Thomas L. Vincent

ORGANIZATION OF PREDATOR-PREY COMMUNITIES AS AN EVOLUTIONARY GAME

We consider a simple predator‐prey model of coevolution. By allowing coevolution both within and between trophic levels the model breaks the traditional dichotomy between coevolution among competitors and coevolution between a prey and its predator. By allowing the diversity of prey and predator species to emerge as a property of the evolutionarily stable strategies (ESS), the model breaks another constraint of most approaches to coevolution that consider as fixed the number of coevolving species. The number of species comprising the ESS is influenced by a parameter that determines the predators niche breadth. Depending upon the parameters value the ESS may contain: 1) one prey and one predator species, 2) two prey and one predator, 3) two prey and two predators, 4) three prey and two predators, 5) three prey and three predators, etc. Evolutionarily, these different ESSs all emerge from the same model. Ecologically, however, these ESSs result in very different patterns of community organization. In some communities the predator species are ecologically keystone in that their removal results in extinctions among the prey species. In others, the removal of a predator species has no significant impact on the prey community. These varied ecological roles for the predator species contrasts sharply with the essential evolutionary role of the predators in promoting prey species diversity. The ghost of predation past in which a predators insignificant ecological role obscures its essential evolutionary role may be a frequent property of communities of predator and prey.

Evolution of cooperation with shared costs and benefits

The quest to determine how cooperation evolves can be based on evolutionary game theory, in spite of the fact that evolutionarily stable strategies (ESS) for most non-zero-sum games are not cooperative. We analyse the evolution of cooperation for a family of evolutionary games involving shared costs and benefits with a continuum of strategies from non-cooperation to total cooperation. This cost–benefit game allows the cooperator to share in the benefit of a cooperative act, and the recipient to be burdened with a share of the cooperators cost. The cost–benefit game encompasses the Prisoners Dilemma, Snowdrift game and Partial Altruism. The models produce ESS solutions of total cooperation, partial cooperation, non-cooperation and coexistence between cooperation and non-cooperation. Cooperation emerges from an interplay between the nonlinearities in the cost and benefit functions. If benefits increase at a decelerating rate and costs increase at an accelerating rate with the degree of cooperation, then the ESS has an intermediate level of cooperation. The game also exhibits non-ESS points such as unstable minima, convergent-stable minima and unstable maxima. The emergence of cooperative behaviour in this game represents enlightened self-interest, whereas non-cooperative solutions illustrate the Tragedy of the Commons. Games having either a stable maximum or a stable minimum have the property that small changes in the incentive structure (model parameter values) or culture (starting frequencies of strategies) result in correspondingly small changes in the degree of cooperation. Conversely, with unstable maxima or unstable minima, small changes in the incentive structure or culture can result in a switch from non-cooperation to total cooperation (and vice versa). These solutions identify when human or animal societies have the potential for cooperation and whether cooperation is robust or fragile.

Divergent and convergent evolution in metastases suggest treatment strategies based on specific metastatic sites

Cancer cells, although maximally fit at their primary site, typically have lower fitness on the adaptive landscapes offered by the metastatic sites due to organ-specific variations in mesenchymal properties and signaling pathways. Clinically evident metastases will exhibit time-dependent divergence from the phenotypic mean of the primary population as the tumor cells evolve and adapt to their new circumstances. In contrast, tumors from different primary sites evolving on identical metastatic adaptive landscapes exhibit phenotypic convergence so that, for example, metastases in the liver from different primary tumors will evolve toward similar adaptive phenotypes. The combination of evolutionary divergence from the primary cancer phenotype and convergence towards similar adaptive strategies in the same tissue cause significant variations in treatment responses particularly for highly targeted therapies. This suggest that optimal therapies for disseminated cancer must take into account the site(s) of metastatic growth as well as the primary organ.

Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics: List of figures

All of life is a game and evolution by natural selection is no exception. Games have players, strategies, payoffs, and rules. In the game of life, organisms are the players, their heritable traits provide strategies, their births and deaths are the payoffs, and the environment sets the rules. The evolutionary game theory developed in this book provides the tools necessary for understanding many of Nature’s mysteries. These include coevolution, speciation, and extinction as well as the major biological questions regarding fit of form and function, diversity of life, procession of life, and the distribution and abundance of life. Mathematics for the evolutionary game are developed based on Darwin’s postulates leading to the concept of a fitness generating function (G-function). The G-function is a tool that simplifies notation and plays an important role in the development of the Darwinian dynamics that drive natural selection. Natural selection may result in special outcomes such as the evolutionarily stable strategy or ESS. An ESS maximum principle is formulated and its graphical representation as an adaptive landscape illuminates concepts such as adaptation, Fisher’s Fundamental Theorem of Natural Selection, and the nature of life’s evolutionary game.

Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics: Frontmatter

All of life is a game and evolution by natural selection is no exception. Games have players, strategies, payoffs, and rules. In the game of life, organisms are the players, their heritable traits provide strategies, their births and deaths are the payoffs, and the environment sets the rules. The evolutionary game theory developed in this book provides the tools necessary for understanding many of Nature’s mysteries. These include coevolution, speciation, and extinction as well as the major biological questions regarding fit of form and function, diversity of life, procession of life, and the distribution and abundance of life. Mathematics for the evolutionary game are developed based on Darwin’s postulates leading to the concept of a fitness generating function (G-function). The G-function is a tool that simplifies notation and plays an important role in the development of the Darwinian dynamics that drive natural selection. Natural selection may result in special outcomes such as the evolutionarily stable strategy or ESS. An ESS maximum principle is formulated and its graphical representation as an adaptive landscape illuminates concepts such as adaptation, Fisher’s Fundamental Theorem of Natural Selection, and the nature of life’s evolutionary game.

Necessary Conditions for an Invasion Proof Strategy

The evolutionarily stable strategy (ESS) as first formulated by Maynard Smith is a concept defined in terms of the pay-off functions of the “mutant” and one of the remaining “players”. This paper demonstrates how this optimality concept may be extended to parametric games. Such games involve not only the payoff functions of every player but a model which puts constraints on the state of the system as well. The extended concept is then applied to a special class of “balanced” games. The balanced game not only greatly simplifies the necessary conditions for the extended ESS solution, but it is particularly applicable to ecological systems. Examples are given to illustrate the results.