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1. Agriculture benefits from ecosystem services provided by mobile agents, such as biological pest control by natural enemies and pollination by bees. However, methods that can generate spatially explicit predictions and maps of these ecosystem services based on empirical data are still scarce. 2. Here we propose a generic statistical model to derive kernel functions to characterize the spatial distribution of ecosystem services provided by mobile agents. The model is similar in spirit to a generalized linear model, and uses data of landscape composition and ecosystem services assessed at target sites to estimate parameters of the kernel. The approach is tested in a simulation study and illustrated by an empirical case study on parasitism rates of the diamondback moth Plutella xylostella. 3. The simulation study shows that the scale parameter of the exponential power kernel can be estimated with limited bias, whereas estimation of the shape parameter is difficult. For the case study the model provides biologically relevant estimates for the kernel associated with parasitism of Plutella xylostella. These estimates can be used to generate ecosystem service maps for existing or planned landscapes. The case study reveals that predictions can be sensitive to the parameter values for the width and shape of the kernel, and to the link function used in the statistical model. 4. In the last two decades numerous empirical studies assessed ecosystem services at target sites and related these to the surrounding landscape. Our method can take advantage of these data by estimating underlying kernels that can be used to map the spatial distribution of ecosystem services. However, empirical data that can discriminate between alternative kernel shapes remain critical.

Understanding the relationship between an insect's developmental rate and temperature is crucial to forecast insect phenology under climate change. In the winter moth Operophtera brumata timing of egg-hatching has severe fitness consequences on growth and reproduction as egg-hatching has to match bud burst of the host tree. In the winter moth, as in many insect species, egg development is strongly affected by ambient temperatures. Here we use laboratory experiments to show for the first time that the effect of temperature on developmental rate depends on the stage of development of the eggs. Building on this experimental finding, we present a novel physiological model to describe winter moth egg development in response to temperature. Our model, a modification of the existing Sharpe−Schoolfield biophysical model, incorporates the effect of developmental stage on developmental rate. Next we validate this model using a 13-year data-set from winter moth eggs kept under ambient conditions and compared this validation with a degree day model and with the Sharpe−Schoolfield model, which lacks the interaction between temperature and developmental stage on developmental rate. We show that accounting for the interaction between temperature and developmental stage improved the predictive power of the model and contributed to our understanding of annual variation in winter moth egg phenology. As climate change leads to unequal changes in temperatures throughout the year, a description of insect development in response to realistic patterns of temperature rather than an invariable degree-day approach will help us to better predict future responses of insect phenology, and thereby insect fitness, to climate change.