UNIVERSITY OF SOUTH AFRICA (UNISA)
College of Science, Engineering and Technology
⋄
APM1514 ASSIGNMENT 02
Semester 1 — Due: Monday, 11 May 2026
⋄
Module Code: APM1514
Module Name: Applied Mathematics 1B
Assignment No.: 02
Due Date: 11 May 2026
Semester: Semester 1, 2026
Submitted in partial fulfilment of the requirements for APM1514
at the University of South Africa.
,UNISA | APM1514 Assignment 02
Question 1: Population Growth with Emigration
The questions below analyse a discrete population model with a birth rate b = 1.1 and a death
rate d = 0.5 per person per year.
1.1 Minimum Initial Population for Perpetual Growth (Fixed Emigration)
Question: Assume that 1000 people move out of the country each year. What should the
initial population be to ensure that the population will always increase?
Step 1: Build the population model.
The general discrete population model including births, deaths, and a fixed emigration of 1000
is:
Pn+1 = Pn + b Pn − d Pn − 1000
Substituting b = 1.1 and d = 0.5:
Pn+1 = Pn + 1.1Pn − 0.5Pn − 1000
Pn+1 = (1 + 1.1 − 0.5) Pn − 1000
Pn+1 = 1.6 Pn − 1000
Step 2: State the condition for perpetual growth.
For the population to always increase, the following must hold:
Pn+1 > Pn
Step 3: Substitute the model into the inequality.
1.6 Pn − 1000 > Pn
Step 4: Isolate Pn .
1.6 Pn − Pn > 1000
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, UNISA | APM1514 Assignment 02
0.6 Pn > 1000
Step 5: Divide both sides by 0.6.
1000
Pn >
0.6
Pn > 1666.67
Step 6: State the conclusion.
Since population is a whole number, the initial population must satisfy:
P0 ≥ 1667
Therefore, the minimum initial population required to guarantee that the population always
increases is 1667 people.
Implementation Insight
The model Pn+1 = 1.6Pn − 1000 has an equilibrium at P = 1000
0.6 ≈ 1666.67. Any initial
value strictly above this equilibrium causes the population to grow without bound, since
the multiplier 1.6 > 1.
1.2 Emigration Rate for Constant Population (Proportional Emigration)
Question: Assume that k% of the population present at the end of each year moves out.
What should k be to guarantee that for any initial population size, the population will forever
stay constant?
Step 1: Build the revised model.
When emigration is proportional to the population at rate k (expressed as a decimal), the
model becomes:
Pn+1 = Pn + b Pn − d Pn − k Pn
Substituting b = 1.1 and d = 0.5:
Pn+1 = (1 + 1.1 − 0.5 − k) Pn
Page 2 of 25
College of Science, Engineering and Technology
⋄
APM1514 ASSIGNMENT 02
Semester 1 — Due: Monday, 11 May 2026
⋄
Module Code: APM1514
Module Name: Applied Mathematics 1B
Assignment No.: 02
Due Date: 11 May 2026
Semester: Semester 1, 2026
Submitted in partial fulfilment of the requirements for APM1514
at the University of South Africa.
,UNISA | APM1514 Assignment 02
Question 1: Population Growth with Emigration
The questions below analyse a discrete population model with a birth rate b = 1.1 and a death
rate d = 0.5 per person per year.
1.1 Minimum Initial Population for Perpetual Growth (Fixed Emigration)
Question: Assume that 1000 people move out of the country each year. What should the
initial population be to ensure that the population will always increase?
Step 1: Build the population model.
The general discrete population model including births, deaths, and a fixed emigration of 1000
is:
Pn+1 = Pn + b Pn − d Pn − 1000
Substituting b = 1.1 and d = 0.5:
Pn+1 = Pn + 1.1Pn − 0.5Pn − 1000
Pn+1 = (1 + 1.1 − 0.5) Pn − 1000
Pn+1 = 1.6 Pn − 1000
Step 2: State the condition for perpetual growth.
For the population to always increase, the following must hold:
Pn+1 > Pn
Step 3: Substitute the model into the inequality.
1.6 Pn − 1000 > Pn
Step 4: Isolate Pn .
1.6 Pn − Pn > 1000
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, UNISA | APM1514 Assignment 02
0.6 Pn > 1000
Step 5: Divide both sides by 0.6.
1000
Pn >
0.6
Pn > 1666.67
Step 6: State the conclusion.
Since population is a whole number, the initial population must satisfy:
P0 ≥ 1667
Therefore, the minimum initial population required to guarantee that the population always
increases is 1667 people.
Implementation Insight
The model Pn+1 = 1.6Pn − 1000 has an equilibrium at P = 1000
0.6 ≈ 1666.67. Any initial
value strictly above this equilibrium causes the population to grow without bound, since
the multiplier 1.6 > 1.
1.2 Emigration Rate for Constant Population (Proportional Emigration)
Question: Assume that k% of the population present at the end of each year moves out.
What should k be to guarantee that for any initial population size, the population will forever
stay constant?
Step 1: Build the revised model.
When emigration is proportional to the population at rate k (expressed as a decimal), the
model becomes:
Pn+1 = Pn + b Pn − d Pn − k Pn
Substituting b = 1.1 and d = 0.5:
Pn+1 = (1 + 1.1 − 0.5 − k) Pn
Page 2 of 25
