Site-bond percolation solution to preventing the propagation of $Phytophthora$ zoospores on plantations

We propose a strategy based on the site-bond percolation to minimize the propagation of Phytophthora zoospores on plantations, consisting in introducing physical barriers between neighboring plants. Two clustering processes are distinguished: (i) one of cells with the presence of the pathogen, detected on soil analysis, and (ii) that of diseased plants, revealed from a visual inspection of the plantation. The former is well described by the standard site-bond percolation. In the latter, the percolation threshold is fitted by a Tsallis distribution when no barriers are introduced. We provide, for both cases, the formulas for the minimal barrier density to prevent the emergence of the spanning cluster. Though this work is focused on a specific pathogen, the model presented here can also be applied to prevent the spreading of other pathogens that disseminate, by other means, from one plant to the neighboring ones. Finally, the application of this strategy to three types of commercially important Mexican chili plants is also shown.

Figure 1

Possible barrier locations (solid lines), directions of microorganism propagation (dotted lines), and modification of the nearest-neighbor meaning induced by inoculated cells with a resistant plant in square [(a) and (d)], triangular [(b) and (e)], and honeycomb [(c) and (f)] lattices. Bottom figures show susceptible plants (green triangle) with a neighboring resistant plant in an inoculated cell (red triangle). As a consequence of the microorganism propagation (red arrows), the nearest-neighbor definition (black arrows) is modified since the site with the susceptible plant can now be linked to farther sites (blue arrows).

Figure 2

Examples of possible initial configurations of a system of size $L=10$. (a) Distribution of cells with susceptible (filled triangles) and resistant (empty triangles) plants. (b) Distribution of inoculated cells.

Figure 3

Spanning clusters formed in the (a) soil and (b) plant cases for the initial conditions of Fig. 2 and the list of bonds given in the text. Only the bonds that connect sites are shown (black lines). In the case (b), bonds connecting a resistant plant in an inoculated site to a susceptible plant are represented with dashed lines. Yellow lines show the modification of the nearest-neighbor definition.

Figure 4

(a) Critical curves for cluster formation over infested soil (hollow figures) and infected plants (solid figures) on triangular (triangles), square (squares), and honeycomb (circles) lattices with no barriers. Theoretical curves for the soil case (dashed lines) and the fit to the data for the plant case (continuous lines) are also shown. (b) Simulation (figures) and theoretical (lines) critical curves in the soil case for square (squares), triangular (triangles), and honeycomb (circles) lattices for several values of $I$: 0.0 (black), 0.1 (purple), 0.2 (green), 0.3 (cyan), 0.4 (blue), and 0.5 (red).

Figure 5

Power-law relation among $\chi$, ${\chi }_c^{\text{plant}}$, and $p_w^{\text{plant}}$ in the plant case when $I$ is fixed for (a) square, (b) triangular, and (c) honeycomb lattices. Black solid line is the identity function. The color scale indicates the value of $I$ from $I=0$ (green) up to $I=1$ (blue) in steps of $\mathrm{\Delta }I=0.05$.

Figure 6

Comparison between simulation results (figures) for $p_w^{\text{plant}}$ as a function of the susceptibility and the curve proposed in Eq. (2) (solid lines) for (a) square, (b) triangular, and (c) honeycomb lattices. The color scale indicates the value of $I$ from $I=0$ (green) up to $I=1$ (blue) in steps of $\mathrm{\Delta }I=0.05$.

Figure 7

(a) Critical values $p_w^{\text{plant}}$ for arbol (A), poblano (P), and serrano (S) chili plants sowed with a square lattice arrangement. Vertical lines indicate their susceptibilities: 1.00 (A), 0.89 (P), and 0.60 (S). The solid curves are the same as in Fig. 6. ${\chi }_c^{\text{plant}}(I=1)=0.28883\pm 0.00007$ is the maximum value of a plant's susceptibility that inhibits the formation of a cluster of diseased plants, even in the extreme case where the patogen is present all over the plantation. (b) Values of $p_w^{\text{plant}}$ given by Eq. (2) and data from Table 1 for the arbol (purple), poblano (black), and serrano (red) chili plants on a square lattice.