The Effects of Dispersal

To investigate the spread of lesions through a crop one must include dispersal effects which describe how spores produced in one location arrive at a different location. Dispersal can occur by wind, by raindrops `splattering' [17], or even ballistically by pressurized expulsion from sporangia, and models can range from relatively simple probabilistic descriptions to solution of turbulent diffusion equations in and above the crop [9]. We will adopt here the descriptive, probabilistic approach and introduce a dispersal kernel, $K(x)$, which is the probability of a spore produced at $x=0$ being dispersed to the location $x$. To determine the density of spores, $S(x)$, arriving at a location $x$, given a spatial distribution of spore production, $P(x)$, one evaluates the integral

$$%% \beq S(x) = \int_{-\infty}^{\infty} K(x-y) P(y) \, dy \stackrel{\mbox{\tiny def}}{=}K * P.$$

One may think of this as summing the probabilities that spores produced at location $y$, the number of which is given probabilistically as $P(y) \, dy$, will disperse the distance $(x-y)$ to the location $x$. Mathematically we write this as the convolution, $S = K*P$.

To include dispersal in the age-structured model we need to interpret $N_t^n$ as the spatial density of lesions which are $t$ days old on day $n$ and update the `Spores Arriving' to include the effects of dispersal from all spatial locations. This gives

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

Writing $\vec{V}_n = ( N_1^n, N_2^n, \cdots N_t^n, \cdots )^T$ the spatio-temporal dynamics are governed by a nonlinear Leslie matrix with dispersal operations:

$$\vec{V}_{n+1} = \textbf{B} \circ \left( \textbf{K} * \vec{V}_{n}\right),$$ (2)

where ${\bf B}$ is the infinite dimensional matrix

$$\textbf{B} = \left( \begin{array}{ccccccccccc} 0 & 0 & 0 &0... ...ts & \vdots & \vdots & \vdots & \ddots\\ \end{array}\right),$$ (3)

K is the matrix composed of dispersal kernels,

$$\bf {K} = \left( \begin{array}{ccccccccccc} 0 & 0 & 0 &0 &0... ... \vdots & \vdots & \vdots & \ddots\\ \par\end{array}\right),$$ (4)

and the operation of element-by-element multiplication (Hadamard product) is denoted by `$\circ$', while the convolution `$*$' is taken element by element. The composite constant,

$$R = 2 \pi P_{\mbox{\tiny intcpt}}P_{\mbox{\tiny infect}}\ SI \ \Delta r^2 ,$$ (5)

is the net number of new lesions produced in an unoccupied environment by an $LP+1$-day old lesion (the youngest lesion which is infectious). Nonlinearity is introduced into the system by $P_{\mbox{\tiny unocc}}$, which must be computed on a daily basis for each location using formula (1).