Adaptation & Evolution Under Climate Change

Derivation: Critical Rate of Environmental Change

Consider a population with mean phenotype $\bar{z}$ in an environment where the optimal phenotype is $\theta(t)$, which shifts at rate $k = d\theta/dt$. Fitness is described by a Gaussian function:

$\bar{W}(t) = \exp\!\left[-\frac{(\bar{z}(t) - \theta(t))^2}{2\omega^2}\right]$

where $\omega^2$ is the width of the fitness function. Population growth follows:

$\frac{dN}{dt} = N \left[r_{\max} \bar{W}(t) - 1\right]$

The mean phenotype evolves according to the breeder's equation(Lush, 1937):

$\frac{d\bar{z}}{dt} = h^2 S = \frac{V_A}{\omega^2 + V_P}(\theta - \bar{z})$

where $h^2 = V_A / V_P$ is heritability, $V_A$ is additive genetic variance, $V_P$ is phenotypic variance, and $S$ is the selection differential. At evolutionary equilibrium, the rate of phenotypic change matches the rate of environmental change:

$\frac{d\bar{z}}{dt} = k \quad \Longrightarrow \quad \text{lag} = \theta - \bar{z} = \frac{k(\omega^2 + V_P)}{V_A}$

The population can persist (evolutionary rescue) only if the lag is small enough that mean fitness remains above replacement ($\bar{W} > 1/r_{\max}$). This gives the critical rate of change:

$k_{\text{crit}} = V_A \sqrt{\frac{2 \ln(r_{\max})}{\omega^2 + V_P}}$

If the environment changes faster than $k_{\text{crit}}$, the population cannot adapt and will decline to extinction. Species with high genetic variance ($V_A$), high intrinsic growth rate ($r_{\max}$), and broad fitness functions ($\omega^2$) are most likely to achieve evolutionary rescue.

The Breeder's Equation in Detail

The response to selection $R$ (change in mean phenotype per generation) is:

$R = h^2 \cdot S$

where $S$ is the selection differential(difference between the mean phenotype of selected parents and the overall population mean), and$h^2$ is the narrow-sense heritability.

Equivalently, using the selection gradient$\beta = S / V_P$:

$R = V_A \cdot \beta = V_A \cdot \frac{\partial \ln \bar{W}}{\partial \bar{z}}$

This is the Lande equation (1976), which generalizes the breeder's equation to natural selection. It shows that the rate of evolution is proportional to the additive genetic variance — the “fuel” of evolution.

Typical heritabilities for ecologically important traits:

• • Body size: $h^2 \approx 0.3\text{--}0.5$
• • Flowering/breeding time: $h^2 \approx 0.2\text{--}0.4$
• • Thermal tolerance: $h^2 \approx 0.1\text{--}0.3$
• • Drought resistance: $h^2 \approx 0.1\text{--}0.3$
• • Fitness (overall): $h^2 \approx 0.05\text{--}0.2$

Derivation: Optimal Plasticity

A reaction norm describes the phenotype as a function of environment: $z = a + bE$, where $a$ is the intercept,$b$ is the plasticity slope, and $E$ is an environmental cue. The optimal phenotype in environment $E$ is $z_{\text{opt}}(E)$.

If the cue perfectly predicts the selective environment, the optimal plasticity slope (de Jong, 2005) is:

$b^* = \frac{\text{Cov}(z_{\text{opt}}, E)}{\text{Var}(E)}$

This is simply the regression coefficient of optimal phenotype on environment. When$z_{\text{opt}}$ changes linearly with $E$, perfect plasticity ($b = b^*$) eliminates the need for genetic evolution entirely.

However, plasticity has costs and limits:

• • Production costs: maintaining sensory and regulatory machinery consumes energy.
• • Information costs: unreliable environmental cues lead to maladaptive plasticity. When reliability $\rho < 1$, optimal plasticity is $b^* = \rho \cdot \text{Cov}(z_{\text{opt}}, E)/\text{Var}(E)$.
• • Limit costs: extreme plastic responses may be physiologically constrained.

Including a cost $c$ proportional to plasticity magnitude, net fitness becomes:

$W(b) = \exp\!\left[-\frac{E_z[(z_{\text{opt}} - a - bE)^2]}{2\omega^2}\right] - c|b|$

Trans-generational Plasticity (TGP)

Trans-generational plasticity occurs when parental environment influences offspring phenotype through non-genetic mechanisms. Notable examples:

• • Daphnia (water fleas): mothers exposed to predator cues produce helmeted offspring via epigenetic modification. Heat-stressed mothers produce offspring with upregulated heat-shock proteins (Jablonka & Raz, 2009).
• • Corals: Pocillopora damicornis parents pre-exposed to elevated temperature produce larvae with enhanced thermal tolerance. DNA methylation patterns in stress-response genes are transmitted across generations (Putnam & Gates, 2015).
• • Plants: Arabidopsis exposed to pathogen attack shows transgenerational immune priming via DNA methylation at defense gene loci (Luna et al., 2012).
• • Sticklebacks: fathers reared at elevated temperatures sire offspring with altered gene expression in hundreds of genes related to acclimation (Shama & Wegner, 2014).

The adaptive potential of TGP depends on the autocorrelation of environments across generations. When the parental environment predicts offspring conditions (high autocorrelation), TGP is favoured. Under rapid climate change, if trends are directional, TGP can speed adaptation by 1–2 generations compared to purely genetic evolution (Bonduriansky et al., 2012).

Derivation: Risk–Benefit Analysis of Assisted Migration

The decision framework balances the expected benefit of migration (reduced extinction risk) against the risks (ecological disruption, maladaptation, pathogen transfer). Following Hoegh-Guldberg et al. (2008):

$\text{Net benefit} = P_{\text{ext}}^{\text{no action}} - P_{\text{ext}}^{\text{assisted}} - R_{\text{invasion}} - R_{\text{disease}}$

where $P_{\text{ext}}$ is extinction probability and $R$ terms are risk costs. The benefit is greatest when:

• • The species has high extinction risk without intervention
• • Suitable future habitat exists but is beyond dispersal range
• • The species is unlikely to become invasive
• • No close relatives exist in the target area (low hybridization risk)

Case Study: Florida Panther Genetic Rescue

The Florida panther (Puma concolor coryi) declined to $\sim 20\text{--}25$ individuals by the early 1990s, suffering severe inbreeding depression: cryptorchidism (90%), kinked tails (88%), cardiac defects (67%). Genetic diversity measured by heterozygosity had declined to $H \approx 0.10$(vs. $H \approx 0.40$ in western populations).

In 1995, eight female Texas pumas were introduced to the Florida population. The result was a dramatic genetic rescue:

• • Population tripled to ~120–230 individuals by 2015
• • Heterozygosity increased from $H \approx 0.10$ to $H \approx 0.25$
• • Cryptorchidism dropped from 90% to 0%
• • Kitten survival increased by 300%

The inbreeding load was relieved according to:

$\ln(\bar{W}) = \ln(W_0) - BF$

where $B$ is the number of lethal equivalents per gamete and $F$is the inbreeding coefficient. For Florida panthers, $B \approx 6.0$ and introduction of Texas genes reduced $F$ from $\sim 0.25$to $\sim 0.08$ (Johnson et al., 2010).

Darwin's Finches: Beak Size Selection

The Grants' long-term study of Geospiza fortis on Daphne Major (Galápagos) provides the best-documented example of natural selection in response to climate events.

During the 1977 drought, large hard seeds became the only food available. Birds with larger beaks had higher survival, producing a measurable selection response:

$R = h^2 \cdot S$

Grant & Grant (2006) measured:

• • Selection differential: $S = 0.70$ mm (difference in mean beak depth between survivors and pre-drought population)
• • Heritability: $h^2 = 0.65$
• • Response: $R = 0.65 \times 0.70 = 0.46$ mm increase in mean beak depth in the next generation

After 2004–2005 drought, the arrival of a competitor (G. magnirostris) reversed selection: smaller beaks were now favoured for accessing small seeds, producing$R = -0.22$ mm. This demonstrates character displacement driven by climate-mediated competition.

Additional Examples of Rapid Microevolution

• • Peppered moth (Biston betularia): Industrial pollution darkened tree bark, shifting selection from light to dark morphs (frequency of carbonaria allele rose from <1% to >90% in ~50 generations). Post-Clean Air Act, light morphs recovered — demonstrating reversible selection on a Mendelian trait (Cook et al., 2012).
• • Pink salmon (Oncorhynchus gorbuscha): run timing in Auke Creek, Alaska has shifted 2 weeks earlier over 17 generations (1979–2011), correlated with ocean warming. Kovach et al. (2012) showed this was driven by selection on a genetic basis ($h^2 \approx 0.5$ for run timing).
• • European blackcap (Sylvia atricapilla): birds wintering in UK (shorter migration) now have ~15% higher winter survival than those migrating to Iberia, driving a heritable shift in migratory direction (Berthold et al., 1992).
• • Red squirrel (Tamiasciurus hudsonicus): breeding date advanced 18 days over 10 years in response to earlier spring; ~15% due to evolution, ~85% due to plasticity (Réale et al., 2003).

Key References

• • Berthold, P. et al. (1992). Rapid microevolution of migratory behaviour in a wild bird species. Nature, 360, 668–670.
• • Bonduriansky, R. et al. (2012). The implications of nongenetic inheritance for evolution in changing environments. Evol. Appl., 5(2), 192–201.
• • Cook, L.M. et al. (2012). Selective bird predation on the peppered moth. Biol. J. Linn. Soc., 106(1), 1–7.
• • de Jong, G. (2005). Evolution of phenotypic plasticity: patterns of plasticity and the emergence of ecotypes. New Phytologist, 166(1), 101–118.
• • Gomulkiewicz, R. & Holt, R.D. (1995). When does evolution by natural selection prevent extinction? Evolution, 49(1), 201–207.
• • Grant, P.R. & Grant, B.R. (2006). Evolution of character displacement in Darwin's finches. Science, 313(5784), 224–226.
• • Hoegh-Guldberg, O. et al. (2008). Assisted colonization and rapid climate change. Science, 321(5887), 345–346.
• • Jablonka, E. & Raz, G. (2009). Transgenerational epigenetic inheritance. Q. Rev. Biol., 84(2), 131–176.
• • Johnson, W.E. et al. (2010). Genetic restoration of the Florida panther. Science, 329(5999), 1641–1645.
• • Kovach, R.P. et al. (2012). Genetic change for earlier migration timing in a pink salmon population. Proc. R. Soc. B, 279(1743), 3870–3878.
• • Lande, R. (1976). Natural selection and random genetic drift in phenotypic evolution. Evolution, 30(2), 314–334.
• • Putnam, H.M. & Gates, R.D. (2015). Preconditioning in the reef-building coral Pocillopora damicornis. J. Exp. Biol., 218(15), 2319–2329.
• • Réale, D. et al. (2003). Genetic and plastic responses of a northern mammal to climate change. Proc. R. Soc. B, 270(1515), 591–596.