A declining species and an increasing logistic species

On a mangrove swamp, scientists are investigating the population of mudskippers and crabs. Initially, there are 800 mudskippers and 800 crabs on the mangrove swamp. At time tt years, the number of mudskippers and crabs are M,CM,C respectively.

(a) For the mudskippers, the scientists discovers that every year, the growth rate is 0.4% of the population size and the death rate is 5 mudskippers per year.

(i) Write down a differential equation relating M,tM,t. mark

(ii) Solve the differential equation in part (i) and determine what happens to the mudskipper population in the future. marks

(b) For the crabs, the scientists propose that C,tC,t are related by the differential equation dCdt=8C−0.005C2\frac{dC}{dt}=8C-0.005C^2.

(i) Find the number of crabs in the mangrove swamp when the rate of change of CC is a maximum. marks

(ii) Find CC in terms of tt. Sketch a graph of CC against tt. marks

[2021/TJC/Prelim/I/10]

Total marks: 12

Solution

If this part uses an earlier result, revisit (a)(ii) Solving and interpreting or open all working together.

(b)(i)

Maximum growth rate

For crabs, dCdt=8C−0.005C2\frac{dC}{dt}=8C-0.005C^2. Find CC when the rate of change is maximum.

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Hint

  1. The rate is a downward parabola in CC; its vertex is at the midpoint of the roots.

Answer · (b)(i)

C=800C=800 maximises growth; rate 3200/year.

Explore the idea

Open interactive exploration

The growth rate is largest at C = 800

The rate R(C) = 8C − 0.005C² is a parabola. A line at height 3200 touches it exactly once.

Height of the horizontal linek = 3200
Try an example
Test boundary heights

Buttons use full-precision boundary values; numbers shown here are rounded.

Number of distinct population sizes
1

The maximum rate 3200 occurs only at C = 800.

All listed intersections are within the original drawing window. A touching point counts once; hollow endpoints are excluded.

Coordinates are rounded to 6 significant figures. Inspect a row to highlight that intersection and bring it into view.
Pointxy
8003200
  • Growth rate R(C)
  • A chosen rate
  • Population sizes with that rate
  • Intersection selected in the coordinate table
Key idea

A logistic population grows fastest at half the carrying capacity.

Ideas to try
Find the rate at C=400C=400 and at C=1200C=1200: they are equal.