# How is this population model different from the Logistic model?

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?

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.