Period-3 dominant phase synchronisation of Zelkova serrata: border-collision bifurcation observed in a plant population

The population synchrony of tree seed production has attracted widespread attention in agriculture, forestry and ecosystem management. Oaks usually show synchronisation of irregular or intermittent sequences of acorn production, which is termed ‘masting’. Tree crops such as citrus and pistachio show a clear two-year cycle (period-2) termed ‘alternate bearing’. We identified period-3 dominant phase synchronisation in a population of Zelkova serrata. As ‘period-3’ is known to provide evidence to imply chaos in nonlinear science, the observed period-3 phase synchronisation of Zelkova serrata is an attractive real-world phenomenon that warrants investigation in terms of nonlinear dynamics. Using the Hilbert transform, we proposed a procedure to determine the fractions of periods underlying the survey data and distinguished the on-year (high yield year) and the off-year (low yield year) of the masting. We quantified the effects of pollen coupling, common environmental noise and individual variability on the phase synchronisation and demonstrated how the period-3 synchronisation emerges through a border-collision bifurcation process. In this paper, we propose a model that can describe diverse behaviours of seed production observed in many different tree species by changing its parameters.