Population Growth Models: Exponential vs Logistic Growth

Exponential growth is growth at a constant per capita rate, which gives a J-shaped curve (dN/dt = rN), while logistic growth slows as the population approaches the carrying capacity K, which gives an S-shaped curve (dN/dt = rN(K − N)/K). In brief: exponential growth happens when resources are unlimited, as with bacteria in fresh medium or a species colonizing a new habitat. Logistic growth includes the effect of limited resources. The growth rate is highest at N = K/2. Density-dependent factors, such as disease and competition, regulate population size, while density-independent factors, such as storms, do not depend on the density.

These models are the foundation of population ecology and are used in conservation, fisheries and epidemiology. This guide gives the equations and shapes of both models, explains how to use them with numbers, compares density-dependent and density-independent factors, discusses the limits of the models, works through a numerical example and corrects common mistakes.

What is exponential growth?

When resources are not limiting, each individual reproduces at the same rate, so the number of new individuals added increases as the population grows. The equation for a continuous model is dN/dt = rN, where N is the population size, t is time and r is the per capita growth rate (births minus deaths per individual). The solution is N_t = N₀e^rt, where e ≈ 2.718. For a discrete model with a growth factor λ per time step, N_t = N₀λ^t.

A graph of N against time is a J-shaped curve that rises slowly at first and then steeply. The doubling time is t_d = ln 2 ÷ r ≈ 0.693 ÷ r. For bacteria, r is large and the doubling time is minutes. For elephants, r is about 0.05 per year, so doubling takes about 14 years at most. Exponential growth cannot continue for long: for example, one pair of rabbits breeding without limit would produce a population that would cover the Earth within a few years. Real examples of exponential growth include the early phase of a bacterial culture, an invasive species colonizing a new region and the growth of the world human population in the twentieth century.

What is logistic growth?

In real habitats, resources such as food, space and nesting sites are limited. As the population grows, competition increases, and the per capita growth rate falls. The logistic model adds a term to represent this: dN/dt = rN × (K − N) ÷ K, where K is the carrying capacity. The term (K − N) ÷ K is the fraction of K that is still unused.

• When N is much smaller than K, (K − N) ÷ K is close to 1, and the growth is close to exponential.
• When N = K/2, the number of individuals added per unit of time is the largest (dN/dt = rK/4). This is the maximum sustainable yield for harvesting.
• When N approaches K, (K − N) ÷ K tends to 0, and growth stops. The population levels off at K.
• If N is greater than K, the growth rate is negative, and the population falls back to K.

The resulting graph is an S-shaped (sigmoid) curve with a lag phase, a rapid growth phase, and a stable plateau. Real populations often fluctuate around K, and may overshoot it, especially when there are time delays in the response to crowding, such as in the reproduction of large animals. Carrying capacity is described in the earlier article.

Exponential and logistic growth compared

What are density-dependent and density-independent factors?

Density-dependent factors have a stronger effect as the population density increases. They are the main regulators of population size, and cause negative feedback.

• Competition for food, water, light, space and mates.
• Predation: predators often catch more prey as the prey becomes more abundant.
• Disease and parasites: spread more easily in crowded populations.
• Accumulation of waste, and territorial behavior that limits breeding.

Density-independent factors affect the same proportion of the population regardless of its density: severe weather, floods, fires, droughts, volcanic eruptions and human pollution. They can cause crashes or sudden changes, particularly in small populations and in species with short life cycles.

Worked example: a population of deer

Ten deer are introduced to an island. The per capita growth rate under good conditions is r = 0.30 per year. The island can support K = 500 deer. Compare the two models, year by year, using a discrete step: exponential N_t+1 = N_t + rN_t and logistic N_t+1 = N_t + rN_t(K − N_t)/K.

1. Year 1, exponential: 10 + 0.30 × 10 = 13. Logistic: 10 + 0.30 × 10 × (490 ÷ 500) = 10 + 2.94 = 12.9. The two models are almost the same at low density.
2. Exponential: N_t = 10 × 1.3^t. After 20 years: 10 × 1.3²⁰ = 10 × 190.0 = 1 900, nearly four times the capacity of the island.
3. Logistic: growth is fastest near N = K/2 = 250, which is passed at around year 14 to 15 (279 deer at year 15, with an increase of 37.5 that year). After that, the yearly increase falls as the population approaches 500, and it is only 22.3 in year 20.
4. Maximum growth: dN/dt at N = 250 is 0.30 × 250 × (250 ÷ 500) = 37.5 deer per year.

The island manager can learn from the model: harvesting the population at about half of K gives the largest sustainable harvest, and removing more than the growth rate leads to decline. The values in the table are from the model with these discrete steps, and are rounded. Real deer populations would show a time delay and fluctuate. In a real population, a very hard winter (density-independent) could reduce the numbers suddenly, and the population would grow again.

What are the limits of these models?

• They assume constant r and K, but in nature the environment changes.
• They ignore the age structure, and differences between individuals.
• They assume that the response to density is immediate, but there are usually time lags.
• They treat the population as closed, with no migration.
• They ignore interactions with other species (predators, competitors, food).

More complex models include these factors. Even so, the logistic model remains the standard introduction to population regulation.

How did the models develop?

Thomas Malthus (1798) described the potential for exponential growth of populations. Pierre-François Verhulst published the logistic equation in 1838. Raymond Pearl rediscovered it in the 1920s. Georgyi Gause tested the model on Paramecium populations in the 1930s. Robert May showed in the 1970s that simple discrete logistic equations can produce complex, even chaotic, behavior at high growth rates.

Which mistakes are common?

Mistake 1: Using the J-curve for populations that are limited

Exponential growth cannot go on for long. Resources soon limit growth.

Mistake 2: Thinking that growth is fastest at carrying capacity

The growth is fastest at K/2. At K, the growth rate is zero.

Mistake 3: Confusing the types of factors

Disease and competition are density-dependent. Fire and flood are density-independent.

Mistake 4: Treating K as a fixed number

It changes with the environment, so real populations vary around it.