Rafael Luís

Non-autonomous periodic systems with Allee effects

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.

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

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.

Bifurcation and invariant manifolds of the logistic competition model

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.

Open problems in some competition models

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

Towards a Theory of Periodic Difference Equations and Its Application to Population Dynamics

We present a survey of some of the most updated results on the dynamics of periodic and almost periodic difference equations.

LOCAL BIFURCATION IN ONE-DIMENSIONAL NONAUTONOMOUS PERIODIC DIFFERENCE EQUATIONS

In this article we extend the theory of local bifurcation in one-dimensional autonomous maps to one-dimensional nonautonomous periodic maps. We give the necessary conditions for the main types of l...

An economical model with Allee effect

We formulate a mathematical model based on Marx theory of economics. The profit rate r is considered as a function of both the exploitation rate e and the organic composition of the capital k. This model possesses a new property, commonly used in biology, called the Allee effect, in which the profit rate declines to zero if it falls below a certain threshold. It is represented by the difference equation r n+1 = f a (r n ), which is a family of unimodal maps depending on the parameter a, where a measures the relative growth of the exploitation rate when the profit rate is zero. Moreover, the model predicts a period-doubling bifurcation scenario as the parameter a increases. Finally, we allow the parameter a fluctuate periodically which leads to a periodic non-autonomous difference equations.

A discrete dynamical system for the Marx model

The Marx model for the profit rate r depending on the exploitation rate e and on the organic composition of the capital k is studied using a dynamical approach. Introducing a discrete dynamical system in the model and supposing that both k and e depend on the profit rate of the previous cycle we get a discrete dynamical system for r, , which is a family of unimodal maps depending on the parameter a, where a is the exploitation rate at profit zero. It is interesting to note that the model can model the phenomenon known as Kondratiev waves in economic systems.