S. Elaydi | Trinity University, Texas (original) (raw)
Papers by S. Elaydi
Trends in The Theory and Practice of Non-Linear Analysis, Proceedings of the VIth International Conference on Trends in the Theory and Practice of Non-Linear Analysis
Publisher Summary This chapter discusses exponential dichotomy of nonlinear systems of ordinary d... more Publisher Summary This chapter discusses exponential dichotomy of nonlinear systems of ordinary differential equations. A dichotomy, exponential or ordinary, is a type of conditional stability. A linear differential equation possesses a dichotomy if there exists an invariant splitting or a continuous decomposition of the Euclidean space into stable and unstable subspaces. The concept of dichotomy was first formulated by Massera and Schaffer who demonstrated its effectiveness in dealing with the problems of asymptoticity, boundedness, and admissability in linear differential equations. Martin and Muldowney, respectively, introduced a generalized dichotomy that incorporates both exponential and ordinary dichotomies. It has been shown that the property of exponential dichotomy is rough in the sense that it is not destroyed by small perturbations of the coefficient matrix. The chapter is divided into two parts. The first part is a survey of known results on exponential dichotomies of linear systems. The second part consists of new results on exponential dichotomies of nonlinear systems.
Journal of Difference Equations and Applications, 2016
Nonlinear Analysis: Theory, Methods & Applications, 1984
Difference Equations, Special Functions and Orthogonal Polynomials, 2007
emis.ams.org
... Chief Editors Saber Elaydi (San Antonio, TX) Abraham F. Jalbout (New Orleans, LA) Bassel E. S... more ... Chief Editors Saber Elaydi (San Antonio, TX) Abraham F. Jalbout (New Orleans, LA) Bassel E. Sawaya (Philadelphia, PA) ... The Average Outgoing Quality Abraham F. Jalbout, Hadi Y. Alkahby, Fouad N. Jalbout, Abdalla M. Darwish……………9-15 ...
Physica D: Nonlinear Phenomena, 2004
When a set of nonlinear differential equations is investigated, most of time there is no analytic... more When a set of nonlinear differential equations is investigated, most of time there is no analytical solution and only numerical 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 the time variable is discrete. In numerical investigations a fourth-order Runge-Kutta integration scheme is usually sufficient. Nevertheless, sometimes a set of difference equations may be required and, in this case, standard schemes like the forward Euler, backward Euler or central difference schemes are used. The major problem encountered with these schemes is that they offer numerical solutions equivalent to those of the set of differential equations only for sufficiently small integration time steps. In some cases, it may be of interest to obtain difference equations with the same type of solutions as for the differential equations but with significantly large time steps. Nonstandard schemes as introduced by Mickens [Nonstandard Finite Difference Models of Differential Equations, World Scientific, 1994] allow to obtain more robust difference equations. In this paper, using such nonstandard scheme, we propose some difference equations as discrete analogues of the Rössler system for which it is shown that the dynamics is less dependent on the time step size than when a nonstandard scheme is used. In particular, it has been observed that the solutions to the discrete models are topologically equivalent to the solutions to the Rössler system as long as the time step is less than the threshold value associated with the Nyquist criterion.
Journal of Mathematical Analysis and Applications, 1986
Nonlinear Analysis: Theory, Methods & Applications, 1985
Journal of Biological Dynamics, 2013
We will investigate the stability and invariant manifolds of a new discrete host-parasitoid model... more We will investigate the stability and invariant manifolds of a new discrete host-parasitoid model. It is a generalization of the Beddington-Nicholson-Bailey model. Our study establishes analytically, for the first time, the stability of the coexistence fixed point.
Natural Resource Modeling, 2015
This paper develops a unified way to describe the various generalized discrete-time nonlinear dyn... more This paper develops a unified way to describe the various generalized discrete-time nonlinear dynamical models with density dependence, Allee effects, and parasitoids. We show how the kappa function can be used to describe the probabilities involved in intra-or interspecific encounters, namely, (i) the probability of surviving to the next generation in the absence of parasitoids or Allee effects, (ii) the encounter probability associated with Allee effects, and (iii) the probability of escaping parasitism in the presence of parasitoids. Having introduced a phenomenological framework of modeling via the kappa function, we then provide a realistic mechanism through stochastic encounters, responsible for generating the kappa function to any of the three involved probabilities. The unified modeling through the kappa function yields insights into how abundances influence species interactions. It is now straightforward to use this unified modeling to analyze and investigate its consequences in species dynamics.
Journal of Difference Equations and Applications, 2010
A new class of maps called unimodal Allee maps are introduced. Such maps arise in the study of po... more 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 periodic systems/difference equations when the system consists of two unimodal Allee maps.
Journal of Biological Dynamics, 2011
Our main objective is to study a Ricker-type competition model of two species. We give a complete... more Our main objective is to study a Ricker-type competition model of two species. We give a complete analysis of stability and bifurcation and determine the centre manifolds, as well as stable and unstable manifolds. It is shown that the autonomous Ricker competition model exhibits subcritical bifurcation, bubbles, perioddoubling bifurcation, but no Neimark-Sacker bifurcations. We exhibit the region in the parameter space where the competition exclusion principle applies.
Journal of Biological Dynamics, 2014
We introduce a discrete-time host-parasitoid model with a strong Allee effect on the host. We ada... more We introduce a discrete-time host-parasitoid model with a strong Allee effect on the host. We adapt the Nicholson-Bailey model to have a positive density dependent factor due to the presence of an Allee effect, and a negative density dependence factor due to intraspecific competition. It is shown that there are two scenarios, the first with no interior fixed points and the second with one interior fixed point. In the first scenario, we show that either both host and parasitoid will go to extinction or there are two regions, an extinction region where both species go to extinction and an exclusion region in which the host survives and tends to its carrying capacity. In the second scenario, we show that either both host and parasitoid will go to extinction or there are two regions, an extinction region where both species go to extinction and a coexistence region where both species survive.
Journal of Difference Equations and Applications, 2011
In this paper, we study a new logistic competition model. We will investigate stability and bifur... more 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 centre manifolds, and study their bifurcation. Saddle-node and period-doubling bifurcation route to chaos are exhibited via numerical simulations.
Journal of Biological Dynamics, 2014
A general notion of the Allee effect for higher dimensional triangular maps is proposed. A global... more A general notion of the Allee effect for higher dimensional triangular maps is proposed. A global dynamics theory is established. The theory is applied to multi-species hierarchical models. Then we provide a detailed study of the global dynamics of three-species Ricker competition models with the Allee effect. Regions of extinction, exclusion and coexistence are identified.
Proceedings of the Mathematics Conference
Encyclopaedia of Mathematics, Supplement III, 2002
Applications of Nonstandard Finite Difference Schemes, 2000
Journal of Mathematical Analysis and Applications, 1989
Trends in The Theory and Practice of Non-Linear Analysis, Proceedings of the VIth International Conference on Trends in the Theory and Practice of Non-Linear Analysis
Publisher Summary This chapter discusses exponential dichotomy of nonlinear systems of ordinary d... more Publisher Summary This chapter discusses exponential dichotomy of nonlinear systems of ordinary differential equations. A dichotomy, exponential or ordinary, is a type of conditional stability. A linear differential equation possesses a dichotomy if there exists an invariant splitting or a continuous decomposition of the Euclidean space into stable and unstable subspaces. The concept of dichotomy was first formulated by Massera and Schaffer who demonstrated its effectiveness in dealing with the problems of asymptoticity, boundedness, and admissability in linear differential equations. Martin and Muldowney, respectively, introduced a generalized dichotomy that incorporates both exponential and ordinary dichotomies. It has been shown that the property of exponential dichotomy is rough in the sense that it is not destroyed by small perturbations of the coefficient matrix. The chapter is divided into two parts. The first part is a survey of known results on exponential dichotomies of linear systems. The second part consists of new results on exponential dichotomies of nonlinear systems.
Journal of Difference Equations and Applications, 2016
Nonlinear Analysis: Theory, Methods & Applications, 1984
Difference Equations, Special Functions and Orthogonal Polynomials, 2007
emis.ams.org
... Chief Editors Saber Elaydi (San Antonio, TX) Abraham F. Jalbout (New Orleans, LA) Bassel E. S... more ... Chief Editors Saber Elaydi (San Antonio, TX) Abraham F. Jalbout (New Orleans, LA) Bassel E. Sawaya (Philadelphia, PA) ... The Average Outgoing Quality Abraham F. Jalbout, Hadi Y. Alkahby, Fouad N. Jalbout, Abdalla M. Darwish……………9-15 ...
Physica D: Nonlinear Phenomena, 2004
When a set of nonlinear differential equations is investigated, most of time there is no analytic... more When a set of nonlinear differential equations is investigated, most of time there is no analytical solution and only numerical 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 the time variable is discrete. In numerical investigations a fourth-order Runge-Kutta integration scheme is usually sufficient. Nevertheless, sometimes a set of difference equations may be required and, in this case, standard schemes like the forward Euler, backward Euler or central difference schemes are used. The major problem encountered with these schemes is that they offer numerical solutions equivalent to those of the set of differential equations only for sufficiently small integration time steps. In some cases, it may be of interest to obtain difference equations with the same type of solutions as for the differential equations but with significantly large time steps. Nonstandard schemes as introduced by Mickens [Nonstandard Finite Difference Models of Differential Equations, World Scientific, 1994] allow to obtain more robust difference equations. In this paper, using such nonstandard scheme, we propose some difference equations as discrete analogues of the Rössler system for which it is shown that the dynamics is less dependent on the time step size than when a nonstandard scheme is used. In particular, it has been observed that the solutions to the discrete models are topologically equivalent to the solutions to the Rössler system as long as the time step is less than the threshold value associated with the Nyquist criterion.
Journal of Mathematical Analysis and Applications, 1986
Nonlinear Analysis: Theory, Methods & Applications, 1985
Journal of Biological Dynamics, 2013
We will investigate the stability and invariant manifolds of a new discrete host-parasitoid model... more We will investigate the stability and invariant manifolds of a new discrete host-parasitoid model. It is a generalization of the Beddington-Nicholson-Bailey model. Our study establishes analytically, for the first time, the stability of the coexistence fixed point.
Natural Resource Modeling, 2015
This paper develops a unified way to describe the various generalized discrete-time nonlinear dyn... more This paper develops a unified way to describe the various generalized discrete-time nonlinear dynamical models with density dependence, Allee effects, and parasitoids. We show how the kappa function can be used to describe the probabilities involved in intra-or interspecific encounters, namely, (i) the probability of surviving to the next generation in the absence of parasitoids or Allee effects, (ii) the encounter probability associated with Allee effects, and (iii) the probability of escaping parasitism in the presence of parasitoids. Having introduced a phenomenological framework of modeling via the kappa function, we then provide a realistic mechanism through stochastic encounters, responsible for generating the kappa function to any of the three involved probabilities. The unified modeling through the kappa function yields insights into how abundances influence species interactions. It is now straightforward to use this unified modeling to analyze and investigate its consequences in species dynamics.
Journal of Difference Equations and Applications, 2010
A new class of maps called unimodal Allee maps are introduced. Such maps arise in the study of po... more 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 periodic systems/difference equations when the system consists of two unimodal Allee maps.
Journal of Biological Dynamics, 2011
Our main objective is to study a Ricker-type competition model of two species. We give a complete... more Our main objective is to study a Ricker-type competition model of two species. We give a complete analysis of stability and bifurcation and determine the centre manifolds, as well as stable and unstable manifolds. It is shown that the autonomous Ricker competition model exhibits subcritical bifurcation, bubbles, perioddoubling bifurcation, but no Neimark-Sacker bifurcations. We exhibit the region in the parameter space where the competition exclusion principle applies.
Journal of Biological Dynamics, 2014
We introduce a discrete-time host-parasitoid model with a strong Allee effect on the host. We ada... more We introduce a discrete-time host-parasitoid model with a strong Allee effect on the host. We adapt the Nicholson-Bailey model to have a positive density dependent factor due to the presence of an Allee effect, and a negative density dependence factor due to intraspecific competition. It is shown that there are two scenarios, the first with no interior fixed points and the second with one interior fixed point. In the first scenario, we show that either both host and parasitoid will go to extinction or there are two regions, an extinction region where both species go to extinction and an exclusion region in which the host survives and tends to its carrying capacity. In the second scenario, we show that either both host and parasitoid will go to extinction or there are two regions, an extinction region where both species go to extinction and a coexistence region where both species survive.
Journal of Difference Equations and Applications, 2011
In this paper, we study a new logistic competition model. We will investigate stability and bifur... more 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 centre manifolds, and study their bifurcation. Saddle-node and period-doubling bifurcation route to chaos are exhibited via numerical simulations.
Journal of Biological Dynamics, 2014
A general notion of the Allee effect for higher dimensional triangular maps is proposed. A global... more A general notion of the Allee effect for higher dimensional triangular maps is proposed. A global dynamics theory is established. The theory is applied to multi-species hierarchical models. Then we provide a detailed study of the global dynamics of three-species Ricker competition models with the Allee effect. Regions of extinction, exclusion and coexistence are identified.
Proceedings of the Mathematics Conference
Encyclopaedia of Mathematics, Supplement III, 2002
Applications of Nonstandard Finite Difference Schemes, 2000
Journal of Mathematical Analysis and Applications, 1989