Age Structure of Lesions

Table 1: Parameters and variables of the Phytophthora infestans invasion model. Nominal values are gleaned from [6] as well as estimates provided by field researchers [23], using the rule of thumb that each parent lesion produces about ten daughter lesions in ideal circumstances.

Parameter	Description	Nominal Value (Units)
$SI$	Sporulation Intensity	$10^8$ (Spores/meter$^{2}$/day)
$LP$	Latency Period	5 (days)
$IP$	Infectious Period	1 (day)
$P_{\mbox{\tiny infect}}$	Probability of infection per landed spore	$10^{-2}$
$P_{\mbox{\tiny intcpt}}$	Probability of interception per dispersed spore	$10^{-1}$
$\Delta r$	Radial growth rate of lesions	$4\times 10^{-3}$ (meter/day)
$LAI$	Leaf Area Index	5 (meter$^{2}$ crop/meter$^{2}$ soil)
$\sigma$	Mean dispersal distance from parent lesion	1 (meter)

Variable	Description	(Units)
$t$	Age of Lesion	(days)
$n$	Day of Simulation (independent variable)	(days)
$N_t^n$	Density of Lesions of age $t$ on day $n$	(number/meter$^{2}$)
$A_t$	Area of a lesion of age $t$ days	(meter$^{2}$)
$\Delta A_{t}$	Newly grown area for a $t$-day-old lesion	(meter$^{2}$)

An individual lesion on a leaf grows at a measurable and well-defined radial growth rate, $\Delta r$, per day, and after a certain latency period ($LP =$ five days for P. infestans) the invaded area of the leaf sporulates for a certain number of days ($IP =$ 1 day). The progress of an individual late blight lesion on a single leaf is depicted in Figure 1. During the infectious period spores are released at a given rate, $SI$, per area per day, and these spores disperse. Some fraction of spores which settle from the air are intercepted by leaves (with probability $P_{\mbox{\tiny intcpt}}$), and of these intercepted spores a fraction, $P_{\mbox{\tiny infect}}$, successfully germinates and infects the plant (provided it does not land on area already occupied by a lesion). The parameters of the model and nominal values are listed in Table 1.

When a lesion is $t$ days old, the area that it adds is the difference between the area it is, $A_t = \pi (t \Delta r)^2$ and the area it will become on the next day, $A_{t+1} = \pi (t+1)^2 \Delta r^2$. Thus,

$$%% \beq \Delta A_{t+1} = A_{t+1}-A_t = \pi \Delta r^2 \left[(... ...2 \right] = (2t+1) \pi \Delta r^2 \approx 2 \pi t \Delta r^2.$$

Consequently, when a lesion is six or more days old, the area which is sporulating is the area which was added to the lesion $LP$ days ago. Since $N_t^n$ is the density of lesions of age $t$ days on day $n$, the number of spores produced by these lesions is

$$%% \beq \mbox{Spores Produced } = N_t^n \times SI \times \Del... ..._t^n \times SI \times 2\pi \Delta r^2 (t -LP), \quad (t > LP).$$

This is an idealization based on the assumption that leaves of the plant are much larger than the lesions; stability of our results to relaxation of this assumption will be investigated in later sections.

Assuming that all dispersal happens locally, the number of spores arriving is the number of spores produced and a model for reproduction of lesions can be written

$$\begin{eqnarray*} N_1^{n+1} & = & P_{\mbox{\tiny intcpt}}\times P_{\mbox{\tiny i... ...1^n \\ & \vdots & \\ N_t^{n+1} & = & N_{t-1}^n \\ & \vdots & \end{eqnarray*}$$

The combination of probabilities in the first line is the probability of the composite event that (first) a spore lands on a leaf and is not subsequently knocked off ( $P_{\mbox{\tiny intcpt}}$), that (second) the spore is able to germinate and penetrate the outer skin of the leaf ( $P_{\mbox{\tiny intcpt}}$), and that (third) the spore has landed on leaf area not currently occupied by a lesion ( $P_{\mbox{\tiny unocc}}$). Probabilistic parameters are set using the `rule of thumb' that 1 parent lesion produces a net 10 daughter lesions in the Netherlands in ideal circumstances [23]. The probability of a spore landing on unoccupied leaf area can be calculated from the ratio of the total leaf area and the total area occupied by lesions,

$$P_{\mbox{\tiny unocc}}(N_1^n, N_2^n, \cdots ) = \max\left[\f... ...pi \Delta r^2}{LAI} \sum_{t=1}^\infty N_t^n \ t^2 , 0\right].$$ (1)

The number of spores produced the previous day is given by

$$%% \beq \mbox{ Spores Produced } = SI \times \underbrace{\su... ... \times 2 \pi \Delta r^2 \sum_{t=LP+1}^\infty N_t^n (t -LP ) .$$