Express P in terms of r,k and t.

Question

The population of Aedes mosquitoes which carry the Dengue virus can be modeled by a differential equation which describes the rate of growth of the population. The population growth rate dP/dt is given by dP/dt = rP(1-P/k) , where r is a positive constant and k is the carrying capacity.

Answer 1

dP/dt=rP(1-P/k),

∫dP/(P(1-P/k))=∫rdt,

1/(P(1-P/k))=A/P+B/(1-P/k) where A and B are constants to be found.

A(1-P/k)+BP=1,

A-AP/k+BP=1.

Matching coefficients: A=1; -1/k+B=0, B=1/k.

1/(P(1-P/k))=1/P+1/(k-P).

∫(1/P+1/(k-P))dP=rt+C, where C is a constant.

ln(P)-ln|k-P|=rt+C,

ln|P/(k-P)|=rt+C.

If k<P, then P/(P-k)=e^rt+C, which can also be written: aP/(P-k)=e^rt, where a is a constant (C=-ln(a), a=e^-C. If C=0, a=1).

aP=e^rt(P-k)=Pe^rt-ke^rt,

Pe^rt-aP=ke^rt,

P=ke^rt/(e^rt-a).

Or P=k/(1-ae^-rt).