Let (y_{itk}) denote the WCR count observed for trap i in week t in year k, and assume it to follow a Poisson distribution with parameter (mu _{itk})
$$begin{aligned} y_{itk} | mu _{itk}, sim Poisson(mu _{itk}) end{aligned}$$
(1)
The intensity parameter (mu _{itk}) represents the rate of emergence for a given time period. Instead of allowing it to depend purely on time t, a phenological variable of growing degree days (GDD) is used, as warmer temperatures are required for WCR development25,26,27,28. GDDs reflect the heat accumulation and are defined as an integral of warmth above a base temperature after a given start date:
$$begin{aligned} GDD = int (T(t)-T_{base})dt. end{aligned}$$
(2)
The above integral can be approximated by
$$begin{aligned} GDD = max left( frac{T_{max} – T_{min}}{2} – T_{base}, 0 right) . end{aligned}$$
(3)
Here (T_{min}) is the minimum daily temperature, (T_{max}) is the maximum daily temperature, and (T_{base}) is a set base temperature. In this study, the base temperature was set to (10,^{circ })C, and the starting date was the beginning of April, which marks the start of the growing season in Austria.
The rate of cumulative emergence of the WCR beetle can be described by a Gompertz function. The Gompertz function is a sigmoidal function which describes growth as being slowest at the beginning and the end of a given period and is defined as
$$begin{aligned} f(z_t) = alpha exp (-beta exp (-gamma z_t)). end{aligned}$$
(4)
where (alpha) is the upper asymptote, (beta) is a relative starting value, (gamma) is a growth rate coefficient which affects the slope, and (z_t) are the cumulative growing degree days. In this study, one can consider the asymptote as proxy to the saturation level of WCR population growth. Lower values of (beta) suggest an earlier first emergence in the season, while lower values of (gamma) indicate a longer emergence period. To investigate whether there is an association between climate variables and the emergence dynamics, the Gompertz curve parameters were assumed to linearly depend on climate covariates. In this regression modelling framework, a spatially correlated residual structure can be added in either (alpha), (beta), and/or (gamma) if there is evidence to do so.
To reflect the nature of the emergence dynamics and to preserve the shape of the increasing Gompertz curve, the parameters of the model were restricted to positive values such that (alpha >0), (beta >0), and (gamma >0). The time at inflection or period of highest growth can be obtained by solving Eq. (4) for the value of t at which the concavity of the function changes. The time at inflection is described as:
$$begin{aligned} T_z^* = frac{log (beta )}{gamma } end{aligned}$$
(5)
The Gompertz function describes cumulative emergence. Thus to describe the marginal emergence rate, the derivative of the Gompertz function can be used instead. Consequently, as the WCR trapping data consisted of weekly counts, the rate of emergence (mu _{itk}) is better described by the log of the derivative of the Gompertz function
$$begin{aligned} log (mu _{itk}) = log (alpha _{ik}) + log (gamma _{ik}) + log (beta _{ik}) + gamma _i z_{itk} – beta _{ik} exp (-gamma z_{itk}). end{aligned}$$
(6)
The parameters (alpha _{ik}), (beta _{ik}) and (gamma _{ik}) are site and year specific such that:
$$begin{aligned}&alpha _{ik} sim N(mu _{alpha _{ik}}, tau _{alpha }) end{aligned}$$
(7)
$$begin{aligned}&gamma _{ik} sim N(mu _{gamma _{ik}}, tau _{gamma }) end{aligned}$$
(8)
$$begin{aligned}&beta _{ik} sim N(mu _{beta _{ik}}, tau _{beta }). end{aligned}$$
(9)
Here, (tau _{alpha }), (tau _{beta }), and (tau _{gamma }) are the precision (inverse variance) parameters of the prior distributions for (alpha), (beta) and (gamma) respectively. Moreover, the means of the distributions (mu _{alpha _{ik}}), (mu _{beta _{ik}}), and (mu _{gamma _{ik}}) can be expressed as functions of known covariates:
$$begin{aligned} mu _{alpha _{ik}}= & {} a_{0} + {mathbf {w}}^T X_{alpha _{ik}}, end{aligned}$$
(10)
$$begin{aligned} mu _{beta _{ik}}= & {} b_{0}, end{aligned}$$
(11)
$$begin{aligned} mu _{gamma _{ik}}= & {} g_{0} + {mathbf {u}}^T X_{gamma _{ik}}. end{aligned}$$
(12)
Here (a_{0}) is the intercept, ({mathbf {w}}) is a vector of the regression coefficients, and (X_{alpha _{ik}}) are the location and year specific covariates. The predictors used in the regression of (mu _{alpha _{ik}}) are the average winter temperature, the precipitation sum during winter, the year, the percentage of the agricultural area per Austrian municipality used for cultivating maize crops (maize), and the corresponding centred coordinates of the trap locations; x, y, and their functions (x^2), (y^2), and xy. The parameter (g_{0}) is the intercept for the regression of (mu _{gamma _{ik}}), and u is the corresponding regression coefficient. The predictor used for (mu _{gamma _ik}) is the average yearly spring temperature.
The intercepts and regression coefficients ((mathbf {w}) and (mathbf {u})) were given non-informative normal priors N(0, 0.01). The precision parameters (tau _{alpha }), (tau _{beta }) and (tau _{gamma }) were assigned prior distributions Gamma(0.01, 0.01).
The model was fitted using WinBUGS through the R2WinBUGS package in R29,30,31. The model was run for 20000 iterations, with a burn-in of 10000 iterations, and a thinning rate of five. Convergence was determined by visual assessments of trace plots and marginal posterior densities.
Source: Ecology - nature.com
