Non-linear eigenvalue problems arising from growth maximization of positive linear dynamical systems

We study a growth maximization problem for a continuous time positive linear system with switches. This is motivated by a problem of mathematical biology (modeling growth-fragmentation processes and the PMCA protocol). We show that the growth rate is determined by the non-linear eigenvalue of a max-plus analogue of the Ruelle-Perron-Frobenius operator, or equivalently, by the ergodic constant of a Hamilton-Jacobi (HJ) partial differential equation, the solutions or subsolutions of which yield Barabanov and extremal norms, respectively. We exploit contraction properties of order preserving flows, with respect to Hilbert's projective metric, to show that the non-linear eigenvector of the operator, or the "weak KAM" solution of the HJ equation, does exist. Low dimensional examples are presented, showing that the optimal control can lead to a limit cycle.Comment: 8 page

Spatially-constrained clustering of ecological networks

Spatial ecological networks are widely used to model interactions between georeferenced biological entities (e.g., populations or communities). The analysis of such data often leads to a two-step approach where groups containing similar biological entities are firstly identified and the spatial information is used afterwards to improve the ecological interpretation. We develop an integrative approach to retrieve groups of nodes that are geographically close and ecologically similar. Our model-based spatially-constrained method embeds the geographical information within a regularization framework by adding some constraints to the maximum likelihood estimation of parameters. A simulation study and the analysis of real data demonstrate that our approach is able to detect complex spatial patterns that are ecologically meaningful. The model-based framework allows us to consider external information (e.g., geographic proximities, covariates) in the analysis of ecological networks and appears to be an appealing alternative to consider such data

Deuterated molecules in DM Tau: DCO+, but no HDO

We report the detection of the J=2-1 line of DCO+ in the proto-planetary disk of DM Tau and re-analyze the spectrum covering the 465 GHz transition of HDO in this source, recently published by Ceccarelli et al. (2005). A modelling of the DCO+ line profile with the source parameters derived from high resolution HCO+ observations yields a DCO+/HCO+ abundance ratio of about 0.004, an order of magnitude smaller than that derived in the low mass cores. The re-analysis of the 465 GHz spectrum, using the proper continuum flux (0.5 Jy) and source systemic velocity (6.05 km/s), makes it clear that the absorption features attributed to HDO and C6H are almost certainly unrelated to these species. We show that the line-to-continuum ratio of an absorption line in front of a Keplerian disk can hardly exceed the ratio of the turbulent velocity to the projected rotation velocity at the disk edge, unless the line is optically very thick (tau > 10 000). This ratio is typically 0.1-0.3 in proto-planetary disks and is about 0.15 in DM Tau, much smaller than that for the alleged absorption features. We also show that the detection of H2D+ in DM Tau, previously reported by these authors, is only a 2-sigma detection when the proper velocity is adopted. So far, DCO+ is thus the only deuterated molecule clearly detected in proto-planetary disks

Simulation of fully resolved finite-size particles in a turbulent flow

In this study we are interested on the statistics of finite sized particles. A sustained homogeneous isotropic turbulence with particles is numerically simulated in order to obtain those statistics