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 $t$ years, the number of mudskippers and crabs are $M,C$ 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 $M,t$.1 mark

(ii) Solve the differential equation in part (i) and determine what happens to the mudskipper population in the future.4 marks

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

(i) Find the number of crabs in the mangrove swamp when the rate of change of $C$ is a maximum.2 marks

(ii) Find $C$ in terms of $t$. Sketch a graph of $C$ against $t$.5 marks

[2021/TJC/Prelim/I/10]