Browsing by Author "Merdan, H."

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An algorithm for Hopf bifurcation analysis of a delayed reaction-diffusion model
(Springer, 2017) Kayan, Şeyma; Merdan, H.; 49206
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 coefficients of the characteristic equation corresponding to linearization of the system. Furthermore, an algorithm to obtain the formulas for determining the direction of the Hopf bifurcation, the stability, and period of the periodic solution is given by using the Poincare normal form and the center manifold theorem. Finally, we give several examples and some numerical simulations to show the effectiveness of the algorithm proposed.
Bifurcation analysis of a modified tumor-immune system ınteraction model ınvolving time delay
(EDP Sciences A., 2017) Kayan, Ş.; Merdan, H.; Yafia, R.; Göktepe, S.
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. Choosing the delay parameter as a bifurcation parameter, we first show that Hopf bifurcation occurs. Second, we determine two properties of the periodic solution, namely its direction and stability, by applying the normal form theory and the center manifold reduction for partial functional differential equations. Furthermore, we discuss the effects of diffusion on the dynamics by analyzing a model with constant coefficients and perform some numerical simulations to support the analytical results. The results show that diffusion has an important effects on the dynamics of a mathematical model.
Effects of the random walk and the maturation period in a diffusive predator–prey system with two discrete delays
(2023) Bilazeroğlu, Şeyma; Göktepe, S.; Merdan, H.; 49206
This study aims to present a complete Hopf bifurcation analysis of a model describing the relationship between prey and predator. A ratio-dependent reaction–diffusion system with two discrete time delays operating under Neumann boundary conditions governs the model that represents this competition. The bifurcation parameter for the analysis is a delay parameter that reflects the amount of time needed for the predator to be able to hunt. Bilazeroğlu and Merdan's algorithm (Bilazeroğlu et al., 2021), which is developed by using the center manifold theorem and normal form theory, is used to establish the existence of Hopf bifurcations and also the stability of the bifurcating periodic solutions. The same procedure is used to illustrate some specific bifurcation properties, such as direction, stability, and period. Furthermore, by examining a model with constant coefficients, we also analyze how diffusion and the amount of time needed for prey to mature impact the model's dynamics. To support the obtained analytical results, we also run some numerical simulations. The results indicate that the dynamic of the mathematical model is significantly influenced by diffusion, the amount of time needed for the predator to gain the capacity to hunt, and the amount of time required for prey to reach maturity that the predator can hunt.
Hopf bifurcations in a class of reaction-diffusion equations including two discrete time delays: An algorithm for determining Hopf bifurcation, and its applications
(2021) Bilazeroğlu, Şeyma; Merdan, H.; 49206
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 equilibrium point. Bifurcation analysis is carried out by choosing one of the delay parameter as a bifurcation parameter and fixing the other in its stability interval. Second, some properties of periodic solutions such as direction of Hopf bifurcation and stability of bifurcating periodic solution are studied through the normal form theory and the center manifold reduction for functional partial differential equations. Moreover, an algorithm is developed in order to determine the existence of Hopf bifurcation (and its properties) of variety of system of reaction-diffusion equations that lie in the same class. The benefit of this algorithm is that it puts a very complex and long computations of existence of Hopf bifurcation for each equation in that class into a systematic schema. In other words, this algorithm consists of the conditions and formulae that are useful for completing the existence analysis of Hopf bifurcation by only using coefficients in the characteristic equation of the linearized system. Similarly, it is also useful for determining the direction analysis of the Hopf bifurcation merely by using the coefficients of the second degree Taylor polynomials of functions in the right hand side of the system. Finally, the existence of Hopf bifurcation for three different problems whose governing equations stay in that class is given by utilizing the algorithm derived, and thus the feasibility of the algorithm is presented. © 2020 Elsevier Ltd
Hopf bifurcations in Lengyel-Epstein reaction-diffusion model with discrete time delay
(Springer, 2015) Merdan, H.; Kayan, Şeyma
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 theory and the center manifold reduction for partial functional differential equations.