Table 1.

Non-local modifications of the gradient operator applied to a function v (or v = (v₁, v₂)).

integral operator	examples	references
is placed before $\mathrm{\nabla }$	$(J\star v_1)\mathrm{\nabla }{v}_2$	[58]
is placed inside $\mathrm{\nabla }$	$\mathrm{\nabla }(\hspace{0.17em}J\star v)$	[65]
replaces $\mathrm{\nabla }$	${\mathcal{A}}_{r}v(x)=\frac{1}{r}{\hspace{0.17em}⨍}_{B_r}v(x+\xi )\frac{\xi }{|\xi |}{F}_r(|\xi |)\hspace{0.17em}\text{d}\xi$	adhesion velocity [35,66]
	${\stackrel{\circ }{\mathrm{\nabla }}}_{r}v(x)=\frac{n}{r}{\hspace{0.17em}⨍}_{S_r}v(x+\xi )\xi \hspace{0.17em}{d}_{{\mathcal{S}}_r}$	non-local chemotaxis [67,68]
is applied to $\mathrm{\nabla }$	${\mathcal{T}}_r\mathrm{\nabla }v(x)=\frac{1}{r}{\int }_0^1{\hspace{0.17em}⨍}_{B_r}(\mathrm{\nabla }v(x+s\xi )\cdot \xi )\frac{\xi }{|\xi |}{F}_r(|\xi |)\hspace{0.17em}d\xi \hspace{0.17em}\text{d}s$	[69]
	${\mathcal{S}}_r\mathrm{\nabla }v(x)=\frac{n}{r}{\int }_0^1{\hspace{0.17em}⨍}_{S_r}(\mathrm{\nabla }v(x+s\xi )\cdot \xi )\xi \hspace{0.17em}{d}_{{\mathcal{S}}_r}\text{d}s$	[69]