I am studying a course of Mathematical Biology. I am trying to interp the following ODE:
$$ \frac{dR}{dt} = \beta R-\gamma(1+\frac{R}{K})R $$
My attempt:
$R$ describes a population in time $t$
$\beta R $ is a growth term
$ -\gamma(1+\frac{R}{K})R $ is a decay term. there is no limiting term like logistic growth $ \gamma(1-\frac{R}{K})R $, so what does it mean?
The equation is just a form of the logistic equation written slightly differently from normal: \begin{align} \frac{dR}{dt}&=(\beta-\gamma)R\Big(1-\frac{\gamma R}{K(\beta-\gamma)}\Big)\\ &=rR\Big(1-\frac{R}{K'}\Big)\ , \end{align} where $\ r=\beta-\gamma\ $ is the growth rate, and $\ K'=\big(\frac{\beta}{\gamma}-1\big)K\ $ is the carrying capacity. Without any details of how the model was constructed I don't think it's possible to know what physical quantities the parameters $\ \beta$, $\ \gamma\ $, or $\ K\ $ represent.