Global stability of a quadratic anti-competitive system of rational difference equations in the plane with Allee effects

We investigate global dynamics of the following systems of difference equations (Formula presented) where the parameters a, b are positive numbers and initial conditions x0 and y0 are arbitrary nonnegative numbers. We find all possible dynamical scenario for this system. We show that this system has substantially different behavior than the corresponding linear fractional system

Bifurcation and global dynamics of a leslie-gower type competitive system of rational difference equations with quadratic terms

We investigate global dynamics of the following systems of difference equations xn+1 = xn/(A1 + B1xn + C1yn), yn+1+1 = yn2/(A2 + B2xn + C2ynP), n = 0, 1, ., where the parameters A1, A2, B1, B2, C1, and C2 are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers.This system is a version of the Leslie-Gower competition model for two species. We show that this system has rich dynamics which depends on the part of parametric space

Investigation of the interaction of a turbulent impinging jet and a heated, rotating disk

ACLInternational audienceThe case of a turbulent round jet impinging perpendicularly onto a rotating, heated disc is investigated, in order to understand the mechanisms at the origin of the influence of rotation on the radial wall jet and associated heat transfer. The present study is based on the complementary use of an analysis of the orders of magnitude of the terms of the mean momentum and Reynolds stress transport equations, available experiments and dedicated Reynolds-Averaged Navier--Stokes (RANS) computations with refined turbulence models. The Reynolds number Rej = 14,500, the orifice-to-plate distance H =5 D, where D is the jet-orifice diameter, and the four rotation rates were chosen to match the experiments of Minagawa and Obi [``Development of turbulent impinging jet on a rotating disk,'' Int. J. Heat Fluid Fl. 25, 759-766 (2004)] and comparisons are made with the Nusselt number distribution measured by Popiel and Boguslawski [``Local heat transfer from a rotating disk in an impinging round jet,'' J. Heat Transf. 108, 357-364 (1986)], at a higher Reynolds number. The overestimation of turbulent mixing in the free-jet before the impact on the disk is detrimental to the prediction of the impingement region, in particular of the Nusselt number close to the symmetry axis, but the self-similar wall jet developing along the disk is correctly reproduced by the models. The analysis, experiments and computations show that the rotational effect do not directly affect the outer layer, but only the inner layer of the wall jet. A noteworthy consequence is that entrainment at the outer edge of the wall jet is insensitive to rotation, which explains the dependence of the wall-jet thickness on the inverse of the non-dimensional rotation rate, observed in the experiments and the Reynolds stress model computations, but not reproduced by the eddy-viscosity models, due to the algebraic dependence to the mean flow. The analysis makes moreover possible the identification of a scenario for the appearance of rotational effects when the rotation rate is gradually increased. For weak rotation rates, the rotation-induced boundary layer appears but does not break the self-similar solution observed for the case without rotation. For intermediate rotation rates, the production of the azimuthal Reynolds stress becomes much stronger than other components, leading to a complete modification of the turbulence anisotropy which is reproduced only by Reynolds stress models. For strong rotation rates, centrifugal effects dominate, leading to an acceleration and thinning of the jet, and consequently an increase of turbulent production and heat transfer, reproduced by all the turbulence models