The graph of this function is a 2-D Gaussian that expands because of diffusion and whose center moves in the direction (v1,v2) with speed v = â v2. 1 + v2. 2.
Text S4: Diffusive movement in two dimensions with drift Suppose animals perform random walks with drift in Ω = R2 and let Pm (x1 , x2 , t) be the probability density of an animal being at location x = (x1 , x2 ) ∈ Ω at time t ≥ 0. The initial value problem that determines the evolution of Pm is now an advection-diffusion equation [1] ( ∂ ( ∂2 ∂ ∂2 ) ∂ ) Pm = D + P − v + v Pm , Pm (x1 , x2 , 0) = δ(x1 )δ(x2 ) m 1 2 ∂t ∂x21 ∂x22 ∂x1 ∂x2 Here, vi is the net velocity of animals in the xi -direction. A drift can result for a variety of reasons, including the presence of wind or water, an animal’s migratory behavior, or the influence of an elevational gradient. Depending on the relative magnitude of the diffusion rate D and the advection/drift terms (v1 and v2 ), the movement can be dominated by random motion, directed motion, or both. The solution of the equation above is ( (x − v t)2 + (x − v t)2 ) 1 1 1 2 2 Pm (x1 , x2 , t) = exp − , (x1 , x2 ) ∈ Ω and t > 0 (1) 4πDt 4Dt The graph of this function is a 2-D Gaussian that expands because of diffusion and whose √ center moves in the direction (v1 , v2 ) with speed v = v12 + v22 . Mean, scale, shape, and covariance of Ps : It can be shown using arguments similar to those used before that { ( ) } 2 2 2 −2 6σ v σ 2 µsi = vi µr , σsi = 2Dµr + vi2 σr2 and κsi = 2r 1 − 2 2 + i r µr Dµr Furthermore, the covariance of Ps is v1 v2 σr2 . Here, as in one and two dimensional cases, we have not yet made any assumptions on the full form of Pr . Therefore, the above results on moments of seed dispersal kernel are generally applicable to organisms moving via diffusion with drift with any retention time pattern (Pr ). Form of Ps : To find an expression for Ps , we substitute Eqs (1) and (3*) into Eq (1*) to obtain [2], ) ( ( ) A0 ( xc )a a−1 x1 v1 + x2 v2 x Ps (x1 , x2 ) = x exp Ka−1 xc bD 2D xc √ ( 4bD2 ) 12 , and x = x21 + x22 . Here, A0 is a positive constant (depending only on a), xc = 4D+bv 2 In a polar direction θ (where the angle is taken with respect to the direction of the drift), we approximate Ps at large distances by ( { }) 1 v cos θ B0 ( xc )a a− 3 2 x exp −x − , x ≫ xc Ps (x, θ) ≈ xc bD xc 2D Therefore, we conclude that Ps exhibits power-law with an√exponential cut-off. It is easy to }, and that it approaches see that the cut-off distance xc can be no larger than max{ bD, 2D v one of these values as the corresponding mode of transport becomes dominant. That is, if √ random motion dominates (bv 2 ≪ 4D) then xc ≈ bD and if directed motion dominates (bv 2 ≫ 4D) then xc ≈ 2D . See Fig S(1) for features of Ps for different values of variations v in seed retention times (σr2 ), which is qualitatively similar previous random walk models without drift (Figure 3*(a-b)). 1
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Seed dispersal kernel (Ps)
Seed dispersal kernel (Ps)
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Figure S 1: Diffusion with drift in two dimensions. (a) The seed dispersal kernel as a function of distance from the source tree (|x|) and standard deviation in seed retention time (σr ). The case σr = 0 corresponds to a Gaussian kernel. (b) The seed dispersal kernel at larger distances. The symbol xij (e.g., x01 ) indicates the distance at which a seed dispersal kernel with σr = j (e.g., σr = 1) begins to have more long distance dispersal events than a seed dispersal kernel with σr = i (e.g., σr = 0). Note that x01 < x02 < x12 (x01 ≈ 4.1, x02 ≈ 4.3, x12 ≈ 5.9). Parameters: D = 1.0, µr = 1.0, v1 = 1, and v2 = 0.
References [1] Okubo A (1986) Dynamical aspects of animal grouping: swarms, schools, flocks, and herds. Adv Biophys 22: 1. [2] Wolfram Research Inc (2004) Mathematica, Version 5.2. Champaign, IL: Wolfram Research, Inc.
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