# Exponential Differential Equation of Population Growth Model

I am wondering what the $b$ could biologically represent in the growth model $\frac{dx}{dt}=axe^{-bx}-c(t)x$?

I believe that $a$ represents the growth rate of the population, $c$ represents the harvest rate, and $x$ represents the population size. I am unsure on $b$, it seems to act as a limiting factor, maybe a carrying capacity, but I would like some input.

It might be better to consider the whole term $ae^{-bx}$. You can see that by rewriting: $$\frac{dx}{dt}=x(ae^{-bx}-c(t))$$ so that the growth rate is $ae^{-bx}$. This will have the effect of causing the growth rate to decrease with population size, since $\lim_{x\rightarrow \infty}e^{-bx}=0$ for positive $b$. It's similar to a carrying capacity, but not exactly that; the limiting population (for constant $c$) is $\frac{1}{b}\ln{\left(\frac{a}{c}\right)}$.
My quick thought: Think of $e^{-bx}$ as a crowding term. That is, the natural growth rate ($ax$), is reduced exponentially as $x$ becomes larger. This is the case in algae fighting for light for example. So more specifically $b$ could represent how strong that crowding effect is. What do you think of that?