AP Syllabus focus:
‘Ecologists compare exponential and logistic models to explain how density and resources affect population growth patterns.’
Population size changes reflect both biological potential and environmental limits. Exponential and logistic models are simplified tools ecologists use to interpret real population data and to predict how resource availability and density effects shape growth.
Core idea: two contrasting growth patterns
This graph compares exponential growth (J-shaped curve) with logistic growth (S-shaped curve) on the same population-vs.-time axes. The dashed line marks carrying capacity (K), illustrating how density-dependent limits cause growth to decelerate and level off near K rather than increasing without bound. Source
Exponential growth (unlimited resources)
Exponential growth describes what happens when resources are effectively unlimited and density effects are minimal (often seen in newly colonized habitats or early in a rebound).
Exponential growth: Population growth in which the per capita rate of increase stays constant, producing a J-shaped curve when population size is graphed over time.
Key features:
Constant per capita growth rate: each individual, on average, contributes the same expected net number of offspring over time.
Growth accelerates as population size (N) increases because more individuals reproduce.
Best viewed as a baseline “potential” growth pattern, not a long-term expectation in nature.
Logistic growth (resource limitation and density dependence)
Logistic growth incorporates the reality that resources (space, food, nesting sites) become limiting, and density-dependent factors intensify as the population grows.
Carrying capacity (K): The maximum population size a particular environment can sustain over time, given available resources and other limiting conditions.
Key features:
S-shaped curve: rapid growth early, then slowing, then leveling near K.
Negative feedback: as N rises, competition and other density-dependent limits reduce per capita growth.
Population growth is highest at intermediate N and approaches zero as N approaches K.
Mathematical comparison (what the models assume)
This logistic-growth graph shows population increasing rapidly at first and then approaching a horizontal asymptote representing carrying capacity (often denoted K). It also highlights the inflection point where growth rate is maximal, reinforcing why logistic growth is fastest at intermediate population sizes and slows as the upper limit is approached. Source
= population size (individuals)
$</p><p>r^{-1} \frac{dN}{dt} = rN\left(1-\frac{N}{K}\right) K\left(1-\frac{N}{K}\right)$ = density-dependent reduction in growth (unitless)
Both models track change in population size over time, but they differ in how they treat resource limits:
Exponential model assumes the environment does not impose meaningful constraints on r.
Logistic model assumes r is effectively reduced as N/K increases, reflecting increasing competition and limitation.
How density and resources change growth patterns
What “density effects” mean in practice
As density rises, individuals often experience:
Reduced access to energy and nutrients
Increased competition for space or mates
Faster spread of disease/parasites
Increased predation in crowded, predictable patches
These mechanisms reduce births, increase deaths, or both—shifting observed growth away from exponential expectations and toward logistic-like dynamics.
Interpreting graphs and data using both models
Ecologists compare the models to explain real patterns:
This plot shows measured population size over time for a yeast culture, with points following a rapid early increase and then a plateau. The leveling illustrates how resource limitation and other density-dependent constraints reduce per-capita growth as the population approaches an upper limit (carrying capacity). Source
Early-time data can look exponential because density is low and resources per capita are high.
As limiting resources decline per capita, growth typically decelerates, matching logistic logic.
Fluctuations around K are common because environments vary; K is best treated as an estimate under particular conditions, not a fixed universal constant.
Using model comparison to explain population outcomes
When exponential is a useful approximation
Short time spans
Low-density populations
High resource availability relative to N
Situations with minimal density-dependent regulation
When logistic is a better explanation
Evidence of resource limitation (slowed growth as N rises)
Strong density-dependent changes in birth/death rates
Populations that stabilize (approximately) around a long-term average size
Model comparison is a way to connect mechanisms (resources, density effects) to patterns (J-shaped vs S-shaped growth) and to justify why a population’s trajectory changes as density increases.
FAQ
Use long-term data and infer $K$ as the central tendency of stable abundances.
Often: moving averages or fitting a logistic curve to multiple years.
A negative $r$ indicates per capita deaths exceed births.
The population declines, and logistic density dependence cannot reverse decline if conditions keep $r<0$.
Time lags in density dependence (e.g., delayed reproduction effects).
Resources can be depleted after growth has already accelerated.
Yes; $K$ can shift with resource supply, habitat area, or climate.
In practice, $K$ is context-specific rather than constant.
They compare goodness-of-fit of model predictions to observed $N(t)$.
They also check whether per capita growth declines with $N$ (supporting logistic).
Practice Questions
State one key difference between exponential and logistic population growth models, and identify which model includes carrying capacity. (2 marks)
One valid difference stated (e.g., exponential assumes unlimited resources/constant per capita growth; logistic includes density dependence/resource limitation). (1)
Logistic model identified as including carrying capacity, . (1)
A population shows rapid initial increase, then the growth rate slows as population size rises. Using exponential and logistic models, explain how density and resources could account for this pattern. (5 marks)
Early growth consistent with exponential model when density is low and resources per capita are high. (1)
Logistic model explanation: as increases, resources become limiting and/or competition increases. (1)
Density-dependent factors reduce per capita growth rate (lower births and/or higher deaths). (1)
Reference to carrying capacity as the level where growth approaches zero. (1)
Clear comparison that exponential does not include /does not reduce growth with , whereas logistic does via density dependence (e.g., term like ). (1)
