Vicenç Méndez

Wavefronts in time-delayed reaction-diffusion systems. Theory and comparison to experiment

We review the recent theoretical progress in the formulation and solution of the front speed problem for time-delayed reaction-diffusion systems. Most of the review is focused on hyperbolic equations. They have been widely used in recent years, because they allow for analytical solutions and yield a realistic description of some relevant phenomena. The theoretical methods are applied to a range of applications, including population dynamics, forest fire models, bistable systems and combustion wavefronts. We also present a detailed account of successful predictions of the models, as assessed by comparison to experimental data for some biophysical systems, without making use of any free parameters. Areas where the work reviewed may contribute to future progress are discussed.

Cosmological solutions with nonlinear bulk viscosity

A recently proposed nonlinear transport equation is used to model bulk viscous cosmologies that may be far from equilibrium, as happens during viscous fluid inflation or during reheating. The asymptotic stability of the de Sitter and Friedmann solutions is investigated. The former is stable for bulk viscosity index q 1. New solutions are obtained in the weakly nonlinear regime for q = 1. These solutions are singular and some of them represent a late-time inflationary era.

Nonlinear bulk viscosity and inflation

We develop a nonlinear generalization of the causal linear thermodynamics of bulk viscosity, incorporating the positivity of the entropy production rate and the effective specific entropy. The theory is applied to viscous fluid inflation (which is necessarily far from equilibrium), and we find thermodynamically consistent inflationary solutions, both exponential and power law.

THE ROLE OF THE DELAY TIME IN THE MODELING OF BIOLOGICAL RANGE EXPANSIONS

The time interval between successive migrations of biological species causes a delay time in the reaction-diffusion equations describing their space-time dynamics. This lowers the predicted speed of the waves of advance, as compared to classical models. It has been shown that this delay-time effect improves the modeling of human range expan- sions. Here, we demonstrate that it can also be important for other species. We present two new examples where the predictions of the time-delayed and the classical (Fisher) ap- proaches are compared to experimental data. No free or adjustable parameters are used. We show that the importance of the delay effect depends on the dimensionless product of the initial growth rate and the delay time. We argue that the delay effect should be taken into account in the modeling of range expansions for biological species.

Dynamical evolution of discrete epidemic models

The dynamical evolution of a deterministic epidemic model is studied. We model the system by taking into account an intermediate class of population, say infected, which after a latency period enter into the infectious class. A mortality rate induced by the disease is also considered. The system evolves towards a final state which may develop a catastrophic epidemic if small outbreaks of the disease emerge. The final size, the threshold and the severity of the epidemic are also analyzed and calculated for this model.

Comb-like models for transport along spiny dendrites

We suggest a modification of a comb model to describe anomalous transport in spiny dendrites. Geometry of the comb structure consisting of a one-dimensional backbone and lateral branches makes it possible to describe anomalous diffusion, where dynamics inside fingers corresponds to spines, while the backbone describes diffusion along dendrites. The presented analysis establishes that the fractional dynamics in spiny dendrites is controlled by fractal geometry of the comb structure and fractional kinetics inside the spines. Our results show that the transport along spiny dendrites is subdiffusive and depends on the density of spines in agreement with recent experiments.

Extinction conditions for isolated populations with Allee effect

One of the main ecological phenomenons is the Allee effect [1-3], in which a positive benefit from the presence of conspecifics arises. In this work we describe the dynamical behavior of a population with Allee effect in a finite domain that is surrounded by a completely hostile environment. Using spectral methods to rewrite the local density of habitants we are able to determine the critical patch size and the bifurcation diagram, hence characterizing the stability of possible solutions, for different ways to introduce the Allee effect in the reaction-diffusion equations.

NEW COSMOLOGICAL SOLUTIONS AND STABILITY ANALYSIS IN FULL EXTENDED THERMODYNAMICS

The Einsteins field equations of FRW universes filled with a dissipative fluid described by full theory of causal transport equations are analyzed. New exact solutions are found using a nonlocal transformation on the nonlinear differential equation for the Hubble factor. The stability of the de Sitter and asymptotically Friedmannian solutions are analyzed using Lyapunov function method.

Foraging success under uncertainty: search tradeoffs and optimal space use

Understanding the structural complexity and the main drivers of animal search behaviour is pivotal to foraging ecology. Yet, the role of uncertainty as a generative mechanism of movement patterns is poorly understood. Novel insights from search theory suggest that organisms should collect and assess new information from the environment by producing complex exploratory strategies. Based on an extension of the first passage time theory, and using simple equations and simulations, we unveil the elementary heuristics behind search behaviour. In particular, we show that normal diffusion is not enough for determining optimal exploratory behaviour but anomalous diffusion is required. Searching organisms go through two critical sequential phases (approach and detection) and experience fundamental search tradeoffs that may limit their encounter rates. Using experimental data, we show that biological search includes elements not fully considered in contemporary physical search theory. In particular, the need to consider search movement as a non-stationary process that brings the organism from one informational state to another. For example, the transition from remaining in an area to departing from it may occur through an exploratory state where cognitive search is challenged. Therefore, a more comprehensive view of foraging ecology requires including current perspectives about movement under uncertainty.

Density-dependent dispersal and population aggregation patterns

We have derived reaction-dispersal-aggregation equations from Markovian reaction-random walks with density-dependent jump rate or density-dependent dispersal kernels. From the corresponding diffusion limit we recover well-known reaction-diffusion-aggregation and reaction-diffusion-advection-aggregation equations. It is found that the ratio between the reaction and jump rates controls the onset of spatial patterns. We have analyzed the qualitative properties of the emerging spatial patterns. We have compared the conditions for the possibility of spatial instabilities for reaction-dispersal and reaction-diffusion processes with aggregation and have found that dispersal process is more stabilizing than diffusion. We have obtained a general threshold value for dispersal stability and have analyzed specific examples of biological interest.