Evolutionary Dynamics & Fitness Landscapes
Evolutionary Dynamics & Fitness Landscapes
Evolutionary dynamics fuses population genetics with dynamical systems to track allele frequency changes under selection, mutation, drift, and migration. The replicator equation governs frequency-dependent selection in well-mixed populations: types with above-average fitness increase in frequency, mirroring Lotka-Volterra dynamics.
A fitness landscape maps genotype to reproductive fitness. Populations climb towards local peaks by selection; mutation and drift allow valley crossing — critical for understanding antibiotic resistance, cancer evolution, and protein engineering.
Replicator Equation
For $n$ types with frequency $x_i$ and fitness $f_i(\mathbf{x})$: $$\dot{x}_i=x_i\bigl[f_i(\mathbf{x})-\bar{f}(\mathbf{x})\bigr],\quad \bar{f}=\sum_j x_jf_j.$$ Mean fitness $\bar{f}$ increases monotonically for constant selection.
Price Equation
$$\Delta\bar{z}=\frac{\operatorname{Cov}(w,z)}{\bar{w}}+\frac{E(w\,\Delta z)}{\bar{w}}.$$ Selection acts through covariance of fitness and trait; second term is transmission bias.
Example 1
Allele $A_1$ has $f_1=1.1$, $A_2$ has $f_2=1.0$, $x_1(0)=0.1$. Will $A_1$ fix?
Solution: Yes — $f_1>\bar{f}$ so $\dot{x}_1>0$ and selection drives $x_1\to1$ (ignoring drift).
Example 2
In the hawk-dove game with payoff matrix $A$, what is the ESS?
Solution: The ESS is the asymptotically stable fixed point of the replicator equation — hawks at frequency $V/C$ when $V<C$ (value $V$, cost $C$).
Practice
- State Fisher\'s fundamental theorem of natural selection.
- Why do populations not always reach the global fitness peak?
- Describe how epistasis complicates the fitness-landscape metaphor.
- What role does genetic drift play on a rugged fitness landscape?
Show Answer Key
1. The mean fitness of a population increases at a rate equal to the genetic variance in fitness: $d\bar{w}/dt = \text{Var}(w)$. In the continuous-time version with additive fitness, the rate of increase of mean fitness equals the additive genetic variance.
2. Populations can get trapped at local fitness peaks due to fitness valleys between peaks. Crossing a valley requires passing through less-fit intermediates, which natural selection opposes. Finite populations may cross via genetic drift (stochastic tunneling), but large populations remain trapped.
3. Epistasis means the fitness effect of a mutation depends on the genetic background (other alleles present). This creates a rugged fitness landscape where the optimal combination cannot be found by changing one gene at a time — the landscape is non-additive and may have many local peaks.
4. Genetic drift (random sampling in finite populations) allows populations to explore the landscape stochastically, occasionally crossing fitness valleys. On rugged landscapes, drift is essential for escaping local optima (Sewall Wright's shifting balance theory). Smaller populations drift more and can explore more broadly, but risk extinction.