Difficulty: Intermediate | Prerequisites: Introduction to Ecology study notes (b, d, r, life tables)
This unit builds on per capita rates (b, d, r) from the ecology intro and develops two models of population growth: exponential and logistic. The logistic model introduces carrying capacity (K) and density dependence, which are central to understanding real-world population regulation. If you are not comfortable with r = b – d, review the ecology intro notes first.
Populations grow based on their size (N) and per capita growth rate (r). When r stays constant regardless of population size, growth is exponential and unsustainable. In reality, r usually declines as the population approaches its carrying capacity (K), producing logistic (S-shaped) growth. The fastest rate of population increase happens at K/2, which is the basis for maximum sustainable yield.
rmax (maximum per capita rate of increase)
The highest possible per capita growth rate for a species, achieved when resources are unlimited and there is no competition. In simple terms, this is how fast the population could grow under ideal conditions.
r (per capita rate of increase)
The actual per capita growth rate at a given time, which may be less than rmax due to environmental constraints. r = b – d.
dN/dt
The rate of change in population size over time. This is the quantity you are solving for in growth equations. Think of it as "how many individuals are added (or lost) per unit time."
Exponential growth
Population growth where r remains constant regardless of N. Produces a J-shaped curve. Only occurs when resources are effectively unlimited.
Density independence
When the per capita growth rate (r) does not change as population size (N) changes. Exponential growth is density-independent.
Logistic growth
Population growth where r decreases as N approaches carrying capacity (K). Produces an S-shaped (sigmoid) curve. More realistic than exponential growth.
Density dependence
When the per capita growth rate (r) changes as a function of population size. As N increases, competition for resources intensifies, birth rates drop, death rates rise, and r declines.
K (carrying capacity)
The maximum population size that an environment can sustain indefinitely. When N = K, r = 0 and the population stops growing.
K/2
Half the carrying capacity. This is the population size at which dN/dt (the total rate of population change) is greatest. The inflection point on the logistic curve.
MSY (maximum sustainable yield)
The largest number of individuals that can be harvested from a population without causing it to decline. Achieved by maintaining the population at K/2, where dN/dt is maximised.
The equation is dN/dt = rN. Population change depends on two things: how many individuals there are (N) and how fast each one reproduces net of deaths (r).
r stays constant no matter how large N gets. There is no feedback from crowding.
The curve is J-shaped: slow at first when N is small, then accelerating as the population multiplies.
Exponential growth can occur when a population colonises a new, empty habitat with abundant resources. It cannot continue indefinitely because resources are finite.
As N increases, individuals compete for limited resources (food, space, mates).
Birth rates (b) tend to decrease because individuals get less energy for reproduction.
Death rates (d) tend to increase because of crowding, disease, and starvation.
Since r = b – d, the effect of rising d and falling b is that r decreases as N increases. This is density dependence.
The equation is dN/dt = rN × (K – N) / K.
The term (K – N) / K is the density-dependent brake. When N is small relative to K, this term is close to 1, and growth is nearly exponential. As N approaches K, the term approaches 0, and growth slows to a stop.
When N = K, (K – N) / K = 0, so dN/dt = 0. The population has reached carrying capacity and stops growing.
The curve is S-shaped (sigmoid).
r (per capita rate) is greatest when N is smallest (near 0), because there is minimal competition. r = rmax when N is very small.
r = 0 when N = K.
dN/dt (total population change per unit time) is greatest at N = K/2. This is the inflection point of the S-curve. Below K/2, total growth is accelerating. Above K/2, it is decelerating.
This distinction between r and dN/dt is critical. r is highest when N is near zero; dN/dt is highest at K/2.
Because dN/dt is greatest at K/2, a population maintained at K/2 adds the most new individuals per unit time.
If you harvest exactly that surplus, the population stays at K/2 and keeps producing at its maximum rate. This harvest amount is the MSY.
MSY is used in fisheries and wildlife management to set sustainable catch limits.
Exponential growth equation
dN/dt = rN
Where dN/dt is the rate of population change, r is the per capita rate of increase, and N is population size.
Logistic growth equation
dN/dt = rN × (K – N) / K
The (K – N) / K term reduces growth as N approaches K. When N is near 0, (K – N)/K is close to 1 (nearly exponential). When N = K, it equals 0 (no growth).
Key graph to know: the logistic S-curve
X-axis: time. Y-axis: population size (N).
The curve starts shallow (small N), steepens through the inflection point at K/2, then levels off as it approaches K.
Be able to draw this from memory, label K and K/2, and mark where dN/dt is greatest (at K/2) and where r is greatest (near N = 0).
Key graph to know: r vs. N
For density-independent growth: r is a flat horizontal line (constant).
For density-dependent growth: r starts at rmax when N is near 0 and declines linearly to 0 when N = K.
Fisheries managers use MSY to set annual catch quotas. The goal is to keep fish populations near K/2, where the population replenishes itself most quickly. Overfishing pushes N well below K/2, and the population produces fewer total new fish each year, making recovery slow.
Invasive species colonising a new habitat often show near-exponential growth initially, because resources are abundant and there are few natural competitors or predators. This is why early detection and rapid response are critical in invasive species management.
Students often confuse r and dN/dt. r is the per capita rate (per individual). dN/dt is the total rate of change for the whole population. r is highest near N = 0; dN/dt is highest at K/2.
Students sometimes think carrying capacity (K) is a fixed number for a species. K depends on the environment and can change if resources change, habitat is destroyed, or conditions shift.
A common error is believing that density-independent growth means the population is not affected by anything. Density-independent factors (storms, fires, droughts) do affect populations, but their impact does not scale with N.
Students often assume logistic growth always produces a smooth S-curve. In nature, populations can overshoot K and oscillate before stabilising, or they may never reach a stable K at all.
⚠️ Be able to draw a logistic curve from memory, label K, K/2, and mark where dN/dt is greatest and where r is greatest. This is explicitly listed in the review guide.
⚠️ Know both equations (exponential and logistic) and be able to explain what each term means and what happens when you change it.
⚠️ Understand the biological reasons why b decreases and d increases as N rises. This is a likely short-essay question.
⚠️ Be able to explain why exponential growth cannot continue indefinitely and give two examples of density-dependent factors.
⚠️ Know where exponential growth tends to occur (populations colonising new, empty habitats).
Fill in the blank: The logistic growth equation is dN/dt = rN × ____. ((K – N) / K)
True or false: dN/dt is greatest when N = K. (False. dN/dt is greatest at K/2.)
Fill in the blank: When N = K, the value of r is ____. (0)
True or false: Exponential growth occurs when r changes with population size. (False. Exponential growth occurs when r is constant, i.e. density-independent.)
Fill in the blank: The concept of harvesting a population at K/2 to maximise yield is called ____. (Maximum sustainable yield, MSY)
Q: Why can't exponential growth continue indefinitely? Give two examples of density-dependent factors.
A: Exponential growth requires unlimited resources, which do not exist in nature. As N increases, individuals compete for finite resources. Two density-dependent factors: intraspecific competition for food (more individuals means less food per individual) and increased disease transmission (higher density makes disease spread faster).
Q: Draw and label a logistic growth curve. Where is r = rmax? Where is r = 0? Where is dN/dt greatest?
A: The curve is S-shaped, starting near 0 and levelling off at K. r = rmax when N is near 0 (essentially no competition). r = 0 when N = K (population at carrying capacity). dN/dt is greatest at the inflection point, N = K/2.
Q: Explain why b tends to decrease and d tends to increase as N rises. What does this mean biologically?
A: As population size grows, competition for resources intensifies. Individuals get less food and fewer territories, so they reproduce less (b falls). Meanwhile, starvation, disease, and stress from crowding cause more deaths (d rises). Since r = b – d, r declines as N increases. This is density-dependent regulation.
Q: In what populations does exponential growth tend to occur?
A: In populations that colonise new, empty habitats where resources are abundant and competition is minimal. (Answer: A in the review guide's multiple choice.)
Q: Is density-independent growth possible? Give an example.
A: Yes, temporarily. A population that colonises a new island or is introduced to a habitat with no competitors may grow exponentially for a period, because r does not yet decline with N. Eventually resources become limiting and growth shifts to density-dependent.
This unit depends on the ecology intro's definitions of b, d, r, and life tables. If those are not solid, the growth equations will not make sense.
It feeds directly into the Competition unit: competition is a density-dependent factor that reduces r as N increases. Understanding density dependence here is prerequisite for understanding how competition regulates populations and shapes communities.
population growth, exponential growth, logistic growth, J-curve, S-curve, sigmoid curve, carrying capacity, K, K/2, maximum sustainable yield, MSY, density dependence, density independence, rmax, dN/dt, per capita growth rate, population regulation, resource limitation, BBIO 180, intro biology, University of Washington