In one model of the changing population P(t) of a community, it is assumed that

  \f[\frac{dP}{dt}=\frac{dB}{dt}-\frac{dD}{dt}\f]

where dB/dt and dD/dt are the birth and death rates, respectively.

(a) Solve for P(t) if dB/dt=k1P and dD/dt=k2P.

(b) Analyze the cases k1>k2, k1=k2 and k1<k2.

First we will solve the differential equation

\f[\frac{dP}{dt}=\frac{dB}{dt}-\frac{dD}{dt}\f]

\f[\frac{dP}{dt}=k_1P-k_2P\f]

Simplifying the both sides we have

\f[\frac{1}{P}dP=(k_1-k_2)dt\f]

Integrating the both sides we have

\f[\int {\frac{1}{P}}dP=\int {(k_1-k_2)}dt\f]

\f[ln(P)=(k_1-k_2)t+C\f]

After simplification the solution will be

\f[P(t)=Ae^{(k_1-k_2)t}\f]

where A=e^C

Since initially the population will be P(0) so we can use the initial condition P(0)=P_0 which will give A=P_0

So the solution will be

\f[P(t)=P_0e^{(k_1-k_2)t}\f]

(a) For k1>k2 we have k1-k2>0 so P will goes to infinity. So it means if the birth rate is greater than the death rate then population will be increasing to infinity.

For k1<k2 we have k1-k2<0 so P will goes to 0. So it means if the birth rate is less than the death rate then population will be decreasing and one day nothing will remain.

  1. For k1-k2 we have k1-k2=0 so P will be constant P(0). So it means if the birth rate is equal to the death rate then population will never changed and it will remain constant the initial population P(0).