arXiv subjects

Jorge M Ramirez

Publications and source records attributed to Jorge M Ramirez.

4 recordsLinked to original sources

arXiv2019

Modeling tropotaxis in ant colonies: recruitment and trail formation

We propose an active walker model for the motion of individual ants communicating via chemical signals. It is assumed that communication takes the form of a time-dependent pheromone field that feedbacks into the motion ants through tropotaxis: individuals can sense the gradient of the pheromone concentration field and adjust their orientation accordingly. The individual model takes the form of a Langevin system of equations in polar coordinates driven by two-dimensional Gaussian fluctuations and with orientation changes in response to two pheromone fields: one emanating from the nest, and other actively produced by ants in their nest-bound journey after finding a food source. We explicitly track the evolution of both fields in three dimensions. The proposed tropotaxis model relating the pheromone field to the orientation changes is similar to Weber's law, but depends explicitly only on the gradient of the pheromone concentration. We identify ranges of values for the model parameters that yield the emergence of two key foraging patterns: successful recruitment to newly found sources, and colony-wide trail networks.

q-bio.PE↗

arXiv2018

Dynamics of drainage under stochastic rainfall in river networks

We consider a linearized dynamical system modelling the flow rate of water along the rivers and hillslopes of an arbitrary watershed. The system is perturbed by a random rainfall in the form of a compound Poisson process. The model describes the evolution, at daily time scales, of an interconnected network of linear reservoirs and takes into account the differences in flow celerity between hillslopes and streams as well as their spatial variation. The resulting stochastic process is a piece-wise deterministic Markov process of the Orstein-Uhlembeck type. We provide an explicit formula for the Laplace transform of the invariant density of streamflow in terms of the geophysical parameters of the river network and the statistical properties of the precipitation field. As an application, we include novel formulas for the invariant moments of the streamflow at the watershed's outlet, as well as the asymptotic behavior of extreme discharge events.

math.PR↗

arXiv2011

Green's functions for Sturm-Liouville problems on directed tree graphs

Let $Γ$ be geometric tree graph with $m$ edges and consider the second order Sturm-Liouville operator $Ł[u]=(-pu')'+qu$ acting on functions that are continuous on all of $Γ$, and twice continuously differentiable in the interior of each edge. The functions $p$ and $q$ are assumed uniformly continuous on each edge, and $p$ strictly positive on $Γ$. The problem is to find a solution $f:Γ\to \R$ to the problem $Ł[f] = h$ with $2m$ additional conditions at the nodes of $Γ$. These node conditions include continuity at internal nodes, and jump conditions on the derivatives of $f$ with respect to a positive measure $ρ$. Node conditions are given in the form of linear functionals $ł_1,...,ł_{2m}$ acting on the space of admissible functions. A novel formula is given for the Green's function $G:Γ\times Γ\to \R$ associated to this problem. Namely, the solution to the semi-homogenous problem $Ł[f] = h$, $ł_i[f] =0$ for $i=1,...,2m$ is given by $f(x) = \int_ΓG(x,y) h(y) \ud ρ$.

math.CA↗

arXiv2011

Population persistence under advection-diffusion in river networks

An integro-differential equation on a tree graph is used to model the evolution and spatial distribution of a population of organisms in a river network. Individual organisms become mobile at a constant rate, and disperse according to an advection-diffusion process with coefficients that are constant on the edges of the graph. Appropriate boundary conditions are imposed at the outlet and upstream nodes of the river network. The local rates of population growth/decay and that by which the organisms become mobile, are assumed constant in time and space. Imminent extinction of the population is understood as the situation whereby the zero solution to the integro-differential equation is stable. Lower and upper bounds for the eigenvalues of the dispersion operator, and related Sturm-Liouville problems are found, and therefore sufficient conditions for imminent extinction are given in terms of the physical variables of the problem.

q-bio.PE↗