The monomolecular, logistic and Gompertz equations are discussed in relation to observed disease progress curves of the red ring/little leaf and the charcoal base rot diseases of the oil palm ( Elaeis guineensis Jacq.) at one plantation in Honduras and two in Costa Rica. The prevalent symptom caused by Bursaphelenchus cocophilus in Honduras was the little leaf condition (chronic form of the disease), but in Costa Rica, the acute form of the disease (rapid death of the plant) prevailed. In Honduras, the disease progress in time was better described by the simple monomolecular model, which gave the best description of the data when the full range of disease incidence (low initial and high final disease assessments) was considered (R²: 0.82-0.99). However, when the prevalent symptom was the acute form of the disease (rapid death of the palms) the Gompertz model gave the best description of the data (R²: 0.79-0.98).

The specific interactions (genotype/pathogen/environment) that explain the differences in symptoms (chronic vs. acute forms of the disease) are not understood, but the differences in the epidemiological behavior (simple vs. compound interest models), can be partially explained considering the behavior of the vector insect ( Rhynchophorus palmarum ), and the ability of the plant to "regulate" the multiplication of the nematode.

For the charcoal base rot ( Ustulina deusta ) disease the monomolecular equation gave the best fit (R²: 0.87-0.98) in all plantations studied.

Geostatistics to these data provided a good estimate of the type and the changes in spatial dependence and structure of the disease in oil palm plantations. Experimental variograms computed for the red ring disease from a grid survey appeared to be anisotropic, and showed that palms affected by the disease were strongly aggregated. The variograms for charcoal base rot appeared to be isotropic, and showed that the pathogen is strongly aggregated too. The spherical and exponential models best fit the data respectively.

Disease progress curves help to understand the epidemiological behavior of a pathogen in a population of plants, since such curves combine the effects of the pathogen, the environment and the host. Understanding the disease epidemiology is a basic step to develop better control strategies (Van der Plank 1963, Madden 1980, Campbell and Madden 1990).

There are two basic disease progress curves: those followed by the so-called polyciclic pathogens, and those fallowed by monocyclic pathogens. Van der Plank (1963) propoused the models logistic and monomolecular to describe these two different types of diseases. However, the concept does not imply that the logistic equation only applies to foliar diseases (or that the monomolecular equation is just for root pathogens). On the other hand, the true nature of the pathosystem can not be inferred directly from an analysis of data better fitting to either model (Pfender 1982, Campbell and Madden 1990).

Geostatistics is a tool that helps to describe the spatial distribution of a disease, since it considers the position of each sample. Geostatistics is based on the theory of regionalized variables, and has the premise that the variance of the difference between samples is a function of the distance between them. Spatial variability is measured by determining the average of the square of the differences between pairs of samples separated by a given distance.

Geostatistics was introduced by geologist to quantify the spatial dependence (autocorrelation) in gold mines, and has been used in agro-forestry, agronomy, entomology and phytopatology (Burgess et al. 1981, Chellemi et al. 1988, Delaville et al. 1996, Lecoustre and Reffye 1986, Schotzko and O'Keeffe 1990, van de Lande 1993, Webster and Boag 1992).

Corky basal stem rot is caused by the fungus Ustulina deusta (Hoffm. ex. Fr.) Lind) (Martín 1970, C. M.I. 1972), and was described by Thompson (1963) in Malaysia. This fungus may cause a very extensive damage (rot) in the basal portion of the stem before the palm shows any external symptom. In some palms young leaves may become yellowish, and the petioles of older leaves break near the base. A clear diagnosis of the disease can be made when the characteristic sporophores of the fungi form on the leaf bases at the base of the palm. (Turner 1981; Chinchilla and Richardson 1986).

Symptoms have not been observed in palms younger than six years in Central America, but younger palms (4 years) may be affected in Malaysia (Thompson 1963). The disease may progress rather fast in some areas in palms between 9 and 11 years of age (Umaña and Chinchilla 1991).

The red ring disease caused by the nematode Bursaphelenchus ( Rhadinaphelenchus ) cocophillus used to be the most important phytosanitary problem of the oil palm in Central America, but an effective integrated management has been developed, that when properly applied, can reduced this disease to a secondary importance. Management includes reducing the inoculum sources of the nematode, and the reduction of the population of the insect vector, Rhynchophorus palmarum (Oehlschlager et al. 2002, Chinchilla 2003).

The role of other possible vectors, and other means of transmission such as harvesting practices and leaf pruning seems to lack much practical importance (Bulgarelli et al.1998, Fenwick 1968, Schuiling and van Dinther 1981).

Symptoms of the red ring disease include a progressive yellowing starting from the older leaves and eventual death of the plant within a few months (acute symptoms). However, infected palms may remain alive for several years (chronic symptoms), when the nematode somehow is restricted to the youngest leaves, where it reproduces extensively causing different kinds of malformations on the developing leaves. Such symptoms are called "the little leaf condition". The name of the diseases originated from the symptoms the nematode normally causes in the stems of affected coconut palms. In oil palm, a ring of decolorized tissue (normally brown) may or may not appear when the stem is cut transversally (Chinchilla 1992 ).

Even though, the red ring disease incidence can be reduced significantly, still constitutes an important potential threat for the oil palm industry in tropical America. Considering this, the information collected in the past is still relevant to help to understand the causes of past outbreaks in order to avoid similar situations to occur in the future. In particular, epidemiological information on this disease is scarce. In the case of charcoal base rot disease, it still constitutes a potential threat, particularly for new plantations established in areas with former oil palm plantations. There are no previous epidemiological analysis of this disease in Central America (Chinchilla Richardson 1988).

Data for this study were obtained during the late 80s and early 90s, in three commercial oil palm plantations, located one in the Atlantic coast of Honduras, and two on the Pacific coast of Costa Rica. These plantations are divided in "harvesting lots" of about 30-60 ha each, which were visited monthly to register disease incidence. Depending on lot age and the year when each epidemic started, data available comprises from two to six consecutive years. For some lots it was not possible to determine the exact date for the onset of the epidemic.

During each monthly visit the position of each palm was determined (row and palm within the row), and positioned in a map of the lot. Data for the red ring disease analysis was obtained from 25 harvesting lots in Honduras and 78 in Costa Rica. For the charcoal base rot disease, only 10 lots were considered in each locality.

Given that in no case disease incidence reached 100%, a maximum incidence was established for each disease (K), to avoid an underestimation of incidence (Neher and Campbell 1992).

Incidence was calculated as the proportion of diseased palms in relation to the total number of healthy palms (total palms minus affected palms in the previous survey), to avoid the effect of the eradication of diseased palms. With the incidence data, we calculated the rate of disease increase using the following equations:

Monomolecular

Logistic

Gompertz

Regression analysis was done with Statgraphics. The selection of the best fitting model was done considering the distribution of the residuals, the determination coefficient ( R²), and the standard error of the estimates (Sy: square root of the error sum of squares divided by n-2). The selected equations were those with the most homogeneous distribution of the residuals, the largest R², and the lowest value for Sy. Lots and plantations were compared after standardizing disease progress curves, using the equations:

where "m" is a shape parameter, (0,1 and 2 for the monomolecular, Gompertz and logistic equations), K is maximum incidence and P is the calculated rate of development (before standardizing)

The analysis of the experimental variograms and fitting to the different models was done using GS+ (Gamma Design Software).

In Honduras, where the little leaf condition was prevalent, the equation that better fitted the data on changes in disease incidence in time was the monomolecular ( Table 1 and Table 2 ). Only in one of the lots this equation could not adjust the data. Within the remaining lots (96%), 21% had a disease incidence lower than 1%, 50% had an incidence between 1 and 5%, the remaining (29%) had an incidence higher than 5%.

In Costa Rica, where the prevalent symptom was the classical manifestation of the disease (acute form), the equation that better fitted the data was Gompertz ( Table 1 and Table 3 ). In this case, data from 55% of the lots were adjusted by the equation: 35% of the lots had an incidence lower than 0.5%, 23% had incidences between one and five percent, and 42% had an incidence higher than 5%.

Such differences between localities in the epidemiological behavior of the same disease are not frequent, but can be explained considering the habits of the insect vector, and possibly, the ability of the plant to restrict the systemic invasion of all its tissues. When a plant shows the little leaf symptoms (chronic form of the disease), it seems that the systemic movement of the nematode is somehow restricted, and the highest populations occur only in the developing young leaves within the whorl. Those palms showing typical little leaf symptoms, commonly do not present signs of nematode activity in the stem (no ring and sometimes just some stains randomly distributed), and nematodes, if present, do not look very healthy. The little leaf symptom was practically the only symptom present in all palms affected by the nematode in Honduras, but was less frequent in Costa Rica in the Pacific coast, where the classical form of the disease was prevalent. This could indicate that some unknown environmental conditions may determine the prevalence of either symptom.

Rotting of the tissue is very limited or absent when the nematodes are somehow confined to the area of leaf elongation within the whorl. This makes the plant less attractive to the insect vector, which respond to volatiles coming from such tissues. The present of larvae and adults of the vector are uncommon in palms affected by typical little leaf symptoms, and this is one of the reason why the palms do not die and may remain showing symptoms for years. The final result is a slow disease-rate increase, where incidence depends on the initial population of the vector already infected with the nematode. It was known that the proportion of vector insects infected with the nematode was particularly high in Honduras (Chinchilla et al. 1991). The epidemiological behavior of such conditions followed the monomolecular equation.

The contrary happened in palms showing the classical or acute form of the disease, where the most common situation was to find larvae and adults of the vector in the rotting tissue of the whorl, which in turn brought more insects until the palm died. A high proportion of the insects breed in the affected tissues may acquire the nematode and transmit it to neighbor plants, determining an epidemiological behavior that could be described as a typical compound interest disease (described with the Gompertz equation).

The population of R. palmarum tends to be aggregated within the plantations (Oehlschlager et al. 1995), and this is also a factor that contributes to the epidemiological behavior of the disease caused by Bursaphelenchus cocophillus . The results of the analysis using geostatistics confirmed the aggregated distribution of palms showing the red ring disease ( Table 4 ).

The epidemiological behavior of this disease followed the classical pattern of a soil-borne disease, and as such was described quite well by the monomolecular equation ( Table 5 and Table 6 ). The amount of initial inoculum is determined by the present of large pieces of infected tissue of trees or oil palms buried in the soil in the case of U. deusta (Zadoks and Schien 1979, Pfender 1982). The analysis using geostatistics confirmed an aggregated distribution of diseased palms (Table 4).

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