Logistic growth model and carrying capacity (8.4.3) | AP Biology Notes

AP Syllabus focus:

‘As limits to growth are imposed, populations typically follow a logistic growth model approaching carrying capacity.’

Population size cannot increase indefinitely in real ecosystems.

Side-by-side graphs compare exponential (J-shaped) growth with logistic (S-shaped) growth. The logistic curve rises rapidly at low population size and then slows as limiting factors intensify, approaching an upper boundary (carrying capacity, $K$ ). Source

The logistic growth model explains how resource limitation and other constraints slow growth as density rises, producing an S-shaped growth curve that stabilises near a limit.

Core ideas: logistic growth and carrying capacity

Carrying capacity (K)

Carrying capacity (K): the maximum population size an environment can sustain over time given available resources and conditions.

K is not a fixed property of a species; it reflects the environment and can change with shifts in food, space, shelter, nutrients, water, and accumulation of wastes.

Logistic growth model (S-shaped growth)

Logistic growth model: a population growth pattern in which growth is rapid at low density, then slows as limiting factors intensify, approaching a stable size near K.

A key feature is that the per capita growth rate declines as population size (N) increases, because individuals compete more strongly for the same limiting resources.

This plot shows logistic population growth rate ( $dN/dt$ ) as a function of population size ( $N$ ). Growth rate is zero at $N=0$ and $N=K$ , and it reaches a maximum at $N=K/2$ , highlighting how density dependence suppresses growth as the population nears carrying capacity. Source

The logistic growth equation

The logistic model is commonly expressed as a differential equation describing how population size changes over time.

$\dfrac{dN}{dt} = r_{\max}N\left(1-\dfrac{N}{K}\right)$

$\dfrac{dN}{dt}$ = change in population size per unit time (individuals per unit time)

$N$ = population size (individuals)

$r_{\max}$ = maximum per capita growth rate under ideal conditions (per unit time)

$K$ = carrying capacity (individuals)

When N is much smaller than K, the term $\left(1-\dfrac{N}{K}\right)$ is close to 1, so growth approximates exponential. As N approaches K, the term approaches 0 and growth slows.

Phases and graph interpretation

Early phase: rapid increase

Low density means abundant resources per individual.
Birth rates are relatively high and death rates relatively low.
Growth resembles exponential because environmental limits have minimal effect.

Deceleration phase: density effects strengthen

As N rises, individuals experience:
- reduced access to food or nesting sites
- increased competition within the population
- higher transmission of infectious disease or parasites
- increased stress or territorial conflict
These effects reduce the net increase in population size, bending the curve.

Near equilibrium: approaching K

Net growth approaches zero as births + immigration balance deaths + emigration (in open populations).
The population may fluctuate around K because environments vary and responses can lag behind change.

What “limits to growth” means in this model

In the logistic framework, limits are constraints that intensify with density and reduce population growth as N increases. Practically, K represents the point at which the environment’s total resource supply and conditions support no further long-term increase in N.

Key implications for AP Biology:

Logistic growth links population density to growth rate, unlike unconstrained exponential growth.
K is an ecological outcome of resource availability and environmental quality, not a guaranteed stable endpoint.
The same species can have different K values in different habitats or seasons because limiting resources differ.

FAQ

They often fit a logistic curve to time-series data using statistical regression.

They may use moving averages, exclude disturbance periods, and report a confidence interval rather than a single $K$.

Yes. Local differences in limiting resources (nest sites, food patches, moisture) can create different effective $K$ values.

This can produce multiple local equilibria rather than one ecosystem-wide $K$.

Theoretical $K$ assumes stable conditions and immediate population response.

Realised $K$ reflects variable weather, disturbances, and time lags in reproduction/survival, so it can shift over time.

If births respond to density with a delay, the population can overshoot and then decline.

These dynamics are often modelled by adding delayed density dependence, which can create cycles around $K$.

Logistic equations track change in $N$, but field populations can change due to movement as well as births/deaths.

This can make a population appear to exceed or fall below $K$ even when local resources are unchanged.

Practice Questions

State what happens to the population growth rate as population size $N$ approaches carrying capacity $K$ , and explain why. (2 marks)

Growth rate decreases and tends towards zero as $N \to K$ . (1)
Because density-dependent limiting factors (e.g., competition for resources) increase, reducing per capita net reproduction/survival. (1)

A population shows an S-shaped growth curve over time. Using the logistic growth model, explain the pattern shown, referring to $r_{\max}$ , $N$ , and $K$ . (5 marks)

At low $N$ , resources are abundant; growth is rapid and approximates exponential. (1)
$r_{\max}$ describes the maximum per capita growth rate under ideal conditions and is most closely approached when $N \ll K$ . (1)
As $N$ increases, the per capita growth rate declines due to increasing density-dependent limitations. (1)
Growth rate slows as $N$ approaches $K$ because $\left(1-\dfrac{N}{K}\right)$ becomes small. (1)
Net growth becomes ~0 near $K$ , producing a plateau (with possible small fluctuations). (1)

Try All Topic Practice Questions

Written by:

Dr Shubhi Khandelwal

Qualified Dentist and Expert Science Educator

Shubhi is a seasoned educational specialist with a sharp focus on IB, A-level, GCSE, AP, and MCAT sciences. With 6+ years of expertise, she excels in advanced curriculum guidance and creating precise educational resources, ensuring expert instruction and deep student comprehension of complex science concepts.