Modelling the flowering of four eucalypts species via MTDg with interactions

Phenological indicators (e.g. date of first and last flowering, first arrival of migrating birds) are used as proxies of global climate change (Parry et al. 2008, Hudson et al. 2005; Root et al. 2003). Long-term (1940-1971) synchrony of four Eucalyptus species was recently quantified mathematically at the populationlevel (Keatley et al. 2004, Kim et al. 2008). Recently Hudson et al. (2009) has identified upper and lower temperature thresholds for E. leucoxylon flowering and showed that E. leucoxylon flowering is influenced predominantly by minimum temperature whose effect is highly non-linear. Keatley et al. (2002) reported that changes in temperature are likely to translate to changes in flowering commencement time. The magnitude of these shifts (Keatley et al. 2002) is greater than the average reported in meta analysis studies (Root et al. 2003, Parry et al 2008) but in agreement with the results for some individual species in recent studies (Abu- Asab et al 2001, Pẽnuelas et al. 2002, Fitter and Fitter 2002). The aim of this paper is to study the multivariate relationship between the probability of flowering with 2-states of rain and temperature via a mixture transition distribution (MTD) with a different transition matrix from each lag to present (MTDg) analysis (Berchtold 2004). The idea of the mixture transition distribution model is to consider independently the effect of each lag to the present instead of considering the effect of the combination of lags as in pure Markov chain processes. The assumption behind the MTD model, namely the assumed equality of the transition matrices among different lags, is a strong assumption. Earlier studies by Kim et al. (2005, 2008) extended the MARCH software (Berchtold 2004) for MTD modelling of the flowering of four eucalyptus species (as time series). For this current study, an extended model for MTDg (Berchtold 2004) analysis which accommodates interactions was developed. This work extends both MARCH and the work of Kim et al. (2005, 2008) to allow for differing transition matrices amongst the lags, i.e. the MTDg with interaction model. Our model is different to MARCH in terms of incorporating interactions between the covariates and also in its minimization process, namely AD Model BuilderTM (Fournier 2000), which uses autodifferentiation as a minimization tool. This is shown to be computationally less intensive than MARCH. We thus developed the MTDg model with interactions model to account for changes in the transition matrices amongst the differing lags. Flowering data were sourced from the Box-Ironbark Forest near Maryborough, Victoria, in particular the flowering records of E. leucoxylon, E. microcarpa, E. polyanthemos and E. tricarpa (1940 and 1971). Flowering intensity was calculated by using a rank score (from 0 to 5) based on the quantity and distribution of flowering (Keatley et al. 2004). We used minimum monthly temperature (MinT), maximum monthly temperature (MaxT), mean monthly diurnal temperature (MeanT) and monthly rainfall (Rain) as covariates and their interaction effects in the MTDg model. The MTDg model with interactions showed that the flowering of E. leucoxylon and E. tricarpa behaves similarly with temperature (both flower at low temperature) and both have a positive relationship with flowering intensity 11 months ago. The flowering of E. microcarpa behaves differently in that E. microcarpa flowers at high temperature. The MTDg model found a highly significant interaction between two climate variables, mean temperature and rainfall, for E. polyanthemos, which commences flowering in January/February but peaks soon after commencement in March. Rain has a direct positive impact on E. tricarpa only. The four species studied here are influenced by temperature and rainfall and as a consequence their flowering phenology will change in response to climate change. This in turn will have an impact on species interactions, community structure, and possibly apicultural industry as these species are some of the main producers of honey in Australia.