Single-Species Population Dynamics
From exponential growth to chaos — the mathematics of populations
A Brief History
Exponential Growth and Decay
The simplest model of population change assumes that individuals reproduce at a constant per-capita rate and death is proportional to population size. This leads to the fundamental differential equation of population dynamics:
$$\frac{dN}{dt} = rN$$
where $N(t)$ is population size, $r$ is the intrinsic rate of increase, and the solution is:
$$N(t) = N_0 e^{rt}$$
If $r > 0$, the population grows exponentially. If $r < 0$, it decays exponentially. The doubling time is $\tau = \ln(2) / r$.
This model was famously applied by Malthus (1798) in his Essay on the Principle of Population, where he argued that while human populations grow geometrically, food supplies grow arithmetically—hence the inevitable spectre of famine. Although Malthus underestimated technological adaptation and overestimated fecundity, his mathematical insight was prescient.
The appeal of the exponential model lies in its simplicity: reproduction and death are density-independent, so the per-capita growth rate $r$ is constant. Yet this simplicity is also its fatal flaw. Exponential growth cannot persist indefinitely. Resources are finite, space is limited, and waste accumulates. Any real population must eventually slow its growth. This realization motivates the introduction of density dependence.
Logistic Growth
Verhulst (1838) introduced a simple modification to account for finite resources. He assumed that the per-capita growth rate decreases linearly with population size, reaching zero at the carrying capacity $K$:
$$\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)$$
The term $(1 - N/K)$ represents a density-dependent brake on growth. When $N \ll K$, growth is approximately exponential. When $N \approx K$, growth slows. At $N = K$, growth stops ($dN/dt = 0$).
The equilibria of the logistic model are found by setting $dN/dt = 0$: $N = 0$ (always an equilibrium, but unstable) and $N = K$ (the stable carrying capacity). For any initial population $N_0 > 0$, the solution approaches $K$ as $t \to \infty$, yielding the characteristic S-shaped (sigmoid) growth curve.
The logistic equation was rediscovered independently by Pearl & Reed (1920), who fitted it to census data for the United States population and other species. Their work established logistic growth as a cornerstone of population ecology. The model captures the essential tension between reproduction and resource limitation, and remains widely used in biology and applied ecology (fisheries, pest control, conservation).
Discrete-Time Models
Many organisms have non-overlapping generations: offspring are produced once per season, and parents die or cease reproduction. For such species, a discrete-time map is more natural than a continuous ODE. The general form is:
$$N_{t+1} = f(N_t)$$where $f$ encodes the population dynamics over one generation. The discrete logistic map is obtained by discretising the Verhulst equation:
$$N_{t+1} = rN_t\left(1 - \frac{N_t}{K}\right)$$By rescaling to the dimensionless variable $x_t = N_t / K$, this simplifies to:
$$x_{t+1} = \lambda x_t(1 - x_t)$$where $\lambda$ is the control parameter. This quadratic map is one of the most celebrated objects in dynamical systems theory. For $\lambda < 1$, the population dies out ($x \to 0$). For $1 < \lambda < 3$, there is a stable fixed point at $x^* = 1 - 1/\lambda$. At $\lambda = 3$, this fixed point loses stability and a period-2 cycle is born—the start of a cascade of period-doubling bifurcations: period 4, then 8, then 16, and so on. Beyond a critical value $\lambda_\infty \approx 3.5699\ldots$, the map exhibits chaos: trajectories are aperiodic and sensitive to initial conditions.
May (1976) famously published "Simple mathematical models with very complicated dynamics" in Nature, highlighting the discrete logistic map as a warning to population biologists and ecologists: even the simplest nonlinear model can generate unpredictable behaviour. This revelation opened the door to chaos theory in population ecology and has profound implications for fisheries management and species forecasting.
Bifurcations
A bifurcation is a qualitative change in the dynamics as a parameter varies. In the continuous logistic equation, there is a transcritical bifurcation at $r = 0$: for $r < 0$, the origin is stable (population extinct); for $r > 0$, the origin becomes unstable and the carrying capacity $N = K$ becomes stable. The two equilibria "exchange stability."
In the discrete logistic map, the route to chaos is via period-doubling bifurcations. At $\lambda = 1$ a nontrivial fixed point appears through a transcritical bifurcation; it remains stable until $\lambda = 3$, where it loses stability and a period-2 orbit is born. As $\lambda$ increases further, the period-2 orbit bifurcates into period-4, period-8, and so on, accumulating at $\lambda_\infty \approx 3.5699$. Beyond $\lambda_\infty$, the attractor is chaotic, punctuated by periodic windows (bands of parameter space where periodic behaviour re-emerges).
A different class of bifurcation is the saddle-node bifurcation (or fold bifurcation), where two equilibria—one stable, one unstable—collide and annihilate as a parameter changes. This can lead to sudden shifts in dynamical behaviour.
A compelling ecological example is the spruce budworm model, proposed by Ludwig, Jones & Holling (1978):
$$\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right) - \frac{BN^2}{A^2 + N^2}$$
The first term is logistic growth; the second is a predation/disease term (saturating at large $N$). The parameters $A$ and $B$ control the predation rate. This model exhibits multiple stable equilibria and hysteresis: the system can jump between a low-population and high-population state depending on where it starts and how parameters change. This phenomenon explains why budworm outbreaks can be sudden and hard to reverse.
Explore: Population Growth
Use the controls below to explore continuous and discrete population dynamics. The left panel shows trajectories of the logistic ODE for various initial conditions. The right panel displays the famous bifurcation diagram for the discrete logistic map, revealing the period-doubling route to chaos.
Harvesting
Many populations are harvested or exploited: fish stocks, timber, game animals, etc. The simplest harvesting model adds a constant removal rate $h$ to the logistic equation:
$$\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right) - h$$At equilibrium, $dN/dt = 0$, so $rN(1 - N/K) = h$. For a given $h$, this quadratic equation has 0, 1, or 2 positive solutions depending on the magnitude of $h$. If $h$ exceeds the maximum growth rate, the population is driven to extinction. The maximum sustainable yield (MSY) is the largest harvest rate compatible with a stable population:
$$h_{\text{MSY}} = \frac{rK}{4}$$This occurs when the population is at $N = K/2$. MSY is a cornerstone of fisheries and wildlife management, although it is now known to be a conservative target given uncertainty and environmental variability.
An alternative is proportional harvesting, where the removal rate is proportional to population size:
$$\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right) - EN$$where $E$ is the harvesting effort. This model is more robust to population fluctuations and is commonly used in fisheries science. Clark (1990) provides a comprehensive treatment of optimal harvesting strategies and the bioeconomics of renewable resource management.
The Allee Effect
Most of the models discussed so far assume that populations can recover from any level of decline, no matter how small. However, at very low densities, social or reproductive cooperation may break down. The Allee effect— named after Allee (1931)—captures this phenomenon: per-capita growth rate decreases below a critical threshold population density.
A strong Allee effect can be modeled as:
$$\frac{dN}{dt} = rN\left(\frac{N}{A} - 1\right)\left(1 - \frac{N}{K}\right)$$where $A$ is the Allee threshold or "refuge density." The factor $(N/A - 1)$ is negative when $N < A$, causing the population to decline toward extinction even in the absence of harvesting or predation. Thus there are now two stable equilibria: $N = 0$ (extinction) and $N = K$ (carrying capacity). If a population falls below $A$, it is trapped on a trajectory to extinction—a critical concern in conservation biology.
Courchamp et al. (1999) provide a comprehensive review of the Allee effect across taxa, including examples from birds, mammals, insects, and plants. Allee effects are particularly important for endangered species management: a species near extinction may be "doomed" if its population falls below the critical threshold, even if habitat is restored. This has profound implications for reserve design and reintroduction programs.
Stochastic Effects
All models so far have been deterministic. In reality, populations are subject to randomness. Two main sources of stochasticity can be distinguished:
- Demographic stochasticity: Randomness in birth and death events at the individual level. Even in a stable population, some individuals reproduce and others do not, purely by chance. In small populations, this can lead to extinction even when the per-capita growth rate is positive.
- Environmental stochasticity: Variation in external conditions (weather, food availability, disease) that affect all individuals. This manifests as random fluctuations in parameters like $r$ or $K$.
A classical approach to stochasticity in populations is the birth-death process, where each individual has a small probability of reproducing or dying per unit time. This leads to a Markov chain on the state space of population sizes. For large populations, the birth-death process approximates the deterministic ODE; for small populations, it generates a distribution of extinction times and fixation probabilities.
The connection to the Moran process, discussed in Population Genetics, is direct: the Moran model assumes a fixed population size and tracks the frequency of a gene or strategy, while the birth-death process allows population size to fluctuate. Under neutral selection, allele frequency in the Moran process is a martingale; with selection, there is a deterministic drift toward the fitter type, but individual trajectories remain stochastic.
A continuous-time analogue of stochasticity is given by Langevin equations, which add a noise term to the deterministic ODE:
$$\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right) + \sigma \sqrt{N} \, dW_t$$where $dW_t$ is a Wiener process increment and $\sigma$ is the noise intensity. The factor $\sqrt{N}$ ensures that demographic stochasticity vanishes as the population grows large. Analyses of such stochastic differential equations reveal that environmental noise can significantly shorten extinction times in declining populations, especially those experiencing an Allee effect.
Exercises
Conceptual Questions
- Explain why the exponential growth model $dN/dt = rN$ is unrealistic for any real population over long time periods. What biological factors would cause growth to slow?
- The logistic equation predicts that a population approaching carrying capacity $K$ will stabilise at $N = K/2$ only if $r$ takes a certain critical value. Is this true? Why or why not?
- In the discrete logistic map $x_{t+1} = \lambda x_t(1-x_t)$, what happens to the long-term dynamics as $\lambda$ increases from 1 to 4? Describe the sequence of bifurcations.
- If a population experiences an Allee effect, why is intervention (e.g., translocating individuals to increase density) more likely to help a declining population compared to one with only density-dependent growth?
- How does environmental stochasticity differ from demographic stochasticity in their effects on extinction risk? Which is more dangerous for a large population, and why?
Computer Problems
- Logistic Growth Simulation. Implement the logistic ODE $\frac{dN}{dt} = rN(1 - N/K)$ using a numerical integrator (e.g., RK4) with $r = 0.5$, $K = 100$, and initial condition $N_0 = 10$. Plot $N(t)$ over $[0, 20]$ and verify that it approaches $K$. Repeat with $r = 2.0$ and comment on the rate of approach.
- Period-Doubling Route to Chaos. Implement the discrete logistic map $x_{t+1} = \lambda x_t(1-x_t)$ and compute the bifurcation diagram by varying $\lambda$ from 2.5 to 4.0 in steps of 0.01. For each $\lambda$, iterate 500 times to remove transients, then plot the next 100 iterates. Identify the period-doubling bifurcations and estimate the onset of chaos around $\lambda \approx 3.57$.
- Harvesting and Maximum Sustainable Yield. Given the harvested logistic model $\frac{dN}{dt} = rN(1 - N/K) - h$ with $r = 1$, $K = 1000$, explore equilibrium solutions as a function of harvest rate $h$. Plot equilibrium $N^*$ versus $h$ and identify the maximum sustainable yield $h_{\text{MSY}} = rK/4$. Simulate what happens if $h$ exceeds $h_{\text{MSY}}$ by 10%.
- The Allee Effect and Extinction Thresholds. Implement the Allee-effect model $\frac{dN}{dt} = rN(N/A - 1)(1 - N/K)$ with $r = 1$, $A = 100$, and $K = 500$. Plot the phase portrait ($dN/dt$ versus $N$) and identify the three equilibria. Starting from three initial conditions ($N = 50, 150, 400$), simulate trajectories and show which escape extinction and which collapse to $N = 0$.
- Stochastic Population Dynamics. Implement the Langevin equation $\frac{dN}{dt} = rN(1 - N/K) + \sigma\sqrt{N}\,dW_t$ with $r = 0.5$, $K = 100$, $\sigma = 0.3$, and $N_0 = 80$. Generate 10 independent stochastic trajectories over $[0, 50]$ using the Euler–Milstein method. Plot all trajectories together and compute the mean time to extinction (if any trajectories hit $N < 1$). Compare with the deterministic case.
References
- Allee, W. C. (1931). Animal Aggregations: A Study in General Sociology. University of Chicago Press.
- Clark, C. W. (1990). Mathematical Bioeconomics: The Optimal Management of Renewable Resources (2nd ed.). Wiley-Interscience.
- Courchamp, F., Clutton-Brock, T., & Grenfell, B. (1999). Inverse density dependence and the Allee effect. Trends in Ecology & Evolution, 14(10), 405–410.
- Ludwig, D., Jones, D. D., & Holling, C. S. (1978). Qualitative analysis of insect outbreak systems: the spruce budworm and forest. Journal of Animal Ecology, 47, 315–332.
- Malthus, T. R. (1798). An Essay on the Principle of Population. J. Johnson. (Reprinted many times; Dover 1976 edition recommended.)
- May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261, 459–467.
- Pearl, R., & Reed, L. J. (1920). On the rate of growth of the population of the United States since 1790 and its mathematical representation. Proceedings of the National Academy of Sciences, 6(6), 275–288.
- Verhulst, P.-F. (1838). Notice sur la loi que la population suit dans son accroissement. Correspondance Mathématique et Physique, 10, 113–121.