DoAssignment study guide

F2.2 · Calculate population growth with conceptual and mathematical models

Learn to calculate population growth with conceptual and mathematical models through clear examples and targeted practice.

Ontario Grade 12 Biology

Population Dynamics

Conceptual and mathematical models for changing populations

A pond has 120 fish at the start of a year. During the year, some fish are born, some die, and some move into or out of the pond. The number of fish at the end depends on all of these changes. Population growth is the change in the number of individuals in a population over time. In this lesson, you will connect that idea to familiar ecology from SBI3U, compare two conceptual models, and practise calculations. Each model makes assumptions, so choosing a model requires more than substituting numbers.

What you will learn

Define a population and identify the factors that change its size.
Describe the difference between exponential and logistic population growth.
Calculate population change, percentage growth, and future population size using suitable models.
Explain why a population model is an estimate rather than a guaranteed prediction.

From SBI3U ecology to population change

In ecology, a population is a group of individuals of the same species living in the same area at the same time. For example, all the fish of one species in a pond could be counted as one population. A population is not the same as a community, which includes populations of different species in an area.

Four processes change population size. Births add individuals, and deaths remove them. Immigration means individuals move into the area; emigration means they move out. If a question gives counts for these processes, the net change is the number added minus the number removed.

A positive net change means the population increased during the interval. A negative net change means it decreased. A net change of zero means its size stayed the same over that interval, even though births, deaths, or movement may still have occurred.

\Delta N=(B+I)-(D+E)

Population size is a count of individuals in a defined area.
Births and immigration add individuals; deaths and emigration remove them.
A population can have zero net change while individuals continue to enter and leave.

Two conceptual models of growth

A conceptual model describes a pattern and the conditions that could produce it. It helps us reason about a population before or alongside calculation. It does not prove that a real population will follow that pattern.

In an exponential growth model, population size increases by the same proportion during each equal time interval. If resources are plentiful and conditions remain favourable, each interval can add more individuals than the one before because the growing population has more individuals able to reproduce. The graph has a J-shaped pattern. Exponential growth is a useful model for a limited period, not a promise of unlimited growth.

In a logistic growth model, growth begins quickly when resources are available, then slows as the population becomes larger. Food, space, nesting sites, or other resources can limit further growth. The population may level off near the carrying capacity, written as

K

. Carrying capacity is the population size an environment can support over time under its current conditions. The graph has an S-shaped pattern.

Real populations may rise and fall rather than settle smoothly at one value. Conditions can change, and individuals may move in or out. A conceptual model points to possible patterns; observations are needed to judge how well it fits a particular population.

N_{t+1}=N_t(1+r)

Exponential growth describes a constant proportional increase and a J-shaped pattern.
Logistic growth describes slowing growth as the population approaches environmental limits.
Carrying capacity is not necessarily fixed; it can change when environmental conditions change.

Evidence, rates, and mathematical models

To calculate growth, first identify the time interval and the population counts. A change of 30 individuals over one month is not the same rate of change as 30 individuals over one year. Keep the time unit consistent throughout a calculation.

The percentage growth rate compares the change with the starting population. Dividing by the starting size makes the result meaningful for comparing populations of different sizes. Multiply by 100 to express the rate as a percentage. A percentage rate applies to the stated interval unless the question says otherwise.

For a simple exponential calculation, let

N_0

be the starting population, let

N_t

be the population after

t

equal intervals, and let

r

be the proportional growth rate per interval written as a decimal. The model multiplies the population by the same factor,

1+r

, each interval. For example, a 5% increase corresponds to

r=0.05

The model assumes that the proportional rate stays the same during the intervals being calculated. That assumption may not hold if resources become limited or conditions change. A calculation can be mathematically correct and still be a poor prediction if its assumptions do not fit the population.

Evidence for a population estimate comes from counting or sampling individuals in a defined area and time. A sample is a smaller part of a population used to estimate the whole. Sampling may miss individuals or count some twice. Results therefore have uncertainty, meaning the true population could differ from the estimate. Do not treat a model output as an observed count.

\text{growth rate (\%)}=\frac{N_t-N_0}{N_0}\times100

Percentage growth uses the starting population as the reference.
Convert a percentage to a decimal before using it as a proportional rate.
State the time interval and units, and distinguish a calculation from an observation.

Choosing and checking a model

Use the information in the question to choose a calculation. If you are given births, deaths, immigration, and emigration, calculate the net change. If you are given a starting population and a percentage change over one interval, calculate the new size. If the same proportional rate is said to continue for several equal intervals, use repeated growth.

Check that the answer makes biological sense. A positive rate should produce a larger population in the simple growth model. A negative rate should produce a smaller one. If the result is not a whole number, report an estimate and explain any rounding. Population counts are whole individuals, but model calculations can produce decimal estimates.

A model is a simplified representation. Exponential growth assumes a steady proportional rate. Logistic growth represents limits through a leveling pattern, but its curve alone does not identify exactly which resource is limiting. Neither model replaces careful observation.

N_t=N_0(1+r)^t

Match the model to the information and assumptions in the question.
Check the direction, units, and reasonable size of the result.
Models simplify; evidence is needed to assess how well a model represents a real population.

Comparing conceptual population-growth models

Model	Pattern	Main idea	Useful caution
Exponential	J-shaped	Same proportional increase each equal interval	The rate may not stay constant
Logistic	S-shaped	Growth slows as limits affect the population	The carrying capacity may change

Worked example

Find the net population change

A small bird population begins a season with 84 birds. During the season, 19 are born, 7 immigrate, 12 die, and 5 emigrate. Find the net change and the population at the end of the season.

Sort additions and removals
Births and immigration add birds. Deaths and emigration remove them. Keeping these groups separate prevents a sign error.
$(19+7)-(12+5)$
Calculate net change
There are 26 additions and 17 removals, so the population increases by 9 birds.
$26-17=9$
Find the ending count
Add the net change to the starting population. The result is the count predicted from the stated events.
$84+9=93$

Answer: The net change is an increase of 9 birds. The ending population is 93 birds.

Check: Additions exceed removals, so an ending count greater than 84 is sensible.

Worked example

Calculate percentage growth

A plant population is estimated at 240 individuals at the start of a growing season and 300 at the end. Calculate its percentage growth over that season.

Find the change
Subtract the starting estimate from the ending estimate to find how many individuals were added overall.
$300-240=60$
Compare change with the start
Divide the increase by 240, the starting population. This gives the increase as a fraction of the original size.
$\frac{60}{240}=0.25$
Convert to a percentage
Multiplying the fraction by 100 expresses the result as a percentage. The rate describes this growing season, not automatically every future season.
$0.25\times100=25\%$

Answer: The estimated population growth was 25% over the growing season.

Check: A 25% increase on 240 is 60, which matches the difference between 300 and 240.

Worked example

Project growth over equal intervals

A population has 500 individuals. A model assumes it grows by 8% per year for two years, with the same proportional rate each year. Use the exponential model to estimate the population after two years. Round to the nearest whole individual.

Convert the rate
Write 8% as a decimal by dividing by 100. Each year, the model keeps 100% of the current population and adds 8%.
$r=0.08$
Apply the growth factor
There are two equal yearly intervals. Use the factor for each interval twice, so the second increase acts on the first year's larger population.
$N_2=500(1.08)^2$
Calculate and round
The model gives 583.2 individuals. Since a population count uses whole individuals, round the estimate to the nearest whole number.
$500(1.08)^2=583.2\approx583$

Answer: The estimated population after two years is 583 individuals.

Check: The estimate is greater than 500, as expected for a positive growth rate. It is a projection under the stated assumption, not an observed count.

Common mistakes and how to avoid them

Adding deaths and emigration to the population.

Correction: Treat births and immigration as additions. Subtract deaths and emigration.

Dividing the change by the ending population when calculating percentage growth.

Correction: Use the starting population as the reference because the question asks how much the population changed from its original size.

Treating a percentage as a whole number in the growth equation.

Correction: Convert the percentage to a decimal. For example, use 0.08 for 8%.

Assuming an exponential projection must describe the population in the future.

Correction: The projection depends on a constant proportional rate. Changing conditions or limited resources can make the real pattern different.

Lesson summary

Population size changes through births, deaths, immigration, and emigration.
Exponential growth models a steady proportional increase; logistic growth represents slowing growth near environmental limits.
Percentage growth compares the change with the starting population.
Repeated growth applies the same growth factor once per equal interval.
Calculations are estimates or projections whose usefulness depends on the assumptions and quality of evidence.

Check your understanding

Question 1

A population begins at 150 individuals and ends at 180. What is its percentage growth over the interval?

16.7%
20%
30%
120%

Show answer and explanation

20%

The change is 30. Dividing 30 by the starting population of 150 gives 0.20, or 20%.

Question 2

A population has 200 individuals and grows by 10% in one interval. What is its modelled size after that interval?

Show answer and explanation

210

A 10% increase is 20 individuals. Adding 20 to 200 gives 220. The value 210 would be a 5% increase.

Question 3

Which statement best describes logistic growth?

The population increases by the same number every interval without limit.
The population decreases by a fixed percentage every interval.
Growth slows as the population approaches environmental limits.
The population immediately reaches its carrying capacity.

Show answer and explanation

Growth slows as the population approaches environmental limits.

The logistic model represents growth slowing as environmental limits affect the population. It does not say that the population immediately reaches carrying capacity.

Key terms

Population: Individuals of the same species living in the same area at the same time.
Net change: The additions to a population minus the removals during a stated interval.
Immigration: Movement of individuals into an area.
Emigration: Movement of individuals out of an area.
Exponential growth: A model in which the population increases by the same proportion during each equal time interval.
Logistic growth: A model in which population growth slows as environmental limits affect the population.
Carrying capacity: The population size an environment can support over time under its current conditions.
Sample: A smaller part of a population used to estimate the population as a whole.

Continue through SBI4U

View the complete SBI4U Ontario Grade 12 Biology curriculum and lessons

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Biology (SBI4U), expectation F2.2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question