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.

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.

scholarly journals Dynamical analysis of a fractional-order avian-human influenza epidemic model with logistic growth for avian population

2020 ◽  
Vol 14 ◽  
pp. 174830262096670
Author(s):  
Xingyang Ye ◽  
Shimin Lin ◽  
Chuanju Xu
Keyword(s):  
Equilibrium Point ◽  
Fractional Order ◽  
Epidemic Model ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Logistic Growth ◽  
Influenza Epidemic ◽  
Human Influenza

In this paper, a fractional order avian-human influenza epidemic model with logistic growth for avian population is investigated. The dynamical behavior of this model is discussed. We first establish the existence, uniqueness, non-negativity and positive invariance of the solution. Then we analyze the existence of various equilibrium points, and some sufficient conditions are derived to ensure the global asymptotic stability of the disease free equilibrium point and endemic equilibrium point. Finally, we take some numerical simulations to validate the analytical results.

scholarly journals Dynamical analysis of a giving up smoking model with time delay

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zizhen Zhang ◽  
Ruibin Wei ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory

AbstractIn this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

scholarly journals The Exemplar Approach to Science and Religion

Symposion ◽  
2019 ◽  
Vol 6 (2) ◽  
pp. 183-193
Author(s):  
Seungbae Park ◽  
Keyword(s):  
Sufficient Conditions ◽  
Big Bang ◽  
School Policy ◽  
Science And Religion ◽  
Religious Institutions ◽  
Religious Activities ◽  
Big Bang Theory ◽  
Necessary And Sufficient ◽  
The Big Bang ◽  
The Universe

We can judge whether some activities are scientific or religious, depending on how similar they are to exemplar scientific activities or to exemplar religious activities, even if we cannot specify the necessary and sufficient conditions for science and religion. The absence of the demarcation between science and religion does not justify the school policy of teaching the creationist hypothesis that God created the universe any more than it justifies the religious policy of teaching evolutionary theory, quantum mechanics, and the Big Bang theory in religious institutions.

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.

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