Biology · Introductory biology · Concept
Exponential and logistic population growth
Population growth models predict how the number of individuals N changes over time. Exponential growth assumes a constant per-capita growth rate r, so dN/dt = rN and N(t) = N₀e^(rt): the population grows ever faster, without limit. Logistic growth adds a carrying capacity K, the largest population the environment can sustain, so growth slows as N approaches K and the curve is S-shaped.
The per-capita growth rate
The per-capita growth rate r is births minus deaths per individual per unit time, leaving migration aside. A population of N individuals then grows by rN per unit time. With r = 0.2 per year, 100 individuals add about 20 a year and 1,000 add about 200: the same r gives faster total growth in a larger population.
Exponential growth
With r constant, N(t) = N₀e^(rt), where N₀ is the starting population. The curve is J-shaped, and the population doubles every ln 2/r time units however large it is. Bacteria in fresh medium, or a species arriving in an empty habitat, can grow this way for a while, until resources run short.
Logistic growth
Logistic growth multiplies rN by (1 − N/K), the fraction of the carrying capacity still unused. When N is small that factor is close to 1 and growth looks exponential; as N nears K it approaches 0 and growth stops. Above K the factor is negative, and the population shrinks back toward K.
The logistic curve
Solving the logistic equation gives N(t) = K/(1 + Ae^(−rt)), where A = (K − N₀)/N₀ makes the curve start at N₀. It rises fastest at N = K/2, where the growth rate reaches its largest value, rK/4.
Density dependence
Carrying capacity sums up density-dependent limits: food, space, disease and predation press harder as a population grows. Density-independent events, such as a frost or a flood, cut a population whatever its size, and the logistic model leaves them out.
Assumptions and limits
Both models assume a constant r, and the logistic model a constant K, with no age structure and no immigration or emigration. Real populations respond to crowding with a delay, so they can overshoot K and oscillate or crash, and K itself changes with the environment. Treat each model as a prediction to compare with counts, not as a law.
Common mistakes
- Writing a percentage as r: 5% per year is r = 0.05.
- Expecting logistic growth to be fastest near K: it is fastest at N = K/2.
- Confusing the per-capita rate r with the population’s total growth rate dN/dt = rN.
- Treating K as a fixed trait of a species: it depends on the environment.
Key terms
- Population
- All the individuals of one species living in a given area at a given time. Population size counts them; density divides the count by the area or volume.
- Per-capita rate
- A rate per individual rather than for the whole population, such as births per person per year. A constant per-capita rate adds more individuals as the population grows.
- Exponential population growth
- Growth at a constant per-capita rate, dN/dt = rN, so the population grows faster and faster. It assumes unlimited resources, which never lasts.
- Logistic population growth
- Growth that slows as the population nears the carrying capacity K: dN/dt = rN(1 − N/K). It gives an S-shaped curve and is a simple model of limited resources.
- Carrying capacity
- K, the largest population an environment can support over time, where births and deaths balance. It changes when the environment changes.
- Doubling time
- The time an exponentially growing quantity takes to double: ln 2 / k for a continuous rate k. It does not depend on the starting amount.
Work through an example
A herd of 50 deer has a per-capita growth rate r = 0.4 per year. Predict its size after 5 years with the exponential model, then with the logistic model and a carrying capacity K = 500.
Compare exponential and logistic growth →Sources and scope
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