Bifurcation analysis of a normal form for excitable media: Are stable dynamical alternans on a ring possible?

We present a bifurcation analysis of a normal form for travelling waves in one-dimensional excitable media. The normal form which has been recently proposed on phenomenological grounds is given in form of a differential delay equation. The normal form exhibits a symmetry preserving Hopf bifurcation which may coalesce with a saddle-node in a Bogdanov-Takens point, and a symmetry breaking spatially inhomogeneous pitchfork bifurcation. We study here the Hopf bifurcation for the propagation of a single pulse in a ring by means of a center manifold reduction, and for a wave train by means of a multiscale analysis leading to a real Ginzburg-Landau equation as the corresponding amplitude equation. Both, the center manifold reduction and the multiscale analysis show that the Hopf bifurcation is always subcritical independent of the parameters. This may have links to cardiac alternans which have so far been believed to be stable oscillations emanating from a supercritical bifurcation. We discuss the implications for cardiac alternans and revisit the instability in some excitable media where the oscillations had been believed to be stable. In particular, we show that our condition for the onset of the Hopf bifurcation coincides with the well known restitution condition for cardiac alternans.Comment: to be published in Chao

INTERACTION BETWEEN WATER AND PLANTS: RICH DYNAMICS IN A SIMPLE MODEL

An ordinary differential equation model describing interaction of water and plants in ecosystem is proposed. Despite its simple looking, it is shown that the model possesses surprisingly rich dynamics including multiple stable equilibria, backward bifurcation of positive equilibria, supercritical or subcritical Hopf bifurcations, bubble loop of limit cycles, homoclinic bifurcation and Bogdanov-Takens bifurcation. We classify bifurcation diagrams of the system using the rain-fall rate as bifurcation parameter. In the transition from global stability of bare-soil state for low rain-fall to the global stability of high vegetation state for high rain-fall rate, oscillatory states or multiple equilibrium states can occur, which can be viewed as a new indicator of catastrophic environmental shift

Advanced Nonlinear Dynamics of Population Biology and Epidemiology

abstract: Modern biology and epidemiology have become more and more driven by the need of mathematical models and theory to elucidate general phenomena arising from the complexity of interactions on the numerous spatial, temporal, and hierarchical scales at which biological systems operate and diseases spread. Epidemic modeling and study of disease spread such as gonorrhea, HIV/AIDS, BSE, foot and mouth disease, measles, and rubella have had an impact on public health policy around the world which includes the United Kingdom, The Netherlands, Canada, and the United States. A wide variety of modeling approaches are involved in building up suitable models. Ordinary differential equation models, partial differential equation models, delay differential equation models, stochastic differential equation models, difference equation models, and nonautonomous models are examples of modeling approaches that are useful and capable of providing applicable strategies for the coexistence and conservation of endangered species, to prevent the overexploitation of natural resources, to control disease’s outbreak, and to make optimal dosing polices for the drug administration, and so forth.View the article as published at https://www.hindawi.com/journals/aaa/2014/214514