A decreasing forest model stops at extinction

Botanists are studying how excessive logging is affecting the population of forest trees in a country in South-East Asia. The population of trees in a forest after tt years is denoted by xx, in thousands. One simple model proposed is that the population increases at a rate proportional to its population size. It is known that trees are chopped off at a constant rate of 3 thousand trees per year and the population stays constant when x=px=p.

(i) Find a differential equation involving x,tx,t. marks

(ii) The initial population of trees in a forest is 5 thousand trees. Solve the differential equation in part (i) and show that x=(5−p)e3t/p+px=(5-p)e^{3t/p}+p. marks

(iii) Given that the population of tree decreases in a finite time, state an inequality satisfied by pp. Sketch the graph of xx against tt for this case, showing clearly the coordinates of the graph’s point of intersection with the horizontal axis. Hence explain the meaning of this axial intercept in the context of the question. marks

[2020/CJC/Prelim/II/4]

Total marks: 12

Solution

(i)

Equilibrium calibration

Find a differential equation for xx (thousands) if the population is constant when x=px=p.

Hint

  1. dxdt=kx−3\frac{dx}{dt}=kx-3.

  2. Constant at x=px=p means kp−3=0kp-3=0.

Answer · (i)

x′=3x/p−3x\prime=3x/p-3, p>0p>0.

(ii)

Solving

With x=5x=5 initially, show x=(5−p)e3t/p+px=(5-p)e^{3t/p}+p.

Hint

  1. Separate: ∫dxx−p=3p∫dt\int\frac{dx}{x-p}=\frac3p\int dt.

Answer · (ii)

x=(5−p)e3t/p+px=(5-p)e^{3t/p}+p.

(iii)

Extinction time

The population decreases in finite time. State an inequality for pp, sketch xx against tt and interpret the intercept.

Explore this idea

Hint

  1. Decrease requires 5−p<05-p<0.

  2. Solve x=0x=0.

Tree population before depletion
Tree population before depletionScroll across to see the full diagram.

Answer · (iii)

p>5p>5; extinction at t∗=(p/3)ln⁡[p/(p−5)]t_*=(p/3)\ln[p/(p-5)].

Population interpretation only 0≤t≤t∗0\le t\le t_*.

Explore the idea

The forest for p = 8

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 x at t = 05
Read the solution at t2.6
Try an example
x at t = 2.6
0.046
Rate dx/dt there
-2.983
  • Solution that starts at the purple dot
  • Closed-form answer
Key idea

With p=8p=8 the forest is gone at t∗=83ln⁡83≈2.6t_*=\frac83\ln\frac83\approx2.6 years.

Ideas to try
  1. Slide the starting value above 8: the forest grows instead.
  2. Find where the solution meets x=0x=0.