19,885 research outputs found
Aphids, Ants and Ladybirds: a mathematical model predicting their population dynamics
The interaction between aphids, ants and ladybirds has been investigated from
an ecological point of view since many decades, while there are no attempts to
describe it from a mathematical point of view. This paper introduces a new
mathematical model to describe the within-season population dynamics in an
ecological patch of a system composed by aphids, ants and ladybirds, through a
set of four differential equations. The proposed model is based on the
Kindlmann and Dixon set of differential equations, focused on the prediction of
the aphids-ladybirds population densities, that share a prey-predator
relationship. The population of ants, in mutualistic relationship with aphids
and in interspecific competition with ladybirds, is described according to the
Holland and De Angelis mathematical model, in which the authors faced the
problem of mutualistic interactions in general terms. The set of differential
equations proposed here is discretized by means the Nonstandard Finite
Difference scheme, successfully applied by Gabbriellini to the mutualistic
model. The constructed finite-difference scheme is positivity-preserving and
characterized by four nonhyperbolic steady-states, as highlighted by the
phase-space and time-series analyses. Particular attention is dedicated to the
steady-state most interesting from an ecological point of view, whose
asymptotic stability is demonstrated via the Centre Manifold Theory. The model
allows to numerically confirm that mutualistic relationship effectively
influences the population dynamic, by increasing the peaks of the aphids and
ants population densities. Nonetheless, it is showed that the asymptotical
populations of aphids and ladybirds collapse for any initial condition, unlike
that of ants that, after the peak, settle on a constant asymptotic value
Accurately model the Kuramoto--Sivashinsky dynamics with holistic discretisation
We analyse the nonlinear Kuramoto--Sivashinsky equation to develop accurate
discretisations modeling its dynamics on coarse grids. The analysis is based
upon centre manifold theory so we are assured that the discretisation
accurately models the dynamics and may be constructed systematically. The
theory is applied after dividing the physical domain into small elements by
introducing isolating internal boundaries which are later removed.
Comprehensive numerical solutions and simulations show that the holistic
discretisations excellently reproduce the steady states and the dynamics of the
Kuramoto--Sivashinsky equation. The Kuramoto--Sivashinsky equation is used as
an example to show how holistic discretisation may be successfully applied to
fourth order, nonlinear, spatio-temporal dynamical systems. This novel centre
manifold approach is holistic in the sense that it treats the dynamical
equations as a whole, not just as the sum of separate terms.Comment: Without figures. See
http://www.sci.usq.edu.au/staff/aroberts/ksdoc.pdf to download a version with
the figure
On fast-slow consensus networks with a dynamic weight
We study dynamic networks under an undirected consensus communication
protocol and with one state-dependent weighted edge. We assume that the
aforementioned dynamic edge can take values over the whole real numbers, and
that its behaviour depends on the nodes it connects and on an extrinsic slow
variable. We show that, under mild conditions on the weight, there exists a
reduction such that the dynamics of the network are organized by a
transcritical singularity. As such, we detail a slow passage through a
transcritical singularity for a simple network, and we observe that an exchange
between consensus and clustering of the nodes is possible. In contrast to the
classical planar fast-slow transcritical singularity, the network structure of
the system under consideration induces the presence of a maximal canard. Our
main tool of analysis is the blow-up method. Thus, we also focus on tracking
the effects of the blow-up transformation on the network's structure. We show
that on each blow-up chart one recovers a particular dynamic network related to
the original one. We further indicate a numerical issue produced by the slow
passage through the transcritical singularity
Riding a Spiral Wave: Numerical Simulation of Spiral Waves in a Co-Moving Frame of Reference
We describe an approach to numerical simulation of spiral waves dynamics of
large spatial extent, using small computational grids.Comment: 15 pages, 14 figures, as accepted by Phys Rev E 2010/03/2
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