Prof. Dr. Rafael Luís

Faculdade de Ciências Exatas e da Engenharia, Universidade da Madeira, Funchal, Madeira, Portugal

Prof. Dr. Rafael Luís

Faculdade de Ciências Exatas e da Engenharia, Universidade da Madeira, Funchal, Madeira, Portugal

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Academic Career Length

18Years (2009-Present)

(Calculated once Scilit publication data is synced: current year − year of first publication + 1)

Representative Publications

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17 June 2011 in Journal of Biological Dynamics

Journal Article

Stability of a Ricker-type competition model and the competitive exclusion principle

Rafael Luís

Saber Elaydi

Henrique M. sobre a empregabilidade dos licenciados pela Universidade Agostinho Neto. Oliveira

Our main objective is to study a Ricker-type competition model of two species. We give a complete analysis of stability and bifurcation and determine the centre manifolds, as well as stable and unstable manifolds. It is shown that the autonomous Ricker competition model exhibits subcritical bifurcation, bubbles, period-doubling bifurcation, but no Neimark–Sacker bifurcations. We exhibit the region in the parameter space where the competition exclusion principle applies.

16 September 2010 in Journal of Difference Equations and Applications

Journal Article

Non-autonomous periodic systems with Allee effects

Rafael Luís

Saber Elaydi

Henrique M. sobre a empregabilidade dos licenciados pela Universidade Agostinho Neto. Oliveira

A new class of maps called unimodal Allee maps are introduced. Such maps arise in the study of population dynamics in which the population goes extinct if its size falls below a threshold value. A unimodal Allee map is thus a unimodal map with three fixed points, a zero fixed point, a small positive fixed point, called threshold point, and a bigger positive fixed point, called the carrying capacity. In this paper, the properties and stability of the three fixed points are studied in the setting of non-autonomous periodic dynamical systems or difference equations. Finally, we investigate the bifurcation of periodic systems/difference equations when the system consists of two unimodal Allee maps.

1 January 2014 in Discrete and Continuous Dynamical Systems-Series B

Journal Article

Local stability implies global stability for the planar Ricker competition model

E. Cabral Balreira

Saber Elaydi

Rafael Luís

Under certain analytic and geometric assumptions we show that local stability of the coexistence (positive) fixed point of the planar Ricker competition model implies global stability with respect to the interior of the positive quadrant. This result is a confluence of ideas from Dynamical Systems, Geometry, and Topology that provides a framework to the study of global stability for other planar competition models.

9 October 2017 in Journal of Difference Equations and Applications

Journal Article

Global stability of higher dimensional monotone maps

E. Cabral Balreira

Saber Elaydi

Rafael Luís

We develop a notion of monotonicity for maps defined on Euclidean spaces

R_{+}^{k}

, of arbitrary dimension k. This is a geometric approach that extends the classical notion of planar monotone maps or planar competitive difference equations. For planar maps, we show that our notion and the classical notion of monotonicity are equivalent. In higher dimensions, we establish certain verifiable conditions under which Kolmogorov monotone maps on

R_{+}^{k}

have a globally asymptotically stable fixed point. We apply our results to two competition population models, the Leslie–Gower and the Ricker models of two- and three-species. It is shown that these two models have a unique interior fixed point that is globally asymptotically stable.

13 June 2011 in Journal of Difference Equations and Applications

Journal Article

Bifurcation and invariant manifolds of the logistic competition model

Malgorzata Guzowska

Rafael Luís

Saber Elaydi

In this paper, we study a new logistic competition model. We will investigate stability and bifurcation of the model. In particular, we compute the invariant manifolds, including the important centre manifolds, and study their bifurcation. Saddle-node and period-doubling bifurcation route to chaos are exhibited via numerical simulations.

22 March 2011 in Journal of Difference Equations and Applications

Journal Article

Open problems in some competition models

Saber Elaydi

Rafael Luís

We present open problems and conjectures for some two-dimensional competition models, namely the logistic competition model and a Ricker-type competition model.

About Me

Rafael Luís is a professor of mathematics at the University of Madeira, Portugal, and a researcher at the Center for Mathematical Analysis, Geometry and Dynamical Systems, University of Lisbon, Portugal. He received his Ph.D. in mathematical analysis from the University of Madeira and has extensive expertise in discrete dynamical systems. His research primarily focuses on stability and bifurcation phenomena in discrete-time models arising in biology, population dynamics, and economics. Over the years, he has also contributed to the theory of periodic nonautonomous discrete dynamical systems. He has authored numerous articles in international peer-reviewed journals and regularly serves as a reviewer for leading journals in the field. He has also actively organized and participated in international scientific conferences. He is a member of the International Society of Difference Equations (ISDE). His current research focuses on stability theory, bifurcation analysis, and the qualitative behavior of nonlinear discrete dynamical systems, with applications to ecology and economics.

Prof. Dr. Rafael Luís

Prof. Dr. Rafael Luís

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