I have mathematical modelling as an elective course. Recently I was studying an article here at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1201589/pdf/pnas-0504053102.
I was taught that logistic growth model looks like this: $\frac{dP}{dt}=rP\left(1-\frac{P}{P_{max}}\right).....(1)$
where $P_{max}$ is the carrying capacity.
But here in this article we have
- $\beta_F$= Proliferation rate
- $\kappa_F\beta_F$= Carrying capacity
And the model is given by $\frac{dV_F}{da}=V_F(a)\left(\beta_F-\frac{V_F(a)}{\kappa_F} \right)=V_F(a)\beta_F\left(1-\frac{V_F(a)}{\kappa_F\beta_F}\right).....(2)$
clearly (1) and (2) are similar. so $\kappa_F\beta_F$ = carrying capacity is valid assumption. So, my question is - Is it always true carrying capacity is proportional to proliferation rate? Can you explain this carrying capacity in the given article?