Table 1.

Summary of the models (dropping the local dependencies in $\mathit{\xi }$)

Case	Non-local independent sensing (4)-(5)-(14)	Non-local dependent sensing (4)-(5)-(15)
i) fast $\mathcal{S}$ and slow q	Drift dominated	Drift dominated
	${\displaystyle {\mathbf{U}}_T=c\overline{U}{\int }_{{\mathbb{S}}^{d-1}}\widehat{\mathbf{v}}{\int }_0^R{\gamma }_{\mathcal{S}}(\lambda )\mathcal{S}(\mathit{\xi }+\lambda \widehat{\mathbf{v}})d\lambda {\int }_0^R{\gamma }_q(\lambda )q(\mathit{\xi }+\lambda \widehat{\mathbf{v}},\widehat{\mathbf{v}})d\lambda d\widehat{\mathbf{v}}}$	${\displaystyle {\mathbf{U}}_T=c\overline{U}{\int }_{{\mathbb{S}}^{d-1}}\widehat{\mathbf{v}}{\int }_0^R\gamma (\lambda )\mathcal{S}(\mathit{\xi }+\lambda \widehat{\mathbf{v}})q(\mathit{\xi }+\lambda \widehat{\mathbf{v}},\widehat{\mathbf{v}})d\lambda d\widehat{\mathbf{v}}}$
ii) slow $\mathcal{S}$ and q	Drift-diffusion	Drift-diffusion
	${\mathbb{D}}_T^0=D{\mathbb{D}}_q$	${\mathbb{D}}_T^0=D{\mathbb{D}}_q$
	${\mathbf{U}}_T^1=\overline{U}\left[{\mathrm{\Gamma }}^q,,\mathrm{\nabla },·,{\mathbb{D}}_q,+,,{\mathrm{\Gamma }}^{\mathcal{S}},,{\mathbb{D}}_q,,{\displaystyle \frac{\mathrm{\nabla }\mathcal{S}}{\mathcal{S}}}\right]$	${\mathbf{U}}_T^1=\overline{U}\mathrm{\Gamma }\left[\mathrm{\nabla },·,{\mathbb{D}}_q,+,,{\mathbb{D}}_q,,,{\displaystyle \frac{\mathrm{\nabla }\mathcal{S}}{\mathcal{S}}}\right]$
iii) fast q and slow $\mathcal{S}$	Drift dominated	Drift dominated
	${\displaystyle {\mathbf{U}}_T={\displaystyle \frac{\overline{U}{\mathrm{\Gamma }}_0^{\mathcal{S}}}{c_0}}\mathcal{S}{\int }_{{\mathbb{R}}_{+}}{\gamma }_q(\lambda ){\int }_{{\mathbb{S}}^{d-1}}\widehat{\mathbf{v}}q(\mathit{\xi }+\lambda \widehat{\mathbf{v}},\widehat{\mathbf{v}})d\widehat{\mathbf{v}}d\lambda }$	${\displaystyle {\mathbf{U}}_T:={\displaystyle \frac{\overline{U}}{c_0}}{\int }_{{\mathbb{R}}_{+}}\gamma (\lambda ){\int }_{{\mathbb{S}}^{d-1}}\widehat{\mathbf{v}}q(\mathit{\xi }+\lambda \widehat{\mathbf{v}},\widehat{\mathbf{v}})d\widehat{\mathbf{v}}d\lambda }$
	${\displaystyle +{\displaystyle \frac{\overline{U}{\mathrm{\Gamma }}_1^{\mathcal{S}}}{c_1}}\mathrm{\nabla }\mathcal{S}{\int }_{{\mathbb{R}}_{+}}{\gamma }_q(\lambda ){\int }_{{\mathbb{S}}^{d-1}}\widehat{\mathbf{v}}\otimes \widehat{\mathbf{v}}q(\mathit{\xi }+\lambda \widehat{\mathbf{v}},\widehat{\mathbf{v}})d\widehat{\mathbf{v}}d\lambda }$	${\displaystyle +{\displaystyle \frac{\overline{U}}{c_1}}\mathrm{\nabla }\mathcal{S}{\int }_{{\mathbb{R}}_{+}}\lambda \gamma (\lambda ){\int }_{{\mathbb{S}}^{d-1}}q(\mathit{\xi }+\lambda \widehat{\mathbf{v}},\widehat{\mathbf{v}})\widehat{\mathbf{v}}\otimes \widehat{\mathbf{v}}d\widehat{\mathbf{v}}d\lambda }$
iv) fast $\mathcal{S}$ and slow q	Drift dominated	Drift dominated
	${\displaystyle {\mathbf{U}}_T={\displaystyle \frac{\overline{U}{\mathrm{\Gamma }}_0^q}{c_0}}{\int }_{{\mathbb{S}}^{d-1}}\widehat{\mathbf{v}}q{\int }_0^R{\gamma }_{\mathcal{S}}(\lambda )\mathcal{S}(\mathit{\xi }+\lambda \widehat{\mathbf{v}})d\lambda d\widehat{\mathbf{v}}}$	${\displaystyle {\mathbf{U}}_T:={\displaystyle \frac{\overline{U}}{c_0}}{\int }_{{\mathbb{S}}^{d-1}}\widehat{\mathbf{v}}q{\int }_0^R\gamma (\lambda )\mathcal{S}(\mathit{\xi }+\lambda \widehat{\mathbf{v}})d\lambda d\widehat{\mathbf{v}}}$
	${\displaystyle +{\displaystyle \frac{\overline{U}{\mathrm{\Gamma }}_1^q}{c_1}}{\int }_{{\mathbb{S}}^{d-1}}\widehat{\mathbf{v}}\otimes \widehat{\mathbf{v}}\mathrm{\nabla }q{\int }_0^R{\gamma }_{\mathcal{S}}(\lambda )\mathcal{S}(\mathit{\xi }+\lambda \widehat{\mathbf{v}})d\lambda d\widehat{\mathbf{v}}}$	${\displaystyle +{\displaystyle \frac{\overline{U}}{c_1}}{\int }_{{\mathbb{S}}^{d-1}}\widehat{\mathbf{v}}\otimes \widehat{\mathbf{v}}\mathrm{\nabla }q{\int }_0^R\lambda \gamma (\lambda )\mathcal{S}(\mathit{\xi }+\lambda \widehat{\mathbf{v}})d\lambda d\widehat{\mathbf{v}}}$