**Author details**

geographical area; and earlier studies have shown they overlap significantly [20]. From the flowering behaviour indices of Keatley and Hudson in [23], *E. leucoxylon* and *E. polyanthemos* were shown to have temporally separated months of peak flowering, September and November, respectively; likewise their flowering commencement months May and October, respectively. These two species can occur in the same geographical area and their flowering period. Differentiation of these two species is based on their differing months of peak flowering as well as their separated months of most probable flowering; October and November, respectively. Likewise their flowering commencement months differ, May and October, respec-

The least degree of synchrony (via the B-MTD rules of synchrony, the MTD models and Moran method) is shown in this chapter to occur between *E. leucoxylon* and *E. microcarpa*; then followed by *E. polyanthemos* and *E. microcarpa*. Our results agree with the findings in [20], which established that a cross between *E. leucoxylon* and *E. microcarpa* is impossible. In terms of climatic signatures: the flowering of *E. microcarpa* behaves differently from *E. leucoxylon* and *E. tricarpa*. *E. microcarpa* flowers at higher temperature and its flowering has a significant and positive relationship with flowering a year ago, refer also to the results reported in [23].

*Eucalyptus tricarpa* and *E. polyanthemos* were shown in this chapter also to be asynchronous (discordant or out of phase). This is in agreement with conclusions reported in [2]. The MTDg model found a significant interaction between two climate variables, mean temperature and rainfall on the flowering of *E.*

*polyanthemos*. As flowering is viewed as either 'off' or 'on' this interaction appears to be delineating *E. polyanthemos'* flowering period. It usually commences flowering in late spring—as mean temperature is increasing and rainfall is decreasing and ceases in early summer; just prior to the warmest mean temperature and lowest rainfall. Specific temperature thresholds for commencement and for the cessation of flowering for the four species studied here, have been established, see [5, 7, 8]. For example, *E. microcarpa* was shown to flower at high temperatures, and *E. leucoxylon* and *E. tricarpa* both at lower temperatures. The flowering of *E. polyanthemos* was shown to be impacted by both rainfall and temperature, with increased flowering when conditions were either cool and dry, or hot and wet—indicative of a rainfall

Moran residual analysis and the B-MTD analysis described in this chapter showed that *E. tricarpa* and *E. microcarpa* did not exhibit a significant synchronous nor an asynchronous relationship. However, for this species pairing, the associated sum of the probabilities for transitions A and B (both off/on to one off/on) is 0.591, which is close to the threshold for synchrony of 0.65. Indeed the more sophisticated MTDg modelling approach which incorporates covariates (mean temperature and rainfall) with interactions, showed that *E. microcarpa* and *E. tricarpa* are synchronous, wherein the MTDg model allows for prior lag 1 to lag 12 month flowering

SOM-based clustering [4] and Moran AR (2) tests also found that *E.*

*polyanthemos* was asynchronous to *E. microcarpa* and *E. tricarpa*, in agreement with the extended Kalman filter (EKF)-based synchrony measures in [15, 21]. Note also it was demonstrated in [20] that *E. polyanthemos* and *E. microcarpa* have 25 years with no overlap (with a long term mean synchrony value of 0.29). Note that the more sophisticated MTDg modelling approach which incorporates covariates (mean temperature and rainfall) with interactions, showed that indeed *E. microcarpa* and *E. tricarpa* are synchronous, wherein the MTDg model allows for prior lag 1 to lag

Recently synchronisation of eucalypt flowering is shown to be a complex mechanism that incorporates all the flowering elements—flowering duration, timing of

tively [23].

*Probability, Combinatorics and Control*

by temperature interaction.

effects and climate covariates.

**68**

12 month flowering effects and climate covariates.

Irene Hudson<sup>1</sup> \*, Susan Won Sun Kim<sup>2</sup> and Marie Keatley<sup>3</sup>

1 Department of Mathematical Sciences, Royal Melbourne Institute of Technology, Melbourne, Australia

2 South Australian Health and Medical Research Institute, Adelaide, South Australia

3 School of Ecosystem and Forest Sciences, The University of Melbourne, Melbourne, Australia

\*Address all correspondence to: irene.hudson@rmit.edu.au

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
