Joel S. Brown

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.

Predator-Prey Coevolution as an Evolutionary Game

To model evolution as an evolutionary game, we have used a fitness generating function to define the fitness of any individual in a community bounded by the same evolutionary constraints. Using a single fitness generating function, we have previously investigated the effect of external inputs on a community at an evolutionarily stable strategy (ESS). Of particular interest are the circumstances under which the external input promotes the coexistence of several strategies in a community that otherwise would have a single-strategy ESS. The external inputs can include physiographic changes, human intervention, or the introduction of a new species not modeled by the single fitness generating function. We consider in detail here the situation of introducing a predator into a hitherto unexploited community of prey. In this case, the prey and predators each have a separate set of evolutionary constraints which produce two different fitness generating functions. Necessary conditions for determining the ESS under two or more fitness generating functions are presented. The coevolution of predator and prey is then examined with the aid of frequency-dependent adaptive landscapes, one for each fitness generating function. As a result of disruptive selection imposed by the predator, we obtain an ESS composed of two coexisting prey strategies and a single predator strategy.

The evolutionary response of systems to a changing environment

To model evolution as an evolutionary game, we have used a fitness generating function to define the fitness of any individual in a community bounded by the same evolutionary constraints. Using a single fitness generating function, we have previously investigated the effect of external inputs on a community at an evolutionarily stable strategy (ESS). Of particular interest are the circumstances under which the external input promotes the coexistence of several strategies in a community that otherwise would have a single-strategy ESS. The external inputs can include physiographic changes, human intervention, or the introduction of a new species not modeled by the single fitness generating function. We consider in detail here the situation of introducing a predator into a hitherto unexploited community of prey. In this case, the prey and predators each have a separate set of evolutionary constraints which produce two different fitness generating functions. Necessary conditions for determining the ESS under two or more fitness generating functions are presented. The coevolution of predator and prey is then examined with the aid of frequency-dependent adaptive landscapes, one for each fitness generating function. As a result of disruptive selection imposed by the predator, it is possible, under different niche breadths of the predator, to obtain ESSs composed of one or more coexisting prey strategies and one or more predator strategies.

Ecology of the Metastatic Process

Metastasis is the dispersal and colonization of cancer cells from a primary tumor to a distant organ in the body. Metastases are associated with more than 90% of all cancer deaths. Yet our ability to decrease the rate of metastasis or improve patient survival has not changed in decades. We see cancer cells as experiencing ecological and evolutionary dynamics within their tumor ecosystem. Metastasis occurs when one or several cancer cells enter the blood stream or lymphatic system, survive in circulatory system, emerge in a distant organ, survive, and proliferate. Similar concepts have been studied in ecology for centuries. The metastasizing cancer cells are like exotic species found in nature which traverse long distances and successfully invade distant location. Metastatic progression can and should be viewed through the lens of invasion ecology. As such, metastases become a matter of weighted probabilities. First, no cancer cell is actually under selection to metastasize. Natural selection cannot adapt a cancer cell clade to something it has yet to experience. Second, metastases are not propagule limited. The sheer number of circulating tumor cells means that most organs likely experience a steady inoculum of cancer cells. Third, the successful cancer cell that metastasizes is unlikely to be a random draw from the primary tumor. Important preadaptations include motility, the ability to move in and out of the blood system, self-sufficiency, and ability to evade the immune system. New organs likely provide ample nutrients but lack safety or structure for the inoculum of one or more cancer cells. Fourth, the pairing of donor organs and recipient organs is highly nonrandom. Organs seem to be nested in their propensity to send and receive metastases. Organs with very high ratios of blood supply to normal cell density seem prime recipients of metastases. Food, safety, and the preadaptations of the cancer cells may best explain metastatic progression. Ultimately the goal of this understanding is to provide therapies that reduce the likelihood of metastases, and the control of disseminated cancers once they have occurred. An understanding of the Darwinian dynamics underlying metastasis invites new therapeutic perspective, such as adaptive therapy.

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.