GLOBAL BIFURCATIONS OF DOMAINS OF FEASIBLE TRAJECTORIES: AN ANALYSIS OF A DISCRETE PREDATOR–PREY MODEL

2006 ◽  
Vol 16 (09) ◽  
pp. 2601-2613 ◽  
Author(s):  
EN-GUO GU ◽  
YIBING HUANG
Keyword(s):  
Fixed Point ◽  
Domain Boundary ◽  
Computer Assisted ◽  
Two Dimensional ◽  
Global Bifurcations ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Global Properties ◽  
Singular Sets ◽  
Domain Boundaries

This paper aims to provide some new results, by a computer-assisted study, on some global bifurcations that change the domains of feasible trajectories (bounded discrete trajectories having an ecological sense). It is shown that the domain boundaries of two-dimensional maps can be generally obtained by the union of all rank preimages of two axes. A discrete predator–prey ecosystem is employed to demonstrate how the global properties of persistence in a biological model can be analyzed. The main results of this paper are from the study of some global bifurcations that change the stable manifold of a saddle fixed point belonging to the domain boundary. The basin fractalization is explained by using two types of nonclassical singular sets. The first one, critical curve, separates the plane into two regions having a different number of real inverses (here zero and two). The second one is a line of nondefinition for one of the two inverses of the map, i.e. this inverse has a vanishing denominator on this line.

ON SOME GLOBAL BIFURCATIONS OF THE DOMAINS OF FEASIBLE TRAJECTORIES: AN ANALYSIS OF RECURRENCE EQUATIONS

2005 ◽  
Vol 15 (05) ◽  
pp. 1625-1639 ◽  
Author(s):  
EN-GUO GU ◽  
JIONG RUAN
Keyword(s):  
Fixed Point ◽  
Stable Manifold ◽  
The Other ◽  
Computer Assisted ◽  
Two Dimensional ◽  
Global Bifurcations ◽  
Basin Boundary ◽  
Recurrence Equations ◽  
Critical Curves ◽  
Global Properties

This paper is an attempt to give new results, by a computer-assisted study, on some global bifurcations that change the structure of the domain of feasible trajectories (bounded discrete trajectories having an ecological sense) which can be obtained by the union of all rank preimages of axes. Three two-dimensional recurrence equations (or maps) are analyzed. The two first maps are degenerated invertible maps (i.e. the inverses of them are well defined except a set of zero lebergue measure) for which the basins of attractor are obtained by the backward iteration of a stable manifold of a saddle fixed point belonging to the basin boundary, and the interior domains of feasible trajectories are given by the intersection between the basin of attractor and the first quadrant. The other is a noninvertible map which is investigated by the use of critical curves, a powerful tool for the analysis of global properties of two-dimensional maps.

scholarly journals Discrete-Time Predator-Prey Model with Bifurcations and Chaos

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
K. S. Al-Basyouni ◽  
A. Q. Khan
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Discrete Time ◽  
Control Method ◽  
Linear Stability Theory ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Positive Fixed Point

In this paper, local dynamics, bifurcations and chaos control in a discrete-time predator-prey model have been explored in ℝ + 2 . It is proved that the model has a trivial fixed point for all parametric values and the unique positive fixed point under definite parametric conditions. By the existing linear stability theory, we studied the topological classifications at fixed points. It is explored that at trivial fixed point model does not undergo the flip bifurcation, but flip bifurcation occurs at the unique positive fixed point, and no other bifurcations occur at this point. Numerical simulations are performed not only to demonstrate obtained theoretical results but also to tell the complex behaviors in orbits of period-4, period-6, period-8, period-12, period-17, and period-18. We have computed the Maximum Lyapunov exponents as well as fractal dimension numerically to demonstrate the appearance of chaotic behaviors in the considered model. Further feedback control method is employed to stabilize chaos existing in the model. Finally, existence of periodic points at fixed points for the model is also explored.

Invasion waves for a diffusive predator–prey model with two preys and one predator

2020 ◽  
pp. 2050081
Author(s):  
Xinzhi Ren ◽  
Tianran Zhang ◽  
Xianning Liu
Keyword(s):  
Fixed Point ◽  
Wave Speed ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Traveling Fronts ◽  
Prey Model ◽  
Traveling Front ◽  
Minimal Wave Speed ◽  
Invasion Waves ◽  
Coexistence Equilibrium

In this paper, we study the existence of invasion waves of a diffusive predator–prey model with two preys and one predator. The existence of traveling semi-fronts connecting invasion-free equilibrium with wave speed [Formula: see text] is obtained by Schauder’s fixed-point theorem, where [Formula: see text] is the minimal wave speed. The boundedness of such waves is shown by rescaling method and such waves are proved to connect coexistence equilibrium by LaSalle’s invariance principle. The existence of traveling front with wave speed [Formula: see text] is got by rescaling method and limit arguments. The non-existence of traveling fronts with speed [Formula: see text] is shown by Laplace transform.

scholarly journals Two-dimensional Dynamics of Cubic Maps

2021 ◽  
Vol 45 (03) ◽  
pp. 427-438
Author(s):  
I. DJELLIT ◽  
W. SELMANI
Keyword(s):  
Bifurcation Analysis ◽  
Basins Of Attraction ◽  
Global Structure ◽  
Two Dimensional ◽  
Global Bifurcations ◽  
Global Properties ◽  
Stability And Bifurcation ◽  
Stability And Bifurcation Analysis ◽  
The Creation

We investigate the global properties of two cubic maps on the plane, we try to explain the basic mechanisms of global bifurcations leading to the creation of nonconnected basins of attraction. It is shown that in some certain conditions the global structure of such systems can be simple. The main results here can be seen as an improvement of the results of stability and bifurcation analysis.

scholarly journals Topological kink plasmons on magnetic-domain boundaries

2019 ◽  
Vol 10 (1) ◽  
Author(s):  
Dafei Jin ◽  
Yang Xia ◽  
Thomas Christensen ◽  
Matthew Freeman ◽  
Siqi Wang ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Domain Boundary ◽  
Magnetic Domain ◽  
Two Dimensional ◽  
Unique Form ◽  
Topological Materials ◽  
Chern Numbers ◽  
Domain Boundaries ◽  
Dimensional Electron Gas

Abstract Two-dimensional topological materials bearing time reversal-breaking magnetic fields support protected one-way edge modes. Normally, these edge modes adhere to physical edges where material properties change abruptly. However, even in homogeneous materials, topology still permits a unique form of edge modes – kink modes – residing at the domain boundaries of magnetic fields within the materials. This scenario, despite being predicted in theory, has rarely been demonstrated experimentally. Here, we report our observation of topologically-protected high-frequency kink modes – kink magnetoplasmons (KMPs) – in a GaAs/AlGaAs two-dimensional electron gas (2DEG) system. These KMPs arise at a domain boundary projected from an externally-patterned magnetic field onto a uniform 2DEG. They propagate unidirectionally along the boundary, protected by a difference of gap Chern numbers ($$\pm1$$ ± 1 ) in the two domains. They exhibit large tunability under an applied magnetic field or gate voltage, and clear signatures of nonreciprocity even under weak-coupling to evanescent photons.

scholarly journals Existence, Multiplicity, and Stability of Positive Solutions of a Predator-Prey Model with Dinosaur Functional Response

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Xiaozhou Feng ◽  
Kenan Shi ◽  
Jianhui Tian ◽  
Tongqian Zhang
Keyword(s):  
Fixed Point ◽  
Functional Response ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Index Theory ◽  
Point Index ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Fixed Point Index Theory

We investigate the property of positive solutions of a predator-prey model with Dinosaur functional response under Dirichlet boundary conditions. Firstly, using the comparison principle and fixed point index theory, the sufficient conditions and necessary conditions on coexistence of positive solutions of a predator-prey model with Dinosaur functional response are established. Secondly, by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory, we establish the bifurcation of positive solutions of the model and obtain the stability and multiplicity of the positive solution under certain conditions. Furthermore, the local uniqueness result is studied when b and d are small enough. Finally, we investigate the multiplicity, uniqueness, and stability of positive solutions when k>0 is sufficiently large.

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