Big bang bifurcations in von Bertalanffy’s dynamics with strong and weak Allee effects

The main purpose of this work was to study population dynamic discrete models in which the growth of the population is described by generalized von Bertalanffy's functions, with an adjustment or correction factor of polynomial type. The consideration of this correction factor is made with the aim to introduce the Allee effect. To the class of generalized von Bertalanffy's functions is identified and characterized subclasses of strong and weak Allee's functions and functions with no Allee effect. This classification is founded on the concepts of strong and weak Allee's effects to population growth rates associated. A complete description of the dynamic behavior is given, where we provide necessary conditions for the occurrence of unconditional and essential extinction types. The bifurcation structures of the parameter plane are analyzed regarding the evolution of the Allee limit with the aim to understand how the transition from strong Allee effect to no Allee effect, passing through the weak Allee effect, is realized. To generalized von Bertalanffy's functions with strong and weak Allee effects is identified an Allee's effect region, to which is associated the concepts of chaotic semistability curve and Allee's bifurcation point. We verified that under some sufficient conditions, generalized von Bertalanffy's functions have a particular bifurcation structure: the big bang bifurcations of the so-called box-within-a-box type. To this family of maps, the Allee bifurcation points and the big bang bifurcation points are characterized by the symmetric of Allee's limit and by a null intrinsic growth rate. The present paper is also a significant contribution in the framework of the big bang bifurcation analysis for continuous 1D maps and unveil their relationship with the explosion birth and the extinction phenomena.

Self-scheduling and bidding strategies of thermal units with stochastic emission constraints

This paper is on the self-scheduling problem for a thermal power producer taking part in a pool-based electricity market as a price-taker, having bilateral contracts and emission-constrained. An approach based on stochastic mixed-integer linear programming approach is proposed for solving the self-scheduling problem. Uncertainty regarding electricity price is considered through a set of scenarios computed by simulation and scenario-reduction. Thermal units are modelled by variable costs, start-up costs and technical operating constraints, such as: forbidden operating zones, ramp up/down limits and minimum up/down time limits. A requirement on emission allowances to mitigate carbon footprint is modelled by a stochastic constraint. Supply functions for different emission allowance levels are accessed in order to establish the optimal bidding strategy. A case study is presented to illustrate the usefulness and the proficiency of the proposed approach in supporting biding strategies. (C) 2014 Elsevier Ltd. All rights reserved.