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)(i) Forming the equation or open all working together.

(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

Open interactive exploration

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.