Plant diseases, caused by fungi, bacteria, viruses and other microorganisms, are a leading cause of agricultural crop loss. One of the most important plant diseases in the world, in terms of damage and control costs, is late blight disease in potatoes and tomatoes, caused by the oömycete Phytophthora infestans (Hooker, 1981). Oömycetes are a distinct group of plant pathogens which until recently were regarded as fungi, but have now been classified as a distinct taxon, more related to algae than to fungi. Epidemiologically however, with regard to the spread of disease in plant populations, oömycetes have much in common with fungal pathogens. Their life cycle includes the same steps of infection of a host, formation of biomass `mycelium' in the host, spatial expansion of the affected area `lesion' in the host, and formation and dispersal of dispersal bodies `spores'. For the purpose of this paper, we will therefore speak about fungi when we discuss epidemiological processes that are relevant to both fungi and oömycetes. The oömycete Phytophthora infestans is taken as an example organism because of its practical importance and because its life cycle attributes are well studied.
The host plant range of P. infestans covers at least 90 plant species, most of them members of the plant family Solanaceae [5]. A great deal of research is dedicated towards breeding resistant potato and tomato varieties and to developing new fungicides. Today, crop losses due to potato late blight have been estimated at 10 to 15 percent of the global annual production [1]. The economic value of these losses plus the cost of crop protection amount to 3 billion US dollar annually [4]. Currently, the control of potato late blight depends on the frequent use of fungicides. Despite this chemical input, late blight epidemics are increasingly more difficult to control. A better understanding of the epidemiology of potato late blight is needed to develop new, effective and environmentally friendly control strategies.
Reproductive strategies of fungi,including the taxonomically distinct but ecologically similar oömycetes, are varied in the extreme. A common theme for foliar plant pathogenic fungi is the production of airborne spores from sporulating bodies. Spores are released, spread with wind and/or rain and after landing on (nearby) plant surfaces they potentially cause new lesions. Once a lesion is initiated, the pathogenic fungus colonises the surrounding plant tissue by sending out hyphae and extracting nutrients from this tissue. The lesion grows at a relatively constant radial rate. Several days after a region of tissue has been colonised by hyphae, sporulating bodies develop from the local mycelium and spores can be produced and released for some time. After this period, the local mycelium dies and sporulation stops. In the mean time the colonised area, and therefore the lesion, has expanded.
For P. infestans this general pattern of latency, infectiousness and senescence results in the very typical circular lesions with an infective annulus some distance behind the (invisible) leading edge of the lesion and dead tissue some radial distance behind that. One may think of lesions as the basic infection unit of P. infestans (Zadoks and Schein [27]). Release and spread of spores from an annular sporulating region inside each lesion, followed by infection, is the basic mode of propagation of P. infestans through a crop. Growth of P. infestans is strongly influenced by the daily cycle of temperature, relative humidity of the air and leaf wetness. Similarly, dispersal of spores and initiation of new lesions experiences strong daily forcing from periodicity in temperature, relative humidity of the air, leaf wetness and daily winds.
Below we will propose a discrete-time, continuous-space model for the density of lesions in a crop. In a spatially invariant setting, neglecting boundaries enforced by finite leaf size, the model for the density of lesions becomes an infinite-dimensional Leslie matrix, and when dispersal is included many of the nonzero-entries in the matrix become spatial convolution operators. Analysis of the Leslie matrix indicates that all eigenvalues are bounded by a largest eigenvalue, whose size can be computed analytically. The existence of this largest eigenvalue allows us to compute an asymptotic bound on the rate of invasion of new lesions into uninfected crops. These results are tested for factorially crossed parameter variations and two dispersal kernels; results are compared with known convergence errors due to the power method approximation and acceleration of the front.