# American Institute of Mathematical Sciences

September  2015, 20(7): 1917-1932. doi: 10.3934/dcdsb.2015.20.1917

## Chaos in a model for masting

 1 Department of Applied Mathematics, Chung Yuan Christian University, Chungli, Taiwan 2 Department of Mathematics, National Taiwan University, Taipei, Taiwan

Received  April 2014 Revised  January 2015 Published  July 2015

Isagi et al introduced a model for masting, that is, the intermittent production of flowers and fruit by trees. A tree produces flowers and fruit only when the stored energy exceeds a certain threshold value. If flowers and fruit are not produced, the stored energy increases by a certain fixed amount; if flowers and fruit are produced, the energy is depleted by an amount proportional to the excess stored energy. Thus a one-dimensional model is derived for the amount of stored energy. When the ratio of the amount of energy used for flowering and fruit production in a reproductive year to the excess amount of stored energy before that year is small, the stored energy approaches a constant value as time passes. However when this ratio is large, the amount of stored energy varies unpredictably and as the ratio increases the range of possible values for the stored energy increases also. In this article we describe this chaotic behavior precisely with complete proofs.
Citation: Kaijen Cheng, Kenneth Palmer. Chaos in a model for masting. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1917-1932. doi: 10.3934/dcdsb.2015.20.1917
##### References:
 [1] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos,, Amer. Math. Monthly, 99 (1992), 332.  doi: 10.2307/2324899.  Google Scholar [2] S. Bassein, The dynamics of a family of one-dimensional maps,, Amer. Math. Monthly, 105 (1998), 118.  doi: 10.2307/2589643.  Google Scholar [3] S. M. Chang and H. H. Chen, Applying snapback repellers in resource budget models,, Chaos, 21 (2011).  doi: 10.1063/1.3660662.  Google Scholar [4] R. L. Devaney, An Introduction to Chaotic Dynamical Systems,, $2^{nd}$ edition, (1989).   Google Scholar [5] Y. Isagi, K. Sugimura, A. Sumida and H. Ito, How does masting happen and synchronize,, J. Theor. Biol., 187 (1997), 231.  doi: 10.1006/jtbi.1997.0442.  Google Scholar [6] A. Satake and Y. Iwasa, Pollen-coupling of forest trees: Forming synchronized and periodic reproduction out of chaos,, J. Theor. Biol., 203 (2000), 63.  doi: 10.1006/jtbi.1999.1066.  Google Scholar

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##### References:
 [1] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos,, Amer. Math. Monthly, 99 (1992), 332.  doi: 10.2307/2324899.  Google Scholar [2] S. Bassein, The dynamics of a family of one-dimensional maps,, Amer. Math. Monthly, 105 (1998), 118.  doi: 10.2307/2589643.  Google Scholar [3] S. M. Chang and H. H. Chen, Applying snapback repellers in resource budget models,, Chaos, 21 (2011).  doi: 10.1063/1.3660662.  Google Scholar [4] R. L. Devaney, An Introduction to Chaotic Dynamical Systems,, $2^{nd}$ edition, (1989).   Google Scholar [5] Y. Isagi, K. Sugimura, A. Sumida and H. Ito, How does masting happen and synchronize,, J. Theor. Biol., 187 (1997), 231.  doi: 10.1006/jtbi.1997.0442.  Google Scholar [6] A. Satake and Y. Iwasa, Pollen-coupling of forest trees: Forming synchronized and periodic reproduction out of chaos,, J. Theor. Biol., 203 (2000), 63.  doi: 10.1006/jtbi.1999.1066.  Google Scholar
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