Table 2 Verhulst-like equations with N0 = 1, \(n=\frac{{\tau }_{decay}}{{\tau }_{growth}}\) with n = ½, 1 and 2.

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\(N\left(t\right)=\frac{M{e}^{t/\tau }}{{\left[\left({M}^{1/n}-1\right)+{e}^{t/n\tau }\right]}^{n}}\) (19) \(\dot{N}=\frac{M\left({M}^{1/n}-1\right){e}^{-t/n\tau }}{{\tau \left[1+\left({M}^{1/n}-1\right){e}^{-t/n\tau }\right]}^{n+1}}\) (20)
\({t}_{inf}=n\tau \times ln\left[n\left({\left(M/{N}_{0}\right)}^{1/n}-1\right)\right]\) (21) \({\dot{N}}_{max}=\frac{M}{n\tau }\times \frac{1}{{\left(1+1/n\right)}^{n+1}}\) (22) \(N\left({t}_{inf}\right)=\frac{M}{{\left(1+1/n\right)}^{n}}\) (23)
  ‘fast’ decay; n = 1/2
asymmetric
Verhulst; n = 1; ‘medium’ symmetric ‘slow’ decay; n = 2 asymmetric
\(N\left(t\right)=\) \(\frac{M{e}^{t/\tau }}{{\left[{(M}^{2}-1)+{e}^{2t/\tau }\right]}^{1/2}}\) (19a) \(\frac{M{e}^{t/\tau }}{\left(M-1\right)+{e}^{t/\tau }}\) (19b) \(\frac{M{e}^{t/\tau }}{{\left[\left(\sqrt{M}-1\right)+{e}^{t/2\tau }\right]}^{2}}\) (19c)
\(\dot{N}=\) \(\frac{M({M}^{2}-1){e}^{-2t/\tau }}{\tau {\left(1+({M}^{2}-1){e}^{-2t/\tau }\right)}^{3/2}}\) (20a) \(\frac{M(M-1){e}^{-t/\tau }}{\tau {\left(1+(M-1){e}^{-t/\tau }\right)}^{2}}\) (20b) \(\frac{M(\sqrt{M}-1){e}^{-t/2\tau }}{\tau {\left(1+(\sqrt{M}-1){e}^{-t/2\tau }\right)}^{3}}\) (20c)
\({t}_{inf}\)= \(\frac{\tau }{2}\times ln\left[\frac{{M}^{2}-1}{2}\right]\) (21a) \(\tau \times \mathrm{ln}\left(M-1\right)\) (21b) \(2\tau \times \mathrm{ln}\left(2(\sqrt{M}-1)\right)\) (21c)
\({\dot{N}}_{max}=\) \(\frac{2}{{3}^{3/2}}\times \frac{M}{\tau }\) (22a) \(0.25\times \frac{M}{\tau }\) (22b) \(\frac{4}{27}\times \frac{M}{\tau }\) (22c)
\(N(t_{inf})=\) \(\frac{M}{\sqrt{3}}\) (23a) \(\frac{M}{2}\) (23b) \(\frac{M}{2.25}\) (23c)