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- Stability (10)
- Allee effect (5)
- Global stability (5)
- Asymptotic stability (4)
- Bifurcation (4)
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- Cycles (4)
- Beverton-Holt equation (3)
- Carrying capacity (3)
- Competition Model (3)
- Difference equation (3)
- Population biology (3)
- Threshold point (3)
- Almost periodicity (2)
- Asymptotic equivalence (2)
- Asymptotic expansion (2)
- Attenuance and resonance (2)
- Bochner almost periodic sequences (2)
- Bohr almost periodic sequences (2)
- Chaos (2)
- Competition model (2)
- Dichotomy (2)
- Difference equations (2)
- Dirac particle in one dimension with random mass (2)
- Discrete dynamical systems (2)
- Discretization (2)
- Exploitation rate (2)
- Fitness function (2)
- Fixed point (2)
- Global Attractivity (2)
- Global asymptotic stability (2)
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Articles 1 - 30 of 40
Full-Text Articles in Physical Sciences and Mathematics
Skew-Product Dynamical Dystems: Applications To Difference Equations, Saber Elaydi, Robert Sacker
Skew-Product Dynamical Dystems: Applications To Difference Equations, Saber Elaydi, Robert Sacker
Saber Elaydi
No abstract provided.
Asymptotic Solutions Of A Discrete Schrödinger Equation Arising From A Dirac Equation With Random Mass, Bernd Aulbach, Saber Elaydi, Klaus Ziegler
Asymptotic Solutions Of A Discrete Schrödinger Equation Arising From A Dirac Equation With Random Mass, Bernd Aulbach, Saber Elaydi, Klaus Ziegler
Saber Elaydi
For a Dirac particle in one dimension with random mass, the time evolution for the average wavefunction is considered. Using the supersymmetric representation of the average Green’s function, we derive a fourth order linear difference equation for the low-energy asymptotics of the average wavefunction. This equation is of Poincar´e type, though highly critical and therefore not amenable to standard methods. In this paper we show that, nevertheless, asymptotic expansions of its solutions can be obtained.
Difference Equations Versus Differential Equations, A Possible Equivalence For The Rössler System?, Christophe Letellier, Saber Elaydi, Luis Aguirre, Aziz Alaoui
Difference Equations Versus Differential Equations, A Possible Equivalence For The Rössler System?, Christophe Letellier, Saber Elaydi, Luis Aguirre, Aziz Alaoui
Saber Elaydi
When a set of non linear differential equations is investigated, most of the time there is no analytical solution and only numerial integration techniques can provide accurate numerical solutions. In a general way the process of numerical integration is the replacement of a set of differential equations with a continuous dependence on the time by a model for which these time variable is discrete. When only a numerical solution is researched, a fourth-order Runge-Kutta integration scheme is usually sufficient. Nevertheless, sometimes a set of differential equations may be required and, in this case, standard schemes like the forward Euler, backward …
Is The World Evolving Discretely?, Saber Elaydi
Global Stability Of Periodic Orbits Of Non-Autonomous Difference Equations And Population Biology, Saber Elaydi, Robert Sacker
Global Stability Of Periodic Orbits Of Non-Autonomous Difference Equations And Population Biology, Saber Elaydi, Robert Sacker
Saber Elaydi
Elaydi and Yakubu showed that a globally asymptotically stable(GAS) periodic orbit in an autonomous difference equation must in fact be a fixed point whenever the phase space is connected. In this paper we extend this result to periodic nonautonomous difference equations via the concept of skew-product dynamical systems. We show that for a k-periodic difference equation, if a periodic orbit of period r is GAS, then r must be a divisor of k. In particular sub-harmonic, or long periodic, oscillations cannot occur. Moreover, if r divides k we construct a non-autonomous dynamical system having minimum period k and which has …
Periodic Difference Equations, Population Biology And The Cushing-Henson Conjectures, Saber Elaydi, Robert Sacker
Periodic Difference Equations, Population Biology And The Cushing-Henson Conjectures, Saber Elaydi, Robert Sacker
Saber Elaydi
We show that for a k-periodic difference equation, if a periodic orbit of period r is globally asymptotically stable (GAS), then r must be a divisor of k. In particular sub-harmonic, or long periodic, oscillations cannot occur. Moreover, if r divides k we construct a non-autonomous dynamical system having minimum period k and which has a GAS periodic orbit with minimum period r. Our method uses the technique of skew-product dynamical systems. Our methods are then applied to prove two conjectures of J. Cushing and S. Henson concerning a non-autonomous Beverton-Holt equation which arises in the study of the response …
Stability Of Hyperbolic And Nonhyperbolic Fixed Points Of One-Dimensional Maps, Fozi M. Dannan, Saber Elaydi, Vadim Ponomarenko
Stability Of Hyperbolic And Nonhyperbolic Fixed Points Of One-Dimensional Maps, Fozi M. Dannan, Saber Elaydi, Vadim Ponomarenko
Saber Elaydi
We present a complete theory for the stability of non-hyperbolic fixed points of one-dimensional continuous maps. As well as we give simple criteria for the global stability of general maps without using derivatives.
Nonautonomous Beverton-Holt Equations And The Cushing-Henson Conjectures, Saber Elaydi, Robert Sacker
Nonautonomous Beverton-Holt Equations And The Cushing-Henson Conjectures, Saber Elaydi, Robert Sacker
Saber Elaydi
In [3] Jim Cushing and Shandelle Henson published two conjectures (see Section 3) related to the Beverton-Holt difference equation (with growth parameter exceeding one), which said that the B-H equation with periodically varying coefficients (a) will have a globally asymptotically stable periodic solution and (b) the average of the state variable along the periodic orbit will be strictly less than the average of the carrying capacities of the individual maps. They had previously [2] proved both statements for period 2.
Global Stability Of Cycles: Lotka-Volterra Competition Model With Stocking, Saber Elaydi, Abdul-Aziz Yakubu
Global Stability Of Cycles: Lotka-Volterra Competition Model With Stocking, Saber Elaydi, Abdul-Aziz Yakubu
Saber Elaydi
In this article, we prove that in connected metric spaces k - cycles are not globally attracting (where k>2). We apply this result to a two species discrete-time Lotka-Volterra competion model with stocking. In particular, we show that an k-cycle cannot be the ultimate life-history of evolution of all population sizes. This solves Yakubu's conjecture but the question on the structure of the boundary of the basins of attraction of the locally stable n-cycles is still open.
An Extension Of Sharkovsky’S Theorem To Periodic Difference Equations, Ziyad Alsharawi, James Angelos, Saber Elaydi, Leela Rakesh
An Extension Of Sharkovsky’S Theorem To Periodic Difference Equations, Ziyad Alsharawi, James Angelos, Saber Elaydi, Leela Rakesh
Saber Elaydi
We present an extension of Sharkovsky’s Theorem and its converse to periodic difference equations. In addition, we provide a simple method for constructing a p-periodic difference equation having an r-periodic geometric cycle with or without stability properties.
Asymptotic Stability Of Linear Difference Equations Of Advanced Type, Fozi Dannan, Saber Elaydi
Asymptotic Stability Of Linear Difference Equations Of Advanced Type, Fozi Dannan, Saber Elaydi
Saber Elaydi
Necessary and sufficient conditions are obtained for the asymptotic stability of difference equations of advanced typen of the form x(n) - ax(n+1) + bx(n+k) = 0, n = 0, 1, .. where a and b are arbitrary real numbers and k > 1. For a = 1, we establish an analogue of a result by Levin and May.
Basin Of Attraction Of Periodic Orbits Of Maps On The Real Line, Saber Elaydi, Robert Sacker
Basin Of Attraction Of Periodic Orbits Of Maps On The Real Line, Saber Elaydi, Robert Sacker
Saber Elaydi
We prove a conjecture by Elaydi and Yakubu which states that the basin of attraction of an attracting 2 k -cycle of the Ricker's map is where E is the set of all eventually 2 r -periodic points. The result is then extended to a more general class of continuous maps on the real line.
Poincaré Types Solutions Of Systems Of Difference Equations, Raghib Abu-Saris, Saber Elaydi, Sophia Jang
Poincaré Types Solutions Of Systems Of Difference Equations, Raghib Abu-Saris, Saber Elaydi, Sophia Jang
Saber Elaydi
No abstract provided.
Global Dynamics Of Triangular Maps, Eduardo Balreira, Saber Elaydi, Rafael Luis
Global Dynamics Of Triangular Maps, Eduardo Balreira, Saber Elaydi, Rafael Luis
Saber Elaydi
We consider continuous triangular maps on IN, where I is a compact interval in the Euclidean space R. We show, under some conditions, that the orbit of every point in a triangular map converges to a fixed point if and only if there is no periodic orbit of prime period two. As a consequence we obtain a result on global stability, namely, if there are no periodic orbits of prime period 2 and the triangular map has a unique fixed point, then the fixed point is globally asymptotically stable. We also discuss examples and applications of our results to competition …
Local Stability Implies Global Stability For The Planar Ricker Competition Model, Eduardo Balreira, Saber Elaydi, Rafael Luis
Local Stability Implies Global Stability For The Planar Ricker Competition Model, Eduardo Balreira, Saber Elaydi, Rafael Luis
Saber Elaydi
Under certain analytic and geometric assumptions we show that local stability of the coexistence (positive) fixed point of the planar Ricker competition model implies global stability with respect to the interior of the positive quadrant. This result is a confluence of ideas from Dynamical Systems, Geometry, and Topology that provides a framework to the study of global stability for other planar competition models.
Existence And Stability Of Periodic Orbits Of Periodic Difference Equations With Delays, Ziyad Alsharawi, James Angelos, Saber Elaydi
Existence And Stability Of Periodic Orbits Of Periodic Difference Equations With Delays, Ziyad Alsharawi, James Angelos, Saber Elaydi
Saber Elaydi
In this paper, we investigate the existence and stability of periodic orbits of the p-periodic difference equation with delays xn = f(n−1, xn−k). We show that the periodic orbits of this equation depend on the periodic orbits of p autonomous equations when p divides k. When p is not a divisor of k, the periodic orbits depend on the periodic orbits of gcd(p, k) nonautonomous p gcd(p,k) - periodic difference equations. We give formulas for calculating the number of different periodic orbits under certain conditions. In addition, when p and k are relatively prime integers, we introduce what we call …
Population Models In Almost Periodic Environments, Toka Diagana, Saber Elaydi, Abdul-Aziz Yakubu
Population Models In Almost Periodic Environments, Toka Diagana, Saber Elaydi, Abdul-Aziz Yakubu
Saber Elaydi
We establish the basic theory of almost periodic sequences on Z+. Dichotomy techniques are then utilized to find sufficient conditions for the existence of a globally attracting almost periodic solution of a semilinear system of difference equations. These existence results are, subsequently, applied to discretely reproducing populations with and without overlapping generations. Furthermore, we access evidence for attenuance and resonance in almost periodically forced population models.
Stability And Asymptoticity Of Volterra Difference Equations: A Progress Report, Saber Elaydi
Stability And Asymptoticity Of Volterra Difference Equations: A Progress Report, Saber Elaydi
Saber Elaydi
We survey some of the fundamental results on the stability and asymptoticity of linear Volterra difference equations. The method of Z-transform is heavily utilized in equations of convolution type. An example is given to show that uniform asymptotic stability does not necessarily imply exponential stabilty. It is shown that the two notions are equivalent if the kernel decays exponentially. For equations of nonconvolution type, Liapunov functions are used to find explicit criteria for stability. Moreover, the resolvent matrix is defined to produce a variation of constants formula. The study of asymptotic equivalence for difference equations with infinite delay is carried …
On The Asymptotic Stability Of Linear Volterra Difference Equations Of Convolution Type, Saber Elaydi, E Messina, A Vecchio
On The Asymptotic Stability Of Linear Volterra Difference Equations Of Convolution Type, Saber Elaydi, E Messina, A Vecchio
Saber Elaydi
No abstract provided.
On The Stochastic Beverton-Holt Equation With Survival Rates, Paul H. Bezandry, Toka Diagana, Saber Elaydi
On The Stochastic Beverton-Holt Equation With Survival Rates, Paul H. Bezandry, Toka Diagana, Saber Elaydi
Saber Elaydi
The paper studies a Beverton-Holt difference equation, in which both the recruitment function and the survival rate vary randomly. It is then shown that there is a unique invariant density, which is asymptotically stable. Moreover, a basic theory for random mean almost periodic sequence on Z+ is given. Then, some suffcient conditions for the existence of a mean almost periodic solution to the stochastic Beverton-Holt equation are given.
An Economical Model With Allee Effect, Rafael Luis, Saber Elaydi, Henrique Oliveira
An Economical Model With Allee Effect, Rafael Luis, Saber Elaydi, Henrique Oliveira
Saber Elaydi
The Marx model for the profit rate r depending on the exploitation rate e and on the organic composition of the capital k is studied using a dynamical approach. Supposing both e(t) and k(t) are continuous functions of time we derive a law for r(t) in the long term. Depending upon the hypothesis set on the growth of k(t) and e(t) in the long term, r(t) can fall to zero or remain constant. This last case contradicts the classical hypothesis of Marx stating that the profit rate must decrease in the long term. Introducing a discrete dynamical system in the …
Population Models With Allee Effect: A New Model, Saber Elaydi, Robert J. Sacker
Population Models With Allee Effect: A New Model, Saber Elaydi, Robert J. Sacker
Saber Elaydi
In this paper we develop several mathematical models of Allee effects. We start by defining the Allee effect as a phenomenon in which individual fitness increases with increasing density. Based on this biological assumption, we develop several fitness functions that produce corresponding models with Allee effects. In particular, a rational fitness function yields a new mathematical model that is our focus of study. Then we study the dynamics of 2-periodic systems with Allee effects and show the existence of an asymptotically stable 2-periodic carrying capacity.
Non-Autonomous Periodic Systems With Allee Effects, Rafael Luis, Saber Elaydi, Henrique Oliveira
Non-Autonomous Periodic Systems With Allee Effects, Rafael Luis, Saber Elaydi, Henrique Oliveira
Saber Elaydi
A new class of maps called unimodal Allee maps are introduced. Such maps arise in the study of population dynamics in which the population goes extinct if its size falls below a threshold value. A unimodal Allee map is thus a unimodal map with tree fixed points, a zero fixed point, a small positive fixed point, called threshold point, and a bigger positive fixed point, called the carrying capacity. In this paper the properties and stability of the three fixed points are studied in the setting of nonautonomous periodic dynamical systems or difference equations. Finally we investigate the bifurcation of …
Towards A Theory Of Periodic Difference Equations And Its Application To Population Dynamics, Saber Elaydi, Rafael Luis, Henrique Oliveira
Towards A Theory Of Periodic Difference Equations And Its Application To Population Dynamics, Saber Elaydi, Rafael Luis, Henrique Oliveira
Saber Elaydi
This survey contains the most updated results on the dynamics of periodic difference equations or discrete dynamical systems this time. Our focus will be on stability theory, bifurcation theory, and on the effect of periodic forcing on the welfare of the population (attenuance versus resonance). Moreover, the survey alludes to two more types of dynamical systems, namely, almost periodic difference equations and stochastic di®erence equations.
Bifurcation And Invariant Manifolds Of The Logistic Competition Model, Malgorzata Guzowska, Rafael Luis, Saber Elaydi
Bifurcation And Invariant Manifolds Of The Logistic Competition Model, Malgorzata Guzowska, Rafael Luis, Saber Elaydi
Saber Elaydi
In this paper we study a new logistic competition model. We will investigate stability and bifurcation of the model. In particular, we compute the invariant manifolds, including the important center manifolds, and study their bifurcation. Saddle-node and period doubling bifurcation route to chaos is exhibited via numerical simulations.
General Allee Effect In Two-Species Population Biology, G Livadiotis, Saber Elaydi
General Allee Effect In Two-Species Population Biology, G Livadiotis, Saber Elaydi
Saber Elaydi
The main objective of this work is to present a general framework for the notion of the strong Allee effect in population models, including competition, mutualistic, and predator–prey models. The study is restricted to the strong Allee effect caused by an inter-specific interaction. The main feature of the strong Allee effect is that the extinction equilibrium is an attractor. We show how a ‘phase space core’ of three or four equilibria is sufficient to describe the essential dynamics of the interaction between two species that are prone to the Allee effect. We will introduce the notion of semistability in planar …
Difference Equations Versus Differential Equations, A Possible Equivalence For The Rössler System?, Christophe Letellier, Saber Elaydi, Luis A. Aguirre, Aziz Alaoui
Difference Equations Versus Differential Equations, A Possible Equivalence For The Rössler System?, Christophe Letellier, Saber Elaydi, Luis A. Aguirre, Aziz Alaoui
Saber Elaydi
When a set of non linear differential equations is investigated, most of the time there is no analytical solution and only numerial integration techniques can provide accurate numerical solutions. In a general way the process of numerical integration is the replacement of a set of differential equations with a continuous dependence on the time by a model for which these time variable is discrete. When only a numerical solution is researched, a fourth-order Runge-Kutta integration scheme is usually sufficient. Nevertheless, sometimes a set of differential equations may be required and, in this case, standard schemes like the forward Euler, backward …
An Economical Model With Allee Effect, Rafael Luis, Saber Elaydi, Henrique Oliveira
An Economical Model With Allee Effect, Rafael Luis, Saber Elaydi, Henrique Oliveira
Saber Elaydi
The Marx model for the profit rate r depending on the exploitation rate e and on the organic composition of the capital k is studied using a dynamical approach. Supposing both e(t) and k(t) are continuous functions of time we derive a law for r(t) in the long term. Depending upon the hypothesis set on the growth of k(t) and e(t) in the long term, r(t) can fall to zero or remain constant. This last case contradicts the classical hypothesis of Marx stating that the profit rate must decrease in the long term. Introducing a discrete dynamical system in the …
Asymptotic Solutions Of A Discrete Schrödinger Equation Arising From A Dirac Equation With Random Mass, Bernd Aulbach, Saber Elaydi, Klaus Ziegler
Asymptotic Solutions Of A Discrete Schrödinger Equation Arising From A Dirac Equation With Random Mass, Bernd Aulbach, Saber Elaydi, Klaus Ziegler
Saber Elaydi
For a Dirac particle in one dimension with random mass, the time evolution for the average wavefunction is considered. Using the supersymmetric representation of the average Green’s function, we derive a fourth order linear difference equation for the low-energy asymptotics of the average wavefunction. This equation is of Poincar´e type, though highly critical and therefore not amenable to standard methods. In this paper we show that, nevertheless, asymptotic expansions of its solutions can be obtained.
Global Stability Of Cycles: Lotka-Volterra Competition Model With Stocking, Saber Elaydi, Abdul-Aziz Yakubu
Global Stability Of Cycles: Lotka-Volterra Competition Model With Stocking, Saber Elaydi, Abdul-Aziz Yakubu
Saber Elaydi
In this article, we prove that in connected metric spaces k - cycles are not globally attracting (where k>2). We apply this result to a two species discrete-time Lotka-Volterra competion model with stocking. In particular, we show that an k-cycle cannot be the ultimate life-history of evolution of all population sizes. This solves Yakubu's conjecture but the question on the structure of the boundary of the basins of attraction of the locally stable n-cycles is still open.