Evolutionary dynamics

The use of differential equations as a means of describing change over time go back to the invention of calculus by Leibniz and Newton, as well as the work of Jakob Bernoulli soon after. But the application of differential equations to biology came much later. 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, as differential equations must do, and 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 than predator-prey interactions. Changes in population numbers over time are not adequate to fully understand evolutiohary change, even though evolutionary change may be taking place at the same time. This is because population numbers do not describe genetic details. Population genetics Genetics is usually thought to have originated with the experiments of Gregor Mendel but unfortunately his contribution was obscured until what is now known as Mendelian genetics was 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. What population genetics does not include is 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. and others to applications across evolutionary biology. An important concept of evolutionary game theory as applied in biology is the evolutionarily stable strategy or ESS. This is important because, 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." 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. However, there is something missing. Can ESSs actually be reached in evolutionary time? Taylor and Jonker (1978) showed that the dynamics in evolutionary time around ESSs could be described through differential equations. Later, Nowak (1990) showed that ESSs could exist that could never be reached in evolutionary time. These discoveries did not render evolutionary game theory irrelevant, rather they showed two very important points: • It could be used to study frequency-dependent selection of behavioural phenotypes • It could be used to study stability (ESSs) and dynamics in evolutionary time, using differential equations. Linking different studies of evolution Given this large body of techniques already available to study evolutionary change using mathematical techniques, why was evolutionary dynamics, another area of mathematical biology needed and how did it arise? The study of the dynamics of evolutionary game theory away from evolutionary stable strategies showed that it could not explain all aspects of evolutionary change at a phenotypic level. The success of population genetics and quantitative genetics as means of understanding evolution did not offer an opportunity to include population dynamics in evolutionary change. New models were required that drew upon earlier work to link ecology and evolution. ==An example==

An example model for evolutionary dynamics is explained below. This based on the work of Dieckmann and Law (1996), drawing upon the work of other evolutionary researchers prior to them, as will be mentioned. Assumptions Dieckmann and Law • 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. • A model of evolutionary dynamics should include population dynamics at an individual level rather than just focusing on selection on fitness. Dieckmann and Law refer to this as: "a microscopic theory". This assumption has been verified by earlier work. • An evolutionary process should include stochastic elements. Of course population genetics and quantitative genetics have shown successful means of understanding evolution in a deterministic manner. The justification of this assumption in this model is that mutations arise at random at a phenotypic level. In addition since individuals in which mutations arise are discrete, mutants may proceed to 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 propose the following equation: {{NumBlk|:|\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}|}} Equation () describes the rate of change over time of adaptive trait values s_i in an ecological community of i = 1, \cdots , N species. W_i(s_i^\prime, s) measures evolutionary fitness of individuals with trait value s_i^\prime against a background of a resident population with trait values s. k_i(s) scale the rate of evolutionary change. They must be non-negative. Why do k_i(s) have to be non-negative? If one or more is negative, they will invert the fitness values of one or more species with respect to the other species being studied. They can be zero but this will just remove one or more species from the study. Equations of the general form of equation () were proposed before and relate to the concept of the fitness landscape which originated (with Sewall Wright) much earlier. The advantage of equation () is that it allowed the exploration of a wide range of scenarios about the evolutionary dynamics of adaptive phenotypic traits, and this stimulated further applications. ==Applications==

Setting \frac{d}{dt}s_i = 0 enables the location of points where the evolutionary dynamics of trait values are zero. Some of these will be evolutionary stable strategies, assuming they are attainable, Other of these points may be similar to the saddle point of a cubic function where evolutionary change can lead past the point where the evolutionary dynamics of trait values are zero. Other points of zero evolutionary trait dynamics may be fitness minima. If these points are convergence stable, selection can occur at these points (described as disruptive selection) leading to evolutionary divergence of trait values away from the point of zero trait dynamics in multiple directions of trait space. This has been described as evolutionary branching. Evolutionary branching has been studied in order to understand speciation. In classifying these three types of evolutionary points where evolutionary dynamics are zero, it is not possible to describe which type will occur where without setting specific parameters, hence the use of "Some...", "Other... may" in the paragraphs above. For more on classification of these points see especially. Ferriere et al. (2002) addressed ecological questions surrounding the stability of mutualism using a model based on the model of evolutionary dynamics described above. Bonsall et al. (2004) applied related models to the evolution of life history trade-offs. The Red Queen hypothesis of continual evolution of species adapting against opposing species (named after the Red Queen's Race in Through the Looking-Glass) has also been demonstrated using this type of model. Broom and Rychtář (2007) extended a game-theoretic model of kleptoparasitism (the evolution of stealing) so that it could be modelled using differential equations. This enabled them to discover new mixed strategies. Another active area of application has been the evolution of host-parasite or host-pathogen interaction. Independently other researchers applied the concept of evolutionary dynamics at a cellular level, in order to better understand the evolution of stem cells. Inspirations from their work may assist future medical therapies. These are some examples of the areas of application of evolutionary dynamics, a field of research that continues to develop. ==References==