Analysis of oscillatory plant transpiration. II

Abstract

Oscillatory plant transpiration has been observed in numerous plants, and it is now believed to be a common rather than an exceptional phenomenon. It is related directly to cyclic variations of the stomatal aperture and indirectly to the water content of guard, subsidiary and mesophyll cells. At least for small plants the upward transport of water from the roots to the leaves and the surrounding atmosphere can be explained entirely in terms of the biologically passive forces of capillarity and transpiration-cohesion. Biologically active forces involving energy consuming pumping also exist, but in small plants they play a minor rôle in the dynamics of the oscillatory water transport. On the macroscopic level of the whole plant the water transport can thus be formulated in terms of an analogue hydraulic-elastic model, by modifying somewhat that formulated in 1972 by Cowan. Differential equations corresponding to this modified model are established and conditions for the existence of periodic solutions are derived. It turns out that these conditions are not very severe mathematically. A considerable freedom is left in particular in the choice of the non-linear dependence between the stomatal aperture and the water content of guard, subsidiary and mesophyll cells. In a given plant there is of course no such freedom, because this dependence is entirely fixed by the mechanical structure of the stomatal complex. Several types of operation of the stomatal complex are possible, because mathematically there is more than one possibility to match the oscillatory waveforms to the experimentally observed ones.

Self‐sustained oscillations in plant water regulation: Induction of bifurcations and anomalous rhythmicity

“…Most of the mathematical models of transpiration are three‐dimensional, nonlinear models (Cowan, 1972; Delwiche & Cooke, 1977; Gumowski, 1981, 1983; Garcia, 1985). The three dynamic elements correspond to the guard, subsidiary and mesophyll cell volumes.…”

“…In principle, these models should be capable of simulating the complicated dynamics demonstrated experimentally. Although period‐2 behaviour can be simulated in variants of some of the models (Gumowski, 1983; Johnsson & Prytz, 2002), period‐3 and higher period oscillations have, to date, not been simulated. Chaotic behaviour has not been observed in model simulations but it has been proposed that, in theory, such behaviour is possible (Gumowski, 1983).…”

“…The period doubling encountered in Avena leaves has been the transition from period‐1 to period‐2 oscillations (Johnsson & Prytz, 2002). Period doubling has also been found in simulations of a mathematical model of the water regulatory system indicating that, in theory, period doubling can arise in such systems (Gumowski, 1983).…”

Rhythms in Plants

“…Gumowski 1981Gumowski , 1983. Coupled models have a correspondingly higher number of dynamic variables.…”

“…A nonlinear model with three variables can show period-doubling and chaotic behavior (Strogatz 1994). Complicated, non-sinusoidal waveforms have been encountered experimentally (Johnsson 1976, Johnsson andPrytz 2002) and some efforts to model their shape have been published (Gumowski 1981(Gumowski , 1983. If time delays are introduced in the reactions, the overall model becomes even more apt to show oscillations.…”