Evolutionary invasion analysis

Evolutionary invasion analysis, also known as adaptive dynamics, is a set of techniques for studying long-term phenotypical evolution developed during the 1990s. It incorporates the concept of frequency dependence from game theory but allows for more realistic ecological descriptions, as the traits vary continuously and gives rise to a non-linear invasion fitness (the classical fitness concept is not directly applicable to situations with frequency dependence).

Introduction and background

The basic principle of evolution, survival of the fittest, was outlined by thenaturalist Charles Darwin in his 1859 book On the origin ofspecies. Though controversial at the time, the central ideas remainvirtually unchanged to this date, even though much more is now knownabout the biological basis of inheritance. Darwin expressed hisarguments verbally, but many attempts have since then been made toformalise the theory of evolution. The perhaps most well known are
population genetics which aim to model thebiological basis of inheritance but usually at the expense ofecological detail, quantitative genetics whichincorporates quantitative traits influenced by genes at many loci andevolutionary game theory which ignores geneticdetail but incorporates a high degree of ecological realism, inparticular that the success of any given strategy depends on thefrequency at which strategies are played in the population, a conceptknown as frequency dependence.

Adaptive Dynamics is a set of techniques developed during the 1990sfor understanding the long-term consequences of small mutations in thetraits expressing the phenotype. They link population dynamics to evolutionary dynamics and incorporate and generalises thefundamental idea of frequency dependent selection from game theory.The number of papers using Adaptive Dynamics techniques is increasingsteadily as Adaptive Dynamics is gaining ground as a versatile toolfor evolutionary modelling.

Fundamental ideas

Two fundamental ideas of Adaptive Dynamics are that the residentpopulation can be assumed to be in a dynamical equilibrium when newmutants appear, and that the eventual fate of such mutants can beinferred from their initial growth rate when rare in the environmentconsisting of the resident. This rate is known as the invasionexponent when measured as the initial exponential growth rate ofmutants, and as the basic reproductive number when it measuresthe expected total number of offspring that a mutant individual willproduce in a lifetime. It can be thought of, and is indeed sometimesalso referred to, as the invasion fitness of mutants. In order to make useof these ideas we require a mathematical model that explicitlyincorporates the traits undergoing evolutionary change. The modelshould describe both the environment and the population dynamics giventhe environment, but in many cases the variable part of theenvironment consists only of the demography of the currentpopulation. We then determine the invasion exponent, the initialgrowth rate of a mutant invading the environment consisting of theresident. Depending on the model, this can be trivial or verydifficult, but once determined, the Adaptive Dynamics techniques can beapplied independent of the model structure.

=Monomorphic evolution=

A population consisting of individuals with the same trait is calledmonomorphic. If not explicitly stated differently, we will assume that thetrait is a real number, and we will write r and m for the traitvalue of the monomorphic resident population and that of an invadingmutant, respectively.

Invasion exponent and selection gradient

The invasion exponent $S\_r(m)$ is defined as the expected growthrate of an initially rare mutant in the environment set by theresident, which simply means the frequency of each phenotype (traitvalue) whenever this suffices to infer all other aspects of theequilibrium environment, such as the demographic composition and theavailability of resources. For each r the invasion exponent can bethought of as the fitness landscape experienced by an initially raremutant. The landscape changes with each successful invasion, as is the case in evolutionary game theory, butin contrast with the classical view of evolution as an optimisationprocess towards ever higher fitness.

We will always assume that the resident is at its demographicattractor, and as a consequence $S\_r(r)\; =\; 0$ for all r,as otherwise the population would grow indefinitely.

The selection gradient is defined as the slope of the invasionexponent at $m=r$, $S\_r\text{'}(r)$. If the sign of the invasion exponent is positive (negative) mutants withslightly higher (lower) trait values may successfully invade. This follows from the linear approximation

::$S\_r(m)\; approx\; S\_r\text{'}(r)\; (m\; -\; r)$

which holds whenever $m\; approx\; r$.

Pairwise-invasibility plots

The invasion exponent represents the fitness landscape as experiencedby a rare mutant. In a large (infinite) population only mutants withtrait values $m$ for which $S\_r(m)$ is positive are able tosuccessfully invade. The generic outcome of an invasion is that themutant replaces the resident, and the fitness landscape as experiencedby a rare mutant changes. To determine the outcome of the resultingseries of invasions pairwise-invasibility plots (PIPs) are often used.These show for each resident trait value $r$ all mutant trait values$m$ for which $S\_r(m)$ is positive. Note that$S\_r(m)$ is zero at the diagonal $m=r$. In PIPs the fitness landscapes asexperienced by a rare mutant correspond tothe vertical lines where the resident trait value $r$ is constant.

Evolutionarily singular strategies

The selection gradient $S\_r\text{'}(r)$ determines the direction ofevolutionary change. If it is positive (negative) a mutant with aslightly higher (lower) trait-value will generically invade andreplace the resident. But what will happen if $S\_r\text{'}(r)$ vanishes?Seemingly evolution should come to a halt at such a point. While thisis a possible outcome, the general situation is more complex. Traitsor strategies $r^*$ for which $S\_\{r^*\}\text{'}(r^*)=0$, are known asevolutionarily singular strategies. Near such points the fitnesslandscape as experienced by a rare mutant is locally `flat'. There are three qualitativelydifferent ways in which this can occur. First, a degenerate case similar to the qubic wherefinite evolutionary steps would lead past the local 'flatness'. Second, a fitness maximum which is known as an evolutionarily stable strategy (ESS) and which, once established, cannot be invaded by nearbymutants. Third, a fitness minimumwhere disruptive selection will occur and the population branch intotwo morphs. This process is known as evolutionary branching. In a pairwise invasibility plot the singular strategies are found where theboundary of the region of positive invasion fitness intersects thediagonal.

Singular strategies can be located and classified once theselection gradient is known. To locate singular strategies, it issufficient to find the points for which the selection gradientvanishes, i.e. to find $r^*$ such that $S\text{'}\_\{r^*\}(r^*)\; =\; 0$. These canbe classified then using the second derivative test from basiccalculus. If the second derivative evaluated at $r^*$ is negative(positive) the strategy represents a local fitness maximum (minimum).Hence, for an evolutionarily stable strategy $r^*$ we have

::

If this does not hold the strategy is evolutionarily unstable and,provided that it also convergence stable, evolutionary branching willeventually occur. For a singular strategy $r^*$ to be convergencestable monomorphic populations with slightly lower or slightly highertrait values must be invadable by mutants with trait values closer to$r^*$. That this can happen the selection gradient $S\_r\text{'}(r)$ in aneighbourhood of $r^*$ must be positive for and negative for$r\; >\; r^*$. This means that the slope of $S\_r\text{'}(r)$ as a function of $r$at $r^*$ is negative, or equivalently

::

The criterion for convergence stability given above can also beexpressed using second derivatives of the invasion exponent, and theclassification can be refined to span more than the simple casesconsidered here.

=Polymorphic evolution=

The normal outcome of a successful invasion is that the mutantreplaces the resident. However, other outcomes are also possible; in particular both the resident and the mutantmay persist, and the population then becomes dimorphic. Assuming that atrait persists in the population if and only if its expectedgrowth-rate when rare is positive, the condition for coexistence amongtwo traits $r\_1$ and $r\_2$ is

::$S\_\{r\_1\}\; (r\_2)\; >\; 0$

and

::$S\_\{r\_2\}\; (r\_1)\; >\; 0,$

where $r\_1$ and $r\_2$ are often referred to as morphs.Such a pair is a protected dimorphism. The set of all protecteddimorphisms is known as the region of coexistence. Graphically,the region consists of the overlapping parts when a pair-wiseinvasibility plot is mirrored over the diagonal

Invasion exponent and selection gradients in polymorphic populations

The invasion exponent is generalised to dimorphic populations in astraightforward manner, as the expected growth rate $S\_\{r\_1,\; r\_2\}(m)$of a rare mutant in the environment set by the two morphs $r\_1$ and$r\_2$. The slope of the local fitness landscape for a mutant close to$r\_1$ or $r\_2$ is now given by the selection gradients

::$S\_\{r\_1,\; r\_2\}\text{'}(r\_1)$

and

::$S\_\{r\_1,\; r\_2\}\text{'}(r\_2)$

In practise, it is often difficult to determine the dimorphicselection gradient and invasion exponent analytically, and one oftenhas to resort to numerical computations.

Evolutionary branching

The emergence of protected dimorphism near singular points during thecourse of evolution is not unusual, but its significance depends onwhether selection is stabilising or disruptive. In the latter case,the traits of the two morphs will diverge in a process often referred to as evolutionary branching. Geritz 1998 presents a compellingargument that disruptive selection only occurs near fitness minima. Tounderstand this heuristically consider a dimorphic population $r\_1$and $r\_2$ near a singular point. By continuity

::$S\_r(m)\; approx\; S\_\{r\_1,r\_2\}(m)$

and, since

::$S\_\{r\_1,\; r\_2\}(r\_1)\; =\; S\_\{r\_1,\; r\_2\}(r\_2)\; =\; 0,$

thefitness landscape for the dimorphic population must be a perturbationof that for a monomorphic resident near the singular strategy.

Trait evolution plots

Evolution after branching is illustrated using trait evolutionplots. These show the region of coexistence, the direction ofevolutionary change and whether points where points where theselection gradient vanishes are fitness maxima or minima. Evolutionmay well lead the dimorphic population outside the region ofcoexistence, in which case one morph is extinct and the populationonce again becomes monomorphic.

=Other uses=

Adaptive dynamics effectively combines game theory and population dynamics. As such, it can be very useful in investigating how evolution affects the dynamics of populations. One interesting finding to come out of this is that individual-level adaptation can sometimes result in the extinction of the whole population/species, a phenomenon known as evolutionary suicide.

=See also=
*Evolution
*Game theory
*Population genetics
*Quantitative genetics

=External links=
* [http://adtoolkit.sourceforge.net The Hitchhiker's guide to Adaptive Dynamics] on which the first version of this article was based (GFDL).
* [http://mathstat.helsinki.fi/~kisdi/addyn.htm Adaptive Dynamics Papers] a comprehensive list of papers about Adaptive Dynamics.