Evolutionary dynamics

This article is about modelling evolutionary change using differential equations. For a related topic, see Evolutionary invasion analysis.

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Evolutionary dynamics is a branch of mathematical evolutionary biology that developed from research using differential equations to model both genetic and phenotypic change. ^[1] Thus it differs from population genetics or quantitative genetics that focus on genetic change, and from population dynamics that describes change in population size over time, but does not include genetic change. Evolutionary game theory first applied to biology by Maynard Smith and Price ^[2] introduced an important connection between ecology and evolution by showing the importance of frequency-dependent selection, but it did not initially provide a flexible link to population dynamic change.

In the 1990s researchers began to understand the opportunity for linking ecological and genetic models using differential equations resulting in evolutionary dynamics. ^[3] Some researchers prefer the terms adaptive dynamics ^[4] or evolutionary invasion analysis. ^[5] The common feature of this work is the use of differential equations to model evolutionary change in a manner that can take into account ecological concepts and phenotypic as well as genetic change.

Origins

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

Main article: 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. ^[6] 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 populaton numbers do not describe genetic details.

Population genetics

Main article: 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. ^[7] 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. ^{[7] : 119} What population genetics does not include is ecological features such as population dynamics.

Quantitative genetics

Main article: Quantitative genetics

Where the evolution of continuous heritatable traits is studied quantitative genetics provides a means of understanding them in a mathematical manner. ^{[8] [9]} Like population genetics it does not take into account population dynamics.

Evolutionary game theory

Main article: Evolutionary game theory

Maynard Smith and Price applied a method originally from economics to the strategy of animal conflict in 1973. ^[2] This was evolutionary game theory, since then very much extended by Maynard Smith ^[10] and others ^[11] to applications across evolutionary biology. An important concept of evolutionary game theory as applied in biology is the evolutionary 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." ^{[10] : 11} (In this definition Maynard Smith referred to it as an "evolutionarily stable strategy" ^{[10] : 11} but the remainder of this reference makes it clear that this is identical to an evolutionary stable strategy.)

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. ^[12] Later, Nowak (1990) ^[13] 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:

1. It could be used to study frequency-dependent selection of behavioural phenotypes
2. 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 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. ^[1]

An example

An example model for evolutionary dynamics is explained below. This based on the work of Dieckmann and Law (1996), ^[14] drawing upon the work of other evolutionary researchers prior to them, as will be mentioned.

Assumptions

Dieckmann and Law ^[14] 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; Dieckmann and Law refer to species throughout) can be understood in relation to the dynamics of its environment. Other researchers on coevolution had identified this. ^[15]
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. ^{[16] [17]}
3. A model of evolutionary dynamics should include population dynamics at an indivdual level rather than just focusing on selection on fitness. Dieckmann and Law refer to this as: "a microscopic theory". ^{[14] : 580} This assumption has been verified by earlier work. ^[18]
4. 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 indivduals 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). ^[19]

Summary equation

Based on their assumptions above, Dieckmann and Law ^{[14] : 581} propose 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 ${\displaystyle s}$. ${\displaystyle k_{i}(s)}$ scale the rate of evolutionary change. They must be non-negative.

Why do ${\displaystyle k_{i}(s)}$ 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 ( 1 ) were proposed before ^{[20] [21]} 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 about the evolutionary dynamics of adaptive phenotypic traits, and this stimulated further applications.

Applications

Setting $${\displaystyle {\frac {d}{dt}}s_{i}=0}$$ 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, ^[21] that is, that they possess convergence stability. ^[22]

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. ^[23]

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 ^{[4] : 310-314} and ^[24] especially. ^{[24] : 65}

What are these models applied to? Speciation has already been mentioned. ^[25] Ferriere et al. (2002) addressed ecological questions surrounding the stability of mutualism using a model based on the model of evolutionary dynamics described above. ^[26] Bonsall et al. (2004) applied related models to the evolution of life history trade-offs. ^[27] 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. ^[28]

Broom and Rychtář (2007) extended a game-theoretic model of kleptoparasitism (the evolution of stealing) so that it could be modelled using differential equations. ^[29] This enabled them to discover new mixed strategies. Another active area of application has been the evolution of host-parasite or host-pathogen interaction. ^{[30] [31]}

Independently other researchers applied the concept of evolutionary dynamics at a cellular level, in order to better understand the evolution of stem cells. ^[32] 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

1. Dieckmann, Odo; Christiansen, Freddy B; Law, Richard, eds. (1996). "Special Issue: Evolutionary Dynamics". Journal of Mathematical Biology. 54 (5/6): 483–688.
2. Maynard Smith, J; Price, G R (1973). "The logic of animal conflict". Nature. 246 (5427): 15–18. Bibcode:1973Natur.246...15S. doi:10.1038/246015a0.
3. Nowak, Martin A. (2006). "Evolutionary Dynamics:Exploring the Equations of Life". Cambridge, MA: Harvard University Press. ISBN   9780674023383.
4. Brännström, Åke; Johansson, Jacob; von Festenberg, Niels (2013). "The Hitchhiker's Guide to Adaptive Dynamics". Games. 4 (3): 304–328. doi: 10.3390/g4030304 .
5. Kokko, Hanna (2007). "Chapter 7: Self-consistent games and evolutionary invasion analysis". Modelling for Field Biologists and Other Interesting People. Cambridge, UK: Cambridge University Press. pp. 140–162. ISBN   9780521538565.
6. Begon, Michael; Townsend, Colin R; Harper, John L (2006). "Chapter 10.2.1 The Lotka-Volterra model". Ecology: From Individuals to Ecosystems (4th ed.). Oxford: Blackwell Publishing Ltd. pp. 298–299. ISBN   9781405111171.
7. Roughgarden, J (1979). Theory of Population Genetics and Evolutionary Ecology: An Introduction. New York, NY: Macmillan. ISBN   9780029488515.
8. Falconer, D S (1989). Introduction to Quantitative Genetics. Harlow, UK: Longman. ISBN   9780582016422.
9. Roff, Derek A. (1997). Evolutionary Quantitative Genetics. New York, NY: Chapman & Hall. ISBN   9780412129711.
10. Maynard Smith, John (1982). Evolution and the Theory of Games. Cambridge, UK: Cambridge University Press. ISBN   9780521288842.
11. Hofbauer, Josef; Sigmund, Karl (1998). Evolutionary Games and Population Dynamics. Cambridge, UK: Cambridge University Press. ISBN   9780521625708.
12. Taylor, P D; Jonker, L B (1978). "Evolutionary stable strategies and game dynamics". Mathematical Biosciences. 40 (1–2): 145–156. doi:10.1016/0025-5564(78)90077-9.
13. Nowak, M (1990). "An evolutionarily stable strategy may be inaccessible". Journal of Theoretical Biology. 142 (2): 237–241. Bibcode:1990JThBi.142..237N. doi:10.1016/S0022-5193(05)80224-3. PMID   2352434.
14. Dieckmann, U; Law, R (1996). "The dynamical theory of evolution: a derivation from stochastic ecological processes". Journal of Mathematical Biology. 34 (5–6): 579–612. doi:10.1007/BF02409751. PMID   8691086.
15. Futuyma, Douglas J; Slatkin, Montgomery, eds. (1983). Coevolution. Sunderland, MA: Sinauer Associates. ISBN   9780878932283.
16. Eshel, I (1983). "Evolutionary and continuous stability". Journal of Theoretical Biology. 103 (1): 99–111. Bibcode:1983JThBi.103...99E. doi:10.1016/0022-5193(83)90201-1.
17. Taylor, P D (1989). "Evolutionary stability in one-parameter mdels under weak selection". Theoretical Population Biology. 36 (2): 125–143. Bibcode:1989TPBio..36..125T. doi:10.1016/0040-5809(89)90025-7.
18. Abrams, P A; Matsuda, H; Harada, Y (1993). "Evolutionarily unstable fitness maxima and stable fitness maxima of continous traits". Evolutionary Ecology. 7 (5): 465–487. Bibcode:1993EvEco...7..465A. doi:10.1007/BF01237642.
19. Fisher, R A (1958). The Genetical Theory of Natural Selection. New York, NY: Dover Publications.
20. Brown, J L; Vincent, T L (1987). "Coevolution as an evolutionary game". Evolution. 41 (1): 66–79. Bibcode:1987Evolu..41...66B. doi:10.1111/j.1558-5646.1987.tb05771.x. PMID   28563763.
21. Takada, J; Kigami, J (1991). "The dynamic attainability of ESS in evolutionary games". Journal of Mathematical Biology. 29 (6): 513–529. doi:10.1007/BF00164049. PMID   1895020.
22. Matessi, C; Di Pasuale, C (1996). "Long-term evolution of multi-locus traits". Journal of Mathematical Biology. 34 (5/6): 613–653. doi:10.1007/BF02409752. PMID   8691087.
23. van Doorn, G S; Edelaar, P; Weissing, F J (2009). "On the Origin of Species by Natural and Sexual Selection". Science. 326 (5960): 1704–1707. Bibcode:2009Sci...326.1704V. doi:10.1126/science.1181661. hdl:10261/36477. PMID   19965377.
24. Diekmann, Odo. "A Beginner's Guide to Adaptive Dynamics". Publications - Prof. dr. O. (Odo) Diekmann. Universiteit Utrecht. Retrieved 19 January 2026.
25. Doebelli, M; Dieckmann, U (2003). "Speciation along environmental gradients". Nature. 421 (6920): 259–264. Bibcode:2003Natur.421..259D. doi:10.1038/nature01274. PMID   12529641.
26. Ferriere, Régis; Bronstein, Judith L; Rinaldi, Serio; Law, Richard; Gaudochon, Mathias (2002). "Cheating and the evolutionary stability of mutualisms". Proceedings of the Royal Society B. 269 (1493): 773–780. doi:10.1098/rspb.2001.1900. PMC   1690960 . PMID   11958708.
27. Bonsall, M E; Jansen, V A A; Hassell, M P (2004). "Life History Trade-Offs Assemble Ecological Guilds". Science. 306 (5693): 111–114. doi:10.1126/science.1100680. PMID   15459391.
28. Kisdi, É; Jacobs, F J A; Geritz, S A H (2001). "Red Queen Evolution by Cycles of Evolutionary Branching and Extinction". Selection. 2 (1–2): 161–176. doi:10.1556/Select.2.2001.1-2.12.
29. Broom, M.; Rychtář, J. (2007). "The Evolution of a Kleptoparasitic System under Adaptive Dynamics". Journal of Mathematical Biology. 54 (2): 151–177. doi:10.1007/s00285-006-0005-2. PMID   16724229.
30. Miller, M.R.; White, A.; Boots, M. (2005). "The evolution of host resistance: Tolerance and control as distinct strategies". Journal of Theoretical Biology. 236 (2): 198–207. Bibcode:2005JThBi.236..198M. doi:10.1016/j.jtbi.2005.03.005. PMID   16005309.
31. Nowak, Martin A.; May, R.M. (23 November 2000). Virus dynamics: Mathematical principles of immunology and virology. Oxford, UK: Oxford University Press. ISBN   9780198504177.
32. Tannenbaum, Emmanuel; Sherley, James L; Shakhnovich, Eugene I (2005). "Evolutionary dynamics of adult stem cells: comparison of random and immortal-strand segregation mechanisms". Physical Review E. 71 (4) 041914. arXiv: q-bio/0411048 . Bibcode:2005PhRvE..71d1914T. doi:10.1103/PhysRevE.71.041914. PMID   15903708. S2CID   11529637.

External links

• Nowak, M. (undated) Evolutionary Game Dynamics from Clay Mathematics Institute ^[a]
• Kisdi, É. (2023) Adaptive Dynamics Papers. ^[b]

Notes

1. This majority of these slides from a lecture refer to evolutionary game theory, only the latter portion refer to the topic of this article evolutionary dynamics.
2. Although the title of this webpage refers to adaptive dynamics, many of the papers on this page refer to evolutionary dynamics, indeed some are cited here.