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

 $$N\left(t\right)=\frac{M{e}^{t/\tau }}{{\left[\left({M}^{1/n}-1\right)+{e}^{t/n\tau }\right]}^{n}}$$ (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)