Review of the Minimum Wave Speed Calculation

We summarize here (and adopt the notation of) arguments presented by Neubert and Caswell [15] for finite Leslie matrices with dispersal, which are in turn based on results of Weinberger [25,26], Kot et al. [12,13] and Neubert et al. [16]. Estimating the speed of the wave of invasion, or front, turns on analyzing the linearization of (2). For sufficiently small $N_t^n$ (for example, in advance of the main infestation), $P_{\mbox{\tiny unocc}}\approx 1$ and the dynamics can be written

$$\vec{V}_{n+1} = \textbf{A} \circ \left( \textbf{K}* \vec{V}_{n}\right) ,$$ (6)

where ${\bf A}$ is the linearization of $\textbf{B}$,

$$%% \beq {\bf A} = \lim_{P_{\mbox{\tiny unocc}}\to 1} \textbf{B}.$$

Sufficiently far in advance of the front, the spatial shape of solutions may be approximated

$$%% \beq \vec{V}_{n} \sim e^{-sx} \vec{w},$$

where $\vec{w}$ is a vector describing the relative abundances in different age-classes of lesions, each of which drops off exponentially at a rate, $s$, in the direction, $x$, in advance of the front. If a front has formed and is traveling at a distance $v$ per iteration, then

$$%% \beq \vec{V}_{n+1}(x) = \vec{V}_{n} (x-v) \sim e^{sv-sx} \vec{w},$$

and substituting into (6),

$$e^{sv-sx} \vec{w} = \left[ \textbf{A} \circ \left( \textbf{K... ...^{-sx} \left[ \textbf{A} \circ \textbf{M} (s)\right] \vec{w} .$$ (7)

Here ${\bf M}(s)$ is the moment-generating matrix computed by element-by-element integration of

$${\bf M}(s) = \int_{-\infty}^{\infty} e^{s y} {\bf K}(y ) \, dy.$$ (8)

To see why, consider one of the nonzero elements of K in the first row:

$$%% \beq K*e^{-sx} = \int_{-\infty}^{\infty} e^{-s (x-y)} K(y)... ...}\int_{-\infty}^{\infty} e^{ s y} K(y) \, dy = e^{-sx} M(s) ,$$

where $M(s)$ is the (scalar) moment generating function for the dispersal kernel $K$.

Cancelling common factors in (7) gives an eigenvalue problem

$$e^{sv} \ \vec{w} = \left[ \textbf{A} \circ \textbf{M} (s) \r... ...ec{w} \stackrel{\mbox{\tiny def}}{=} \textbf{H} (s) \ \vec{w}.$$ (9)

Suppose $\bf {H} (s)$ has (countable) eigenvalues $\lambda_1(s), \lambda_2(s), \cdots$, non-increasingly ordered by magnitude. The minimum wave speed conjecture is that the speed of the wave of invasion is smaller than $v^*$, where

$$v^* = \min_{0<s <\hat{s}} \left[\frac1s \ln \left( \lambda_1(s) \right) \right],$$ (10)

where $\hat{s}$ is the maximum $s$ for which all elements of ${\bf M}(s)$ are defined.

There are two perspectives to take on the applicability and influence of $v^*$, the minimum wave speed perspective and the dynamic perspective. From the minimum wave speed perspective, $v^*$ provides an overestimate of all possible speeds for fronts arising from compactly supported initial conditions. The argument can be summarized as follows. Given that the nonlinear growth rate is non-negative but always bounded above by the linear growth rate, it is clear that the norm of solutions to the nonlinear system is bounded by the norm of solutions to the linear system. Any finite, compactly supported initial condition can be bounded above by some spatial translate of $e^{-sx} \ \vec{w}$, for all $s$. Since the linearized dynamics maps exponential solutions to exponential solutions, the nonlinear evolution from compact initial conditions is bounded above by the linear evolution of suitably translated $e^{-sx} \ \vec{w}$, independent of $s$. These can be written (as above) as translating solutions with given speeds. The slowest of these must therefore provide an over-estimate of the progress of nonlinearly-evolving fronts with compact initial data. For finite matrices this argument was quite elegantly stated recently by Neubert and Caswell [15]. In many, but not all, cases it can also be shown that fronts accelerate to the minimum speed, in which case it becomes the asymptotic speed of fronts.

A related, dynamic perspective suggests that the `minimum' speed should be the asymptotic front speed. This perspective harks back to Kolmogorov et. al. [11], but was stated in the context of dynamics by Dee and Langer [3] and Powell et. al. [19,18]. In a travelling frame of reference, $z=x-nv$, the solution to the linearized equation can be written

$$\vec{V}_n ={\cal F}{\cal T}^{-1} \left[ e^{-i n v k} \textbf{H}(ik) \hat{\vec {V}}_0 \right],$$ (11)

where ${\cal F}{\cal T}^{-1}$ denotes the inverse Fourier transform, $\hat{\vec {V}}_0$ is the Fourier transform of the initial data and H is as in the discussion above, but evaluated with the substitution $s\rightarrow ik$. Asymptotically, using the power method, the integrand in (11) can be written

$$e^{-i n v k} \textbf{H}(ik) \hat{\vec {V}}_0 = a_1 e^{-i n v... ...(ik)\right) -ivk\right\} \right] \hat{\vec{e}}_1 (k) + \cdots,$$

where $\lambda_1$ is the largest magnitude eigenvalue and $\hat{\vec{e}}_1$ the associated eigenvector. Thus

$$\vec{V}_n \approx \frac1{2\pi} \int_{-\infty}^\infty e^{i k ... ...da_1(ik)\right) -ivk\right\} \right] \hat{\vec{e}}_1 (k) \ dk.$$ (12)

The integral in (12) can be evaluated by method of steepest descents to get a further asymptotic approximation; the stationary point is given by the root of

$$\frac{d}{dk} \left[ \ln\left(\lambda_1(ik)\right) -ivk \right] \stackrel{\mbox{\tiny set}}{=}0.$$ (13)

If $k^*$ is the stationary point solving (13), an associated speed for the travelling frame of reference, $v^*$, is chosen so that the wave neither grows nor shrinks in this frame of reference, that is

$$\mbox{Real } \left[ \ln\left(\lambda_1(ik^*)\right) -iv^*k^* \right] = 0.$$ (14)

Working through the algebra, one finds that the solutions to (13, 14) correspond exactly to (10), using the substitution $ik^*\rightarrow s^*$.

As pointed out by Dee and Langer and later by Powell et al., these equations have a dynamic interpretation. The quantity being maximized in (13) is the exponential growth rate of a particular Fourier mode in a frame of reference travelling with speed $v$. Thus, the stationary point, $k^*$, is that mode which has maximal growth rate; the method of steepest descents becomes a statement that the asymptotic front solution is that solution which grows from the most unstable Fourier mode in the ensemble describing the initial data. The choice of $v^*$ via (14) is then just diagnosing the speed attached to the most unstable mode using a stationary phase argument. The dynamic viewpoint suggests that the minimal wave speed, $v^*$, should not only be an upper bound, but also the asymptotic speed obvserved, since it is connected to the growth and propagation of the most unstable wave component of the solution. Moreover, an asymptotic form for the solution is predicted,

$$\vec{V}_n (z) \sim e^{i k^*z}\ \frac{a_1(k^*)}{\sqrt{2 n \pi... ... \right)^2 \right]^{-\frac12} \ \hat{\vec{e}}_1 (k^*) + c.c.$$ (15)

Incorporating the factor of $\sqrt{n}$ from the denominator of (15) into the exponent indicates that observed fronts should converge from below to the asymptotic speed, $v^*$, as

$$v_{\mbox{\tiny observed}} = v^* \left( 1- \frac{\ln(n)}{2 n k^*} \right) ,$$ (16)

a result which we will use to analyze our observations below.