Keep population units consistent

Since the 1990s, a group of scientists has started conservation work to prevent the extinction of a rare species of flying fox in the wild Western Australia forests. The number of such species, in thousands, observed in the forests at time tt years after the start of the conservation is denoted by NN. It is known that the death rate of the flying foxes is proportional to the number of flying foxes present and that the birth rate of the flying foxes has been maintained constant at 2000 per year. There were only 1000 flying foxes in the forests at the start of the conservation and the rate of increase of the number of flying foxes was 1500 per year when there were 3000 flying foxes.

(i) Form a differential equation relating N,tN,t and show that it can be reduced to 6dNdt=12−N6\frac{dN}{dt}=12-N. marks

(ii) Solve the differential equation to find NN in terms of tt. marks

(iii) Deduce the minimum number of years in integer, needed for the number of flying foxes to exceed 6000. marks

(iv) Sketch a graph to show how the number of flying foxes in the forests varies with time and suggest the long term behavior of the flying fox population. marks

(v) Suggest a possible limitation of the model in part (i). mark

[2019/RVHS/Prelim/I/10]

Total marks: 13

Solution

(i)

Forming the equation

Form a differential equation and show it reduces to 6dNdt=12−N6\dfrac{dN}{dt}=12-N.

Hint

  1. dNdt=births−deaths=2−kN\frac{dN}{dt}=\text{births}-\text{deaths}=2-kN.

  2. Use N=3N=3, dNdt=1.5\frac{dN}{dt}=1.5 to find kk.

Answer · (i)

6N′=12−N6N\prime=12-N.

(ii)

Solving

Solve for NN in terms of tt, given N=1N=1 at t=0t=0.

Explore this idea

Hint

  1. Separate and integrate, with a modulus.

  2. Use N(0)=1N(0)=1.

Answer · (ii)

N=12−11e−t/6N=12-11e^{-t/6} (thousands).

Explore the idea

The flying fox population

Each dash shows the slope the differential equation gives at that point. Change the starting value and the solution follows the dashes.

Starting value of N at t = 01
Read the solution at t4
Try an example
N at t = 4
6.313
Rate dN/dt there
0.948
  • Solution that starts at the purple dot
  • Closed-form answer
Key idea

The population tends to 12 thousand whatever the start.

Ideas to try
  1. Start above 12 and watch the population fall towards 12.
  2. Read NN at t=4t=4 years.
(iii)

First integer year

Deduce the least whole number of years for NN to exceed 6.

Hint

  1. Solve 12−11e−t/6>612-11e^{-t/6}>6.

Answer · (iii)

44 years.

(iv)

Sketch and long-term behaviour

Sketch NN against tt and describe the long-term behaviour.

Hint

  1. Starts at (0,1)(0,1), increasing, concave, with asymptote N=12N=12.

Flying fox population
Flying fox populationScroll across to see the full diagram.

Answer · (iv)

Rises concavely from 1 toward 12 thousand.

(v)

Limitation

Suggest a limitation of the model.

Hint

  1. Think about assumptions: constant births and proportional deaths.

Answer · (v)

Constant births/proportional deaths ignore changing habitat or mortality.