Table 1.

Six families of 1-dimensional dispersal kernels used in this study, together with their characteristics. The mean distance travelled is obtained from $\delta ={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\left|x\right|\gamma (x)dx}$. Expression Γ() stands for the Gamma function.

Kernel families	Expression	Parameters values	Weight of the tail	Mean distance travelled, δ
Exponential	${\gamma }_1(x)=\frac{1}{2\alpha }\exp \left(-\left|\frac{x}{\alpha }\right|\right)$	α>0	Rapidly varying Exponential	α
Gaussian	${\gamma }_2(x)=\frac{1}{\sqrt{\pi }\alpha }\exp \left(-\frac{x^2}{{\alpha }^2}\right)$	α>0	Rapidly varying Thin-tailed	$\frac{\alpha }{\sqrt{\pi }}$
Power-law	${\gamma }_3(x)=\frac{a-1}{2\alpha }{\left(1+\left|\frac{x}{\alpha }\right|\right)}^{-a}$	α>0 a>1	Regularly varying Fat-tailed	$\frac{\alpha }{a-2}$
Exponential power	${\gamma }_4(x)=\frac{c}{2\alpha \Gamma (1/c)}\exp \left(-{\left|\frac{x}{\alpha }\right|}^c\right)$	α,c>0	Rapidly varying Thin-tailed for c>1 Fat-tailed for c<1 Exponential for c = 1	$\alpha \frac{\Gamma (2/c)}{\Gamma (1/c)}$
2Dt	${\gamma }_5(x)=\frac{\Gamma (a)}{\alpha \sqrt{\pi }\Gamma \left(a-\frac{1}{2}\right)}{\left(1+\frac{x^2}{{\alpha }^2}\right)}^{-a}$	α>0 a>1/2	Regularly varying Fat-tailed	$\frac{\alpha }{\sqrt{\pi }(a-1)}\frac{\Gamma (a)}{\Gamma \left(a-\frac{1}{2}\right)}$
Mixture of two Gaussians	${\gamma }_6(x)=p\left(\frac{1}{\sqrt{\pi }{\alpha }_1}{e}^{-x^2/{\alpha }_1^2}\right)+\left(1-p\right)\left(\frac{1}{\sqrt{\pi }{\alpha }_2}{e}^{-x^2/{\alpha }_2^2}\right)$	α1≠α2 p>0	Rapidly varying Thin-tailed (but leptokurtic)	$p\frac{{\alpha }_1}{\sqrt{\pi }}+\left(1-p\right)\frac{{\alpha }_2}{\sqrt{\pi }}$