On a discrete selection–mutation model

Abstract

A discrete-time population model in which individuals are distributed over a discrete phenotypic trait-space is studied. It is shown that, for an irreducible mutation matrix $\Gamma$, if mutation is small, then an interior equilibrium exists, that is globally asymptotically stable in $R_{+}^n\setminus \left\{0\right\}$, while for arbitrary large mutation, each trait persists uniformly. For the model reduced to only two traits, conditions for the global stability of the interior equilibrium are provided. When structure is introduced in the model, namely when mutation matrix $\Gamma$ has block-diagonal form, with each diagonal block being irreducible, competitive exclusion among traits is analysed and sufficient conditions are given for one trait to drive all the other traits to extinction.

Keywords::

• selection–mutation
• robust uniform persistence
• coexistence
• competitive exclusion

Acknowledgements

The work of A.S. Ackleh is partially supported by the National Science Foundation grant # DMS-1312963. The work of R.J.S. is partially supported by the University of Southern California Dornsife School of Letters Arts and Sciences Faculty Development Grant. The work of P.L. Salceanu is partially supported by the Louisiana Board of Regents grant # LEQSF(2012-15)-RD-A-29.