UNIVERSITY OF SOUTH AFRICA
College of Science, Engineering and Technology
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APM1514: Applied Mathematics 1B
Assignment 02 — Semester 1, 2026
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
APM1514
Module Code:
Applied Mathematics 1B
Module Name:
Assignment 02
Assignment:
11 May 2026
Due Date:
Submitted in partial fulfilment of the requirements for APM1514 — UNISA 2026
,UNISA | APM1514 Assignment 02
Question 1: Population Model with Emigration
Population dynamics can be modelled as a discrete recurrence relation where the net change
per period depends on birth rate, death rate, and migration. Here, with birth rate b = 1.1 and
death rate d = 0.5 per person per year, the net natural growth rate is 1 + 1.1 − 0.5 = 1.6.
Question 1.1: Minimum Initial Population for Perpetual Growth
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?
Solution:
The population model combines natural growth and a fixed annual emigration of 1000 people.
Starting from Pn , the population at the end of the next year 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 = 1.6Pn − 1000
For the population to always increase, we need:
Pn+1 > Pn
Substituting the model expression:
1.6Pn − 1000 > Pn
Subtracting Pn from both sides:
Page 2 of 26
, UNISA | APM1514 Assignment 02
0.6Pn > 1000
Dividing both sides by 0.6:
1000
Pn >
0.6
Pn > 1666.6
Therefore the initial population must satisfy P0 > 1666.67. Since population is a whole num-
ber:
P0 ≥ 1667
Critical Consideration
The condition Pn > 1666.67 must hold at every step, not only at t = 0. Since the model
is linear and the growth factor is 1.6 > 1, any initial value above the equilibrium point
(which is 1000
0.6 ≈ 1666.67) will produce a sequence that increases without bound. Start-
ing below the equilibrium causes the sequence to decrease toward zero and eventually
become negative.
Question 1.2: Emigration Rate for a Constant Population
Question: Assume that instead, 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?
Solution:
When emigration is proportional to the current population, the model becomes:
k
Pn+1 = Pn + b Pn − d Pn − Pn
100
Substituting b = 1.1 and d = 0.5:
Page 3 of 26
College of Science, Engineering and Technology
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
APM1514: Applied Mathematics 1B
Assignment 02 — Semester 1, 2026
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
APM1514
Module Code:
Applied Mathematics 1B
Module Name:
Assignment 02
Assignment:
11 May 2026
Due Date:
Submitted in partial fulfilment of the requirements for APM1514 — UNISA 2026
,UNISA | APM1514 Assignment 02
Question 1: Population Model with Emigration
Population dynamics can be modelled as a discrete recurrence relation where the net change
per period depends on birth rate, death rate, and migration. Here, with birth rate b = 1.1 and
death rate d = 0.5 per person per year, the net natural growth rate is 1 + 1.1 − 0.5 = 1.6.
Question 1.1: Minimum Initial Population for Perpetual Growth
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?
Solution:
The population model combines natural growth and a fixed annual emigration of 1000 people.
Starting from Pn , the population at the end of the next year 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 = 1.6Pn − 1000
For the population to always increase, we need:
Pn+1 > Pn
Substituting the model expression:
1.6Pn − 1000 > Pn
Subtracting Pn from both sides:
Page 2 of 26
, UNISA | APM1514 Assignment 02
0.6Pn > 1000
Dividing both sides by 0.6:
1000
Pn >
0.6
Pn > 1666.6
Therefore the initial population must satisfy P0 > 1666.67. Since population is a whole num-
ber:
P0 ≥ 1667
Critical Consideration
The condition Pn > 1666.67 must hold at every step, not only at t = 0. Since the model
is linear and the growth factor is 1.6 > 1, any initial value above the equilibrium point
(which is 1000
0.6 ≈ 1666.67) will produce a sequence that increases without bound. Start-
ing below the equilibrium causes the sequence to decrease toward zero and eventually
become negative.
Question 1.2: Emigration Rate for a Constant Population
Question: Assume that instead, 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?
Solution:
When emigration is proportional to the current population, the model becomes:
k
Pn+1 = Pn + b Pn − d Pn − Pn
100
Substituting b = 1.1 and d = 0.5:
Page 3 of 26
