Considering the Gompertz equation, $\frac{dN}{dt}=r_oe^{-\alpha t}N$, and the logistic growth equation, $\frac{dN}{dt}=rN(1-\frac{N}{k})$. How is the carrying capacity from Gompertz equation differ from that of the logistic growth? What about the inflection point?
I'm having coding problems uploading the solution curves for both equations, but this link contains a good sample of what it should look like, Logistic and Gompertz solution curve
After finding solutions for both, I obtain for the Gompertz, $N(t)=N_0\exp(\frac{r_0}{\alpha}[1-e^{-\alpha t}])$, and for logistic growth, $N(t)=\frac{k}{1+(\frac{k}{N_0}-1)e^{-rt}}$. So taking the limit we obtain the carrying for capacity for Gompertz, $N^*=N_0e^{\frac{r_0}{\alpha}}$, and for logistic growth it is obviously, $N^*=k$. From here I am confused as to how to proceed in showing how they differ or not.
For the inflection point, we have, $N=\frac{k}{2}$, for logistic growth, and, $N=\frac{1}{\alpha}$ for the Gompertz, but yet again I'm not sure how to compare the two to see how they differ.
The reason for all the madness should yield the derivation of the new form of the Gompertz equation, $\frac{dN}{dt}=\alpha N\ln(\frac{k}{N})$.