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The population $P(t)$ of a biological species can be modelled by $$ \frac{dP}{dt}=P(\beta - \delta P) $$ subject to $P(0)=P_0$ where $\beta$ is the birth rate and $\delta$ is the death rate. Assuming $P>0$, find the population $P(t)$ for three cases:

  1. $P_0<\hat{P}$
  2. $P_0=\hat{P}$
  3. $P_0>\hat{P}$

where $\hat{P}=^{\beta}/_{\delta}$

So far, I have:

We see the equations is separable and so $$ \int\frac{dP}{P(\beta - \delta P)}=\int dt $$ then in partial fractions we have $$ \int\frac{dP}{\beta P}+\int\frac{\delta}{\beta (\beta - \delta P)}dP=\int dt $$ integrating both sides $$ \frac{1}{\beta}\ln P+\frac{\delta}{\beta}\ln(\beta - \delta P)=t + k $$ $$ \ln P+\delta\ln(\beta - \delta P)=\beta t + k $$ $$ P\cdot(\beta - \delta P)^\delta=ke^{\beta t} $$ so at $t=0$ we have $$ P_0\cdot(\beta - \delta P_0)^\delta=k $$ hence $$ P\cdot(\beta - \delta P)^\delta=P_0\cdot(\beta - \delta P_0)^\delta\cdot e^{\beta t} $$

I don't really see how to apply the three cases to this equation; I mean it seems that $P_0=\hat{P}$ would mean $P(t)$ is identically 0 but I don't think this makes any sense in terms of the original problem. So my reasoning must be wrong..? And for the other two cases I don't see how I would express $P(t)$ generally.

b3174802
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1 Answers1

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However, writing the solution as $$ P(t) = \frac{\beta}{\delta}\frac{1}{1-\frac{P_0-\frac{\beta}{\delta}}{P_0}\mathrm{e}^{-\beta t}} $$ It should be clear what happens to the solution

$\textbf{update}$ $$ \int \frac{dP}{P\left(\frac{\beta}{\delta} - P\right)} = \int \delta dt =\delta t + \lambda $$ integrals of the form $$ \int \frac{dx}{x(b-x)} = \frac{\ln x - \ln (x-b)}{b} $$ thus the integral becomes $$ \frac{1}{\frac{\beta}{\delta}}\ln\left(\frac{P}{P - \frac{\beta}{\delta}}\right) = \delta t + \lambda\\ \frac{P}{P - \frac{\beta}{\delta}} = \mathrm{e}^{\frac{\beta}{\delta}\left(\delta t + \lambda\right)} = A\mathrm{e}^{\beta t} $$ I.C leasd to $$ \frac{P_0}{P_0 - \frac{\beta}{\delta}} = A $$ re-arranging leads to the first equation.

$\textbf{Note that I had sign wrong in the first equation which has been rectified}$

Chinny84
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