Big Bang Bifurcation Analysis and Allee Effect in Generic Growth Functions

2016 ◽  
Vol 26 (06) ◽  
pp. 1650108 ◽  
Author(s):  
J. Leonel Rocha ◽  
Abdel-Kaddous Taha ◽  
D. Fournier-Prunaret
Keyword(s):  
Bifurcation Analysis ◽  
Allee Effect ◽  
Dynamical Behavior ◽  
Big Bang ◽  
Biological Research ◽  
Growth Equation ◽  
Growth Functions ◽  
Bifurcation Structures ◽  
Parameter Values ◽  
The Big Bang

The main purpose of this work is to study the dynamics and bifurcation properties of generic growth functions, which are defined by the population size functions of the generic growth equation. This family of unimodal maps naturally incorporates a principal focus of ecological and biological research: the Allee effect. The analysis of this kind of extinction phenomenon allows to identify a class of Allee’s functions and characterize the corresponding Allee’s effect region and Allee’s bifurcation curve. The bifurcation analysis is founded on the performance of fold and flip bifurcations. The dynamical behavior is rich with abundant complex bifurcation structures, the big bang bifurcations of the so-called “box-within-a-box” fractal type being the most outstanding. Moreover, these bifurcation cascades converge to different big bang bifurcation curves with distinct kinds of boxes, where for the corresponding parameter values several attractors are associated. To the best of our knowledge, these results represent an original contribution to clarify the big bang bifurcation analysis of continuous 1D maps.

Dynamical Analysis and Big Bang Bifurcations of 1D and 2D Gompertz's Growth Functions

2016 ◽  
Vol 26 (11) ◽  
pp. 1630030 ◽  
Author(s):  
J. Leonel Rocha ◽  
Abdel-Kaddous Taha ◽  
D. Fournier-Prunaret
Keyword(s):  
Population Size ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Big Bang ◽  
Dynamical Analysis ◽  
Logistic Growth ◽  
Growth Equation ◽  
Growth Functions ◽  
Continuous Embedding ◽  
The Big Bang

In this paper, we study the dynamics and bifurcation properties of a three-parameter family of 1D Gompertz's growth functions, which are defined by the population size functions of the Gompertz logistic growth equation. The dynamical behavior is complex leading to a diversified bifurcation structure, leading to the big bang bifurcations of the so-called “box-within-a-box” fractal type. We provide and discuss sufficient conditions for the existence of these bifurcation cascades for 1D Gompertz's growth functions. Moreover, this work concerns the description of some bifurcation properties of a Hénon's map type embedding: a “continuous” embedding of 1D Gompertz's growth functions into a 2D diffeomorphism. More particularly, properties that characterize the big bang bifurcations are considered in relation with this coupling of two population size functions, varying the embedding parameter. The existence of communication areas of crossroad area type or swallowtails are identified for this 2D diffeomorphism.

Statement Concerning the Supplementary Volume of the Encyclopedia of Philosophy

Philosophy ◽  
1998 ◽  
Vol 73 (1) ◽  
pp. 122-124
Author(s):  
PAUL EDWARDS
Keyword(s):  
Good Deal ◽  
Big Bang ◽  
Bertrand Russell ◽  
Front Page ◽  
Biological Research ◽  
Simone De Beauvoir ◽  
Alphabetical Order ◽  
Modern Physics ◽  
The Times ◽  
The Big Bang

The Macmillan Reference Company and Prentice Hall International recently released a volume entitled ‘Supplement of the Encyclopedia of Philosophy’. As the editor-in-chief of the original eight-volume Encyclopedia I wish to explain why I must disassociate from this Supplement.The Supplement does contain many valuable articles by recognized philosophers, but it violates the spirit of the original work in one important respect. An article in the Oxford Companion to Philosophy accurately describes the Encyclopedia as a ‘massive Enlightenment work’ and similar descriptions were offered in a front-page review in the Times Literary Supplement of London (September 14, 1967) by Anthony (now Lord) Quinton. My associates and I edited the Encyclopedia in the spirit of Voltaire and Diderot, of Hume and Bertrand Russell. We tried to be fair to religious and metaphysical philosophers, but a good deal of space was devoted to radical thinkers and movements that had been frequently neglected or mishandled in earlier reference works. Furthermore, philosophers whom we regarded as obscurantists, while their ideas were never misrepresented, received the kind of critical treatment we thought appropriate. This spirit has not been preserved in the Supplement. There are some interesting and balanced articles on religious topics, but the highly significant biological research, reported in the writings of Stephen J. Gould and Richard Dawkins, which undermines one major form of the design argument, is not even mentioned. The ‘big bang’ is briefly mentioned (p. 143), but there is no reference to the work of Adolf Grünbaum, Steven Weinberg and other scientists and philosophers showing that neither the big bang nor any other cosmological theory of modern physics support a First Cause. More seriously, a number of contemporary writers, mostly German and French, who are regarded with suspicion if not outright contempt by most analytic philosophers are given extensive and even enthusiastic coverage. In alphabetical order they are Hannah Arendt, Simone de Beauvoir, Jacques Derrida, Michel Foucault, Hans-Georg Gadamer, and Paul Ricoeur (five articles on Ricoeur). It may be argued that, whatever the defect of their work, these figures have achieved such prominence that articles about them are warranted. Perhaps so, but what we get are totally uncritical pieces.

Bifurcation Analysis of the γ-Ricker Population Model Using the Lambert W Function

2020 ◽  
Vol 30 (07) ◽  
pp. 2050108 ◽  
Author(s):  
J. Leonel Rocha ◽  
Abdel-Kaddous Taha
Keyword(s):  
Population Model ◽  
Allee Effect ◽  
Pitchfork Bifurcation ◽  
Big Bang ◽  
Lambert W Function ◽  
Dynamical Study ◽  
Existence And Stability ◽  
Fixed Point Equation ◽  
Bifurcation Structures ◽  
Global And Local

In this work, we present the dynamical study and the bifurcation structures of the [Formula: see text]-Ricker population model. Resorting to the Lambert [Formula: see text] function, the analytical solutions of the positive fixed point equation for the [Formula: see text]-Ricker population model are explicitly presented and conditions for the existence and stability of these fixed points are established. The main focus of this work is the definition and characterization of the Allee effect bifurcation for the [Formula: see text]-Ricker population model, which is not a pitchfork bifurcation. Consequently, we prove that the phenomenon of Allee effect for the [Formula: see text]-Ricker population model is associated with the asymptotic behavior of the Lambert [Formula: see text] function in a neighborhood of zero. The theoretical results describe the global and local bifurcations of the [Formula: see text]-Ricker population model, using the Lambert [Formula: see text] function in the presence and absence of the Allee effect. The Allee effect, snapback repeller and big bang bifurcations are investigated in the parameters space considered. Numerical studies are included.

A view of the universe before the big bang

2006 ◽  
Vol 190 ◽  
pp. 15-15
Author(s):  
D CASTELVECCHI
Keyword(s):  
Big Bang ◽  
The Big Bang ◽  
The Universe

The First Galaxies in the Universe

2013 ◽  
Author(s):  
Abraham Loeb ◽  
Steven R. Furlanetto
Keyword(s):  
Galaxy Evolution ◽  
Big Bang ◽  
First Stars ◽  
Distant Galaxies ◽  
Formation And Evolution ◽  
Early Galaxies ◽  
First Galaxies ◽  
Large Telescopes ◽  
New Generation ◽  
The Big Bang

This book provides a comprehensive, self-contained introduction to one of the most exciting frontiers in astrophysics today: the quest to understand how the oldest and most distant galaxies in our universe first formed. Until now, most research on this question has been theoretical, but the next few years will bring about a new generation of large telescopes that promise to supply a flood of data about the infant universe during its first billion years after the big bang. This book bridges the gap between theory and observation. It is an invaluable reference for students and researchers on early galaxies. The book starts from basic physical principles before moving on to more advanced material. Topics include the gravitational growth of structure, the intergalactic medium, the formation and evolution of the first stars and black holes, feedback and galaxy evolution, reionization, 21-cm cosmology, and more.

The Planet in a Pebble

2010 ◽  
Author(s):  
Jan Zalasiewicz
Keyword(s):  
Solar System ◽  
Volcanic Eruptions ◽  
Big Bang ◽  
Forensic Analysis ◽  
Mineral Matter ◽  
Supernova Explosions ◽  
Deep Underground ◽  
The Big Bang ◽  
The Universe

This is the story of a single pebble. It is just a normal pebble, as you might pick up on holiday - on a beach in Wales, say. Its history, though, carries us into abyssal depths of time, and across the farthest reaches of space. This is a narrative of the Earth's long and dramatic history, as gleaned from a single pebble. It begins as the pebble-particles form amid unimaginable violence in distal realms of the Universe, in the Big Bang and in supernova explosions and continues amid the construction of the Solar System. Jan Zalasiewicz shows the almost incredible complexity present in such a small and apparently mundane object. Many events in the Earth's ancient past can be deciphered from a pebble: volcanic eruptions; the lives and deaths of extinct animals and plants; the alien nature of long-vanished oceans; and transformations deep underground, including the creations of fool's gold and of oil. Zalasiewicz demonstrates how geologists reach deep into the Earth's past by forensic analysis of even the tiniest amounts of mineral matter. Many stories are crammed into each and every pebble around us. It may be small, and ordinary, this pebble - but it is also an eloquent part of our Earth's extraordinary, never-ending story.

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