Dr. Orhan Ozgur Aybar

Title:
Distinct Roles of Fear and Density-Dependent Carrying Capacity in Predator-Prey Dynamics: A Continuous and Discrete-Time Analysis

Abstract:
We investigate a predator-prey system in which prey growth is simultaneously influenced by two biologically distinct mechanisms: a fear effect associated with perceived predation risk and a prey-density-dependent carrying capacity of the form (K(x)=k+\beta x). By studying continuous- and discrete-time formulations in parallel, we show that these two mechanisms play fundamentally different dynamical roles.

For the continuous-time system, we establish positivity and boundedness of solutions, characterize the equilibria, and analyze Hopf and transcritical bifurcations. Increasing the habitat-feedback parameter (\beta) destabilizes the coexistence equilibrium and promotes oscillatory behavior, while the fear parameter (f) leaves the continuous Hopf threshold unchanged but monotonically suppresses the equilibrium predator density.

The corresponding discrete-time model exhibits a richer dynamical structure. Fear lowers the Neimark-Sacker bifurcation threshold, demonstrating that its effect on stability differs substantially between continuous and discrete formulations. Numerical investigations further reveal quasi-periodic dynamics, mode-locked periodic regions, Arnold-tongue structures, and multistability with finely interdigitated basins of attraction.

These results demonstrate that density-dependent environmental feedback primarily controls stability and oscillation amplitude, whereas fear predominantly regulates predator abundance and has a formulation-dependent influence on stability. The study therefore highlights the importance of distinguishing ecological mechanisms from the mathematical time representation used to model them.

dr. İlknur Kuşbeyzi Aybar

Title:
Oscillatory dynamics in the adaptive exponential integrate-and-fire model

Abstract:
In this study, we investigate the local oscillatory dynamics of the Adaptive Exponential Integrate-and-Fire (AdEx) model. The exponential nonlinearity is approximated by quadratic and cubic polynomials with controlled error bounds, allowing an analytically tractable formulation. We derive explicit conditions for the existence and stability of equilibria and characterize Hopf bifurcations analytically. The first Lyapunov coefficient is obtained in closed form to determine the criticality of the Hopf bifurcation. We also derive leading-order period coefficients describing the amplitude dependence of periodic solutions near the bifurcation. For the cubic approximation, parameter regimes with one or three equilibria and the associated multistability transitions are characterized. Finally, the analytical predictions and local validity of the polynomial approximations are assessed against the full exponential AdEx model.

dr. Sándor Kovács

Title:
Delay-Induced Instability in a Mathematical Model of SARS-CoV-2 Transmission
(Authors: Sándor Kovács, Mónika Madai, Szilvia György)

Abstract:
Essential oils may delay the transmission of airborne viruses and, as a consequence, slow their spread. This effect may also be relevant in the case of SARS-CoV-2. We investigate the role of such delays in a mathematical model of viral transmission, considering two different ways of incorporating a time delay into the model. Our analysis shows that the two approaches can lead to different dynamical behaviour: in one case, the equilibria remain stable for any delay, while in the other, a Hopf bifurcation occurs when the delay reaches a critical value.