Invasion exponent and selection gradient

The invasion exponent ${\displaystyle S_{r}(m)}$ is defined as the expected growth rate of an initially rare mutant in the environment set by the resident (r), which means the frequency of each phenotype (trait value) whenever this suffices to infer all other aspects of the equilibrium environment, such as the demographic composition and the availability of resources. For each r, the invasion exponent can be thought of as the fitness landscape experienced by an initially rare mutant. The landscape changes with each successful invasion, as is the case in evolutionary game theory, but in contrast with the classical view of evolution as an optimisation process towards ever higher fitness.

We will always assume that the resident is at its demographic attractor, and as a consequence ${\displaystyle S_{r}(r)=0}$ for all r, as otherwise the population would grow indefinitely.

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

${\displaystyle S_{r}(m)\approx S_{r}(r)+S_{r}'(r)(m-r)}$

which holds whenever ${\displaystyle m\approx r}$.

Pairwise-invasibility plots

The invasion exponent represents the fitness landscape as experienced by a rare mutant. In a large (infinite) population only mutants with trait values ${\displaystyle m}$ for which ${\displaystyle S_{r}(m)}$ is positive are able to successfully invade. The generic outcome of an invasion is that the mutant replaces the resident, and the fitness landscape as experienced by a rare mutant changes. To determine the outcome of the resulting series of invasions pairwise-invasibility plots (PIPs) are often used. These show for each resident trait value ${\displaystyle r}$ all mutant trait values ${\displaystyle m}$ for which ${\displaystyle S_{r}(m)}$ is positive. Note that ${\displaystyle S_{r}(m)}$ is zero at the diagonal ${\displaystyle m=r}$. In PIPs the fitness landscapes as experienced by a rare mutant correspond to the vertical lines where the resident trait value ${\displaystyle r}$ is constant.

Evolutionarily singular strategies

The selection gradient ${\displaystyle S_{r}'(r)}$ determines the direction of evolutionary change. If it is positive (negative) a mutant with a slightly higher (lower) trait-value will generically invade and replace the resident. But what will happen if ${\displaystyle S_{r}'(r)}$ vanishes? Seemingly evolution should come to a halt at such a point. While this is a possible outcome, the general situation is more complex. Traits or strategies ${\displaystyle r^{*}}$ for which ${\displaystyle S_{r^{*}}'(r^{*})=0}$, are known as evolutionarily singular strategies. Near such points the fitness landscape as experienced by a rare mutant is locally `flat'. There are three qualitatively different ways in which this can occur. First, a degenerate case similar to the saddle point of a qubic function where finite 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 nearby mutants. Third, a fitness minimum where disruptive selection will occur and the population branch into two morphs. This process is known as evolutionary branching. In a pairwise invasibility plot the singular strategies are found where the boundary of the region of positive invasion fitness intersects the diagonal.

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

${\displaystyle S_{r^{*}}''(r^{*})<0}$

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

${\displaystyle {\frac {d}{dr}}S_{r}'(r){\Big |}_{r=r^{*}}<0.}$

The criterion for convergence stability given above can also be expressed using second derivatives of the invasion exponent, and the classification can be refined to span more than the simple cases considered here.

Invasion exponent and selection gradients in polymorphic populations

The invasion exponent is generalised to dimorphic populations straightforwardly, as the expected growth rate ${\displaystyle S_{r_{1},r_{2}}(m)}$ of a rare mutant in the environment set by the two morphs ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$. The slope of the local fitness landscape for a mutant close to ${\displaystyle r_{1}}$ or ${\displaystyle r_{2}}$ is now given by the selection gradients

${\displaystyle S_{r_{1},r_{2}}'(r_{1})}$

and

${\displaystyle S_{r_{1},r_{2}}'(r_{2})}$

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

Evolutionary branching

The emergence of protected dimorphism near singular points during the course of evolution is not unusual, but its significance depends on whether 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 compelling argument that disruptive selection only occurs near fitness minima. To understand this heuristically, consider a dimorphic population ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ near a singular point. By continuity

${\displaystyle S_{r}(m)\approx S_{r_{1},r_{2}}(m)}$

and, since

${\displaystyle S_{r_{1},r_{2}}(r_{1})=S_{r_{1},r_{2}}(r_{2})=0,}$

the fitness landscape for the dimorphic population must be a perturbation of that for a monomorphic resident near the singular strategy.

Trait evolution plots

Evolution after branching is illustrated using trait evolution plots. These show the region of coexistence, the direction of evolutionary change and whether points where the selection gradient vanishes are fitness maxima or minima. Evolution may well lead the dimorphic population outside the region of coexistence, in which case one morph is extinct and the population once again becomes monomorphic.