Center for Mathematical Analysis, Geometry and Dynamical Systems

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Organizational Unit

Publications

The effects of evolution on the stability of competing species

Publication . Elaydi, S.; Kang, Y.; Luís, Rafael

Based on evolutionary game theory and Darwinian evolution, we propose and study discrete-time competition models of two species where at least one species has an evolving trait that affects their intra-specific, but not their inter-specific competition coefficients. By using perturbation theory, and the theory of the limiting equations of non-autonomous discrete dynamical systems, we obtain global stability results. Our theoretical results indicate that evolution may promote and/or suppress the stability of the coexistence equilibrium depending on the environment. This relies crucially on the speed of evolution and on how the intra-specific competition coefficient depends on the evolving trait. In general, equilibrium destabiliza tion occurs when α > 2, when the speed of evolution is sufficiently slow. In this case, we conclude that evolution selects against com plex dynamics. However, when evolution proceeds at a faster pace, destabilization can occur when α < 2. In this case, if the competition coefficient is highly sensitive to changes in the trait v, destabilization and complex dynamics occur. Moreover, destabilization may lead to either a period-doubling bifurcation, as in the non-evolutionary Ricker equation, or to a Neimark-Sacker bifurcation.

2022Journal article

Open access

Geometry and Global Stability of 2D Periodic Monotone Maps

Publication . Balreira, E. Cabral; Luís, Rafael

We establish conditions to ensure global stability of a competitive periodic system from hypotheses on individual maps. We study planar competitive maps of Kolgomorov type. We show how conditions for global stability for individual maps will remain invariant under composition and hence establish a globally stable cycle. Our main theoretical contribution is to show that stability for monotone non-autonomous periodic maps can be reduced to a problem of global injectivity. We provide analytic conditions that can be checked and illustrate our results with important competition models such as the planar Leslie-Gower and Ricker maps.

2021Journal article

Open access

Linear stability Conditions for a first order n-dimensional mapping

Publication . Luís, Rafael

In this paper we present an alternative way to compute the coefficients of a character istic polynomial of a matrix via the trace, determinant and the sum of the minors that may be useful in determining the local stability conditions for mappings.

2021Journal article

Open access