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.

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.

Global Dynamics of a Holling-II Amensalism System with Nonlinear Growth Rate and Allee Effect on the First Species

2021 ◽  
Vol 31 (03) ◽  
pp. 2150050
Author(s):  
Demou Luo ◽  
Qiru Wang
Keyword(s):  
Growth Rate ◽  
Allee Effect ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Nonlinear Growth ◽  
Trivial Equilibrium ◽  
Existence And Stability ◽  
Stable Equilibria ◽  
Theoretical Results

Of concern is the global dynamics of a two-species Holling-II amensalism system with nonlinear growth rate. The existence and stability of trivial equilibrium, semi-trivial equilibria, interior equilibria and infinite singularity are studied. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, the global dynamics of the model is performed. Next, we incorporate Allee effect on the first species and offer a new analysis of equilibria and bifurcation discussion of the model. Finally, several numerical examples are performed to verify our theoretical results.

scholarly journals On the stability of Einstein universe in f(R, T, Rμν Tμν) gravity

2018 ◽  
Vol 33 (36) ◽  
pp. 1850216 ◽  
Author(s):  
M. Sharif ◽  
Arfa Waseem
Keyword(s):  
State Parameter ◽  
Big Bang ◽  
Graphical Analysis ◽  
Momentum Tensor ◽  
Field Equations ◽  
Einstein Universe ◽  
Existence And Stability ◽  
The Stability ◽  
The Big Bang ◽  
Big Bang Singularity

This paper investigates the existence and stability of Einstein universe in the context of f(R, T, Q) gravity, where Q = R[Formula: see text] T[Formula: see text]. Considering linear homogeneous perturbations around scale factor and energy density, we formulate static as well as perturbed field equations. We parametrize the stability regions corresponding to conserved as well as non-conserved energy–momentum tensor using linear equation of state parameter for particular models of this gravity. The graphical analysis concludes that for a suitable choice of parameters, stable regions of the Einstein universe are obtained which indicates that the big bang singularity can be avoided successfully by the emergent mechanism in non-minimal matter-curvature coupled gravity.

A DARWINIAN DYNAMIC MODEL FOR THE EVOLUTION OF POST-REPRODUCTION SURVIVAL

2021 ◽  
pp. 1-18
Author(s):  
J. M. CUSHING ◽  
KATHRYN STEFANKO
Keyword(s):  
Natural Selection ◽  
Dynamic Model ◽  
Discrete Time ◽  
Population Model ◽  
Bifurcation Theory ◽  
Allee Effect ◽  
Phenotypic Trait ◽  
Trade Off ◽  
The Stability ◽  
Low Dimensional

We derive and study a Darwinian dynamic model based on a low-dimensional discrete- time population model focused on two features: density-dependent fertility and a trade-off between inherent (density free) fertility and post-reproduction survival. Both features are assumed to be dependent on a phenotypic trait subject to natural selection. The model tracks the dynamics of the population coupled with that of the population mean trait. We study the stability properties of equilibria by means of bifurcation theory. Whether post-reproduction survival at equilibrium is low or high is shown, in this model, to depend significantly on the nature of the trait dependence of the density effects. An Allee effect can also play a significant role.

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