Equation of the Month

Equation of the Month

A blog run by the

Theoretical Population Ecology and Evolution Group,

Biology Dept.,

Lund University

The purpose of this blog is to emphasize the role of theory for our understanding of natural, biological systems. We do so by highlighting specific pieces of theory, usually expressed as mathematical 'equations', and describing their origin, interpretation and relevance.

Tuesday, June 28, 2011

The Canonical Equation of Adaptive Dynamics

What it means
The equation describes how the value of an ecological trait (z) evolves depending on the per capita mutation rate (μ), the variance of mutation effects (σ2), the population size at equilibrium (N* ) and the selection gradient (the last factor). W(z', z) is called invasion fitness and is measured as the per capita growth rate of a morph with trait value z' in an environment where a morph with a trait value z dominates (the resident trait). Note that the derivative in the last factor, i.e. the slope of the invasion fitness, is taken with respect to the mutant trait z' and evaluated at the resident trait value z.

Implications and importance
The equation is derived for mutation-limited evolution in large, monomorphic, asexual populations (Dieckmann and Law 1996). Changes in the trait value are assumed to be small and occur as successful mutant populations establish and replace the resident population. It is biologically straightforward to see why the different factors in the equation affect the rate of evolution. To start with, the product of and N* dictates how often mutations arise in the population. The factor 1/2 appears in the equation because under directional selection half of the mutations in a one-dimensional trait are bound to go in the 'wrong' direction. Higher variance in the mutation effects (σ;2) increases the rate of evolution by making the mutational steps longer. The slope of the invasion fitness around the resident trait value indicates how much fitness increases (or decreases) with a small mutational step. Evolution will be faster with a steeper slope since the likelihood of a successful invasion increases when the relative fitness advantage of the mutant over the resident is high. This slope also affects the direction of evolution such that z evolves towards higher values when the slope is positive and vice versa. Evolution will come to a halt when the slope of the invasion fitness is zero.

The canonical equation of adaptive dynamics, and generalisations of it, is especially useful for dealing with frequency-dependent selection and situations where the ecological feedback environment is affected by the evolutionary change. It has strong connections to evolutionary game theory and can for example be used to study gradual evolution to an Evolutionary Stable Strategy, ESS, or to evolutionary branching points (Geritz et al 1998, McGill and Brown 2007). Applications include food web evolution, speciation, fisheries management and the evolution of cooperation.

The canonical equation is related to other approaches to describe gradual evolution such as quantitative genetics or strategy dynamics. The approaches differ mainly in assumptions about the genetic variation (e.g. mutation-limited evolution vs. standing genetic variation) and whether or not ecological feedback is affecting evolution or not (changing vs. fixed adaptive landscape).

Jacob Johansson

Further reading:
Dieckmann U. and Law R. 1996. The dynamical theory of coevolution: A derivation from stochastic ecological processes. Journal of Mathematical Biology 34: 579–612

Geritz, S., É. Kisdi, E., G. Meszéna, G., and J. A. J. Metz, 1998. Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evolutionary Ecology 12: 35-57.

Champagnat, N., Ferrière, R., Ben Arous, G. (2001) The canonical equation of adaptive dynamics: a mathematical view. Selection 2, 73-83 .

Waxman, D. and Gavrilets, S. 2005. 20 Questions on Adaptive Dynamics. Journal of Evolutionary Biology 18: 1139-1154

McGill, B., and J. Brown. 2007. Evolutionary game theory and adaptive dynamics of continuous traits. Annual Review of Ecology, Evolution and Systematics 38: 403-435.

Species-Area relationship

S = cAz

What it means

The equation states the relationship between an area (A) and the expected number of extant species (S) within that area. The constants c and z define the shape of the nonlinear relationship (Rosenzweig 2000). With an increasing area the expected number of species inhabiting that area is also increasing at a rate mainly dictated by z.

Where does it come from?

Originally the relationship, presented above, was theoretically derived from a species-abundance framework (Preston 1962). Given the assumption of a lognormal distribution of species abundances in a community, Preston derived the equation and calculated the z-value to be 0.27. This provided an empirically testable theory of biodiversity in island biogeography as well as mainland regions of different size.

Explanation and implications

The species-area relationship (mainly described by the exponent z above) can be explained by fundamental eco-evolutionary processes such as migration, speciation and extinctions which ultimately are driven by mechanisms such as niche availability, density dependence and species ranges (McGlade 1999). Although all mechanisms possibly are ubiquitous, some may be more important under certain conditions than others.. For example, large geographical areas include more diverse habitats, and hence more niches, facilitating high species diversity. In addition the degree of migration to and from the island, dictated by island area and isolation, has been identified as an important factor affecting the relationship.
In mainland areas with similar conditions the relationship can be explained by population size and geographical range of the species (McGlade 1999). As geographical area is decreased, population sizes and species ranges also decrease. This may give rise to an increase in extinction rate. Conversely, increasing population size and range facilitate speciation as large populations with large ranges often contain large genetic variation and are split into allopatry more often.

It has been shown that the coefficient c is often dependent on the taxon and biogeographical region, whereas z is more stable and has been estimated to fall between 0.20-0.35 for mainland biogeography and 0.12-0.17 for island biogeography (MacArthur 1969). These parameter estimations often fall below the theoretical value derived by Preston. Lower z-values than predicted can, for example, indicate high immigration of transient species from surrounding areas. Conversely, large z-values may indicate large islands or geographical areas which include several biomes whose species can evolve as independent assemblages. The species-area relationship has often been used in conservation biology (Krebs 1999), but not always without problems (see e.g., He & Hubbell 2011)

Mikael Pontarp

Further reading
He, F. & Hubbell, S.P. (2011) Species-area always overestimate extinction rate from habitat loss. Nature, 473

Krebs, C.J. (1999) Ecological methodology. Addison-Welsy Educational Publishers. Menlo Park

MacArthur, H.R & Wilson, E.O. (1968) The theory of island Biogeography. Princeton university press. Princeton

McGlade, J. (1999) Advanced ecological theory. Blackwell sience. London

Preston .F.W. (1962) The canonical distribution of commonness and rarity. Ecology, 43

Rosenzweig, M.L (2000) Species diversity in space and time. Cambridge University Press. Cambridge.