Text S3: Diffusive movement in two dimensions. Suppose animals perform random walks in Ω = R2 (i.e., two dimensions) and let. Pm(x1,x2,t) be the probability ...
Text S3: Diffusive movement in two dimensions Suppose animals perform random walks in Ω = R2 (i.e., two dimensions) 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 [1] ( ∂2 ∂ ∂2 ) Pm , Pm = D + ∂t ∂x21 ∂x22
Pm (x1 , x2 , 0) = δ(x1 )δ(x2 )
(1)
and its solution is ( x2 + x2 ) 1 2 Pm (x1 , x2 , t) = exp − 1 , 4πDt 4Dt
(x1 , x2 ) ∈ Ω and t > 0
(2)
The graph of this function is a 2-D Gaussian that expands because of diffusion. Mean of Ps : It follows from Eq (4) of Text S1 and the fact that Pm is an even function of xi that ∫ ∞ ∫ ∞(∫ ∞ (∫ ∞ ) ) µsi = µmi (t)Pr (t) dt = xi Pm dxi dxj Pr (t) dt = 0 (3) 0
−∞
0
−∞
Scale of Ps : Substituting Eqs (2), (3), and (1*) into Eq (??) we obtain ∫ ∞(∫ ∞ ∫ ∞ ∫ ∞ ) 2 2 σsi = xi Pm dx1 dx2 Pr (t) dt = (2Dt)Pr (t) dt = 2Dµr −∞
0
−∞
(4)
0
We define the total scale of Ps to be σs2
=
2 ∑
2 σsi = 4Dµr
i=1
Shape and total kurtosis of Ps : Substituting Eqs (2), (3), (4), and (1*) into Eq (1) of Text S1 produces (see Text S2 for similar algebraic steps) ∫ ∞∫ ∞ 1 3σr2 4 κsi = 4 x Ps dx1 dx2 − 3 = 2 σsi −∞ −∞ i µr We define the total kurtosis of Ps to be κs =
2 ∑
κsi =
i=1
6σr2 µ2r
Covariance of Ps : As Pm is constant for fixed x21 + x22 , the covariance of Ps is ∫ ∞ ∫ ∞(∫ ∞ ∫ ∞ ) xi xj Pm dx1 dx2 Pr (t) dt = 0 · Pr (t) dt = 0 σs12 = 0
−∞
−∞
−∞
Note that, until here, we have not made any assumptions on the full form of Pr . Therefore, as in for the one dimensional case, the above results are generally applicable to any retention time pattern for a diffusively moving organism in two dimensions. 1
Form of Ps obtain [2],
To find an expression for Ps , we substitute Eqs (2) and (3*) into Eq (1*) to A0 Ps (x1 , x2 ) = a+1 xa−1 Ka−1 xc
(
x xc
) ,
(x1 , x2 ) ∈ Ω
Here, A0 is a positive constant (depending only on a), xc = large distances, we have the approximation Ps (x) ≈
B0 a+ 12
xa− 2 e− xc , 3
x
√ √ bD, and x = x21 + x22 . At
x ≫ xc
xc
As in one dimensional case of Eqs (6*) and (7*), Ps exhibits power-law with an exponential cut-off.
References [1] Okubo A, Levin SA (2001) Diffusion and ecological problems: modern perspectives. New York: Springer-Verlag. [2] Wolfram Research Inc (2004) Mathematica, Version 5.2. Champaign, IL: Wolfram Research, Inc.
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