Malthusian Exponential Growth
The simplest population model assumes each individual contributes equally to reproduction regardless of density. With $N(t)$ as population size and $r$ as the intrinsic rate of natural increase:
\[\dot{N} = rN\]This is the Malthusian growth equation. The solution is:
\[N(t) = N_0 e^{rt}\]where $N_0 = N(0)$ is the initial population size. The parameter $r = b - d$ is the difference between per capita birth rate $b$ and death rate $d$.
When $r > 0$, the population grows without bound — clearly unrealistic. Real populations are bounded by food, space, disease, and predation. The per capita growth rate $\dot{N}/N = r$ is constant, independent of $N$, which is the key unrealistic assumption.
The Logistic Equation
Verhulst added the simplest possible density dependence: the per capita growth rate declines linearly with $N$, reaching zero at the carrying capacity $K$:
\[\frac{\dot{N}}{N} = r\!\left(1 - \frac{N}{K}\right)\]This gives the logistic differential equation:
\[\dot{N} = rN\!\left(1 - \frac{N}{K}\right)\]The term $(1 - N/K)$ acts as a braking factor. When $N \ll K$, growth is approximately exponential. When $N \to K$, $\dot{N} \to 0$. When $N > K$, $\dot{N} < 0$ and the population declines.
The equilibria are $N^* = 0$ (unstable) and $N^* = K$ (stable). Stability follows from $d\dot{N}/dN = r(1 - 2N/K)$, which equals $r > 0$ at $N=0$ and $-r < 0$ at $N=K$.
Analytical Solution
The logistic ODE is separable. Rewriting:
\[\frac{dN}{N(1 - N/K)} = r\,dt\]Using partial fractions:
\[\frac{1}{N(1 - N/K)} = \frac{1}{N} + \frac{1/K}{1 - N/K}\]Integrating both sides:
\[\ln N - \ln(1 - N/K) = rt + C\]Solving for $N(t)$ with initial condition $N(0) = N_0$:
\[\boxed{N(t) = \frac{K}{1 + \left(\dfrac{K}{N_0} - 1\right)e^{-rt}}}\]Key properties of this solution:
| Property | Value |
|---|---|
| Asymptote as $t \to \infty$ | $K$ |
| Inflection point (maximum growth rate) | $N = K/2$ |
| Time to inflection (from $N_0 < K/2$) | $t^* = \frac{1}{r}\ln!\frac{K - N_0}{N_0}$ |
| Maximum $\dot{N}$ | $rK/4$ |
The inflection at $N = K/2$ is ecologically important: it corresponds to maximum sustainable yield (MSY) in fisheries management. Harvesting to maintain $N = K/2$ theoretically maximizes the long-run catch.
Discrete Logistic Map and Chaos
Replacing continuous time with discrete generations gives the discrete logistic map (May 1976):
\[N_{t+1} = r N_t\!\left(1 - \frac{N_t}{K}\right)\]Rescaling with $x_t = N_t/K$:
\[x_{t+1} = r x_t(1 - x_t)\]This deceptively simple map exhibits a full range of dynamical behaviors depending on $r$:
| Growth Rate $r$ | Behavior |
|---|---|
| $0 < r \leq 1$ | Monotone convergence to $x^* = 1 - 1/r$ |
| $1 < r \leq 3$ | Damped oscillations to stable equilibrium |
| $3 < r \leq 3.449$ | Period-2 limit cycle |
| $3.449 < r \leq 3.544$ | Period-4 cycle |
| $r > 3.57$ | Chaos (period-doubling cascade complete) |
| $r = 4$ | Full chaos on $[0,1]$, ergodic |
The bifurcation cascade follows the Feigenbaum constant $\delta \approx 4.669$: the ratio of successive bifurcation intervals converges to $\delta$. For $r = 4$, the invariant density is:
\[\rho(x) = \frac{1}{\pi\sqrt{x(1-x)}}\]This U-shaped distribution means the chaotic trajectory spends more time near 0 and 1 than near 0.5. The Lyapunov exponent $\lambda_L = \ln 2 > 0$ confirms sensitive dependence on initial conditions.
Allee Effects
The standard logistic assumes per capita growth rate is maximized at low density. The Allee effect (Allee 1931) occurs when per capita growth rate decreases at low density due to:
- Difficulty finding mates (especially in sexually reproducing species)
- Loss of cooperative hunting or predator dilution
- Reduced social thermoregulation
The strong Allee effect (component) introduces a critical threshold $A$ ($0 < A < K$):
\[\dot{N} = rN\!\left(\frac{N}{A} - 1\right)\!\left(1 - \frac{N}{K}\right)\]This cubic model has three equilibria: $N=0$ (stable), $N=A$ (unstable — the Allee threshold), and $N=K$ (stable). Populations below $A$ decline to extinction despite positive $r$. The Allee threshold acts as a minimum viable population size from a purely deterministic standpoint.
The weak Allee effect does not create an unstable equilibrium but slows growth at low density without causing decline:
\[\dot{N} = rN\!\left(1 - \frac{N}{K}\right)\!\left(\frac{N}{N + A}\right)\]Here per capita growth approaches zero as $N \to 0$ rather than going negative, avoiding the extinction basin.
Theta-Logistic Generalization
The theta-logistic model (Gilpin and Ayala 1973) allows nonlinear density dependence:
\[\dot{N} = rN\!\left[1 - \left(\frac{N}{K}\right)^\theta\right]\]The standard logistic corresponds to $\theta = 1$. When $\theta < 1$, density dependence is strongest at low densities (concave relationship). When $\theta > 1$, density dependence is felt primarily near $K$. The parameter $\theta$ can be estimated from time series data.
The solution to the theta-logistic is:
\[N(t) = K\!\left[1 + \left(\left(\frac{N_0}{K}\right)^{-\theta} - 1\right)e^{-r\theta t}\right]^{-1/\theta}\]Model selection between logistic ($\theta=1$) and theta-logistic can be performed using AIC comparing fits to population time series.
Fitting to Data
Fitting the logistic to a population time series with observations $N_1, N_2, \ldots, N_T$ at times $t_1, \ldots, t_T$ typically uses nonlinear least squares:
\[\hat{\theta} = \arg\min_{\theta} \sum_{i=1}^T \left[N_i - N(t_i; r, K, N_0)\right]^2\]Alternatively, rearranging the discrete logistic in difference form:
\[\frac{N_{t+1} - N_t}{N_t} = r - \frac{r}{K}N_t\]This is a linear regression of per capita growth rate on $N_t$, yielding OLS estimates $\hat{r}$ (intercept) and $\hat{r}/K$ (slope). Confidence intervals on $K = \hat{r}/\hat{\text{slope}}$ require the delta method or bootstrapping since $K$ is a ratio of estimated quantities.
A common diagnostic is the Pella-Tomlinson plot: per capita growth rate $\Delta N_t / N_t$ vs $N_t$, which should be linear with negative slope under the logistic assumption.