# American Institute of Mathematical Sciences

March  2016, 21(2): 655-676. doi: 10.3934/dcdsb.2016.21.655

## Evolutionary dynamics of a multi-trait semelparous model

 1 Interdisciplinary Program in Applied Mathematics, University of Arizona, 617 N Santa Rita, Tucson, Arizona 85721, United States 2 Interdisciplinary Program in Applied Mathematics and Department of Mathematics, University of Arizona, 617 N Santa Rita, Tucson, Arizona 85721, United States

Received  December 2014 Revised  April 2015 Published  November 2015

We consider a multi-trait evolutionary (game theoretic) version of a two class (juvenile-adult) semelparous Leslie model. We prove the existence of both a continuum of positive equilibria and a continuum of synchronous 2-cycles as the result of a bifurcation that occurs from the extinction equilibrium when the net reproductive number $R_{0}$ increases through $1$. We give criteria for the direction of bifurcation and for the stability or instability of each bifurcating branch. Semelparous Leslie models have imprimitive projection matrices. As a result (unlike matrix models with primitive projection matrices) the direction of bifurcation does not solely determine the stability of a bifurcating continuum. Only forward bifurcating branches can be stable and which of the two is stable depends on the intensity of between-class competitive interactions. These results generalize earlier results for single trait models. We give an example that illustrates how the dynamic alternative can change when the number of evolving traits changes from one to two.
Citation: Amy Veprauskas, J. M. Cushing. Evolutionary dynamics of a multi-trait semelparous model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 655-676. doi: 10.3934/dcdsb.2016.21.655
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