Table 3.

Sensitivity derivatives. In all expressions, an overbar denotes a reduced quantity; in particular, $\overline{u}=u\text{/}w$, $\overline{v}=v\text{/}w$ and ${\overline{v}}_{\ast }=v_{\ast }\text{/}w$. ${\overline{v}}_{\mathrm{c}}$ and ${\overline{v}}_2$ have been defined in (4.14) and (4.21); ${\overline{S}}_{D0}^{(b)}$ has been defined in (4.28). Expressions in the second and the fourth columns have been modified with the help of (3.25). If increasing the diameter does not increase the basic metabolic rate, α in the fifth row should be set to zero. Expressions in the third and fifth columns are approximations of the respective expressions to their lefts under the assumption that ${\overline{v}}_c^4\gg 1$ and $u^3\ll w^3$.

	${\displaystyle \frac{x}{v_{\ast }}}{\displaystyle \frac{\mathrm{\partial }{v}_{\ast }}{\mathrm{\partial }x}}$		${\displaystyle \frac{x}{C_{\ast }}}{\displaystyle \frac{\mathrm{\partial }{C}_{\ast }}{\mathrm{\partial }x}}$		${\displaystyle \frac{x}{C}}{\displaystyle \frac{\mathrm{\partial }C}{\mathrm{\partial }x}}$ or ${\displaystyle \frac{x}{P}}{\displaystyle \frac{\mathrm{\partial }P}{\mathrm{\partial }x}}$
x	exact	approximation	exact	approximation	exact
β	${\displaystyle \frac{2{\overline{u}}^4}{3{\overline{v}}_{\ast }+4{\overline{u}}^4}}$	${\displaystyle \frac{2^{\phantom{\int }}}{3_{\phantom{\int }}}}{\overline{u}}^4$	${\displaystyle \frac{2{\overline{u}}^4}{3{\overline{v}}_{\ast }+2{\overline{u}}^4}}$	${\displaystyle \frac{2}{3}}{\overline{u}}^4$	${\displaystyle \frac{2{\overline{u}}^4}{2\overline{v}+{\overline{v}}^4+{\overline{u}}^4}}$
τ	${\displaystyle \frac{k_{\tau }}{\tau }}{\displaystyle \frac{{\overline{v}}_{\ast }}{3{\overline{v}}_{\ast }+4{\overline{u}}^4}}$	${\displaystyle \frac{{k_{\tau }}^{\phantom{\int }}}{{3\tau }_{\phantom{\int }}}}$	${\displaystyle \frac{k_{\tau }}{\tau }}{\displaystyle \frac{2{\overline{v}}_{\ast }}{3{\overline{v}}_{\ast }+2{\overline{u}}^4}}$	${\displaystyle \frac{2{k_{\tau }}^{\phantom{\int }}}{3{\tau }_{\phantom{\int }}}}$	${\displaystyle \frac{k_{\tau }}{\tau }}{\displaystyle \frac{2\overline{v}}{2\overline{v}+{\overline{v}}^4+{\overline{u}}^4}}$
b	$-{\displaystyle \frac{2}{{\overline{v}}_c^4}}{\displaystyle \frac{{\overline{v}}_{\ast }+{\overline{u}}^4({\overline{v}}_c^4+1)}{3{\overline{v}}_{\ast }+4{\overline{u}}^4}}$	$-{\displaystyle \frac{2}{3}}{\displaystyle \frac{1+{\overline{u}}^4{\overline{v}}_c^4}{{\overline{v}}_c^4}}$	${\displaystyle \frac{2}{{\overline{v}}_c^4}}{\displaystyle \frac{{\overline{v}}_{\ast }^4-{\overline{u}}^4{\overline{v}}_c^4}{3{\overline{v}}_{\ast }+2{\overline{u}}^4}}$	${\displaystyle \frac{2^{\phantom{\int }}}{3_{\phantom{\int }}}}{\displaystyle \frac{1-{\overline{u}}^4{\overline{v}}_c^4}{{\overline{v}}_c^4}}$	${\displaystyle \frac{2}{{\overline{v}}_c^4}}{\displaystyle \frac{{\overline{v}}^4-{\overline{u}}^4{\overline{v}}_c^4}{2\overline{v}+{\overline{v}}^4+{\overline{u}}^4}}$
c0	$-{\displaystyle \frac{2^{\phantom{\int }}}{{\overline{v}}_2^4}}{\displaystyle \frac{{\overline{v}}_{\ast }+{\overline{u}}^4}{3{\overline{v}}_{\ast }+4{\overline{u}}^4}}$	$-{\displaystyle \frac{2}{3{\overline{v}}_2^4}}$	${\displaystyle \frac{2}{{\overline{v}}_2^4}}{\displaystyle \frac{{\overline{v}}_{\ast }+{\overline{u}}^4}{3{\overline{v}}_{\ast }+2{\overline{u}}_{\phantom{\int }}^4}}$	${\displaystyle \frac{2}{3{\overline{v}}_2^4}}$	${\displaystyle \frac{2}{{\overline{v}}_2^4}}{\displaystyle \frac{{\overline{v}}^4}{2\overline{v}+{\overline{v}}^4+{\overline{u}}^4}}$
d	${\displaystyle \frac{(2\alpha \hspace{0.17em}-\hspace{0.17em}{\overline{S}}_{D0}^{(b)}){\overline{v}}_{\ast }}{3{\overline{v}}_{\ast }\hspace{0.17em}+\hspace{0.17em}4{\overline{u}}^4}}\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{(4\hspace{0.17em}-\hspace{0.17em}{\overline{S}}_{D0}^{(b)}){\overline{u}}^4}{3{\overline{v}}_{\ast }\hspace{0.17em}+\hspace{0.17em}4{\overline{u}}^4}}$	${\displaystyle \frac{2\alpha \hspace{0.17em}-\hspace{0.17em}{\overline{S}}_{D0}^{{(b)}^{\phantom{\int }}}}{3_{\phantom{\int }}}}$	${\displaystyle \frac{(4\alpha \hspace{0.17em}+\hspace{0.17em}{\overline{S}}_{D0}^{(b)}){\overline{v}}_{\ast }}{3{\overline{v}}_{\ast }\hspace{0.17em}+\hspace{0.17em}2{\overline{u}}^4}}\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{(4\hspace{0.17em}+\hspace{0.17em}{\overline{S}}_{D0}^{(b)}){\overline{u}}^4}{3{\overline{v}}_{\ast }\hspace{0.17em}+\hspace{0.17em}2{\overline{u}}^4}}$	${\displaystyle \frac{4\alpha \hspace{0.17em}+\hspace{0.17em}{\overline{S}}_{D0}^{(b)}}{3}}$	${\displaystyle \frac{4\alpha \overline{v}\hspace{0.17em}+\hspace{0.17em}4{\overline{u}}^4\hspace{0.17em}+\hspace{0.17em}{\overline{v}}^4{\overline{S}}_{D0}^{(b)}}{2\overline{v}\hspace{0.17em}+\hspace{0.17em}{\overline{v}}^4\hspace{0.17em}+\hspace{0.17em}{\overline{u}}^4}}$
l	${\displaystyle \frac{(3\alpha -2){\overline{v}}_{\ast }+2{\overline{u}}^4}{3{\overline{v}}_{\ast }+4{\overline{u}}^4}}$	$\alpha -{\displaystyle \frac{2^{\phantom{\int }}}{3_{\phantom{\int }}}}$	${\displaystyle \frac{(6\alpha +2){\overline{v}}_{\ast }+6{\overline{u}}^4}{3{\overline{v}}_{\ast }+2{\overline{u}}^4}}$	$2\alpha +{\displaystyle \frac{2}{3}}$	${\displaystyle \frac{6\alpha \overline{v}+4{\overline{u}}^4+2{\overline{v}}^4}{2\overline{v}+{\overline{v}}^4+{\overline{u}}^4}}$