Organisms and Populations · Lesson 5 of 11
Logistic Growth, Carrying Capacity and Resource Limits
“Why ecological resources impose limits; Reading the sigmoid growth curve; Building the Verhulst–Pearl logistic equation; Worked logistic calculations with biological interpretation; The fastest absolute increase and model assumptions; Selecting a growth model from evidence”
• Explain how limited resources modify exponential growth. • Identify the lag, acceleration, deceleration and saturation phases of a sigmoid curve. • Interpret carrying capacity and derive the logistic growth expression. • Compare the exponential and logistic models and their limits. • Perform quantitative calculations using logistic growth rates.
Why ecological resources impose limits
No population can multiply indefinitely within a finite habitat. Individuals require energy, nutrients, water, space and suitable physical conditions. At low abundance, these resources may be plentiful relative to demand, so per-capita reproduction remains high. As abundance rises, resource competition, disease transmission, crowding or waste accumulation can reduce reproduction or increase mortality. Consequently growth often slows before indefinite exponential multiplication would be possible. This broad relationship motivates density-dependent population models.
Carrying capacity K is the population size that an environment can sustain in the particular circumstances represented by a model. It is not a permanent property of the species alone. A dry season, habitat restoration, loss of prey or change in nesting sites can alter K. Two habitats can support different numbers of the same species, and the same habitat can have different effective carrying capacities at different times. The carrying capacity is therefore a contextual quantity associated with available resources and ecological interactions.
Reading the sigmoid growth curve
The simplest logistic model produces a sigmoidal, or S-shaped, curve when population N is plotted vertically against time. Its early portion begins gently if the starting population is very small, then steepens as more individuals reproduce. The middle section can rise rapidly while enough resources remain. As abundance approaches K, growth progressively slows, and the curve approaches a horizontal plateau. The original chapter describes a lag phase followed by acceleration, deceleration and an asymptote near carrying capacity.
The words lag, acceleration and deceleration describe the overall curve, but they should not be confused with separate compulsory biological stages. If the initial population is already near K, an obvious lag or steep acceleration may not occur. If a population starts above K, the simplest model predicts decline toward K. Real populations may oscillate around a shifting carrying capacity due to seasons, delayed feedbacks, epidemics or external disturbances, rather than tracing one perfect S-shaped line.
Building the Verhulst–Pearl logistic equation
The exponential model states dN/dt=rN. To represent declining per-capita growth as resources become limiting, the logistic model multiplies rN by the factor (K−N)/K, equivalent to 1−N/K. Thus dN/dt=rN(1−N/K). The ratio N/K is the fraction of capacity currently represented by the population, and the parenthetical factor describes the remaining fraction in this idealised model. The expression is a mathematical approximation of density dependence, not a literal measurement of food remaining.
When N is much smaller than K, N/K is close to zero and the density factor is close to one, so dN/dt≈rN and logistic growth resembles exponential growth. As N rises toward K, the factor approaches zero and net growth slows. At N=K, the predicted instantaneous net change is zero. When N exceeds K and r is positive, 1−N/K becomes negative, predicting a decline. These limiting cases provide a useful check that the equation reflects its intended ecological mechanism.
Worked logistic calculations with biological interpretation
Suppose a population has N=200 individuals, intrinsic net rate r=0.30 year⁻¹ and carrying capacity K=1000 individuals. The density factor is 1−200/1000=0.80. The instantaneous growth rate is dN/dt = 0.30×200×0.80 = 48 individuals per year at that moment. Under the exponential model at the same N and r, it would be 0.30×200 = 60 individuals per year. Thus density limitation reduces the predicted instantaneous increase by 12 individuals per year. The arithmetic does not mean exactly 48 births will occur because deaths and subsequent changes in N are included in the net rate.
Now take N=900 with the same r and K. The factor becomes 1−0.9 = 0.1, giving dN/dt=0.30×900×0.10=27 individuals per year. Although there are more organisms than at N=200, their net increase is smaller because the density factor is much lower. At N=1000 the predicted net rate is zero; at N=1200 it is 0.30×1200×(−0.2)=−72 individuals per year. The negative sign means the model predicts a decline toward carrying capacity, not that organisms are being counted as negative individuals.
The fastest absolute increase and model assumptions
In the simple logistic equation the absolute net growth rate rN(1−N/K) is greatest at N=K/2, assuming positive and constant r and K. This follows because the product N(1−N/K) is a concave quadratic with its maximum halfway between its two zeros, N=0 and N=K. For K=1000 and r=0.30 year⁻¹, N=500 gives a maximum instantaneous increase of 0.30×500×0.5 = 75 individuals per year. Above that point, larger population size is outweighed by increasing density limitation.
The logistic model assumes, among other simplifications, a fixed carrying capacity, proportional density feedback, a smoothly changing population and no major lag between population changes and their ecological consequences. Real organisms may have delayed reproduction, age-specific mortality, seasonal resources, immigration and nonlinear competitive effects. Such populations can overshoot K or fluctuate rather than approaching it monotonically. A model's value lies in making assumptions and consequences explicit so observations can confirm or challenge them.
Selecting a growth model from evidence
A J-shaped exponential curve is appropriate as a short-term approximation when the net per-capita rate is nearly constant and resource effects are weak. An S-shaped logistic curve is useful where the net rate systematically declines as N increases and growth approaches a relatively stable capacity. Neither curve can be identified reliably from only two observations, and neither should be selected merely because the graph looks roughly upward-sloping. Measurements across multiple times and conditions reveal whether a model's assumptions are tenable.
An application of this reasoning is to collect population data over many years, plot abundance against time with labelled axes and compare the observed pattern with expected exponential or logistic behaviour. Human population history is shaped by changing fertility, mortality, healthcare, migration and social conditions, so a single fixed-r or fixed-K model may not describe a century of data. The mathematical concepts help analyse trends without implying that natural populations or people always follow one simple curve.
| Feature | Exponential model | Logistic model |
|---|---|---|
| Equation | dN/dt = rN | dN/dt = rN(1−N/K) |
| Growth curve | J-shaped for positive r | S-shaped when N begins below K |
| Resource treatment | No explicit limitation | Growth weakens with density |
| Long-term behaviour | Unlimited if r stays positive | Approaches K under fixed ideal assumptions |
| Key caution | Overestimates sustained growth | K and feedback are not always constant |
Quick Quiz
Quiz
What does carrying capacity K represent in the simple logistic model?
If N is much smaller than K, what does logistic growth resemble?
For N=200, K=1000 and r=0.3 year⁻¹, what is the logistic instantaneous growth rate?
In the simple logistic model with positive r, at what population is absolute growth greatest?
What does the model predict if N is greater than K?
Practice Problems
- Derive the logistic model from exponential growth and interpret the density factor.
- A population of 400 organisms lives in habitat with K=1000 and r=0.2 year⁻¹. Find its instantaneous growth rate.
- Find the population size producing maximal net growth when K=800 and calculate that rate for r=0.4 year⁻¹.
- An animal population begins above K. Explain the predicted change and an ecological reason the prediction may not follow a smooth curve.
- Describe an S-shaped growth curve with labelled axes and compare it with a J-shaped curve.
Start with dN/dt=rN, where r is net per-capita increase in ideal low-density conditions. Introduce a density-dependent factor that is near one at low N and falls linearly to zero at carrying capacity K: (K−N)/K = 1−N/K. Multiplying yields dN/dt=rN(1−N/K). This factor is an idealised representation of decreasing per-capita net growth due to resource and interaction limits.
Key Takeaways
• Finite resources and ecological interactions can reduce per-capita net growth. • Carrying capacity K depends on the habitat and may change over time. • The logistic equation is dN/dt = rN(1−N/K). • When N is much smaller than K, logistic and exponential growth are similar. • At N=K the simple logistic net growth rate is zero, and above K it becomes negative. • Absolute logistic growth is maximal at N=K/2 when r and K are constant and positive. • Natural populations may fluctuate around a changing carrying capacity rather than showing a perfect sigmoid.