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If I have this equation:

$p'(t)-p(t)\alpha =0$

I can say that $p$ is a function that represents the size of a population at time t. The rate at which the population grows is constant. The solution will show that the size of the population is proportional to the initial size.

If I have this equation:

$p'(t)-p(t)f(t) =0$

I can say the rate at which the population grows is determined by $f$. The size of the population is still proportional to the initial size.

But if I have:

$p'(t)-p(t)\alpha = h(t)$

It's difficult to determine from the solution what role the initial size of the population, $p(0)$, has. The solution is:

$p(t)=\bigl(\int e^{-\alpha t}h(t)\ dt + c\bigr)\ e^{\alpha t}$

So my question is this: if I'm interpreting these differential equations as growth functions, what does $c$ represent in the last equation? In the previous equations, $c=p(0)$.

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    $p(0)=c$ still means that $c$ is the initial size of the population if $p(t)$ represents the size of the population at time $t$.2012-09-12
  • 0
    For constant $h(t)$ (breeding plus immigration), there is a fairly natural interpretation as a "virtual" population, since $p(t)$ is a constant plus an exponential term.2012-09-12

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