* Author to whom correspondence should be addressed. E-mail: satake@bio-math10.biology.kyushu-u.ac.jp J. theor. Biol. (2000) 203, 63}84 doi:10.1006/jtbi.1999.1066, available online at http://www.idealibrary.com on Pollen Coupling of Forest Trees: Forming Synchronized and Periodic Reproduction out of Chaos AKIKO SATAKE* AND YOH IWASA Department of Biology, Faculty of Science, Kyushu ; niversity, Fukuoka 812-8581, Japan (Received on 23 February 1999, Accepted in revised form on 12 December 1999) Many of the tree species in mature forests show masting; their reproductive activity has a large variance between years and is often synchronized between di!erent individuals. In this paper, we analyse a globally coupled map model in which trees accumulate photosynthate every year, produce #owers when the energy reserve level exceeds a threshold, and set seeds and fruits at a rate limited by pollen availability. Without pollen limitation, the trees in the forest show independent chaotic #uctuation. Coupling of trees via pollen exchange results in reproduction being synchronized partially or completely over the forest. The whole forest shows diverse dynamical behaviors determined by the values of two essential parameters; the depletion coe$cient k and the coupling strength . We "nd perfectly synchronized periodic reproduc- tion, synchronized reproduction with a chaotic time series, clustering phenomena, and chaotic reproduction of trees without synchronization over individuals. There are many parameter windows in which synchronized reproduction of trees shows a stable periodic #uctuation. For perfectly synchronized forests, we can calculate all the Lyapunov exponents analytically. They show that synchronized reproduction of all the trees in the forest can only occur when trees #ower at low (but positive) levels in a signi"cant fraction of years, resulting in small fruit sets due to outcrossed pollen limitation. This is consistent with the observation that the distinction between mast years and non-mast years is often not clear cut.  2000 Academic Press Introduction Many forest trees reproduce intermittently, rather than at a constant rate. A number of #owers and fruits are produced during one year (called a mast year) but very little reproductive activity occurs during the several subsequent years until the next mast year. This &&synchronous highly variable seed production among years by a population of plants'' (Kelly, 1994) is called masting. The distinction between mast years and non-mast years is often not clear cut, but Herrera et al. (1998) rejected unimordality of the annual reproductive output of trees in favor of bimordal- ity. Perfect periodicity in reproduction is rarely observed (Kelly, 1994), and the interval between masting is rather irregular (Silvertown, 1980). Several explanations have been proposed for the masting phenomena. The earliest was the resource-matching hypothesis *seed output might vary because the plant's available re- sources vary (Bu K sgen & Mu K nch, 1929). A positive correlation between weather conditions and seed production level has been observed (Sharp & Chisman, 1961; Sharp & Sprague, 1967; Sork et al., 1993). However, the variance between years 0022}5193/00/060063#22 $35.00/0  2000 Academic Press