Fronts from complex two-dimensional dispersal kernels: Theory and application to Reid’s paradox

Note that Eq. (2) does not include saturation. We have checked that including saturation in the 2D simulations, the front speeds are the same. Obviously the same happens in the CSRW and DSRW approaches after linearization.

According to field observations in sites close to those where the dispersal kernel was measured, the fecundity $f$ of this species is of the order of $104seedsdispersed∕treeyr$⁠, and its postdispersal seed-to-adult survival probability is $s≃0.06%$ (see Ref. 7 and Refs. 25 and 26 therein). Thus we estimate the net reproductive rate, $R0=fs$⁠, to be in the range of $6–60seeds∕treeyr$⁠. The age at first reproduction (generation time) of the same species is $T≃20yr$ (Ref. 18, Table 1).

The numerical simulations agree with Eq. (2) but not with Eq. (1) with, e.g., $R[p(x,y,t)]=R0p(x,y,t)$⁠.

The minimum speed [Eqs. (3)–(5) for the CSRW; Eq. (11) for the DSRW] is that of the front: this is seen by comparing to simulations (full curve in Fig. 1 for the CSRW; rhombs in Fig. 2 for the DSRW).

The CSRW uses $φ(Δ)$⁠, which is the kernel found in Ref. 7. But the simulations and DSRW use $ϕ(Δ)$⁠. Otherwise the agreement in Figs. 1 and 2 cannot be attained. For the DSRW, see especially Eq. (9).

Several days of computing time are not enough for the bimodal kernel; in contrast, for the kernel $φS(Δ)$ the necessary computing time is about $30min$ (these results have been obtained for a virtual grid of $10002$ nodes and using a personal computer with an Intel Pentium III, $393kbyte$ RAM and $1133MHz$⁠).

For histogram data, in Ref. 34 it has been argued that $c1D<c2D$ for some specific kernels and parameter values. However, negative-displacement jumps $(Δx<0)$ in one dimension were apparently omitted [because Eq. (12) instead of (13) in Ref. 34 was apparently used to compute the 1D speeds in Table 1 there]. As the general proof in Sec. VI shows, one will always find that $c2D<c1D$⁠, provided of course that jumps with $Δx<0$ are taken into account in one dimension. For nonhistogram data, the authors of Ref. 34 report that $c2D<c1D$ for some specific kernels and parameter values, in agreement with the general proof in Sec. VI in the present paper [but also in this case, their Eq. (9) instead of (8) in Ref. 34 should have been applied, again to take into account jumps with $Δx<0$].