Demographic vs Environmental Stochasticity
Consider a population of size $N$ with per capita birth rate $b$ and death rate $d$. The intrinsic rate is $r = b - d$.
Demographic stochasticity arises because each individual’s fate is random, even if $b$ and $d$ are fixed. For a birth-death process, the variance in population change over a short interval $\Delta t$ is:
\[\text{Var}(\Delta N) = (b + d)\,N\,\Delta t\]The coefficient of variation of $\Delta N/N$ scales as $N^{-1/2}$, so demographic stochasticity matters most when $N$ is small.
Environmental stochasticity is variation in the vital rates themselves across years due to weather, food availability, disease, etc. If $r$ is random with mean $\bar{r}$ and variance $\sigma_e^2$, then even large populations feel this noise. The variance in $\Delta N/N$ scales as $N^0$ (independent of $N$), meaning environmental stochasticity is the dominant source of variance for large populations.
Diffusion Approximation
For large $N$, the integer-valued birth-death process can be approximated by a continuous-time stochastic differential equation (SDE). For a population with mean growth rate $\mu(N)$ and variance rate $v(N)$, the Itô SDE is:
\[dN = \mu(N)\,dt + \sqrt{v(N)}\,dW_t\]where $W_t$ is standard Brownian motion. For logistic growth with environmental stochasticity:
\[\mu(N) = rN\!\left(1 - \frac{N}{K}\right), \qquad v(N) = \sigma_e^2 N^2 + \sigma_d^2 N\]The $N^2$ term is environmental stochasticity; the $N$ term is demographic stochasticity.
The stochastic logistic in the diffusion limit has a stationary distribution (when it exists). Working with $X = \ln N$ via Itô’s lemma transforms the equation to:
\[dX = \left[\bar{r} - \frac{\sigma_e^2}{2} - \frac{e^X}{K}\left(r + \frac{\sigma_d^2}{2}\right)\right]dt + \sqrt{\sigma_e^2 + \sigma_d^2 e^{-X}}\,dW_t\]The $-\sigma_e^2/2$ correction is the Itô correction: stochastic growth requires $\bar{r} > \sigma_e^2/2$ for long-run persistence, not merely $\bar{r} > 0$.
Extinction Probability and Time
For a population on the interval $(0, K)$ with absorbing barrier at $0$ (extinction), the mean time to extinction from initial size $N_0$ is given by the solution to:
\[\mu(N)\frac{dT}{dN} + \frac{v(N)}{2}\frac{d^2 T}{dN^2} = -1\]with boundary conditions $T(0) = 0$ and $T’(K) = 0$. For the simple case $\mu(N) = rN$ and $v(N) = \sigma_e^2 N^2$ (pure environmental stochasticity in log-space), the mean extinction time scales as:
\[\mathbb{E}[T_\text{ext}] \approx \frac{K^{2r/\sigma_e^2}}{r} \cdot C\]This super-exponential scaling in $K$ means that modest increases in carrying capacity can dramatically reduce extinction risk, a result with direct implications for minimum viable population (MVP) analysis.
Quasi-Stationary Distributions
Before extinction, a stochastically fluctuating population tends to hover near a quasi-stationary distribution (QSD) — the conditional distribution of population size given non-extinction. For the birth-death chain with states ${0, 1, \ldots, N_\text{max}}$ where $0$ is absorbing, the QSD $\mathbf{q}$ satisfies:
\[\mathbf{q} \mathbf{Q} = -\theta \mathbf{q}\]where $\mathbf{Q}$ is the generator of the chain restricted to transient states and $\theta$ is the rate of absorption (extinction). From the QSD, the mean time to extinction is $\mathbb{E}[T_\text{ext}] = 1/\theta$.
| Population metric | Demographic stochasticity | Environmental stochasticity |
|---|---|---|
| Variance in $\Delta N$ | $\propto N$ | $\propto N^2$ |
| Dominant for | Small populations | Large populations |
| Persistence condition | $r > 0$ | $r > \sigma_e^2/2$ |
| $T_\text{ext}$ scaling | $\propto e^N$ | $\propto K^{2r/\sigma_e^2}$ |
Moran Effect and Spatial Synchrony
Environmental stochasticity is often spatially correlated: all populations in a region experience similar weather. The Moran effect states that spatially separated populations with the same density dependence but correlated environmental noise will have abundance time series with cross-correlation equal to the environmental correlation. For two populations with correlated noise $\rho_\epsilon$:
\[\text{Corr}(N_1(t), N_2(t)) \approx \rho_\epsilon \cdot \frac{\sigma_e^2}{\sigma_e^2 + \sigma_d^2/\bar{N}}\]Spatial synchrony amplifies regional extinction risk: if all subpopulations crash simultaneously, there are no sources for recolonization.