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

Abstract

Bimodal dispersal probability distributions with characteristic distances differing by several orders of magnitude have been derived and favorably compared to observations by Nathan et al. ͓Nature ͑London͒ 418, 409 ͑2002͔͒. For such bimodal kernels, we show that two-dimensional molecular dynamics computer simulations are unable to yield accurate front speeds. Analytically, the usual continuous-space random walks ͑CSRWs͒ are applied to two dimensions. We also introduce discrete-space random walks and use them to check the CSRW results ͑because of the inefficiency of the numerical simulations͒. The physical results reported are shown to predict front speeds high enough to possibly explain Reid's paradox of rapid tree migration. We also show that, for a time-ordered evolution equation, fronts are always slower in two dimensions than in one dimension and that this difference is important both for unimodal and for bimodal kernels.

Progress in front propagation research

“…In contrast, the effect of a finite jump time (τ = 0) is to introduce a delay in the whole flux J . Equation (11) for the non-delayed limit τ = 0 is well known to arise from biased random walks [50]. This simple macroscopic derivation of equation (9) from the set (10) is appealing because of its simplicity.…”

“…such that parent trees reproduce only once and then die) [108,109]. But previous results in 1D space show that substantially more complicated models (with overlapping generations) do not change the order of magnitude of the front speed [110] 7 , so the approximate equation (157) has been applied in recent work [11,12,25,35].…”

“…We also discuss the recent explanation of Reid's paradox of rapid forest recolonizations using bimodal kernels with long-distance dispersal in two-dimensional space [11,35].…”

“…In the last six years, many new analytical results on front propagation have been published, dealing with sequential reaction and dispersion [11], fronts from biased random walks [12][13][14][15], age-structured systems [16], distributed delays [17][18][19][20][21][22], dispersive variability [22,23], interacting species [24,25], anomalous diffusion fronts [26,27], dispersal kernel effects [28], convective systems [29][30][31], etc. In this review we present these theoretical advances in a single, unified framework.…”

“…combustion flames) and fastFourier transforms (which require less computing time for integro-difference equations), we also review recent numerical approaches based on reactive random walks on grids (which are more reasonable for biophysical systems in which there is a reproduction process) [11].…”