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

(a)(i)

Forming the equation

Write a differential equation for mudskippers: growth 0.4% of the population per year, 5 deaths per year.

Hint

  1. dMdt=growth−deaths\frac{dM}{dt}=\text{growth}-\text{deaths}.

Answer · (a)(i)

M′=0.004M−5M\prime=0.004M-5.

(a)(ii)

Solving and interpreting

Solve for MM (M=800M=800 initially) and describe the future population.

Explore this idea

Hint

  1. M′=0.004(M−1250)M'=0.004(M-1250).

Answer · (a)(ii)

M=1250−450e0.004tM=1250-450e^{0.004t}; modeled extinction≈255.41 years.

Stop the population model at extinction.

Explore the idea

The mudskippers

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 M at t = 0800
Read the solution at t255.4
Try an example
M at t = 255.4
2.062
Rate dM/dt there
-4.992
  • Solution that starts at the purple dot
  • Closed-form answer
Key idea

Below 1250 the population falls to extinction; above 1250 it grows without bound.

Ideas to try
  1. Start above 1250: the population grows instead.
  2. Find when the curve reaches 0.
(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.

Explore this idea

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

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.
(b)(ii)

Logistic solution

Find CC in terms of tt (with C=800C=800 initially) and sketch it.

Explore this idea

Hint

  1. dCdt=0.005C(1600−C)\frac{dC}{dt}=0.005C(1600-C).

  2. Partial fractions.

Crab population
Crab populationScroll across to see the full diagram.

Answer · (b)(ii)

C=1600/(1+e−8t)C=1600/(1+e^{-8t}); rises from 800 toward 1600, concave down for t>0t>0.

Explore the idea

The crabs

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 C at t = 0800
Read the solution at t0.3
Try an example
C at t = 0.3
1467
Rate dC/dt there
976.205
  • Solution that starts at the purple dot
  • Closed-form answer
Key idea

The full logistic family is S-shaped. From the given starting value 800, the physical curve for t≥0t\ge0 begins at the inflection and is concave down as it approaches 1600.

Ideas to try
  1. Start at 400 or 1200: the curve moves towards 1600.
  2. Is the growth faster below or above 800?