Hüseyin Merdan

TOBB University of Economics and Technology · Department of Mathematics

We analyze Hopf bifurcation and its properties of a class of system of reaction-diffusion equations involving two discrete time delays. First, we discuss the existence of periodic solutions of this class under Neumann boundary conditions, and determine the required conditions on parameters of the system at which Hopf bifurcation arises near equilib...

We present a mathematical model for a market involving two stocks which are traded within a single homogeneous group of investors who have similar motivations and strategies for trading. It is assumed that the market consists of a fixed amount of cash and stocks (additions in time are not allowed, so the system is closed) and that the trading group...

We consider a general first-order scalar difference equation with and without Allee effect. The model without Allee effect represents asexual reproduction of a species while the model including Allee effect represents sexual reproduction. We analyze global stabilities of both models analytically and compare the results obtained. Numerical simulation...

We present an algorithm for determining the existence of a Hopf bifurcation of a system of delayed reaction–diffusion equations with the Neumann boundary conditions. The conditions on parameters of the system that a Hopf bifurcation occurs as the delay parameter passes through a critical value are determined. These conditions depend on the coeffici...

We study stability and Hopf bifurcation analysis of a model that refers to the competition between the immune system and an aggressive host such as a tumor. The model which describes this competition is governed by a reaction-diffusion system including time delay under the Neumann boundary conditions, and is based on Kuznetsov-Taylor's model. Choos...

We study the stability and Hopf bifurcation analysis of a coupled two-neuron system involving both discrete and distributed delays. First, we analyze stability of equilibrium point. Choosing delay term as a bifurcation parameter, we also show that Hopf bifurcation occurs under some conditions when the bifurcation parameter passes through a critical...

We present an algorithm for determining the existence of Hopf bifurcations of a system of general delayed reaction-diffusion equations with the Neumann boundary conditions. The conditions on parameters of the system that a Hopf bifurcation occurs as the delay parameter passes through a critical value are determined. These conditions depend on the c...

We study the stability and Hopf bifurcation analysis of an asset pricing model that is based on the model introduced by Caginalp and Balenovich, under the assumption of a fixed amount of cash and stock in the system. First, we analyze stability of equilibrium points. Choosing the momentum coefficient as a bifurcation parameter, we also show that Ho...

In this paper, we give a detailed Hopf bifurcation analysis of a recurrent neural network system involving both discrete and distributed delays. Choosing the sum of the discrete delay terms as a bifurcation parameter the existence of Hopf bifurcation is demonstrated. In particular, the formulae determining the direction of the bifurcations and the...

We investigate bifurcations of the Lengyel-Epstein reaction-diffusion model involving time delay under the Neumann boundary conditions. We first give stability and Hopf bifurcation analysis of the ODE models including delay associated with this model. Later, we extend these analysis to the PDE model. We determine conditions on parameters of both mo...

The book covers nonlinear physical problems and mathematical modeling, including molecular biology, genetics, neurosciences, artificial intelligence with classical problems in mechanics and astronomy and physics. The chapters present nonlinear mathematical modeling in life science and physics through nonlinear differential equations, nonlinear disc...

The model analyzed in this paper is based on the model set forth by Aziz Alaoui et al. [Aziz Alaoui & Daher Okiye, 2003; Nindjin et al., 2006] with time delay, which describes the competition between the predator and prey. This model incorporates a modified version of the Leslie-Gower functional response as well as that of Beddington-DeAngelis. In...

In this paper we give a detailed Hopf bifurcation analysis of a ratio-dependent predator–prey system involving two different discrete delays. By analyzing the characteristic equation associated with the model, its linear stability is investigated. Choosing delay terms as bifurcation parameters the existence of Hopf bifurcations is demonstrated. Sta...

We investigate bifurcations of the Lengyel-Epstein reaction-diffusion model involving time delay under the Neumann boundary conditions. Choosing the delay parameter as a bifurcation parameter, we show that Hopf bifurcation occurs. We also determine two properties of the Hopf bifurcation, namely direction and stability, by applying the normal form t...

The aim of this paper is to give a detailed analysis of Hopf bifurcation of a ratio-dependent predator-prey system involving two discrete delays. A delay parameter is chosen as bifurcation parameter for the analysis. Stability of the bifurcating periodic solutions is determined by using the center manifold theorem and the normal form theory introdu...

The effect of the high/low liquidity in the market on the asset price forecasting is studied by deriving a system of ordinary differential equations. The model is an extension of that introduced by Caginalp and Merdan for the system involving a single asset traded by heterogenous groups. Derivation is based on the finiteness of assets (rather than...

In this paper, we have considered a system of delay differential equations. The system without delayed arises in the Lengyel–Epstein model. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. Linear stability is investigated and existence of Hopf bifurcation is demonstrated via analyzing the associated characteristic...

This work represents Hopf bifurcation analysis of a general non-linear differential equation involving time delay. A special form of this equation is the Hutchinson–Wright equation which is a mile stone in the mathematical modeling of population dynamics and mathematical biology. Taking the delay parameter as a bifurcation parameter, Hopf bifurcati...

The aim of this paper is to investigate bifurcations of the Lengyel- Epstein reaction di¤usion model involving delay under Neumann bound- ary conditions. The bifurcation analysis of the model shows that Hopf bifurcation occurs by regarding the delay as the bifurcation parameter. Using the normal form theory and the center manifold reduction for par...

In this paper, we investigate stability conditions of equilibrium points of a general delay difference population model with and without Allee effects which occur at low population density. The analysis demonstrates that Allee effects have both stabilizing and destabilizing effects on population dynamics including time delay.

The stability conditions of equilibrium points of the population model Xt+1 = lambda X(t)f(Xt-3) with and without Allee effects are investigated. It is assumed that the Allee effect occurs at low population density. Analysis and numerical simulations show that Allee effects have both stabilizing and destabilizing effects on population dynamics with...

The stability conditions of equilibrium points of the population model Xt+1 = λXtf (Xt−3) with and without Allee effects are investigated. It is assumed that the Allee effect occurs at low population density. Analysis and numerical simulations show that Allee effects have both stabilizing and destabilizing effects on population dynamics with delay.

The stability conditions of equilibrium points of the population model Xt+1 = λXtf (Xt−3) with and without Allee effects are investigated. It is assumed that the Allee effect occurs at low population density. Analysis and numerical simulations show that Allee effects have both stabilizing and destabilizing effects on population dynamics with delay.

The stability conditions of equilibrium points of the population model Xt+1 = λXtf (Xt−3) with and without Allee effects are investigated. It is assumed that the Allee effect occurs at low population density. Analysis and numerical simulations show that Allee effects have both stabilizing and destabilizing effects on population dynamics with delay.

The stability conditions of equilibrium points of the population model Xt+1 = λXtf (Xt−3) with and without Allee effects are investigated. It is assumed that the Allee effect occurs at low population density. Analysis and numerical simulations show that Allee effects have both stabilizing and destabilizing effects on population dynamics with delay.

Asset price dynamics is studied by using a system of ordinary differential equations which is derived by utilizing a new excess demand function introduced by Caginalp [4] for a market involving more information on demand and supply for a stock rather than their values at a particular price. Derivation is based on the finiteness of assets (rather th...

We present a stability analysis of steady-state solutions of a continuous-time predator– prey population dynamics model subject to Allee effects on the prey population which occur at low population density. Numerical simulations show that the system subject to an Allee effect takes a much longer time to reach its stable steady-state solution. This...

We present a stability analysis of steady-state solutions of a continuous-time predator–prey population dynamics model subject to Allee effects on the prey population which occur at low population density. Numerical simulations show that the system subject to an Allee effect takes a much longer time to reach its stable steady-state solution. This r...

This paper presents the stability analysis of equilibrium points of a general continuous time population dynamics model involving predation subject to an Allee effect which occurs at low population density. The mathematical results and numerical simulations show that the system subject to an Allee effect takes much longer time to reach its stable s...

This paper presents the stability analysis of equilibrium points of a general discrete-time population dynamics involving predation with and without Allee effects which occur at low population density. The mathematical analysis and numerical simulations show that the Allee effect has a stabilizing role on the local stability of the positive equilib...

This paper presents the stability analysis of equilibrium points of a continuous population dynamics with delay under the Allee effect which occurs at low population density. The mathematical results and numerical simulations show the stabilizing role of the Allee effects on the stability of the equilibrium point of this population dynamics. Editor...

In this paper, we study the stability analysis of equilibrium points of population dynamics with delay when the Alice effect occurs at low population density. Mainly, our mathematical results and numerical simulations point to the stabilizing effect of the Allee effects on population dynamics with delay. (C) 2006 Elsevier Ltd. All rights reserved.

A system of ordinary differential equations is used to study the price dynamics of an asset under various conditions. One of these involves the introduction of new information that is interpreted differently by two groups. Another studies the price change due to a change in the number of shares. The steady state is examined under these conditions t...

We study the temporal evolution of an interface separating two phases for its large-time behavior by adapting renormalization group methods and scaling theory. We consider a full two-phase model in the quasi-static regime and implement a renormalization procedure in order to calculate the characteristic length of a self-similar system, R(t), that i...

Renormalization group and scaling theory have been used to determine the large time growth exponent for the characteristic length, R(t), of an interface in the form R(t) ∼ tβ. The exponent β is different in the two cases: quasi-static, in which the time derivative in the heat equation is suppressed, and the fully dynamic system. This paper examines...

Renormalization group (RG) methods are described for determining the key ex- ponents related to the decay of solutions to nonlinear parabolic dierential equations. Higher order (in the small coecient of the nonlinearity) methods are developed. Exact solutions and theorems in

The application of renormalization techniques to interface problems is considered after a brief review of the methodology. We study the standard sharp interface problem in the quasi-static limit (time derivative set to zero in the heat equation) for large time. The characteristic length, R(t), behaves as t β where β has values in the continuous spe...

Scaling and renormalization group (RG) methods are used to study parabolic equations with a small nonlinear term and find the decay exponents. The determination of decay exponents is viewed as an asymptoti-cally self similar process that facilitates an RG approach. These RG methods are extended to higher order in the small coefficient of the nonlin...

Scaling and renormalization group (RG) methods are used to study parabolic equations with a small nonlinear term and find the decay exponents. The determination of decay exponents is viewed as an asymptotically self similar process that facilitates an RG approach. These RG methods are extended to higher order in the small coe#cient of the nonlinear...

The stability conditions of equilibrium points of the population model Xt+1 = Xtf(Xt−3) with and without Allee effects are investigated. It is assumed that the Allee effect occurs at low population density. Analysis and numerical simulations show that Allee effects have both stabilizing and destabilizing effects on population dynamics with delay.

In this paper, we have considered a system of delay differential equations. The system without delayed arises in the Lengyel–Epstein model. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. Linear stability is investigated and existence of Hopf bifurcation is demonstrated via analyzing the associated characteristic...