractors and Cyclic Evolutionary Regimes
    (pp. 138-152)

    We show in this chapter that resource-consumer communities can be characterized by multiple evolutionary attractors and cyclic evolutionary regimes. We consider a standard resource-consumer demographic model with one adaptive trait for each population and derive the corresponding AD canonical equation, namely an evolutionary model composed of two ODEs, one for the resource trait and one for the consumer trait. Then, we perform a complete bifurcation analysis of the evolutionary model with respect to various demographic and environmental parameters and show that up to three evolutionary attractors can be present. Moreover, the evolutionary dynamics can easily promote resource diversity, as well...

  9. Chapter Six Catastrophes of Evolutionary Regimes
    (pp. 153-171)

    We show in this chapter that the set of unviable evolutionary states, namely the set of all ancestral conditions giving rise to evolutionary trajectories leading to the extinction of one or more populations, depends discontinuously upon demographic and environmental parameters. In particular, we show that the discontinuities of the unviable set are produced by catastrophes of evolutionary attractors, namely by bifurcations at which microscopic parameter perturbations beget transients toward macroscopically different evolutionary regimes. Not all catastrophes produce discontinuities of the unviable set. However, the discontinuity is guaranteed whenever the catastrophic bifurcation involves the critical values of the adaptive traits at...

  10. Chapter Seven Branching-Extinction Evolutionary Cycles
    (pp. 172-185)

    We show in this chapter that evolving communities can have a quite peculiar evolutionary attractor, called a branching-extinction evolutionary cycle. First, an adaptive trait characterizing a monomorphic species evolves toward a branching point, where the species turns dimorphic by splitting into two resident populations. Then, the two resident traits coevolve until one of the two populations goes extinct. Finally, the remaining population evolves back to the branching point, thus closing the evolutionary cycle. All this is shown by studying the evolution of cannibalistic traits in consumer populations and by focusing on the role of environmental richness. The evolution of “dwarf”...

  11. Chapter Eight Demographic Bistability and Evolutionary Reversals
    (pp. 186-203)

    In this chapter we show that when the evolutionary state of the community does not uniquely determine the resident demographic attractor, mutation-selection processes may force the resident populations to switch between alternative demographic attractors and cause abrupt changes in the selection pressure acting on the community. As a result, evolutionary cycles can develop even in the extreme case of a community composed of a single resident population characterized by a single adaptive trait. Indeed, periodic switches between alternative demographic equilibria may induce the periodic reversal of the rate of evolutionary change and force the trait to endlessly oscillate. We consider...

  12. Chapter Nine Slow-Fast Populations Dynamics and Evolutionary Ridges
    (pp. 204-230)

    We show in this chapter that contrasting timescales in the demographic dynamics of the resident populations may generate sharp transitions from stationary to cyclic demographic dynamics, resulting in abrupt variations of the selection pressure acting on the community. Such variations raise so-called evolutionary ridges in trait space, where resident populations are poised between stationary and cyclic coexistence. The main result is that evolutionary trajectories may slide along evolutionary ridges and even be trapped at special points called evolutionary pseudo-equilibria. The novel phenomena of evolutionary sliding and confinement of traits at evolutionary pseudo-equilibria should be generic to all communities in which...

  13. Chapter Ten The First Example of Evolutionary Chaos
    (pp. 231-242)

    We present in this chapter the first example of chaotic Red Queen dynamics. We consider a Lotka-Volterra tritrophic food chain composed of a resource, its consumer, and a predator species, each characterized by a single adaptive trait, and we show that for suitable modeling and parameter choices the evolutionary trajectories of the corresponding AD canonical equation approach an evolutionary strange attractor in the three-dimensional trait space. Most of this chapter is taken from Dercole and Rinaldi (2008).

    Until now we have examined numerous examples of evolutionary dynamics, which, however, were all concerned with the evolution of at most two traits....

  14. Appendix A. Second-order Dynamical Systems and Their Bifurcations
    (pp. 243-271)
  15. Appendix B. The Invasion Implies Substitution Theorem
    (pp. 272-276)
  16. Appendix C. The Probability of Escaping Accidental Extinction
    (pp. 277-280)
  17. Appendix D. The Branching Conditions
    (pp. 281-286)
  18. Bibliography
    (pp. 287-324)
  19. Index
    (pp. 325-333)