Effective medium approximation
For a linearized version of Fisher , we first obtain the effect of disorder on invasion velocity using Effective Medium Approximation (EMA). Using the EMA approach, we can replace the spatially irregular diffusion constant with a uniform one in which the effective diffusivity is everywhere equal to (D_{e}) and can replace (D_0(1+xi f(x))) in Eq. (2)44. To obtain (D_{e}), one needs to discretize Eq. (2) using a finite-difference method, which leads to the following equation for the population density39:
$$begin{aligned} frac{partial C_i(t)}{partial t}=sum _{jin {i}}W_{ij}[C_j(t)-C_i(t)]+RC_i(t) ;, end{aligned}$$
(4)
where j belongs to nearest neighbors of i and (W_{ij}=D_{ij}/delta ^2) stands for the density flow rate between units i and j with distance of (delta). Due to spatially irregular diffusion constant we have (W_{ij}=W_0(1+xi f_{ij})). Following44, one has
$$begin{aligned} D_e=bigg ( int _{0}^{infty } frac{g(w) dw}{w} bigg )^{-1} end{aligned}$$
(5)
where g(w) is probability density function for (W_{ij}). Applying this approach to our case leads to
$$begin{aligned} D_{e}=D_0(1-|xi |^2/3) end{aligned}$$
(6)
where (|xi |) stands for dimensionless magnitude of (xi) ((xi ^2) has a physical dimension of length, meter).
Perturbation analysis for invasion front fluctuations
The first step towards the study of dynamics of a propagating front is linearizing (3) by neglecting the (C^2) term in an environment with effect diffusion constant, (D_e), as:
$$begin{aligned} frac{partial C}{partial t}=RC+frac{partial }{partial x} big ((D_e + xi f(x)) frac{partial }{partial x} Cbig ) end{aligned}$$
(7)
This is based on the fact that near the front, the cell density is (C ll 1). In other words, focusing on the dynamics of the front position, automatically grants us the possibility of linearizing (3).
The construction of the solution can be proceeded according to a valuable insight given in the classic paper43, where the particle density is written as follows
$$begin{aligned} C(zeta ,t) approx C_0(zeta +eta (t),t)+ delta C_1(zeta ,t) end{aligned}$$
(8)
where C is written in the comoving frame and (zeta = x-vt). It is assumed that (delta C_1 ll 1) and in the same order as the perturbing function. So that terms containing f(x) and (delta C_1) can be neglected. Furthermore, (C_0) is assumed to satisfy the linearized Eq. (3) with (xi =0), i.e.
$$begin{aligned} frac{partial C_0(zeta ,t)}{partial t}-hat{Gamma } C_0(zeta )=,& {} frac{partial C_0(zeta ,t)}{partial t} nonumber &-bigg (D_e frac{d^2}{dzeta ^2} + vfrac{d}{dzeta } + Rbigg )C_0(zeta ,t) = 0 end{aligned}$$
(9)
Which has the following solution
$$begin{aligned} C_0(zeta ,t) = frac{1}{sqrt{4pi D_e t}}e^{-frac{1}{2}sqrt{frac{R}{D_e}}zeta }e^{-frac{zeta ^2}{4D_e t}} end{aligned}$$
(10)
The first term in (8) describes the effects of the perturbing function f(x) on the position of the propagating front, while the second term shows the change in the shape of the front. This approach has also been employed and well explained in a recent paper17. As shown in17,43, to determine the effective diffusion coefficient for the fluctuating front, it is sufficient to solve (3) using (8) for (eta (t)). Note also that since we are interested in the dynamics of the system in long times ((t gg frac{1}{R})), (v_e) can be assumed to be equal to (2sqrt{R D_e})45. Plugging (8) in Eq. (9) expressed in comoving coordinates and considering (xi =bar{xi }/D_e) yields
$$begin{aligned} frac{partial delta C_1}{partial t} – hat{Gamma }delta C_1 + dot{eta (t)} C_0(zeta ,t) = bar{xi } bigg (f(zeta ) C’_0(zeta ,t)bigg )’ end{aligned}$$
(11)
Noting that the operator (hat{Gamma }) is not self-adjoint (The adjoint of (hat{Gamma }) is: (hat{Gamma ^dagger }=D_e dfrac{d^2}{dzeta ^2} – v_e dfrac{d}{dzeta } + R)) and following43, we multiply Eq. (11) from the left in the eigenfunction of (hat{Gamma ^dagger }) with 0 eigenvalue (which is (e^{sqrt{frac{R}{D_e}}zeta })) and integrate. Thus,
$$begin{aligned} {,} & int _{-infty }^{infty } e^{sqrt{frac{R}{D_e}}zeta }frac{partial delta C_1(zeta ,t)}{partial t} dzeta + dot{eta (t)}int _{-infty }^{infty }e^{sqrt{frac{R}{D_e}}zeta }C’_0(zeta ,t) dzeta nonumber &quad = bar{xi } int _{-infty }^{infty } e^{sqrt{frac{R}{D_e}}zeta }bigg (f(zeta ) C’_0(zeta ,t)bigg )’dzeta end{aligned}$$
(12)
Which yields
$$begin{aligned} dot{eta }(t) =bar{xi } dfrac{int _{-infty }^{infty } e^{sqrt{frac{R}{D_e}}zeta }bigg (f(zeta ) C’_0(zeta ,t)bigg )’ dzeta }{int _{-infty }^{infty }e^{sqrt{frac{R}{D_e}}zeta }C’_0(zeta ,t) dzeta } end{aligned}$$
(13)
Which can further be simplified into
$$begin{aligned} dot{eta }(t)=,& {} bar{xi }dfrac{int _{-infty }^{infty } e^{sqrt{frac{R}{D_e}}zeta }f(zeta ) C’_0(zeta ,t) dzeta }{int _{-infty }^{infty }e^{sqrt{frac{R}{D_e}}zeta }C_0(zeta ,t) dzeta } nonumber =,& {} bar{xi } e^{-frac{R t}{4}}int _{-infty }^{infty } e^{sqrt{frac{R}{D_e}}zeta }f(zeta ) C’_0(zeta ,t) dzeta end{aligned}$$
(14)
Or equivalently,
$$begin{aligned} eta (t) =bar{xi } int _{0}^{t} dtau e^{-frac{Rtau }{4}} int _{-infty }^{infty } dzeta e^{sqrt{frac{R}{D_e}}zeta }f(zeta ) C’_0(zeta ,tau ) end{aligned}$$
(15)
According to17, the effective diffusion would be given by
$$begin{aligned} D_{C} = dfrac{langle eta ^2(t) rangle }{2t} end{aligned}$$
(16)
Or
$$begin{aligned} D_{C}=,& {} frac{bar{xi }^2}{2t}int _{0}^{t}d{T_1}int _{0}^{t}d{T_2}int _{-infty }^{infty } C’_0(zeta ,T_1)C’_0(zeta ,T_2) nonumber × e^{-frac{R T_1}{4}}e^{-frac{R T_2}{4}}e^{2sqrt{frac{R}{D_e}}zeta } dzeta end{aligned}$$
(17)
where we have performed an ensemble average over (eta ^2(t)) using the fact that (langle f(x)f(y) rangle = delta (x-y)). A numerical calculation of (16) can be readily computed using any mathematical software. However, valuable insight can still be obtained from (17), if we use dimensionless parameters (tau _i=frac{T_i}{t}) and (sigma =sqrt{frac{R}{D_0}}zeta). In other words
$$begin{aligned} D_{C}=,& {} bar{xi }^2 dfrac{sqrt{R}}{32pi D^{3/2}_0}int _{0}^{1}dtau _1int _{0}^{1}dtau _2int _{-infty }^{infty }dsigma nonumber × dfrac{left( 1+dfrac{sigma }{Rttau _1}right) left( 1+dfrac{sigma }{Rttau _2}right) }{sqrt{tau _1tau _2}}e^{-frac{sigma ^2}{4Rttau _1}}e^{-frac{sigma ^2}{4Rttau _2}}e^{sigma }e^{-R tfrac{(tau _1+tau _2)}{4}} end{aligned}$$
(18)
Equation (18) gives the effective diffusion coefficient for the stochastic behavior of the front. For a diffusive behavior, we would expect this effective diffusion coefficient to tend to a constant at large times. At large times, we can approximate the integral as follows,
$$begin{aligned} D_{C} approx bar{xi }^2 dfrac{sqrt{R}}{32pi D^{3/2}_e}int _{0}^{1}dtau _1int _{0}^{1}dtau _2int _{-infty }^{infty }dsigma dfrac{1}{sqrt{tau _1tau _2}}e^{-frac{sigma ^2}{4Rttau _1}} e^{-frac{sigma ^2}{4Rttau _2}}e^{sigma }e^{-R tfrac{(tau _1+tau _2)}{4}} end{aligned}$$
(19)
Luckily, Eq. (19) can be evaluated exactly to yield
$$begin{aligned} D_{C}&approx bar{xi }^2 dfrac{sqrt{R}}{32pi D^{3/2}_e} nonumber &quad frac{2 pi (2 R t-1) text {Erf}left( frac{sqrt{R t}}{2}right) -2 pi e^{2 R t} text {Erf}left( frac{3 sqrt{R t}}{2}right) +2 pi e^{2 R t} text {Erf}left( sqrt{2} sqrt{R t}right) +8 sqrt{pi } e^{-frac{1}{4} (R t)} sqrt{R t}-4 sqrt{2 pi } sqrt{R t}}{R t} end{aligned}$$
(20)
Where Erf(x) is the error function. As (trightarrow infty) this gives the following simple relation for the diffusion constant for the wave front
$$begin{aligned} D_{C}(trightarrow infty ) = bar{xi }^2 dfrac{sqrt{R}}{8 D^{3/2}_e} end{aligned}$$
(21)
Substituting (xi =bar{xi }/D_e) and (D_e=D_0(1-|xi |^2/3)), we will get the following beautiful equation for the effective diffusion constant of the front at large times:
$$begin{aligned} D_{C}(trightarrow infty ) =dfrac{1}{8} xi ^2 sqrt{R D_0(1- |xi |^2/3)}. end{aligned}$$
(22)
Source: Ecology - nature.com
