Parameter estimation for distributed delay based population models from laboratory data : egg hatching of Oulema duftschmidi (Redtenbacher) (Coleoptera, Chrysomelidae) as an example

The cohort development of poikilotherms under favorable temperature conditions may be described by the time-invariant distributed delay models that require three parameters, i.e. the thermal threshold T0, the thermal constant W and the variability parameter H representing distributed maturation times, also known as time distributed or stochastic development. Here, a parsimonious method is developed that uses stage-frequency matrices obtained under controlled conditions. The analysis of these matrices permits the estimation of the three parameters, while the incorporation of both developmental threshold and thermal constant into the time-invariant distributed delay model permits the representation of stochastic cohort development under constant temperatures. The evaluated parameters can also be used in time-varying distributed delay models that are particularly useful for a wide range of fluctuating temperature conditions. We consider stage-frequency matrices obtained from cohorts of eggs and larvae of Oulema duftschmidi Redthenbacher at different constant temperatures. A visual inspection of the matrices clearly shows the temperature dependent as well as the time distributed passage from the egg to the larval stage. Under the current range of temperatures, a linear model for developmental rate satisfactorily describes egg development, and estimates the thermal threshold (T0 = 11.2°C), the thermal constant (W = 81.3 day-degrees) and the delay order parameter (H = 70). The resulting model can be used to represent the development of cohorts of O. duftschmidi eggs in a favorable temperature range.