Replicator equations
The replicator equation converts fitness differences into continuous changes in strategy frequencies. It is one of the central deterministic models of evolutionary game dynamics.
The general equation
Let \(x_i\) be the frequency of strategy \(i\), with
\[x_i\ge0,\qquad \sum_{i=1}^{m}x_i=1.\]If \(f_i(\mathbf x)\) is its fitness and
\[\bar f(\mathbf x)=\sum_{j=1}^{m}x_jf_j(\mathbf x),\]is the population mean fitness, then
\[\boxed{\frac{dx_i}{dt}=x_i\left(f_i(\mathbf x)-\bar f(\mathbf x)\right)}.\]Interpretation of the sign
If
\[f_i>\bar f,\]then \(dx_i/dt>0\) and strategy \(i\) increases in frequency. If
\[f_i<\bar f,\]then it decreases. If \(f_i=\bar f\), its instantaneous frequency change is zero.
Why the factor \(x_i\) appears
The fitness advantage is multiplied by the current frequency. A strategy that is absent has \(x_i=0\), so
\[\frac{dx_i}{dt}=0.\]Therefore an absent strategy cannot appear spontaneously under the basic replicator equation. Mutation or immigration must be added if new strategies are to enter.
Frequencies remain normalised
Summing the equations gives
\[\sum_i\frac{dx_i}{dt}=\sum_i x_if_i-\bar f\sum_i x_i.\]Since
\[\sum_i x_if_i=\bar f\qquad\text{and}\qquad\sum_i x_i=1,\]we obtain
\[\boxed{\sum_i\frac{dx_i}{dt}=0}.\]Hence, if the frequencies initially sum to one, they continue to sum to one.
Connection with payoff matrices
For an evolutionary game with payoff matrix \(A\), take
\[f_i(\mathbf x)=(A\mathbf x)_i.\]The mean fitness is
\[\bar f=\mathbf x^TA\mathbf x.\]The replicator system becomes
\[\boxed{\frac{dx_i}{dt}=x_i\left[(A\mathbf x)_i-\mathbf x^TA\mathbf x\right]}.\]Two strategies
Let \(x\) be the frequency of strategy \(A\), so strategy \(B\) has frequency \(1-x\). Then
\[\bar f=xf_A+(1-x)f_B.\]The equation for \(x\) is
\[\frac{dx}{dt}=x(f_A-\bar f).\]Substituting the mean fitness gives
\[f_A-\bar f=(1-x)(f_A-f_B),\]so
\[\boxed{\frac{dx}{dt}=x(1-x)(f_A-f_B)}.\]This compact form shows that the direction of evolution is determined by the fitness difference between the two strategies.
Two-strategy payoff matrix
For
\[A=\begin{pmatrix}a&b\\c&d\end{pmatrix},\]the fitnesses are
\[f_A=ax+b(1-x),\]\[f_B=cx+d(1-x).\]Therefore
\[\boxed{\frac{dx}{dt}=x(1-x)\left[(a-c)x+(b-d)(1-x)\right]}.\]Equilibria
An equilibrium satisfies \(dx/dt=0\). The two boundary equilibria are always
\[\boxed{x=0,\qquad x=1}.\]An internal equilibrium exists when
\[f_A=f_B.\]Provided \(a-b-c+d\ne0\), this gives
\[\boxed{x^*=\frac{d-b}{a-b-c+d}}.\]It is an internal biological equilibrium only when \(0 For a one-dimensional two-strategy system, stability can be understood by checking the sign of \(dx/dt\) on either side of an equilibrium. If trajectories point toward the equilibrium from both sides, it is locally stable. If they point away, it is unstable. IfStability from the direction field
Dominance
throughout the interior, so \(A\) moves toward fixation.
The reverse holds if \(B\) always has greater fitness.
Stable coexistence
If \(A\) is favoured when rare but disadvantaged when common, the internal equilibrium can be stable. This is negative frequency-dependent selection.
Both strategies then persist in the deterministic model.
Bistability
If each strategy performs better when common, the internal equilibrium is unstable. Initial conditions on one side move toward \(A\), while those on the other side move toward \(B\).
The internal equilibrium then acts as a threshold separating two basins of attraction.
Hawk–Dove dynamics
For the Hawk–Dove payoff matrix
\[A=\begin{pmatrix}\frac{V-C}{2}&V\\0&\frac V2\end{pmatrix},\qquad C>V>0,\]the internal equilibrium is
\[\boxed{x^*=\frac VC}.\]Because Hawk is favoured below this frequency and Dove is favoured above it, the equilibrium is stable.
Fixed fitness as a special case
If the two strategies have constant fitnesses \(f_A\) and \(f_B\), then
\[\frac{dx}{dt}=x(1-x)(f_A-f_B).\]If the difference \(s=f_A-f_B\) is constant,
\[\boxed{\frac{dx}{dt}=s x(1-x)},\]which has the same mathematical form as logistic growth, although \(x\) here is a frequency constrained between zero and one rather than a population size.
Relative rather than absolute fitness
Adding the same quantity \(c(\mathbf x)\) to every strategy fitness does not change the dynamics because
\[(f_i+c)-(\bar f+c)=f_i-\bar f.\]Thus only fitness differences matter in the basic replicator equation.
Population size is not modelled
The variables \(x_i\) are frequencies. The replicator equation does not by itself describe whether the total number of organisms is growing or declining.
Models that couple ecological population size to evolutionary frequencies are needed when demographic feedback matters.
Boundary invariance
If a strategy frequency reaches zero, it remains zero in the mutation-free model. More generally, each face of the frequency simplex is invariant.
This is mathematically useful but biologically means that mutation, immigration or innovation must be modelled explicitly when new types can arise.
Replicator–mutator extension
Mutation between strategies can be represented by a mutation matrix \(Q=(q_{ji})\), where \(q_{ji}\) is the probability that reproduction by type \(j\) produces type \(i\). One common form is
\[\boxed{\frac{dx_i}{dt}=\sum_j x_jf_jq_{ji}-x_i\bar f}.\]Unlike the basic replicator equation, this system can introduce a strategy that is currently absent if mutation into it is possible.
Deterministic approximation
Replicator dynamics describe smooth deterministic frequency change. Finite populations also experience random drift, so stochastic evolutionary processes may be more appropriate when population size is small or fixation probabilities are the main question.
From game dynamics to evolving traits
Replicator equations usually assume a fixed set of available strategies. Evolution can also be modelled when the trait itself varies continuously and new nearby variants arise by mutation.
This motivates adaptive dynamics, the next lesson.