Table 2 Four families of 2-dimensional dispersal kernels used in this study, together with their characteristics. The mean distance travelled is obtained from $\delta ={\displaystyle \underset{0}{\overset{+\infty }{\int }}\gamma (r)2\pi rdr}$. Expression Γ() stands for the Gamma function.

Exponential

${\gamma }_1(x,y)=\frac{1}{2\pi {\alpha }^2}\exp \left(-\frac{r}{\alpha }\right)$

α >0

Rapidly varying

Exponential

2α

Gaussian

${\gamma }_2(x,y)=\frac{1}{\pi {\alpha }^2}\exp \left(-\frac{x^2+y^2}{{\alpha }^2}\right)$

α >0

Rapidly varying

Thin-tailed

$\frac{\alpha \sqrt{\pi }}{2}$

Power-law

${\gamma }_3(x,y)=\frac{(a-2)(a-1)}{2\pi {\alpha }^2}{\left(1+\frac{r}{\alpha }\right)}^{-a}$

α>0

a>2

Regularly varying

Fat-tailed

$\frac{2\alpha }{a-3}$

Exponential power

${\gamma }_4(x,y)=\frac{c}{2\pi {\alpha }^2\Gamma (2/c)}\exp \left(-{\left(\frac{r}{\alpha }\right)}^c\right)$

α,c>0

Rapidly varying

Thin-tailed for c>1

Fat-tailed for c<1

Exponential for c = 1

$\alpha \frac{\Gamma (3/c)}{\Gamma (2/c)}$