Evolutionary dynamics

Population dynamics

Alfred J. Lotka and Vito Volterra developed independently the Lotka-Volterra equations, a pair of differential equations producing a simple descriptive model of the population dynamic interaction of a predator and a prey species. This took place in the early twentieth century. The model includes a parameter representing time, alongside other parameters representing the population numbers of the two species. The basic Lotka-Volterra equations can be extended to represent interactions between more species or populations, and to represent competitive interactions other than predator-prey interactions.

Changes in population numbers over time are not adequate to fully understand evolutionary change, even though evolutionary change may be taking place at the same time, since population numbers do not describe genetic details.

Population genetics

Genetics is usually thought to have originated with the experiments of Gregor Mendel, whose contributions were obscured until what is now known as Mendelian genetics were rediscovered, providing the basis for modern population genetics. Population genetics enables the mathematical modelling of discrete heritable characteristics. Although simple assumptions start with discrete changes over evolutionary time, more complicated assumptions require the solution of differential equations.^: 119 Population genetics does not include ecological features such as population dynamics.

Quantitative genetics

Where the evolution of continuous heritatable traits is studied, quantitative genetics provides a means of understanding them in a mathematical manner. Like population genetics, it does not take into account population dynamics.

Evolutionary game theory

Maynard Smith and Price applied a method originally from economics to the strategy of animal conflict in 1973. This resulted in the creation of evolutionary game theory. The name stems from its many applications across evolutionary biology. An important concept of evolutionary game theory is the evolutionarily stable strategy or ESS. As Maynard Smith defined it: "An ESS is a strategy such that, if all the members of a population adopt it, then no mutant strategy could invade the population under the influence of natural selection.^: 11

Thus, evolutionary game theory provides a means of describing the outcomes of evolution through behavioural strategies, without having to consider population genetics or quantitative genetics. Taylor and Jonker (1978) showed that the dynamics of ESSs, over an evolutionary time frame, could be described through differential equations. Later, Nowak (1990) showed that certain ESSs could never be reached in evolutionary time.

These discoveries demonstrated two use cases of Evolutionary game theory:

1. The study of frequency-dependent selection of behavioural phenotypes.
2. The study of stability (ESSs) and dynamics in evolutionary time, using differential equations.

Linking different studies of evolution

The study of the dynamics of evolutionary game theory, despite the power of understanding evolutionary stable strategies, could not explain all aspects of evolutionary change at a phenotypic level. Meanwhile, the otherwise successful study of population genetics and quantitative genetics did not offer an opportunity to include population dynamics in evolutionary change. Thus, evolutionary dynamics was formulated as another link between ecology and evolution.

Assumptions

Dieckmann and Law based their work on the following assumptions:

1. Evolution needs to be considered in a coevolutionary context. This is so that evolutionary dynamics of a species (or population within a species) can be understood in relation to the dynamics of its environment. Other researchers on coevolution had identified this.
2. A mathematical model of evolution needs to be dynamical, that is, it needs to represent change over time. Although evolutionary stable strategies are important in understanding evolutionary outcomes, they raise questions about attainability and dynamics around them.
3. A model of evolutionary dynamics should include population dynamics at an individual level rather than just focusing on selection for fitness. Dieckmann and Law refer to this as: "a microscopic theory".^: 580 This assumption had been verified by earlier work.
4. An evolutionary process should include stochastic elements. While population genetics and quantitative genetics have shown successful means of understanding evolution in a deterministic manner, this assumption allows the model to account for the fact that mutations arise at random at a phenotypic level. In addition, since individuals in which mutations arise are discrete, mutants may proceed to undergo sudden random extinction because of the nature of the individual in which they exist. This latter aspect of stochasticity goes back to Fisher (1958).

Summary equation

Based on their assumptions above, Dieckmann and Law^: 581 proposed the following equation:

${\displaystyle {\frac {d}{dt}}s_{i}=k_{i}(s)\cdot {\frac {\partial }{\partial s_{i}^{\prime }}}W_{i}(s_{i}^{\prime },s){\Big |}_{s_{i}^{\prime }=s_{i}}}$

1

Equation (1) describes the rate of change over time of adaptive trait values ${\displaystyle s_{i}}$ in an ecological community of ${\displaystyle i=1,\cdots ,N}$ species.

${\displaystyle W_{i}(s_{i}^{\prime },s)}$ measures evolutionary fitness of individuals with trait value ${\displaystyle s_{i}^{\prime }}$ against a background of a resident population with trait values ${\displaystyle s}$. Each ${\displaystyle k_{i}(s)}$ is a non-negative scaling coeffcient for the rate of evolutionary change.

Equations of the general form of equation (1) were proposed prior, and relate to the concept of the fitness landscape which originated (with Sewall Wright) much earlier.

The advantage of equation (1) is that it allowed the exploration of a wide range of scenarios concerning the evolutionary dynamics of adaptive phenotypic traits, stimulating use in further applications.