Inhibitor-Induced Wavetrains and Spiral Waves in an Extended FitzHugh-Nagumo Model of Nerve Cell Dynamics.

It is well known that the FitzHugh-Nagumo model is one of the simplified forms of the four-variable Hodgkin-Huxley model that can reflect most of the significant phenomena of nerve cell action potential. However, this model cannot capture the irregular action potentials of sufficiently large periods in a one-parameter family of solutions. Motivated by this, we propose a modified FitzHugh-Nagumo reaction-diffusion system by changing its recovery kinetics. First, we investigate the parameter regime to know the existence of the wavetrains. Second, we conceive the occurrence of Eckhaus bifurcations of solutions that divide the solution region into two parts. The essential spectra at different grid points explore the occurrence of bifurcations of the waves. We find that the wavetrains of sufficiently large periods cross the stability boundary. This characteristic phenomenon is absent in the standard FitzHugh-Nagumo model. Finally, we observe a reasonable agreement between the direct PDE simulations and the solutions in the traveling wave ODEs. Furthermore, the model exhibits spiral wave for monotone and non-monotone cases that agrees with the waves observed in cellular activity.

Numerical bifurcation analysis and pattern formation in a minimal reaction-diffusion model for vegetation.

Model-aided understanding of the mechanism of vegetation patterns and desertification is one of the burning issues in the management of sustainable ecosystems. A pioneering model of vegetation patterns was proposed by C. A. Klausmeier in 1999 (Klausmeier, 1999) that involves a downhill flow of water. In this paper, we study the diffusive Klausmeier model that can describe the flow of water in flat terrain incorporating a diffusive flow of water. It consists of a two-component reaction-diffusion system for water and plant biomass. The paper presents a numerical bifurcation analysis of stationary solutions of the diffusive Klausmeier model extensively. We numerically investigate the occurrence of diffusion-driven instability and how this depends on the parameters of the model. Finally, the model predicts some field observed vegetation patterns in a semiarid environment, e.g. spot, stripe (labyrinth), and gap patterns in the transitions from bare soil at low precipitation to homogeneous vegetation at high precipitation. Furthermore, we introduce a two-component reaction-diffusion model considering a bilinear interaction of plant and water instead of their cubic interaction. It is inspected that no diffusion-driven instability occurs as if vegetation patterns can be generated. This confirms that the diffusive Klausmeier model is the minimal reaction-diffusion model for the occurrence of vegetation patterns from the viewpoint of a two-component reaction-diffusion system.

Modeling the dispersal effect to reduce the infection of COVID-19 in Bangladesh.

In this paper, we propose a four compartmental model to understand the dynamics of infectious disease COVID-19. We show the boundedness and non-negativity of solutions of the model. We analytically calculate the basic reproduction number of the model and perform the stability analysis at the equilibrium points to understand the epidemic and endemic cases based on the basic reproduction number. Our analytical results show that disease free equilibrium point is asymptotically stable (unstable) and endemic equilibrium point is unstable (asymptotically stable) if the basic reproduction number is less than (greater than) unity. The dispersal rate of the infected population and the social awareness control parameter are the main focus of this study. In our model, these parameters play a vital role to control the spread of COVID-19. Our results reveal that regional lockdown and social awareness (e.g., wearing a face mask, washing hands, social distancing) can reduce the pandemic of the current outbreak of novel coronavirus in a most densely populated country like Bangladesh.

Alternans and Spiral Breakup in an Excitable Reaction-Diffusion System: A Simulation Study.

The determination of the mechanisms of spiral breakup in excitable media is still an open problem for researchers. In the context of cardiac electrophysiological activities, spiral breakup exhibits complex spatiotemporal pattern known as ventricular fibrillation. The latter is the major cause of sudden cardiac deaths all over the world. In this paper, we numerically study the instability of periodic planar traveling wave solution in two dimensions. The emergence of stable spiral pattern is observed in the considered model. This pattern occurs when the heart is malfunctioning (i.e., ventricular tachycardia). We show that the spiral wave breakup is a consequence of the transverse instability of the planar traveling wave solutions. The alternans, that is, the oscillation of pulse widths, is observed in our simulation results. Moreover, we calculate the widths of spiral pulses numerically and observe that the stable spiral pattern bifurcates to an oscillatory wave pattern in a one-parameter family of solutions. The spiral breakup occurs far below the bifurcation when the maximum and the minimum excited states become more distinct, and hence the alternans becomes more pronounced.