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I have this following population model: $$ \frac{dX}{dt} = f(X) = pX(1-\frac{X}{Y})(\frac{X}{Z} -1) $$

and I have to compare it to the following Logistic Growth model: $$ \frac{dX}{dt} = f(X) = pX(1-\frac{X}{Y})$$ where p = growth rate and Y is the carrying capacity of the population?

What does the term $(\frac{X}{Z} -1)$ do in order to differ it from the standard Logistic model and what kind of populations could I model with this altered version?

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For small $X$, your model with an $\frac{X}{Z}-1$ factor, hereafter the $Z$-model, obtained $\dot{X}\approx -pX$ so small populations exponentially decay into extinction, whereas in the logistic model they exponentially grow until they're no longer small because $\dot{X}\approx pX$.

If $X$ were much larger than $Y$, on the logistic model $\dot{X}\approx-\frac{pX^2}{Y}$ would lead to population decay, which is also true of the $Z$ model's approximation $\dot{X}\approx-\frac{pX^3}{YZ}$. These asymptotic results are somewhat different though, in that the former implies $\frac{1}{X}$ grows approximately linearly, while the latter has this behaviour for $\frac{1}{X^2}$ instead, so that the decay is slower.

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