Identification of the initial population of a nonlinear predator-prey system backwards in time

We study for the first time the ill-posed backward problem for a contaminated nonlinear predator-prey system whose velocities of migration depend on the total average populations in the considered space domain. We propose a new regularized problem for which we are able to prove its unique solvability in Theorem 1. Moreover, under some mild assumptions on the true solution, we give useful and rigorous error estimates and convergence rates in both the L² –and H² – norms in Theorems 2 and 3, respectively. Furthermore, numerical simulations are performed to illustrate the accuracy and stability of the regularized solution

The morpho-kinematics of the circumstellar envelope around the AGB star EP Aqr

ALMA observations of CO(1-0) and CO(2-1) emissions of the circumstellar envelope of EP Aqr, an oxygen-rich AGB star, are reported. A thorough analysis of their properties is presented using an original method based on the separation of the data-cube into a low velocity component associated with an equatorial outflow and a faster component associated with a bipolar outflow. A number of important and new results are obtained concerning the distribution in space of the effective emissivity, the temperature, the density and the flux of matter. A mass loss rate of (1.6$\pm$0.4)10$^{-7}$ solar masses per year is measured. The main parameters defining the morphology and kinematics of the envelope are evaluated and uncertainties inherent to de-projection are critically discussed. Detailed properties of the equatorial region of the envelope are presented including a measurement of the line width and a precise description of the observed inhomogeneity of both morphology and kinematics. In particular, in addition to the presence of a previously observed spiral enhancement of the morphology at very small Doppler velocities, a similarly significant but uncorrelated circular enhancement of the expansion velocity is revealed, both close to the limit of sensitivity. The results of the analysis place significant constraints on the parameters of models proposing descriptions of the mass loss mechanism, but cannot choose among them with confidence.Comment: 26 pages, 31 figures, accepted for publication in MNRA