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- Keyword
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- Stability (6)
- Allee effect (4)
- Competition Model (3)
- Cycles (3)
- Global stability (3)
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- Asymptotic stability (2)
- Beverton-Holt equation (2)
- Bifurcation (2)
- Carrying capacity (2)
- Fitness function (2)
- Fixed point (2)
- Global Attractivity (2)
- Lotka Volterra (2)
- Non-hyperbolic (2)
- Schwarzian (2)
- Stocking (2)
- Threshold point (2)
- Almost periodicity (1)
- Asymptotic equivalence (1)
- Asymptotic expansion (1)
- Attenuance and resonance (1)
- Beverton-Holt (1)
- Beverton-Holt equations (1)
- Bochner almost periodic sequences (1)
- Bohr almost periodic sequences (1)
- Center Manifolds (1)
- Chaos (1)
- Competition models (1)
- Cushing-Henson (1)
- Cushing-Henson conjecture (1)
Articles 1 - 20 of 20
Full-Text Articles in Physical Sciences and Mathematics
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.
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.
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 …
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.
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.
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.
Nonautonomous Beverton-Holt Equations And The Cushing-Henson Conjectures, Saber Elaydi, Robert J. Sacker
Nonautonomous Beverton-Holt Equations And The Cushing-Henson Conjectures, Saber Elaydi, Robert J. 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.
Difference Equations From Discretization Of A Continuous Epidemic Model With Immigration Of Infectives, Sophia Jang, Saber Elaydi
Difference Equations From Discretization Of A Continuous Epidemic Model With Immigration Of Infectives, Sophia Jang, Saber Elaydi
Saber Elaydi
A continuous-time epidemic model with immigration of infectives is introduced. Systems of difference equations obtained from the continuous-time model by using nonstandard discretization technique are presented. Comparisons between the continuous-time model and its discrete counter-part are made.
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.
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.
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.
Skew-Product Dynamical Dystems: Applications To Difference Equations, Saber Elaydi, Robert J. Sacker
Skew-Product Dynamical Dystems: Applications To Difference Equations, Saber Elaydi, Robert J. Sacker
Saber Elaydi
No abstract provided.
Is The World Evolving Discretely?, Saber Elaydi
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 …
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.