97 research outputs found
Three-points interfacial quadrature for geometrical source terms on nonuniform grids
International audienceThis paper deals with numerical (finite volume) approximations, on nonuniform meshes, for ordinary differential equations with parameter-dependent fields. Appropriate discretizations are constructed over the space of parameters, in order to guarantee the consistency in presence of variable cells' size, for which -error estimates, , are proven. Besides, a suitable notion of (weak) regularity for nonuniform meshes is introduced in the most general case, to compensate possibly reduced consistency conditions, and the optimality of the convergence rates with respect to the regularity assumptions on the problem's data is precisely discussed. This analysis attempts to provide a basic theoretical framework for the numerical simulation on unstructured grids (also generated by adaptive algorithms) of a wide class of mathematical models for real systems (geophysical flows, biological and chemical processes, population dynamics)
The Speed of Fronts of the Reaction Diffusion Equation
We study the speed of propagation of fronts for the scalar reaction-diffusion
equation \, with . We give a new integral
variational principle for the speed of the fronts joining the state to
. No assumptions are made on the reaction term other than those
needed to guarantee the existence of the front. Therefore our results apply to
the classical case in , to the bistable case and to cases in
which has more than one internal zero in .Comment: 7 pages Revtex, 1 figure not include
Modelling Effects of Rapid Evolution on Persistence and Stability in Structured Predator-Prey Systems
In this paper we explore the eco-evolutionary dynamics of a predator-prey model, where the prey population is structured according to a certain life history trait. The trait distribution within the prey population is the result of interplay between genetic inheritance and mutation, as well as selectivity in the consumption of prey by the predator. The evolutionary processes are considered to take place on the same time scale as ecological dynamics, i.e. we consider the evolution to be rapid. Previously published results show that population structuring and rapid evolution in such predator-prey system can stabilize an otherwise globally unstable dynamics even with an unlimited carrying capacity of prey. However, those findings were only based on direct numerical simulation of equations and obtained for particular parameterizations of model functions, which obviously calls into question the correctness and generality of the previous results. The main objective of the current study is to treat the model analytically and consider various parameterizations of predator selectivity and inheritance kernel. We investigate the existence of a coexistence stationary state in the model and carry out stability analysis of this state. We derive expressions for the Hopf bifurcation curve which can be used for constructing bifurcation diagrams in the parameter space without the need for a direct numerical simulation of the underlying integro-differential equations. We analytically show the possibility of stabilization of a globally unstable predator-prey system with prey structuring. We prove that the coexistence stationary state is stable when the saturation in the predation term is low. Finally, for a class of kernels describing genetic inheritance and mutation we show that stability of the predator-prey interaction will require a selectivity of predation according to the life trait
OptiJ: Open-source optical projection tomography of large organ samples
The three-dimensional imaging of mesoscopic samples with Optical Projection Tomography (OPT) has become a powerful tool for biomedical phenotyping studies. OPT uses visible light to visualize the 3D morphology of large transparent samples. To enable a wider application of OPT, we present OptiJ, a low-cost, fully open-source OPT system capable of imaging large transparent specimens up to 13 mm tall and 8 mm deep with 50 µm resolution. OptiJ is based on off-the-shelf, easy-to-assemble optical components and an ImageJ plugin library for OPT data reconstruction. The software includes novel correction routines for uneven illumination and sample jitter in addition to CPU/GPU accelerated reconstruction for large datasets. We demonstrate the use of OptiJ to image and reconstruct cleared lung lobes from adult mice. We provide a detailed set of instructions to set up and use the OptiJ framework. Our hardware and software design are modular and easy to implement, allowing for further open microscopy developments for imaging large organ samples
Pushed and pulled fronts in a discrete reaction-diffusion equation
We consider the propagation of wave fronts connecting unstable and stable uniform solutions to a discrete reaction-diffusion equation on a one-dimensional integer lattice. The dependence of the wavespeed on the coupling strength µ between lattice points and on a detuning parameter (α) appearing in a nonlinear forcing is investigated thoroughly. Via asymptotic and numerical studies, the speed both of 'pulled' fronts (whereby the wavespeed can be characterised by the linear behaviour at the leading edge of the wave) and of 'pushed' fronts (for which the nonlinear dynamics of the entire front determine the wavespeed) is investigated in detail. The asymptotic and numerical techniques employed complement each other in highlighting the transition between pushed and pulled fronts under variations of µ and α
Renormalization Group Theory And Variational Calculations For Propagating Fronts
We study the propagation of uniformly translating fronts into a linearly
unstable state, both analytically and numerically. We introduce a perturbative
renormalization group (RG) approach to compute the change in the propagation
speed when the fronts are perturbed by structural modification of their
governing equations. This approach is successful when the fronts are
structurally stable, and allows us to select uniquely the (numerical)
experimentally observable propagation speed. For convenience and completeness,
the structural stability argument is also briefly described. We point out that
the solvability condition widely used in studying dynamics of nonequilibrium
systems is equivalent to the assumption of physical renormalizability. We also
implement a variational principle, due to Hadeler and Rothe, which provides a
very good upper bound and, in some cases, even exact results on the propagation
speeds, and which identifies the transition from ` linear'- to `
nonlinear-marginal-stability' as parameters in the governing equation are
varied.Comment: 34 pages, plain tex with uiucmac.tex. Also available by anonymous ftp
to gijoe.mrl.uiuc.edu (128.174.119.153), file /pub/front_RG.tex (or .ps.Z
Pushed traveling fronts in monostable equations with monotone delayed reaction
We study the existence and uniqueness of wavefronts to the scalar
reaction-diffusion equations with monotone delayed reaction term and . We are mostly interested in the situation when the graph of is not
dominated by its tangent line at zero, i.e. when the condition , is not satisfied. It is well known that, in such a case, a
special type of rapidly decreasing wavefronts (pushed fronts) can appear in
non-delayed equations (i.e. with ). One of our main goals here is to
establish a similar result for . We prove the existence of the minimal
speed of propagation, the uniqueness of wavefronts (up to a translation) and
describe their asymptotics at . We also present a new uniqueness
result for a class of nonlocal lattice equations.Comment: 17 pages, submitte
Evolutionary dynamics of Lewis signaling games: signaling systems vs. partial pooling
Transfer of information between senders and receivers, of one kind or another, is essential to all life. David Lewis introduced a game theoretic model of the simplest case, where one sender and one receiver have pure common interest. How hard or easy is it for evolution to achieve information transfer in Lewis signaling?. The answers involve surprising subtleties. We discuss some if these in terms of evolutionary dynamics in both finite and infinite populations, with and without mutation
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