Difficulty: Introductory to Intermediate | Prerequisites: Basic maths (multiplication, summation). Familiarity with the concept of exponential growth is helpful.
Population ecology asks a deceptively simple question: why do populations grow, shrink, or stay the same size? The answer involves growth rates, resource limits, and the interplay between birth and death. Life tables quantify survival and reproduction at each age, and growth models (exponential and logistic) describe how populations change over time. This material is central to ecology and has direct applications in conservation, wildlife management, and understanding disease outbreaks.
You will need to be comfortable with two key scenarios from the exam: a leopard frog population in a wetland pond (illustrating logistic growth and carrying capacity) and a hummingbird life table (illustrating survivorship, fecundity, and net reproductive rate).
Populations grow exponentially when resources are unlimited but shift to logistic growth as they approach the carrying capacity (K), where resources run out and growth stalls. Life tables track how survival decreases and reproduction increases with age, and the net reproductive rate (R0) tells you whether a population is growing, stable, or shrinking. Density-dependent factors (like competition for food or breeding sites) are what cause populations to level off at K.
Carrying capacity (K)
The maximum population size that a given environment can sustain indefinitely, given the available resources. Think of it as the population ceiling set by the environment.
Exponential growth
Population growth that occurs at a constant per capita rate, producing a J-shaped curve. In simple terms, the population doubles, then doubles again, then doubles again, with no limit in sight. This only happens when resources are unlimited.
Logistic growth
Population growth that slows as the population approaches carrying capacity, producing an S-shaped (sigmoid) curve. Think of it as exponential growth with a brake that gets stronger as the population gets bigger.
Per capita growth rate (r)
The rate of population increase per individual per unit time. When r is positive the population grows; when r is zero the population is stable; when r is negative the population shrinks.
Maximum per capita growth rate (r_max)
The highest possible per capita growth rate for a species, achieved under ideal conditions with unlimited resources. In simple terms, it is the species' growth potential when nothing holds it back.
Density-dependent factor
Any factor that affects population growth in a way that scales with population density. Examples: competition for food, competition for breeding sites, disease transmission, predation pressure. These are what cause the logistic slowdown as N approaches K.
Density-independent factor
Any factor that affects population growth regardless of how many individuals there are. Examples: a flood, a freeze, a wildfire, a volcanic eruption. These hit the population equally whether it is large or small.
Life table
A table that records survivorship (lx) and fecundity (mx) for each age class in a population. Think of it as a demographic spreadsheet that tracks how many survive and how many offspring they produce at each age.
Survivorship (lx)
The proportion of individuals born that survive to age x. It starts at 1.0 (all individuals alive at birth) and decreases with age.
Fecundity (mx)
The average number of offspring produced by an individual of age x. It often increases with age as organisms mature and become more reproductively capable.
Net reproductive rate (R0)
The average number of offspring produced by an individual over its entire lifetime, calculated as the sum of lx times mx for all age classes. If R0 > 1, the population is growing. If R0 = 1, it is stable. If R0 < 1, it is shrinking.
Life history tradeoff
The principle that organisms cannot maximise all aspects of survival and reproduction simultaneously. Energy spent on reproduction is energy not available for survival, and vice versa. In the hummingbird data, survivorship drops while fecundity rises with age, illustrating this tradeoff.
Initial population (N) = 50 frogs. Maximum per capita growth rate (r_max) = 0.2 per month. Population stabilises at approximately 500 individuals after 24 months.
The carrying capacity (K) of the wetland pond is 500 individuals. This is the population at which resources (breeding sites in the reeds, food in the centre of the pond) become limiting.
The logistic growth equation is: dN/dt = r_max × N × (K - N) / K
The term (K - N) / K is the key. When N is small relative to K, this term is close to 1 and growth is nearly exponential. As N approaches K, this term approaches 0 and growth slows to a halt.
At K = 500 and N = 500: (500 - 500) / 500 = 0. Growth rate is zero. The population is stable.
The hummingbird life table provides four age classes (x = 0, 1, 2, 3) with survivorship (lx) and fecundity (mx) at each age.
Survivorship (lx) is the probability of surviving from birth to age x: lx at age 0 = 1.0, age 1 = 0.5, age 2 = 0.2, age 3 = 0.05.
Fecundity (mx) is the number of offspring produced at each age: mx at age 0 = 0.0, age 1 = 1.0, age 2 = 2.0, age 3 = 3.0.
The probability that a newborn survives to age 2 is simply lx at age 2 = 0.2 (that is, 20%).
R0 = sum of (lx × mx) for all age classes.
Age 0: 1.0 × 0.0 = 0.0
Age 1: 0.5 × 1.0 = 0.5
Age 2: 0.2 × 2.0 = 0.4
Age 3: 0.05 × 3.0 = 0.15
R0 = 0.0 + 0.5 + 0.4 + 0.15 = 1.05
R0 = 1.05 means the population is growing (R0 > 1), but only just barely.
In the hummingbird data, survivorship decreases with age (from 1.0 to 0.05) while fecundity increases (from 0.0 to 3.0).
This illustrates the life history tradeoff: as organisms invest more in reproduction, their probability of surviving drops. Resources allocated to producing offspring come at the cost of survival.
Organisms that survive to older ages tend to reproduce more, but very few individuals reach those ages.
Density-dependent examples (from the frog scenario): increased competition for limited breeding sites in the reeds as the population grows. This factor intensifies as N rises and is what causes the population to stabilise at K.
Density-independent examples: a massive flood washing away egg masses, a sudden winter freeze killing 20% of frogs regardless of population size, solar radiation levels. These affect the population without regard to how dense it is.
The stabilisation of the frog population at 500 is driven by density-dependent factors, because the effect scales with population size.
dN/dt = r_max × N × (K - N) / K
Where:
dN/dt = change in population size over time
r_max = maximum per capita growth rate (0.2 per month for the frogs)
N = current population size
K = carrying capacity (500 for the frogs)
(K - N) / K = the proportion of carrying capacity still available
R0 = Σ (lx × mx) for all age classes x
Age (x) | lx | mx | lx × mx |
|---|---|---|---|
0 | 1.0 | 0.0 | 0.0 |
1 | 0.5 | 1.0 | 0.5 |
2 | 0.2 | 2.0 | 0.4 |
3 | 0.05 | 3.0 | 0.15 |
Total | R0 = 1.05 |
R0 > 1, so the population is growing.
N | (K - N) / K | Interpretation |
|---|---|---|
50 | 0.90 | Nearly exponential growth |
250 | 0.50 | Growth at half its maximum rate |
450 | 0.10 | Growth nearly stalled |
500 | 0.00 | No growth, population at K |
Logistic growth models are used in fisheries management to set sustainable harvest levels: you want to harvest at the point of maximum growth (around N = K/2), not when the population is near K and barely growing. Life tables underpin actuarial science (insurance) and are used in conservation biology to model endangered species recovery. Understanding density-dependent factors helps wildlife managers predict when populations will self-regulate and when intervention is needed.
Students often confuse carrying capacity (K) with the growth rate (r). K is a population size (a number of individuals), not a rate. In the frog scenario, K = 500 individuals, not 0.2 per month or 24 months.
Students sometimes think exponential growth continues forever in natural populations. It does not. All real populations eventually encounter resource limits and shift toward logistic growth.
A common error is calculating R0 by simply adding all fecundity values. You must multiply each fecundity (mx) by its corresponding survivorship (lx) before summing. Not all individuals survive to reproduce.
Students may think density-independent factors are irrelevant to population regulation. They are relevant; they can cause sudden population crashes. But they do not cause the gradual levelling off at K, which is driven by density-dependent factors.
⚠️ Know the logistic growth equation and be able to explain what happens to (K - N) / K as N approaches K. This is tested frequently.
⚠️ Be able to calculate R0 from a life table. Show your working: multiply lx by mx for each row, then sum.
⚠️ Distinguish density-dependent from density-independent factors with specific examples. "Competition for breeding sites" is density-dependent. "A flood" is density-independent.
⚠️ The carrying capacity question is straightforward but commonly missed: K = the population at which growth stabilises, read directly from the scenario (500 individuals for the frogs).
⚠️ Experimental design questions may ask you to design a study testing whether food limitation or breeding site limitation determines K. Know how to state a null hypothesis, identify variables, and describe control vs experimental groups.
True or False: Carrying capacity (K) is a rate of growth. (False, it is a population size.)
Fill in the blank: When N = K, the value of (K - N) / K equals ______. (0)
True or False: R0 = 1.05 means the population is shrinking. (False, it is growing.)
Fill in the blank: The probability of a newborn hummingbird surviving to age 2 is ______. (0.2)
True or False: A winter freeze that kills 20% of all frogs regardless of density is a density-dependent factor. (False, it is density-independent.)
Q: Based on the life table, what is the net reproductive rate (R0) of the hummingbird population, and is the population growing or shrinking?
A: R0 = (1.0 × 0.0) + (0.5 × 1.0) + (0.2 × 2.0) + (0.05 × 3.0) = 0 + 0.5 + 0.4 + 0.15 = 1.05. Since R0 > 1, the population is growing.
Q: Based on the leopard frog case study, what is the carrying capacity (K) of the wetland pond?
A: 500 individuals. The population stabilises at approximately 500 after 24 months.
Q: As the frog population approaches 500 individuals, what happens to (K - N) / K in the logistic growth equation?
A: It approaches 0.0. As N gets closer to K, the numerator (K - N) shrinks toward zero, so the whole fraction approaches zero and population growth effectively stops.
Q: Which of the following is a density-dependent factor: increased competition for breeding sites, a massive flood, solar radiation, or a winter freeze?
A: Increased competition for limited breeding sites in the reeds. It intensifies as population density rises.
Q: Explain the concept of life history tradeoffs using the hummingbird data.
A: Life history tradeoffs reflect the fact that organisms allocate finite energy between survival and reproduction. In the hummingbird data, survivorship (lx) drops sharply from 1.0 at birth to 0.05 at age 3, while fecundity (mx) rises from 0.0 to 3.0 over the same period. Older individuals that have survived invest heavily in reproduction, but the probability of reaching those older ages is very low. Energy used for reproduction is not available for maintaining the body, so increased fecundity comes at the cost of reduced survival.
Q: Design an experiment to test whether food limitation or breeding site limitation stabilises the frog population at 500.
A: Null hypothesis: food supplementation will not change the carrying capacity. Independent variable: food availability (supplemented vs not). Dependent variable: population size over time. Experimental group: ponds receiving supplemental food in the centre. Control group: identical ponds with no supplemental food, same breeding site availability. If supplementing food raises K above 500, then food was the limiting factor. If K remains at 500 despite extra food, breeding site limitation is the more likely cause.
Density-dependent factors tie directly to interspecific competition (the niches topic): competition for resources is both an ecological interaction and a driver of population regulation. The logistic model connects to evolution, because natural selection acts on heritable variation within populations, and population dynamics determine effective population size, which influences genetic drift. Conservation biology uses these models to predict population viability under habitat loss and climate change.
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