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Effect of conidial dispersal of the fungal pathogen Entomophaga maimaiga (Zygomycetes: Entomophthorales) on survival of its gypsy moth (Lepidoptera: Lymantriidae) host
Biological Control 29 (2004) 138–144 www.elsevier.com/locate/ybcon
Effect of conidial dispersal of the fungal pathogen Entomophaga maimaiga (Zygomycetes: Entomophthorales) on survival of its gypsy moth (Lepidoptera: Lymantriidae) host Ronald M. Weseloh* Department of Entomology, Connecticut Agricultural Experiment Station, New Haven, CT 06511, USA Received 20 February 2003; accepted 5 June 2003
Abstract Several researchers have developed a one-generational computer model that simulates infection prevalence of gypsy moth, Lymantria dispar, caterpillars by its fungal pathogen, Entomophaga maimaiga. Inputs required are temperature, humidity, and rainfall records, a measure of fungus resting spore load in the soil, and an estimate of gypsy moth larval density. In a previous study, the model accurately tracked fungal-induced host mortality as long as airborne fungal conidia were allowed to disperse freely over a local area. In 2002, dispersal of conidia and its influence on the impact of the fungus on the gypsy moth was investigated. Gypsy moth densities and fungus resting spore loads were measured in 15 plots within a 3 km area. In 7 of the plots, prevalence of fungal disease was determined weekly by collecting and rearing gypsy moth larvae. Different strategies were used to disperse conidia within the model, and resulting simulated prevalence rates were compared to actual data. Model output was most accurate when airborne conidia were permitted to disperse equally to all plots. Thus, to accurately assess the impact of the fungus in one location, it is necessary to take into account fungal activity throughout the local area. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Entomophaga maimaiga; Lymantria dispar; Conidial dispersal; Fungal pathogen; Infection prevalence; Simulation model; Weather
1. Introduction The very effective gypsy moth, Lymantria dispar L., natural enemy, Entomophaga maimaiga Humber, Shimazu & Soper, has been conspicuously present in North America since 1989 (Andreadis and Weseloh, 1990). Hajek et al. (1990) showed it to be identical to the fungal pathogen of the same name in Asia, and various scenarios have been discussed as to how it arrived in North America (Hajek et al., 1995; Weseloh, 1998). Researchers have made substantial progress in uncovering the pathogenÕs biology, epidemiology, environmental requirements and effect on the gypsy moth (see review by Hajek, 1999). Entomophaga maimaiga passes the winter as azygospores (i.e., resting spores), in forest litter. These resting spores may survive for 10 years or more (Wese* Fax: 1-203-974-8502. E-mail address: [email protected].
1049-9644/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S1049-9644(03)00135-X
loh and Andreadis, 2002). As gypsy moths begin hatching in the spring, moist conditions cause the germination of some resting spores to produce airborne conidia and subsequent infection of larvae. Resting spores continue to germinate, and the resulting conidia infect caterpillars throughout the time when larvae are present so long as the forest floor is moist. The fungus develops in infected larvae and after a larva dies, the fungus may sporulate if very humid conditions prevail, as during or after a rainstorm. The resulting conidia are released into the air and can cause infection in other larvae if they contact larval integuments. For infection by conidia to occur, very humid conditions must prevail throughout the sporulation, transportation, and germination process. Conidia produced by sporulating fungus from host cadavers are largely responsible for the oftenmassive infections of gypsy moth larvae that occur when caterpillars are large. Near the end of the larval period, more and more fungi in dead larvae produce resting spores rather than conidia. These resting spores stay
R.M. Weseloh / Biological Control 29 (2004) 138–144
within the cadavers, which accumulate on tree trunks and eventually weather off to accumulate in the soil where the spores survive from year to year. Weather conditions are very important for infection by conidia to occur. The fungus in infected caterpillars does not produce conidia until caterpillar death, and then only for about 8 days at decreasing rates (Hajek and Soper, 1992), and only if sufficient precipitation occurs to raise the relative humidity to near 100% (Hajek, 1999). Also, fungal rate of development is dependent on temperature. Thus, temperature conditions and precipitation timing that result in infected caterpillars dying when humidity is high are critical for substantial gypsy moth mortality. Weseloh et al. (1993) developed a computer model incorporating specific weather records that simulates the interaction between E. maimaiga and its host. Weseloh (1999, 2002) has since refined this model. For the Weseloh (2002) study, in order for the model to accurately reflect actual prevalence rates of the fungus in nature, it was necessary to disperse conidia among the 3 groups of local plots from which data had been gathered. The large dispersal effect was unexpected. I have since used these results to investigate both local and long-range dispersal of this fungus, especially pertaining to the rapidity with which the fungus spread just after it was first found in North America (Weseloh, 2003). I carried out the present study, with independent data, to further document local dispersal and obtain more information about dispersal characteristics.
2. Materials and methods 2.1. Model description Weseloh et al. (1993), and Weseloh (1999, 2002), described the model fully, so only a brief description will be given here. See the Appendix A for more details. The model was written in C and run on a Gateway 2000 computer. Gypsy moth larvae hatch on dates determined from a normal distribution with mean (SD) 7.5 (2.5) days (Weseloh, 1999), with the beginning hatch date determined from observations of natural hatch. Larvae that hatch on a given date form a healthy cohort that develops according to a degree-day model with a threshold temperature of 12 °C. These larvae can be infected by resting spores in the soil depending on the degree of soil moisture (estimated as a function of daily rainfall) and the number of resting spores in the soil (estimated via a soil bioassay described below). All larvae from a healthy cohort that become infected on a given date form a separate infected cohort. Larvae in an infected cohort develop at the same rate as healthy larvae, but the fungus develops within them at a rate determined from data obtained by Hajek et al. (1993).
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When an infected cohort dies, the contained fungi produce airborne conidia in amounts that are determined by larval instar, proportion of day with 98% humidity or above, and time elapsed since caterpillar death. At this point the model disperses the conidia in ways that are outlined below. Larvae become infected via these airborne conidia depending on conidial concentration. The model tracks the number of larvae, their instars, and the number infected for each day that larvae are present. Output is given as daily fungal prevalence and/or seasonal larval survival rates. 2.2. Data collection As presently constituted, the model requires continuous recordings of temperature and humidity, daily rainfall, a measure of resting spore load in the soil, and an estimate of gypsy moth density as inputs. Model fungal prevalence can be compared to actual prevalence of fungus in forest larvae. In 2002 15 plots were placed at Mansfield Hollow State Park in Connecticut for study (see Fig. 1). The plots were located around Mansfield Hollow Lake and close to open field areas, and the topography was quite flat. This was the same area used in 2001 (Weseloh, 2002), and included the reuse of four 2001 plots. To measure resting spore abundance, soil samples were collected from just under the leaf litter at each of 10 oak trees per plot in April 2002. The soil samples from a plot were mixed together, taken to the laboratory and set up in 30 ml subsamples (1–2 per plot) in 2.5 14.0 cm diam. plastic Petri dishes. The soil was kept moist at 17 °C and 15 h light per day. After 1 week, a 13 cm circle of aluminum window screen was laid on top of the soil and 25 laboratory-reared gypsy moth larvae (1st or 2nd instars) were placed on top of the screen. After 24 h, larvae were removed and reared individually in 30 ml plastic cream cups filled 1/4 full with gypsy moth diet (Southland Products, Lake Village, Arkansas) to determine infection. Larvae were checked for 2 weeks, and any dead larvae were dissected and examined under a compound microscope to determine cause of death. The percent of larvae that died from fungus from each plot was used as a direct input to the model. Gypsy moth density was also determined at each plot on 20 May, 2002, by recording the number of gypsy moth larvae on 25, 20 cm terminals of favored food plants (mainly various species of oaks Quercus) or witch hazel (Hamamelis virginiana L.). Seven of the plots were chosen for more intensive study (circles in Fig. 1). In these plots, larval terminal counts were also taken the week before and the week after 20 May. These counts were averaged to estimate gypsy moth density. During each of 8 weeks when gypsy moth larvae were present (14 May to 26 June), an attempt was made to collect at least 100 larvae at each of
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2.3. Modeling conidial dispersal
Fig. 1. Representations of the forest plots in Mansfield Hollow used in this study. The asterisks are the locations of the 8 plots where resting spore load and gypsy moth density only were sampled. Weather instruments were placed and gypsy moth larvae collected to determine fungus infection prevalence in the 7 intensive plots represented by the circles. Weather data from M7, represented by the largest circle, was used to run the model at the 8 least-intensively studied plots. In graphs A and B, zero gypsy moth numbers and resting spore loads were assumed in the outlined areas representing the lake and fields. Delaunay triangulation as the interpolation procedure was used in graphs A and B. In graphs C and D, polygons of influence were used as the interpolation procedure (see text). Graphs A and C represent interpolation outcomes for soil bioassays, and graphs B and D the same for larval density. The site represented is 3.94 km wide and 4.39 km long.
these plots per week. This was not always possible, but usually at least 30 were collected. Each larva was immediately placed in a 30 ml clear plastic cup that was 1/4 full of gypsy moth diet, returned to the laboratory, and reared for 2 weeks. As before, any larvae that died were dissected and examined under a compound microscope to determine cause of death. To record weather data, a temperature and humidity recording data logger (Hobo Pro Rh/Temp, Onset Computer Corp, Pocasset, MA) and a data logging rain gauge (Onset) were placed in each of the seven intensive plots for the time that gypsy moth larvae were active (late April to early July). The temperature-humidity loggers took readings every 10 min and the tippingbucket rain gauges recorded an event every time 0.25 mm of rain accumulated.
Various methods were used for modeling the dispersal of conidia in the Mansfield Hollow area. For all 15 plots, data were available on resting spore load in the soil and gypsy moth density, and I assumed that weather conditions in the non-monitored plots (represented by the asterisks in Fig. 1) would be similar to those in the monitored plots. Thus, using the weather records obtained from the plot M7 (largest circle in Fig. 1) as input for the non-monitored plots, outputs were obtained from all plots, even though prevalence rates were only sampled in the 7 intensive plots. This situation permitted me to obtain estimates of conidial numbers in all plots, and then redistribute them to see how this affected model output in the intensive plots. Conidia were redistributed in 3 ways: (1) Simple averaging (2) Delaunay triangle interpolation, and (3) polygonal interpolation. For simple averaging on days when conidia were produced, the number of conidia at a plot of interest and all other plots out to a given distance from the first plot was averaged. The model used this averaged value of conidial numbers to cause infection at the chosen plot. For Delaunay triangle interpolation the area of Mansfield Hollow was divided into forested and non-forested areas, the latter being open fields and a lake (outlined areas in Figs. 1A and B). Because there were no gypsy moths in the latter areas, resting spore load and larval density were assigned values of 0.0 to points located 63 m apart. These points were then treated as equivalent to the points where the 15 plots were located, and the entire area was divided into Delaunay triangles and the values of resting spore load and gypsy moth larval density over a 128 128 grid (44 m between points) determined by linear interpolation (see Isaaks and Srivastava (1989) for an explanation of Delaunay triangular interpolation). For polygonal interpolation, data were used from the 15 plots only. Using the geographic information system IDRISI (Eastman, 1993), inverse distance weighting with an exponent of 20 was used to interpolate to a 64 64 grid, (88 m between points). Twenty was chosen as an exponent simply because it was high enough to make the resulting interpolation very close to what would occur by dividing the region into polygons each of which contains the area closest to a given point and assigning the value at each point to all parts of the encircling polygon (as in Figs. 1C and D, see Isaaks and Srivastava (1989) for more information on such polygons of influence). Any larger number would have worked as well. These two interpolation measures were used because they did not change values at the known points. Also, the usually preferred method of kriging could not readily be applied because variograms were not consistent enough to have confidence in the method. Delaunay triangulation and polynomial interpolation not only preserve the real data but also
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represent extremes in interpolation methods (linear vs. step, respectively). Thus, I felt the procedures would be useful in the present context. Having obtained grids of resting spore load and gypsy moth density using the 2 different interpolation methods, a ‘‘kinked’’ linear dispersal distribution, or ‘‘kernel’’ (Weseloh, 2003) was used to redistribute conidia. This distribution is a 2-segment linear construct starting with a y-intercept at 1 and going down to a y value of 0.01 at a given distance from the point of interest (called the inclusion radius), and then from 0.01 at that given distance to y ¼ 0 at 100 km. As explained in Weseloh (2003), I used this kernel as a kind of ‘‘leptokurtic,’’ or fat-tailed, distribution that appeared to give a good fit for dispersal of E. maimaiga conidia. Thus, at each point of the appropriate interpolation grids, the pointÕs value of resting spore load and gypsy moth density, as well as weather data obtained from plot M7, were used as model inputs to calculate conidial production for a given date. Then the model redistributed the conidia using the dispersal kernel to give a weighted average value at each point that could be used to infect healthy larvae. To compare results for these different redistribution methods, sums of squares of differences were calculated between the modeled prevalence output at the locations of each of the 7 intensive plots and the actual infection prevalence rates obtained from forest sampling at these same plots.
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Fig. 2. Comparisons between proportions of caterpillars in each instar on different dates as determined from collection of forest caterpillars (solid circles) from the 7 intensively sampled plots and model output for plot M7 (line).
3. Results 3.1. Preliminary result: instar distribution The model keeps track of instars in healthy cohorts, making it possible in a preliminary result to compare actual development rates with the model degree-day mechanism. The degree-day model developed by Weseloh (2002) with a threshold value of 12 °C appears to be adequate for simulating development of larvae (Fig. 2). In most cases, the dates of first appearance of the various instars in the forest were similar to model predictions, but instars persisted longer in the forest than in the model. This may have been a consequence of our inability to distinguish healthy from parasitized or otherwise unhealthy larvae in the forest, because in some cases unhealthy larvae develop slower than healthy ones. 3.2. Comparisons between model and forest infection prevalences When I ran the model using input data from the 7 intensive plots without allowing for dispersal of conidia, infection prevalences predicted by model output had only mediocre correspondences to actual infection
prevalence at these plots (Fig. 3). In 2002, infection prevalences in Mansfield Hollow were only high for 1 collection period (17–20 June). The model outputs generally mimic this spike in time but not amplitude, and predicted prevalence was too low for plots that had low gypsy moth numbers (M1 and M5). However, if conidia from all 15 plots mix freely using the simple averaging algorithm with an inclusion radius of 20.0 km (so that one averaged conidial concentration is applied to all plots), model and actual infection prevalences are much closer (Fig. 4). Therefore, conidial dispersal at the local level is important for accurate model simulation, a conclusion also reached by Weseloh (2002). I also made comparisons of model infection prevalence and actual infection prevalence (via sums of squares of differences) using different dispersal methods and inclusion radii (Fig. 5). Simple averaging gave better fits than Delaunay triangle interpolation at all radii, and was better than polynomial interpolation above 1 km. It is surprising that Delaunay triangle interpolation, which arguably is the best technique of the 3 for accurately simulating the known spatial structure at the site, did the worst. Both simple averaging and polygonal inter-
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Fig. 3. Comparisons between infection prevalence on different dates in the different plots as obtained from collections of larvae (solid circles, brackets indicate 95% binomial confidence intervals), and model output, assuming no local dispersal of conidia (line).
Fig. 4. Comparisons between infection prevalence on different dates in the different plots as obtained from collections of larvae (solid circles, brackets indicate 95% binomial confidence intervals), and model output, assuming that local conidial dispersal occurs via the simple average algorithm with an inclusion radius of 20.0 km.
polation had minimal sums of squares at about 1.7 km, but only simple averaging stayed near its minimum at larger inclusion radii. It appears that simple averaging of conidial numbers over about a 2 km area produces accurate simulation results. 3.3. Importance of resting spore load and gypsy moth density in auxiliary plots The conclusion from the above section implies that it is necessary to sample resting spore load and gypsy moth larval density at several places within a 2 km area of the location of interest in order to obtain meaningful results from the model. Accordingly, I wanted to determine for auxiliary plots if variations in resting spore load are more or less important than variations in gypsy moth density for affecting model results. I ran the model under the simple averaging algorithm using an inclusion radius of 2.0 km, and input soil bioassay values of 5.0 or 50.0%, but used the actual sampled values for gypsy moth input density. Then I used sample bioassay values
Fig. 5. Sum of squares of difference between forest prevalence rates and model outputs for different inclusion radii using the three conidial dispersal mechanisms discussed in the text.
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and ran the model with either 0.1 or 1.0 larvae/terminal. These values approximated the ranges of resting spore load (1.0–63.27%) and density (0.04–0.947 larvae/terminal) found in plots in 2002. The total sum of squares of differences between daily infection prevalence in the 7 intensive plots for model runs with 5.0 or 50.0% soil bioassay values was 213,343. When density was varied between 0.1 and 1.0, the sum of squares was 569,168. Thus, order of magnitude variations in larval density was more important in causing variations in simulated fungal infection prevalence than order of magnitude variations in resting spore loads.
4. Discussion I used data acquired in 2001 and previous years to develop the E. maimaiga fungus model and to find parameter values that give good fits to forest infection prevalence. I also determined that averaging conidial distribution among plots gave a better fit to data than not allowing for such dispersal (Weseloh, 2002). The same model fit data very well when used without change in 2002 if I averaged conidia numbers from other local plots. Although in 2002 plots were in the same forest used to refine the model in 2001, weather conditions and infection prevalences were different in the 2 years. Thus, the effect of local dispersal on infection rates appears to be consistent. I do not know why simple averaging gave better results than the 2 other procedures for dispersing conidia. After all, it seems reasonable to expect that the probability of conidia dispersing between plots will decline as the distance between plots increases. To a certain extent, this is true for all dispersal procedures, as even simple averaging gave the best fits for inclusion radii around 2 km. But using a monotonically declining dispersal kernel does not work nearly as well as simple mixing. One possibility is that on the scale of a forest, conidia do not arise from point sources. Much of the forested areas around each of the plots had infected gypsy moths that would have contributed dispersing conidia. Local areas having relatively low numbers of gypsy moths would receive more conidia than were produced there, and vice versa for high population areas. The relative flatness of the area would also contribute to homogeneous dispersal of conidia. The overall effect would be one of averaging conidial concentrations over the local area. The excellent fits obtained using simple averaging means that it is not necessary to interpolate around sample points, even if extensive non-forested areas are present. Because of conidial dispersal the local effect of the fungus is essentially homogeneous. This simplifies sampling and analysis considerably. However, conidial dispersal also means that to obtain accurate model results at one point, it is necessary to sample for resting
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spore load and gypsy moth density at additional local points. This study shows that averaging over 15 points is adequate, but possibly fewer would be needed. (Weseloh (2002) found that averaging over 6 local point was sufficient.) Resting spore load does not need to be determined as accurately as gypsy moth density and can even be estimated by the number of years since defoliation by the gypsy moth has occurred (Weseloh and Andreadis, 2002). Taking terminal counts when gypsy moth larvae are young can be logistically difficult, but density may be estimated with some precision by counting gypsy moth egg masses earlier in the year (Weseloh, 2003). Dispersal of conidia helps explain the rapid spread of E. maimaiga after its discovery in North America in 1989 (Weseloh, 2003). It also helps explain data showing insensitivity of infection prevalence to host density (Hajek, 1999). Pronounced dispersal of infective conidia may be an important factor enabling E. maimaiga to control its host over wide areas.
Acknowledgments I thank Lori Brown and Morgan Lowry of the Connecticut Agricultural Experiment Station for their help in carrying out this study, and John A. Tanner and other personnel at the USDA APHIS Methods Development Laboratory at Otis, Massachusetts for providing gypsy moth eggs. This research was supported, in part, by a Special Technology Development Program Grant from the USDA Forest Service. Appendix A The model is completely described by Weseloh (2002). This Appendix A presents the critical equations. The model has a time step of 1 day, so the equations are date-specific. However, I omit time-related subscripts to simplify the equations. From data by Hajek et al. (1993), the rates of development of the fungus per day (r) in caterpillars held at different temperatures (T ) from 7.5 to 26 °C is r ¼ 0:162 þ 0:032 T 0:000752 T 2 Using this equation, the model calculates daily cumulative fungal development rates so that the dates when infected cohorts die (i.e., cumulative rate ¼ 1.0) can be determined. The probability of infection by resting spores (Pr ) on a day is 2
Pr ¼ 0:0035 R½sinð0:01431 MÞ ; where R is the percent of larvae infected in a soil bioassay, and M is a forest moisture index defined as the total amount of rain, in mm, accumulated over the
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present day and the last 4 days, with no more than 10 mm of rain allowed per day. After E. maimaiga kills a larva, the number of conidia produced per m3 of forest space (i.e., including the volume from top of canopy to ground) per day (c) varies with instar: Instar 2: c ¼ 11; 639 Ni H expð0:36 dÞ Instar 3: c ¼ 24; 542 Ni H expð0:763 dÞ Instar 4: c ¼ 32; 832 Ni H expð0:491 dÞ Instar 5+: c ¼ 204; 875 Ni H expð0:859 dÞ In the above equations, Ni is the number of larvae in an infected cohort of an appropriate instar, H is the proportion of the day for which humidity is above 98%, and d is the number of days since caterpillar death from fungus, up to a maximum of 8 days after which c ¼ 0. The rate at which healthy larvae become infected with conidia depends on conidial concentration (c) as well as the size of the larva (s). The size factor used is a function of the degree-days of development (D) of each healthy cohort from hatch: s ¼ D2 =ð91613:9 þ 0:6798 D2 Þ The probability, Pc , that a 5th instar will become infected when there are c conidia in a m3 volume is c
Pc ¼ 1 ð1 0:002 sÞ ; where s is the above mentioned size factor. The total probability of infection (PT ) from both resting spores (Pr ) and conidia (Pc ) is PT ¼ Pr þ Pc Pr Pc because the two probabilities are independent. The model multiplies this total probability by the number of healthy caterpillars in a cohort to give the number of larvae in a new, infected cohort. References Andreadis, T.G., Weseloh, R.M., 1990. Discovery of Entomophaga maimaiga in North American gypsy moth, Lymantria dispar. Proc. Natl. Acad. Sci. USA 87, 2461–2465.
Eastman, J.R., 1993. IDRISI version 4.1 [DOS]. Clark University, Worcester, MA. Hajek, A.E., 1999. Pathology and epizootiology of Entomophaga maimaiga infections in forest Lepidoptera. Microb. Mol. Biol. Rev. 63, 814–835. Hajek, A.E., Humber, R.A., Elkinton, J., 1995. Mysterious origin of Entomophaga maimaiga in North America. Am. Entomol. 41, 31– 42. Hajek, A.E., Humber, R.A., Elkinton, J., May, S.B., Walsh, S.R.A., Silver, J.C., 1990. Allozyme and restriction fragment length polymorphism analyses confirm Entomophaga maimaiga responsible for 1989 epizootics in North American gypsy moth populations. Proc. Natl. Acad. Sci. USA 87, 6979– 6982. Hajek, A.E., Larkin, T.S., Carruthers, R.I., Soper, R.S., 1993. Modeling the dynamics of Entomophaga maimaiga (Zygomycetes: Entomophthorales) epizootics in gypsy moth (Lymantria: Lymantriidae) populations. Environ. Entomol. 22, 1172– 1187. Hajek, A.E., Soper, R.S., 1992. Temporal dynamics of Entomophaga maimaiga after death of gypsy moth (Lepidoptera: Lymantriidae) larval hosts. Environ. Entomol. 21, 129–135. Isaaks, E.H., Srivastava, R.M., 1989. An Introduction to Applied Geostatistics. Oxford University Press, New York, Oxford. Weseloh, R.M., 1998. Possibility for recent origin of the gypsy moth (Lepidoptera: Lymantriidae) fungal pathogen Entomophaga maimaiga (Zygomycetes: Entomophthorales) in North America. Environ. Entomol. 27, 171–177. Weseloh, R.M., 1999. Entomophaga maimaiga (Zygomycete: Entomophthorales) resting spores and biological control of the gypsy moth (Lepidoptera: Lymantriidae). Environ. Entomol. 28, 1162– 1171. Weseloh, R.M., 2002. Modeling the impact of the fungus, Entomophaga maimaiga (Zygomycetes: Entomophthorales) on gypsy moth (Lepidoptera: Lymantriidae): Incorporating infection by conidia. Environ. Entomol. 31, 1071–1084. Weseloh, R.M., 2003. Short and long range dispersal in the gypsy moth (Lepidoptera: Lymantriidae) fungal pathogen, Entomophaga maimaiga (Zygomycetes: Entomophthorales). Environ. Entomol. 32, 111–122. Weseloh, R.M., Andreadis, T.G., 2002. Detecting the titer in forest soils of spores of the gypsy moth (Lepidoptera: Lymantriidae) fungal pathogen, Entomophaga maimaiga (Zygomycetes: Entomophthorales). Can. Entomol. 134, 269–279. Weseloh, R.M., Andreadis, T.G., Onstad, D.W., 1993. Modeling the influence of rainfall and temperature on the phenology of infection of gypsy moth, Lymantria dispar, larvae by the fungus Entomophaga maimaiga. Biol. Control 3, 311–318.
Effect of conidial dispersal of the fungal pathogen Entomophaga maimaiga (Zygomycetes: Entomophthorales) on survival of its gypsy moth (Lepidoptera: Lymantriidae) host Ronald M. Weseloh* Department of Entomology, Connecticut Agricultural Experiment Station, New Haven, CT 06511, USA Received 20 February 2003; accepted 5 June 2003
Abstract Several researchers have developed a one-generational computer model that simulates infection prevalence of gypsy moth, Lymantria dispar, caterpillars by its fungal pathogen, Entomophaga maimaiga. Inputs required are temperature, humidity, and rainfall records, a measure of fungus resting spore load in the soil, and an estimate of gypsy moth larval density. In a previous study, the model accurately tracked fungal-induced host mortality as long as airborne fungal conidia were allowed to disperse freely over a local area. In 2002, dispersal of conidia and its influence on the impact of the fungus on the gypsy moth was investigated. Gypsy moth densities and fungus resting spore loads were measured in 15 plots within a 3 km area. In 7 of the plots, prevalence of fungal disease was determined weekly by collecting and rearing gypsy moth larvae. Different strategies were used to disperse conidia within the model, and resulting simulated prevalence rates were compared to actual data. Model output was most accurate when airborne conidia were permitted to disperse equally to all plots. Thus, to accurately assess the impact of the fungus in one location, it is necessary to take into account fungal activity throughout the local area. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Entomophaga maimaiga; Lymantria dispar; Conidial dispersal; Fungal pathogen; Infection prevalence; Simulation model; Weather
1. Introduction The very effective gypsy moth, Lymantria dispar L., natural enemy, Entomophaga maimaiga Humber, Shimazu & Soper, has been conspicuously present in North America since 1989 (Andreadis and Weseloh, 1990). Hajek et al. (1990) showed it to be identical to the fungal pathogen of the same name in Asia, and various scenarios have been discussed as to how it arrived in North America (Hajek et al., 1995; Weseloh, 1998). Researchers have made substantial progress in uncovering the pathogenÕs biology, epidemiology, environmental requirements and effect on the gypsy moth (see review by Hajek, 1999). Entomophaga maimaiga passes the winter as azygospores (i.e., resting spores), in forest litter. These resting spores may survive for 10 years or more (Wese* Fax: 1-203-974-8502. E-mail address: [email protected].
1049-9644/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S1049-9644(03)00135-X
loh and Andreadis, 2002). As gypsy moths begin hatching in the spring, moist conditions cause the germination of some resting spores to produce airborne conidia and subsequent infection of larvae. Resting spores continue to germinate, and the resulting conidia infect caterpillars throughout the time when larvae are present so long as the forest floor is moist. The fungus develops in infected larvae and after a larva dies, the fungus may sporulate if very humid conditions prevail, as during or after a rainstorm. The resulting conidia are released into the air and can cause infection in other larvae if they contact larval integuments. For infection by conidia to occur, very humid conditions must prevail throughout the sporulation, transportation, and germination process. Conidia produced by sporulating fungus from host cadavers are largely responsible for the oftenmassive infections of gypsy moth larvae that occur when caterpillars are large. Near the end of the larval period, more and more fungi in dead larvae produce resting spores rather than conidia. These resting spores stay
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within the cadavers, which accumulate on tree trunks and eventually weather off to accumulate in the soil where the spores survive from year to year. Weather conditions are very important for infection by conidia to occur. The fungus in infected caterpillars does not produce conidia until caterpillar death, and then only for about 8 days at decreasing rates (Hajek and Soper, 1992), and only if sufficient precipitation occurs to raise the relative humidity to near 100% (Hajek, 1999). Also, fungal rate of development is dependent on temperature. Thus, temperature conditions and precipitation timing that result in infected caterpillars dying when humidity is high are critical for substantial gypsy moth mortality. Weseloh et al. (1993) developed a computer model incorporating specific weather records that simulates the interaction between E. maimaiga and its host. Weseloh (1999, 2002) has since refined this model. For the Weseloh (2002) study, in order for the model to accurately reflect actual prevalence rates of the fungus in nature, it was necessary to disperse conidia among the 3 groups of local plots from which data had been gathered. The large dispersal effect was unexpected. I have since used these results to investigate both local and long-range dispersal of this fungus, especially pertaining to the rapidity with which the fungus spread just after it was first found in North America (Weseloh, 2003). I carried out the present study, with independent data, to further document local dispersal and obtain more information about dispersal characteristics.
2. Materials and methods 2.1. Model description Weseloh et al. (1993), and Weseloh (1999, 2002), described the model fully, so only a brief description will be given here. See the Appendix A for more details. The model was written in C and run on a Gateway 2000 computer. Gypsy moth larvae hatch on dates determined from a normal distribution with mean (SD) 7.5 (2.5) days (Weseloh, 1999), with the beginning hatch date determined from observations of natural hatch. Larvae that hatch on a given date form a healthy cohort that develops according to a degree-day model with a threshold temperature of 12 °C. These larvae can be infected by resting spores in the soil depending on the degree of soil moisture (estimated as a function of daily rainfall) and the number of resting spores in the soil (estimated via a soil bioassay described below). All larvae from a healthy cohort that become infected on a given date form a separate infected cohort. Larvae in an infected cohort develop at the same rate as healthy larvae, but the fungus develops within them at a rate determined from data obtained by Hajek et al. (1993).
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When an infected cohort dies, the contained fungi produce airborne conidia in amounts that are determined by larval instar, proportion of day with 98% humidity or above, and time elapsed since caterpillar death. At this point the model disperses the conidia in ways that are outlined below. Larvae become infected via these airborne conidia depending on conidial concentration. The model tracks the number of larvae, their instars, and the number infected for each day that larvae are present. Output is given as daily fungal prevalence and/or seasonal larval survival rates. 2.2. Data collection As presently constituted, the model requires continuous recordings of temperature and humidity, daily rainfall, a measure of resting spore load in the soil, and an estimate of gypsy moth density as inputs. Model fungal prevalence can be compared to actual prevalence of fungus in forest larvae. In 2002 15 plots were placed at Mansfield Hollow State Park in Connecticut for study (see Fig. 1). The plots were located around Mansfield Hollow Lake and close to open field areas, and the topography was quite flat. This was the same area used in 2001 (Weseloh, 2002), and included the reuse of four 2001 plots. To measure resting spore abundance, soil samples were collected from just under the leaf litter at each of 10 oak trees per plot in April 2002. The soil samples from a plot were mixed together, taken to the laboratory and set up in 30 ml subsamples (1–2 per plot) in 2.5 14.0 cm diam. plastic Petri dishes. The soil was kept moist at 17 °C and 15 h light per day. After 1 week, a 13 cm circle of aluminum window screen was laid on top of the soil and 25 laboratory-reared gypsy moth larvae (1st or 2nd instars) were placed on top of the screen. After 24 h, larvae were removed and reared individually in 30 ml plastic cream cups filled 1/4 full with gypsy moth diet (Southland Products, Lake Village, Arkansas) to determine infection. Larvae were checked for 2 weeks, and any dead larvae were dissected and examined under a compound microscope to determine cause of death. The percent of larvae that died from fungus from each plot was used as a direct input to the model. Gypsy moth density was also determined at each plot on 20 May, 2002, by recording the number of gypsy moth larvae on 25, 20 cm terminals of favored food plants (mainly various species of oaks Quercus) or witch hazel (Hamamelis virginiana L.). Seven of the plots were chosen for more intensive study (circles in Fig. 1). In these plots, larval terminal counts were also taken the week before and the week after 20 May. These counts were averaged to estimate gypsy moth density. During each of 8 weeks when gypsy moth larvae were present (14 May to 26 June), an attempt was made to collect at least 100 larvae at each of
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2.3. Modeling conidial dispersal
Fig. 1. Representations of the forest plots in Mansfield Hollow used in this study. The asterisks are the locations of the 8 plots where resting spore load and gypsy moth density only were sampled. Weather instruments were placed and gypsy moth larvae collected to determine fungus infection prevalence in the 7 intensive plots represented by the circles. Weather data from M7, represented by the largest circle, was used to run the model at the 8 least-intensively studied plots. In graphs A and B, zero gypsy moth numbers and resting spore loads were assumed in the outlined areas representing the lake and fields. Delaunay triangulation as the interpolation procedure was used in graphs A and B. In graphs C and D, polygons of influence were used as the interpolation procedure (see text). Graphs A and C represent interpolation outcomes for soil bioassays, and graphs B and D the same for larval density. The site represented is 3.94 km wide and 4.39 km long.
these plots per week. This was not always possible, but usually at least 30 were collected. Each larva was immediately placed in a 30 ml clear plastic cup that was 1/4 full of gypsy moth diet, returned to the laboratory, and reared for 2 weeks. As before, any larvae that died were dissected and examined under a compound microscope to determine cause of death. To record weather data, a temperature and humidity recording data logger (Hobo Pro Rh/Temp, Onset Computer Corp, Pocasset, MA) and a data logging rain gauge (Onset) were placed in each of the seven intensive plots for the time that gypsy moth larvae were active (late April to early July). The temperature-humidity loggers took readings every 10 min and the tippingbucket rain gauges recorded an event every time 0.25 mm of rain accumulated.
Various methods were used for modeling the dispersal of conidia in the Mansfield Hollow area. For all 15 plots, data were available on resting spore load in the soil and gypsy moth density, and I assumed that weather conditions in the non-monitored plots (represented by the asterisks in Fig. 1) would be similar to those in the monitored plots. Thus, using the weather records obtained from the plot M7 (largest circle in Fig. 1) as input for the non-monitored plots, outputs were obtained from all plots, even though prevalence rates were only sampled in the 7 intensive plots. This situation permitted me to obtain estimates of conidial numbers in all plots, and then redistribute them to see how this affected model output in the intensive plots. Conidia were redistributed in 3 ways: (1) Simple averaging (2) Delaunay triangle interpolation, and (3) polygonal interpolation. For simple averaging on days when conidia were produced, the number of conidia at a plot of interest and all other plots out to a given distance from the first plot was averaged. The model used this averaged value of conidial numbers to cause infection at the chosen plot. For Delaunay triangle interpolation the area of Mansfield Hollow was divided into forested and non-forested areas, the latter being open fields and a lake (outlined areas in Figs. 1A and B). Because there were no gypsy moths in the latter areas, resting spore load and larval density were assigned values of 0.0 to points located 63 m apart. These points were then treated as equivalent to the points where the 15 plots were located, and the entire area was divided into Delaunay triangles and the values of resting spore load and gypsy moth larval density over a 128 128 grid (44 m between points) determined by linear interpolation (see Isaaks and Srivastava (1989) for an explanation of Delaunay triangular interpolation). For polygonal interpolation, data were used from the 15 plots only. Using the geographic information system IDRISI (Eastman, 1993), inverse distance weighting with an exponent of 20 was used to interpolate to a 64 64 grid, (88 m between points). Twenty was chosen as an exponent simply because it was high enough to make the resulting interpolation very close to what would occur by dividing the region into polygons each of which contains the area closest to a given point and assigning the value at each point to all parts of the encircling polygon (as in Figs. 1C and D, see Isaaks and Srivastava (1989) for more information on such polygons of influence). Any larger number would have worked as well. These two interpolation measures were used because they did not change values at the known points. Also, the usually preferred method of kriging could not readily be applied because variograms were not consistent enough to have confidence in the method. Delaunay triangulation and polynomial interpolation not only preserve the real data but also
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represent extremes in interpolation methods (linear vs. step, respectively). Thus, I felt the procedures would be useful in the present context. Having obtained grids of resting spore load and gypsy moth density using the 2 different interpolation methods, a ‘‘kinked’’ linear dispersal distribution, or ‘‘kernel’’ (Weseloh, 2003) was used to redistribute conidia. This distribution is a 2-segment linear construct starting with a y-intercept at 1 and going down to a y value of 0.01 at a given distance from the point of interest (called the inclusion radius), and then from 0.01 at that given distance to y ¼ 0 at 100 km. As explained in Weseloh (2003), I used this kernel as a kind of ‘‘leptokurtic,’’ or fat-tailed, distribution that appeared to give a good fit for dispersal of E. maimaiga conidia. Thus, at each point of the appropriate interpolation grids, the pointÕs value of resting spore load and gypsy moth density, as well as weather data obtained from plot M7, were used as model inputs to calculate conidial production for a given date. Then the model redistributed the conidia using the dispersal kernel to give a weighted average value at each point that could be used to infect healthy larvae. To compare results for these different redistribution methods, sums of squares of differences were calculated between the modeled prevalence output at the locations of each of the 7 intensive plots and the actual infection prevalence rates obtained from forest sampling at these same plots.
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Fig. 2. Comparisons between proportions of caterpillars in each instar on different dates as determined from collection of forest caterpillars (solid circles) from the 7 intensively sampled plots and model output for plot M7 (line).
3. Results 3.1. Preliminary result: instar distribution The model keeps track of instars in healthy cohorts, making it possible in a preliminary result to compare actual development rates with the model degree-day mechanism. The degree-day model developed by Weseloh (2002) with a threshold value of 12 °C appears to be adequate for simulating development of larvae (Fig. 2). In most cases, the dates of first appearance of the various instars in the forest were similar to model predictions, but instars persisted longer in the forest than in the model. This may have been a consequence of our inability to distinguish healthy from parasitized or otherwise unhealthy larvae in the forest, because in some cases unhealthy larvae develop slower than healthy ones. 3.2. Comparisons between model and forest infection prevalences When I ran the model using input data from the 7 intensive plots without allowing for dispersal of conidia, infection prevalences predicted by model output had only mediocre correspondences to actual infection
prevalence at these plots (Fig. 3). In 2002, infection prevalences in Mansfield Hollow were only high for 1 collection period (17–20 June). The model outputs generally mimic this spike in time but not amplitude, and predicted prevalence was too low for plots that had low gypsy moth numbers (M1 and M5). However, if conidia from all 15 plots mix freely using the simple averaging algorithm with an inclusion radius of 20.0 km (so that one averaged conidial concentration is applied to all plots), model and actual infection prevalences are much closer (Fig. 4). Therefore, conidial dispersal at the local level is important for accurate model simulation, a conclusion also reached by Weseloh (2002). I also made comparisons of model infection prevalence and actual infection prevalence (via sums of squares of differences) using different dispersal methods and inclusion radii (Fig. 5). Simple averaging gave better fits than Delaunay triangle interpolation at all radii, and was better than polynomial interpolation above 1 km. It is surprising that Delaunay triangle interpolation, which arguably is the best technique of the 3 for accurately simulating the known spatial structure at the site, did the worst. Both simple averaging and polygonal inter-
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Fig. 3. Comparisons between infection prevalence on different dates in the different plots as obtained from collections of larvae (solid circles, brackets indicate 95% binomial confidence intervals), and model output, assuming no local dispersal of conidia (line).
Fig. 4. Comparisons between infection prevalence on different dates in the different plots as obtained from collections of larvae (solid circles, brackets indicate 95% binomial confidence intervals), and model output, assuming that local conidial dispersal occurs via the simple average algorithm with an inclusion radius of 20.0 km.
polation had minimal sums of squares at about 1.7 km, but only simple averaging stayed near its minimum at larger inclusion radii. It appears that simple averaging of conidial numbers over about a 2 km area produces accurate simulation results. 3.3. Importance of resting spore load and gypsy moth density in auxiliary plots The conclusion from the above section implies that it is necessary to sample resting spore load and gypsy moth larval density at several places within a 2 km area of the location of interest in order to obtain meaningful results from the model. Accordingly, I wanted to determine for auxiliary plots if variations in resting spore load are more or less important than variations in gypsy moth density for affecting model results. I ran the model under the simple averaging algorithm using an inclusion radius of 2.0 km, and input soil bioassay values of 5.0 or 50.0%, but used the actual sampled values for gypsy moth input density. Then I used sample bioassay values
Fig. 5. Sum of squares of difference between forest prevalence rates and model outputs for different inclusion radii using the three conidial dispersal mechanisms discussed in the text.
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and ran the model with either 0.1 or 1.0 larvae/terminal. These values approximated the ranges of resting spore load (1.0–63.27%) and density (0.04–0.947 larvae/terminal) found in plots in 2002. The total sum of squares of differences between daily infection prevalence in the 7 intensive plots for model runs with 5.0 or 50.0% soil bioassay values was 213,343. When density was varied between 0.1 and 1.0, the sum of squares was 569,168. Thus, order of magnitude variations in larval density was more important in causing variations in simulated fungal infection prevalence than order of magnitude variations in resting spore loads.
4. Discussion I used data acquired in 2001 and previous years to develop the E. maimaiga fungus model and to find parameter values that give good fits to forest infection prevalence. I also determined that averaging conidial distribution among plots gave a better fit to data than not allowing for such dispersal (Weseloh, 2002). The same model fit data very well when used without change in 2002 if I averaged conidia numbers from other local plots. Although in 2002 plots were in the same forest used to refine the model in 2001, weather conditions and infection prevalences were different in the 2 years. Thus, the effect of local dispersal on infection rates appears to be consistent. I do not know why simple averaging gave better results than the 2 other procedures for dispersing conidia. After all, it seems reasonable to expect that the probability of conidia dispersing between plots will decline as the distance between plots increases. To a certain extent, this is true for all dispersal procedures, as even simple averaging gave the best fits for inclusion radii around 2 km. But using a monotonically declining dispersal kernel does not work nearly as well as simple mixing. One possibility is that on the scale of a forest, conidia do not arise from point sources. Much of the forested areas around each of the plots had infected gypsy moths that would have contributed dispersing conidia. Local areas having relatively low numbers of gypsy moths would receive more conidia than were produced there, and vice versa for high population areas. The relative flatness of the area would also contribute to homogeneous dispersal of conidia. The overall effect would be one of averaging conidial concentrations over the local area. The excellent fits obtained using simple averaging means that it is not necessary to interpolate around sample points, even if extensive non-forested areas are present. Because of conidial dispersal the local effect of the fungus is essentially homogeneous. This simplifies sampling and analysis considerably. However, conidial dispersal also means that to obtain accurate model results at one point, it is necessary to sample for resting
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spore load and gypsy moth density at additional local points. This study shows that averaging over 15 points is adequate, but possibly fewer would be needed. (Weseloh (2002) found that averaging over 6 local point was sufficient.) Resting spore load does not need to be determined as accurately as gypsy moth density and can even be estimated by the number of years since defoliation by the gypsy moth has occurred (Weseloh and Andreadis, 2002). Taking terminal counts when gypsy moth larvae are young can be logistically difficult, but density may be estimated with some precision by counting gypsy moth egg masses earlier in the year (Weseloh, 2003). Dispersal of conidia helps explain the rapid spread of E. maimaiga after its discovery in North America in 1989 (Weseloh, 2003). It also helps explain data showing insensitivity of infection prevalence to host density (Hajek, 1999). Pronounced dispersal of infective conidia may be an important factor enabling E. maimaiga to control its host over wide areas.
Acknowledgments I thank Lori Brown and Morgan Lowry of the Connecticut Agricultural Experiment Station for their help in carrying out this study, and John A. Tanner and other personnel at the USDA APHIS Methods Development Laboratory at Otis, Massachusetts for providing gypsy moth eggs. This research was supported, in part, by a Special Technology Development Program Grant from the USDA Forest Service. Appendix A The model is completely described by Weseloh (2002). This Appendix A presents the critical equations. The model has a time step of 1 day, so the equations are date-specific. However, I omit time-related subscripts to simplify the equations. From data by Hajek et al. (1993), the rates of development of the fungus per day (r) in caterpillars held at different temperatures (T ) from 7.5 to 26 °C is r ¼ 0:162 þ 0:032 T 0:000752 T 2 Using this equation, the model calculates daily cumulative fungal development rates so that the dates when infected cohorts die (i.e., cumulative rate ¼ 1.0) can be determined. The probability of infection by resting spores (Pr ) on a day is 2
Pr ¼ 0:0035 R½sinð0:01431 MÞ ; where R is the percent of larvae infected in a soil bioassay, and M is a forest moisture index defined as the total amount of rain, in mm, accumulated over the
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present day and the last 4 days, with no more than 10 mm of rain allowed per day. After E. maimaiga kills a larva, the number of conidia produced per m3 of forest space (i.e., including the volume from top of canopy to ground) per day (c) varies with instar: Instar 2: c ¼ 11; 639 Ni H expð0:36 dÞ Instar 3: c ¼ 24; 542 Ni H expð0:763 dÞ Instar 4: c ¼ 32; 832 Ni H expð0:491 dÞ Instar 5+: c ¼ 204; 875 Ni H expð0:859 dÞ In the above equations, Ni is the number of larvae in an infected cohort of an appropriate instar, H is the proportion of the day for which humidity is above 98%, and d is the number of days since caterpillar death from fungus, up to a maximum of 8 days after which c ¼ 0. The rate at which healthy larvae become infected with conidia depends on conidial concentration (c) as well as the size of the larva (s). The size factor used is a function of the degree-days of development (D) of each healthy cohort from hatch: s ¼ D2 =ð91613:9 þ 0:6798 D2 Þ The probability, Pc , that a 5th instar will become infected when there are c conidia in a m3 volume is c
Pc ¼ 1 ð1 0:002 sÞ ; where s is the above mentioned size factor. The total probability of infection (PT ) from both resting spores (Pr ) and conidia (Pc ) is PT ¼ Pr þ Pc Pr Pc because the two probabilities are independent. The model multiplies this total probability by the number of healthy caterpillars in a cohort to give the number of larvae in a new, infected cohort. References Andreadis, T.G., Weseloh, R.M., 1990. Discovery of Entomophaga maimaiga in North American gypsy moth, Lymantria dispar. Proc. Natl. Acad. Sci. USA 87, 2461–2465.
Eastman, J.R., 1993. IDRISI version 4.1 [DOS]. Clark University, Worcester, MA. Hajek, A.E., 1999. Pathology and epizootiology of Entomophaga maimaiga infections in forest Lepidoptera. Microb. Mol. Biol. Rev. 63, 814–835. Hajek, A.E., Humber, R.A., Elkinton, J., 1995. Mysterious origin of Entomophaga maimaiga in North America. Am. Entomol. 41, 31– 42. Hajek, A.E., Humber, R.A., Elkinton, J., May, S.B., Walsh, S.R.A., Silver, J.C., 1990. Allozyme and restriction fragment length polymorphism analyses confirm Entomophaga maimaiga responsible for 1989 epizootics in North American gypsy moth populations. Proc. Natl. Acad. Sci. USA 87, 6979– 6982. Hajek, A.E., Larkin, T.S., Carruthers, R.I., Soper, R.S., 1993. Modeling the dynamics of Entomophaga maimaiga (Zygomycetes: Entomophthorales) epizootics in gypsy moth (Lymantria: Lymantriidae) populations. Environ. Entomol. 22, 1172– 1187. Hajek, A.E., Soper, R.S., 1992. Temporal dynamics of Entomophaga maimaiga after death of gypsy moth (Lepidoptera: Lymantriidae) larval hosts. Environ. Entomol. 21, 129–135. Isaaks, E.H., Srivastava, R.M., 1989. An Introduction to Applied Geostatistics. Oxford University Press, New York, Oxford. Weseloh, R.M., 1998. Possibility for recent origin of the gypsy moth (Lepidoptera: Lymantriidae) fungal pathogen Entomophaga maimaiga (Zygomycetes: Entomophthorales) in North America. Environ. Entomol. 27, 171–177. Weseloh, R.M., 1999. Entomophaga maimaiga (Zygomycete: Entomophthorales) resting spores and biological control of the gypsy moth (Lepidoptera: Lymantriidae). Environ. Entomol. 28, 1162– 1171. Weseloh, R.M., 2002. Modeling the impact of the fungus, Entomophaga maimaiga (Zygomycetes: Entomophthorales) on gypsy moth (Lepidoptera: Lymantriidae): Incorporating infection by conidia. Environ. Entomol. 31, 1071–1084. Weseloh, R.M., 2003. Short and long range dispersal in the gypsy moth (Lepidoptera: Lymantriidae) fungal pathogen, Entomophaga maimaiga (Zygomycetes: Entomophthorales). Environ. Entomol. 32, 111–122. Weseloh, R.M., Andreadis, T.G., 2002. Detecting the titer in forest soils of spores of the gypsy moth (Lepidoptera: Lymantriidae) fungal pathogen, Entomophaga maimaiga (Zygomycetes: Entomophthorales). Can. Entomol. 134, 269–279. Weseloh, R.M., Andreadis, T.G., Onstad, D.W., 1993. Modeling the influence of rainfall and temperature on the phenology of infection of gypsy moth, Lymantria dispar, larvae by the fungus Entomophaga maimaiga. Biol. Control 3, 311–318.