345 research outputs found
Prevalent Behavior of Strongly Order Preserving Semiflows
Classical results in the theory of monotone semiflows give sufficient
conditions for the generic solution to converge toward an equilibrium or
towards the set of equilibria (quasiconvergence). In this paper, we provide new
formulations of these results in terms of the measure-theoretic notion of
prevalence. For monotone reaction-diffusion systems with Neumann boundary
conditions on convex domains, we show that the set of continuous initial data
corresponding to solutions that converge to a spatially homogeneous equilibrium
is prevalent. We also extend a previous generic convergence result to allow its
use on Sobolev spaces. Careful attention is given to the measurability of the
various sets involved.Comment: 18 page
Numerical Continuation and Bifurcation Analysis in a Harvested Predator-Prey Model with Time Delay using DDE-Biftool
Time delay has been incorporated in models to reflect certain physical or
biological meaning. The theory of delay differential equations (DDEs), which
has seen extensive growth in the last seventy years or so, can be used to
examine the effects of time delay in the dynamical behavior of systems being
considered. Numerical tools to study DDEs have played a significant role not
only in illustrating theoretical results but also in discovering interesting
dynamics of the model. DDE-Biftool, which is a Matlab package for numerical
continuation and numerical bifurcation analysis of DDEs, is one of the most
utilized and popular numerical tools for DDEs. In this paper, we present a
guide to using the latest version of DDE-Biftool targeted to researchers who
are new to the study of time delay systems. A short discussion of an example
application, which is a harvested predator-prey model with a single discrete
time delay, will be presented first. We then implement this example model in
DDE-Biftool, pointing out features where beginners need to be cautious. We end
with a comparison of our theoretical and numerical results
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