We have shown in this paper how to rationally extend the methodology of Neubert and Caswell [15], incorporating age structure and dispersal into an integrodifference population model, to the infinite dimensional case of crop disease lesions propagating through a crop. Even in the extremely unstable case of late potato blight the agreement between analytic results and simulations are well within expected error tolerances. The minimum wave speed, given by the infinite dimensional version of 10, is an upper bound for rates of invasion progation and seems to be the asymptotic speed selected for waves of invasion, as suggested by the dynamic interpretations of Dee and Langer [3] and Powell et al. [18,19]. In fact, results in this paper suggest that the dynamic interpration has the additional virtue of accurately describing the rate of convergence to the asymptotic wave speed.
We have also described a modelling approach for lesion-based foliar diseases, which may find potential application in any sort of invasion process where growing patches are the basic unit of infection. Examples include cheat grass in the American West [22], the pathogenic fungi Botrytis spp. which cause `fire disease' in flower crops and infect field and greenhouse vegetables, small fruits, ornamental plants, flower bulbs and forest tree seedlings world wide [10], and insects such as the Southern Pine Beetle (which create `spot' infections in patches of pine forest [20]) or gypsy moth, which seems to invade by via spots [21]. By no means is this the first attempt at modelling spread of late blight and fungal pathogens (see, for example van den Bosch et al. [24] and Pielaat and van den Bosch [17]), nor (more generally) age structured spread in general (see Hengeveld [7] and Shigesada and Kawasaki [22] for reviews). But it is the first attempt that we know of to put the concept of an ever-growing stage structure on the relatively firm and simple ground of a Leslie matrix formulation.
The biggest drawback in the modelling approach used here is the difficulty
in accounting for two factors: finite leaf size and coalescence of
lesions. The first factor is not too difficult to imagine incorporating, though
possible tedious. At the coarsest level when lesions grow to the average
size of a leaf they can grow no more, which would amount to truncating the
nonlinear Leslie matrix at an age class of lesions corresponding to the
area of the largest leaves. At a slightly less coarse scale, in plants
with a size distribution of leaves, one would need to estimate the probability
of a lesion using up all available area by the probability that it had landed
on a leaf of its current size. This would give a transition probability of
smaller than one for lesions larger than a certain size, which would drop to
zero for lesions at the largest leaf size. Finally, when lesions grow on
finite substrates they will eventualy encounter boundaries, and while they
may continue to grow the new growth area will no longer be annular.
Consequently some estimate of the probability of observing annular growth of
area 
, which would then alter the rate at which new lesions are
formed. In all of these cases, realistic incorporation of realistic leaf
sizes would ruin the special matrix structure which allowed for analytic
calculation of the largest magnitude eigenvalue, though the theory predicting
the existence of a single, largest eigenvalue would remain in place.
The second factor, coalescence of lesions on a leaf, would be somewhat more difficult to address. Shigesada and Kawasaki [22] outline a procedure for approximating the rate at which patches of an invasive species run into one another. The basic idea is to model the process as a summation, so that when two circular patches encounter one another they are approximated as a new patch of size equal to the sum of the previous patches. Knowing the distribution of distances at which patches are established ad their radial growth rates, one can predict the mean time until patches encounter one another. In our age-structure framework this would manifest as a new kind of transition probability: among all age classes would be a class of transitions which would allow a lesion of a given size to sum with a lesion of any other size and create a new lesion in a size class equal to the sum of the two. The resulting transition matrix would be lower triangular (except for the top row, representing the production of new lesions), and have nonzero entries up to the point where maximum dispersal distance and radial growth no longer allow for two lesions to coalesce (i.e. when the size of the lesion is greater than its capacity for dispersal). At this stage new theoretical difficulties are bound to be encountered; the infinite dimensional version of the size class/dispersal formalism was relatively easy to describe in the current case due to the simple form of the Leslie matrix involved.
Both of these factors, however, clearly reduce the growth rate of the
lesion population and therefore would slow down the wave of invasion.
Consequently we may expect that 
as calculated above to remain an upper
bound for the speed of invasions. It is then particularly useful since the
entire calculation can be performed analytically, given the simple algebraic
form of the largest eigenvalue. This may allow for relatively simple
evaluation of invasion threat and control for this important crop disease.