Modelling: assumptions, limits and interpretation

Why this earns Outstanding

"Independent reflection and extrapolation" is one of the Outstanding criteria — and it is the part most students never write.
A model is only as good as its assumptions. Saying what they are, and what breaks them, is exactly the reflection being asked for.

State the assumptions you have made — constant rate, no losses, the shape is a perfect cone, the population grows continuously.
State the domain of validity — over what values of the variable does the model make sense?
Comment on behaviour at the edges — what happens as $x \to 0$ , or as $t \to \infty$ ? Does it tend to a limit, or grow without bound?
Say where it fails. Exponential growth models fail once a resource runs out; a linear model fails outside the range of the data.
Extrapolate carefully and flag it: predicting inside the range is interpolation, outside it is a much weaker claim.

"This assumes the rate stays constant, which is reasonable over a short interval but not over a year."
"As $t \to \infty$ the model tends to $L$ , so $L$ behaves as a long-run ceiling."
"The model gives a negative value for $x > 10$ , which is physically impossible, so its valid domain is $0 \le x \le 10$ ."

Worked Example

The number of fish in a lake is modelled by

$P(t) = \frac{5000}{1 + 4e^{-0.6t}}$

where $t$ is in years since monitoring began.

Find the long-run population, the initial population, and comment on the suitability of the model.

Answer:

Initial population. Setting $t = 0$ , $e^{0} = 1$ , so

$P(0) = \frac{5000}{1 + 4} = 1000$

There were 1000 fish when monitoring began.

Long-run population. As $t \to \infty$ , the exponent $-0.6t \to -\infty$ , so $e^{-0.6t} \to 0$ . Therefore

$P(t) \to \frac{5000}{1 + 0} = 5000$

The population approaches a ceiling of 5000 fish, which it never quite reaches.

Comment on suitability.

The model assumes a fixed carrying capacity of 5000 and a constant growth parameter — reasonable in a stable lake with no intervention.
It assumes growth is continuous and smooth, whereas fish breed seasonally, so the real curve would rise in steps.
It takes no account of shocks — disease, drought, or fishing would move the population off this curve entirely.
It is most trustworthy over the middle range of $t$ . Extrapolating decades ahead assumes conditions never change, which is a much weaker claim than the fit over the monitored period.

"Independent reflection and extrapolation" is one of the Outstanding criteria — and it is the part most students never write.
A model is only as good as its assumptions. Saying what they are, and what breaks them, is exactly the reflection being asked for.

State the assumptions you have made — constant rate, no losses, the shape is a perfect cone, the population grows continuously.
State the domain of validity — over what values of the variable does the model make sense?
Comment on behaviour at the edges — what happens as $x \to 0$ , or as $t \to \infty$ ? Does it tend to a limit, or grow without bound?
Say where it fails. Exponential growth models fail once a resource runs out; a linear model fails outside the range of the data.
Extrapolate carefully and flag it: predicting inside the range is interpolation, outside it is a much weaker claim.

"This assumes the rate stays constant, which is reasonable over a short interval but not over a year."
"As $t \to \infty$ the model tends to $L$ , so $L$ behaves as a long-run ceiling."
"The model gives a negative value for $x > 10$ , which is physically impossible, so its valid domain is $0 \le x \le 10$ ."