AP Syllabus focus:
‘Population growth can be modeled mathematically using birth and death rates and change in population size over time.’
Population ecologists use simple equations to translate births, deaths, and changes in abundance into quantitative models. These parameters help compare populations, evaluate assumptions, and interpret how fast a population is increasing or decreasing over time.
Core quantities and parameters
Population size and time intervals
Population growth equations track how population size changes across a defined time interval (for example, days, months, or years). Clear time boundaries matter because counts of births and deaths depend on the census window.
Population size (N): The number of individuals in a population at a specified time.
When N is measured at two times, the change can be expressed as an absolute difference or as a rate that includes time.
Births and deaths as drivers of change
At the population level, two demographic processes determine the direction of change:
Births (B): individuals added
Deaths (D): individuals lost
Net change is determined by the balance of these processes, not by either process alone. Immigration and emigration can also affect N, but the simplest AP Bio models often focus on births and deaths to build foundational intuition.
Discrete-time growth equations
Net change and growth rate over an interval
Discrete models treat growth as occurring between two census points.
This graph compares geometric (discrete-time) growth with exponential (continuous-time) growth for the same starting population, showing that continuous compounding leads to a slightly faster increase in over time. It visually reinforces the difference between interval-based change (discrete censuses) and instantaneous change modeled with . Source
This is common when data are collected periodically (e.g., annual surveys).
= change in population size over the interval (individuals)
= population size (individuals)
= number of births during the interval (individuals)
= number of deaths during the interval (individuals)
This relationship is often turned into a rate by dividing by time, which allows comparisons across studies with different interval lengths.
= length of the time interval (time units)
= population growth rate over the interval (individuals per time)
Because B and D can scale with population size, absolute rates are hard to compare across populations of different sizes. That motivates per capita parameters.
Per capita rates and the parameter r
Standardising by population size
Per capita measures express births or deaths per individual per unit time, supporting comparisons across populations.
Per capita birth rate (b): Average number of births per individual per unit time.
A matching death parameter is commonly used alongside b to describe losses.
Per capita death rate (d): Average number of deaths per individual per unit time.
These rates combine into a single parameter that captures overall tendency to grow or shrink under the observed conditions.
The per capita growth rate
The parameter r summarizes net per capita change:
If b > d, then r is positive and N tends to increase.
If b < d, then r is negative and N tends to decrease.
If b = d, then r is zero and N is stable (over that interval).
= per capita population growth rate (per time)
= per capita birth rate (births per individual per time)
= per capita death rate (deaths per individual per time)
Continuous-time form and key assumptions
Linking r to change in N
When births and deaths are treated as occurring continuously (a useful idealisation), population change is written as a rate proportional to N.
Exponential growth produces a J-shaped curve when population size is plotted against time, reflecting the differential form under unlimited resources. The paired logistic curve adds resource limitation (carrying capacity) and shows how growth slows as increases. Source
This form is central because it connects an individual-level average (r) to whole-population change.
= instantaneous rate of change in population size (individuals per time)
= per capita population growth rate (per time)
= population size at time (individuals)
Interpreting this equation requires recognising its assumptions:
r is constant over the time period (no rapid environmental shifts affecting b or d).
Individuals are treated as demographically similar (age structure and sex ratio effects are ignored).
The population is effectively closed to migration unless additional terms are added.
Measurement reflects a defined spatial boundary so N, B, and D refer to the same population.
Interpreting parameter values from data
When working with empirical data, keep the mapping from observations to parameters clear:
B and D are counts tied to a specific interval.
b and d are rates per individual, so they depend on how N is estimated for that interval (often using an average N across the interval).
r is a net rate, so similar r values can arise from very different combinations of b and d.
FAQ
They often use an average population size (e.g., mean of starting and ending $N$) as the denominator.
If growth is rapid, shorter census intervals can reduce bias in estimated $b$, $d$, and $r$.
Because $r$ is a net quantity: high $b$ and high $d$ can yield the same $r$ as low $b$ and low $d$.
This can imply very different life histories and sensitivities to disturbance despite identical net growth.
$\Delta N/\Delta t$ uses changes between two census points (discrete time).
$ dN/dt $ treats change as continuous and instantaneous, which is an idealisation that can simplify modelling when events occur throughout the interval.
If many individuals are pre-reproductive, observed $b$ may be low even if adult fecundity is high.
Age-specific birth and death rates can therefore differ markedly from whole-population averages $b$ and $d$.
When immigration or emigration is substantial relative to births and deaths.
In that case, net change is better represented by $\Delta N = (B - D) + (I - E)$, where $I$ is immigrants and $E$ is emigrants.
Practice Questions
A population has 140 births and 95 deaths in one year. State for the year and whether the population increased or decreased. (2 marks)
(1)
Population increased (1)
Explain how the parameters , , , , , and are related in population growth models, and describe two assumptions commonly made when using to model growth. (5 marks)
Defines/links as population size at a given time (1)
States net change as (1)
Explains and are per capita rates (births/deaths per individual per unit time) (1)
States and interprets sign of (growth/decline) (1)
Gives two valid assumptions (any two): constant over interval; closed population (no migration); individuals demographically similar/no age structure; continuous change approximation (1)
