Apologies if this question is already known - I have three population ($x(t),y(t)$ and $z(t)$ evalutated at discrete time steps. If we define the sum of these as $T(t) = x(t) + y(t) + z(t)$, you can write
$x(t+1) = ax(t)T(t)$
$y(t+1) = by(t)T(t)$
$z(t+1) = cz(t)T(t)$
where $a$, $b$ and $c$ are constants. For now, $T(t)$ is a constant. If so, one can solve this as a recursive series. However, for a variety of reasons we'd prefer to make this a series of ODEs if possible. Naively, from Taylor's theorem, we know that a function can be written as a sum of the form
$x(t + \delta t) = x(t) + \delta t x'(t) + .....$
If we let $\delta t = 1$, and truncate the series after the first derivative, then we might write these as a system of ODEs after re-arrangement to get..
$x'(t) \approx x(t)(aT(t) - 1)$
$y'(t) \approx y(t)(bT(t) - 1)$
$z'(t) \approx z(t)(cT(t) - 1)$
Solving this seems to give answers close to the exact solution (see image attached, these are acceptably close for our purposes) but I'd like to quantify the error if I can - I considered using a version of Lagrange Remainder formula, but since the form here is cobbled together from a discrete series I don't think there's any way of doing this?
The other question I have involves our motivation for switching to ODEs - it would be great if we could consider situations where $T(t)$ might not be constant, and there may be growth or decay with time, for example if a growth term $g$ was added to $x$ so that
$x'(t) \approx x(t)(aT(t) - 1) + g$ .
There might also be situations when $a,b,$ and $c$ could be themselves functions of time. Is it acceptable to make the simplification here, and what is the best way to quantify the error that arises from such approximation?
