daniel campos

University of Cambridge, Faculty of Modern and Medieval Languages, Undergraduate

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Papers by daniel campos

Nonstationary dynamics of encounters: Mean valuable territory covered by a random searcher

Physical Review E, 2017

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Optimal search strategies of space-time coupled random walkers with finite lifetimes

Physical Review E, 2015

We present a simple paradigm for detection of an immobile target by a space-time coupled random w... more We present a simple paradigm for detection of an immobile target by a space-time coupled random walker with a finite lifetime. The motion of the walker is characterized by linear displacements at a fixed speed and exponentially distributed duration, interrupted by random changes in the direction of motion and resumption of motion in the new direction with the same speed. We call these walkers "mortal creepers." A mortal creeper may die at any time during its motion according to an exponential decay law characterized by a finite mean death rate ω m. While still alive, the creeper has a finite mean frequency ω of change of the direction of motion. In particular, we consider the efficiency of the target search process, characterized by the probability that the creeper will eventually detect the target. Analytic results confirmed by numerical results show that there is an ω m-dependent optimal frequency ω = ω opt that maximizes the probability of eventual target detection. We work primarily in one-dimensional (d = 1) domains and examine the role of initial conditions and of finite domain sizes. Numerical results in d = 2 domains confirm the existence of an optimal frequency of change of direction, thereby suggesting that the observed effects are robust to changes in dimensionality. In the d = 1 case, explicit expressions for the probability of target detection in the long time limit are given. In the case of an infinite domain, we compute the detection probability for arbitrary times and study its early-and late-time behavior. We further consider the survival probability of the target in the presence of many independent creepers beginning their motion at the same location and at the same time. We also consider a version of the standard "target problem" in which many creepers start at random locations at the same time.

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Characterization of stationary states in random walks with stochastic resetting

Physical Review E, 2016

It is known that introducing a stochastic resetting in a random-walk process can lead to the emer... more It is known that introducing a stochastic resetting in a random-walk process can lead to the emergence of a stationary state. Here we study this point from a general perspective through the derivation and analysis of mesoscopic (continuous-time random walk) equations for both jump and velocity models with stochastic resetting. In the case of jump models it is shown that stationary states emerge for any shape of the waiting-time and jump length distributions. The existence of such state entails the saturation of the mean square displacement to an universal value that depends on the second moment of the jump distribution and the resetting probability. The transient dynamics towards the stationary state depends on how the waiting time probability density function decays with time. If the moments of the jump distribution are finite then the tail of the stationary distributions is universally exponential, but for Lévy flights these tails decay as a power law whose exponent coincides with that from the jump distribution. For velocity models we observe that the stationary state emerges only if the distribution of flight durations has finite moments of lower order; otherwise, as occurs for Lévy walks, the stationary state does not exist, and the mean square displacement grows ballistically or superdiffusively, depending on the specific shape of the distribution of movement durations.

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Mesoscopic description of random walks on combs

Physical Review E, 2015

Combs are a simple caricature of various types of natural branched structures, which belong to th... more Combs are a simple caricature of various types of natural branched structures, which belong to the category of loopless graphs and consist of a backbone and branches. We study continuous time random walks on combs and present a generic method to obtain their transport properties. The random walk along the branches may be biased, and we account for the effect of the branches by renormalizing the waiting time probability distribution function for the motion along the backbone. We analyze the overall diffusion properties along the backbone and find normal diffusion, anomalous diffusion, and stochastic localization (diffusion failure), respectively, depending on the characteristics of the continuous time random walk along the branches.

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First-passage times in multiscale random walks: The impact of movement scales on search efficiency

Physical review. E, Statistical, nonlinear, and soft matter physics, 2015

An efficient searcher needs to balance properly the trade-off between the exploration of new spat... more An efficient searcher needs to balance properly the trade-off between the exploration of new spatial areas and the exploitation of nearby resources, an idea which is at the core of scale-free Lévy search strategies. Here we study multiscale random walks as an approximation to the scale-free case and derive the exact expressions for their mean-first-passage times in a one-dimensional finite domain. This allows us to provide a complete analytical description of the dynamics driving the situation in which both nearby and faraway targets are available to the searcher, so the exploration-exploitation trade-off does not have a trivial solution. For this situation, we prove that the combination of only two movement scales is able to outperform both ballistic and Lévy strategies. This two-scale strategy involves an optimal discrimination between the nearby and faraway targets which is only possible by adjusting the range of values of the two movement scales to the typical distances between ...

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Stochastic dynamics and logistic population growth

Physical Review E, 2015

The Verhulst model is probably the best known macroscopic rate equation in population ecology. It... more The Verhulst model is probably the best known macroscopic rate equation in population ecology. It depends on two parameters, the intrinsic growth rate and the carrying capacity. These parameters can be estimated for different populations and are related to the reproductive fitness and the competition for limited resources, respectively. We investigate analytically and numerically the simplest possible microscopic scenarios that give rise to the logistic equation in the deterministic mean-field limit. We provide a definition of the two parameters of the Verhulst equation in terms of microscopic parameters. In addition, we derive the conditions for extinction or persistence of the population by employing either the "momentum-space" spectral theory or the "real-space" Wentzel-Kramers-Brillouin (WKB) approximation to determine the probability distribution function and the mean time to extinction of the population. Our analytical results agree well with numerical simulations.

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Reaction-diffusion wave fronts: Multigeneration biological species under climate change

Physical Review E, 2002

A generalization of reaction-diffusion models to multigeneration biological species is presented.... more A generalization of reaction-diffusion models to multigeneration biological species is presented. It is based on more complex random walks than those in previous approaches. The new model is developed analytically up to infinite order. Our predictions for the speed agree to experimental data for several butterfly species better than existing models. The predicted dependence for the speed on the number of generations per year allows us to explain the change in speed observed for a specific invasion.

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Stationary energy probability density of oscillators driven by a random external force

Physical Review E, 2013

We derive rigorous analytical results for the stationary energy probability density function of l... more We derive rigorous analytical results for the stationary energy probability density function of linear and nonlinear oscillators driven by additive Gaussian noise. Our study focuses on two cases: (i) a harmonic oscillator subjected to Gaussian colored noise with an arbitrary correlation function and (ii) nonlinear oscillators with a general potential driven by Gaussian white noise. We also derive analytical expressions for the stationary moments of the energy and investigate the partition of the mean energy between kinetic and potential energy. To illustrate our general results, we consider specifically the case of exponentially correlated noise for (i) and power-law and bistable potentials for (ii). Our theoretical results are substantiated by Langevin simulations.

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Stochastic model for population migration and the growth of human settlements during the Neolithic transition

Physical Review E, 2008

We present a stochastic two-population model that describes the migration and growth of semiseden... more We present a stochastic two-population model that describes the migration and growth of semisedentary foragers and sedentary farmers along a river valley during the Neolithic transition. The main idea is that random migration and transition from a sedentary to a foraging way of life, and backwards, is strongly coupled with the local crop production and associated degradation of land. We derive a nonlinear integral equation for the population density coupled with the equations for the density of soil nutrients and crop production. Our model provides a description of the formation of human settlements along the river valley. The numerical results show that the individual farmers have a tendency for aggregation and clustering. We show that the large-scale pattern is a transient phenomenon which eventually disappears due to land degradation.

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Biased random walks and propagation failure

Physical Review E, 2007

The critical value of the reaction rate able to sustain the propagation of an invasive front is o... more The critical value of the reaction rate able to sustain the propagation of an invasive front is obtained for general non-Markovian biased random walks with reactions. From the Hamilton-Jacobi equation corresponding to the mean field equation we find that the critical reaction rate depends only on the mean waiting time and on the statistical properties of the jump length probability distribution function and is always underestimated by the diffusion approximation. If the reaction rate is larger than the jump frequency, invasion always succeeds, even in the case of maximal bias. Numerical simulations support our analytical predictions.

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Stochastic fluctuations of the transmission rate in the susceptible-infected-susceptible epidemic model

Physical Review E, 2012

We analyze the dynamics of the susceptible-infected-susceptible epidemic model when the transmiss... more We analyze the dynamics of the susceptible-infected-susceptible epidemic model when the transmission rate displays Gaussian white noise fluctuations around its mean value. We obtain analytic expressions for the final size distribution of infectives and the mean infected number of individuals. The model displays a variety of noise-induced transitions as a function of the basic reproductive rate and the noise intensity. We derive a threshold criterion for epidemic invasion in the presence of external noise and determine the mean epidemic duration and the mean epidemic extinction time.

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Analysis of fronts in reaction-dispersal processes

Physical Review E, 2004

The existence of traveling wave front solutions with a minimum speed selected for reaction-disper... more The existence of traveling wave front solutions with a minimum speed selected for reaction-dispersal processes is studied. We obtain a general existence condition in terms of the waiting time and dispersal distance probability distribution functions and we detail this result for situations of ecological interest. In particular, when particles disperse according to jumps of short length and any waiting time probability distribution function, we show that the minimum speed selection for traveling wave fronts is not always possible, so the waiting time and the dispersal distance distributions cannot be arbitrarily chosen.

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Front propagation in reaction-dispersal models with finite jump speed

Physical Review E, 2004

The continuous-time random walk and the Hamilton-Jacobi method are used to reach analytical expre... more The continuous-time random walk and the Hamilton-Jacobi method are used to reach analytical expressions for the speed of traveling fronts in reaction-dispersal models. In this work the waiting time and jump length are assumed to be coupled random variables. The jump length for any jump is selected according to the waiting time at the end of the previous jump, and in consequence jumps of finite speed are performed. We study the effect of finite jump speed of the particles on the speed of the traveling fronts and find that in the parabolic and hyperbolic limits it can exceed the jump speed of the particles. We report analytical expressions for different probability distribution functions. Finally, we introduce the possibility that several particle speeds are allowed, so different dispersal mechanisms can be considered simultaneously.

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Anomalous reaction-transport processes: The dynamics beyond the law of mass action

Physical Review E, 2008

In this paper we reconsider the mass action law ͑MAL͒ for the anomalous reversible reaction A B w... more In this paper we reconsider the mass action law ͑MAL͒ for the anomalous reversible reaction A B with diffusion. We provide a mesoscopic description of this reaction when the transitions between two states A and B are governed by anomalous ͑heavy-tailed͒ waiting-time distributions. We derive the set of mesoscopic integro-differential equations for the mean densities of reacting and diffusing particles in both states. We show that the effective reaction rate memory kernels in these equations and the uniform asymptotic states depend on transport characteristics such as jumping rates. This is in contradiction with the classical picture of MAL. We find that transport can even induce an extinction of the particles such that the density of particles A or B tends asymptotically to zero. We verify analytical results by Monte Carlo simulations and show that the mesoscopic densities exhibit a transient growth before decay.

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$Research paper thumbnail of Description of diffusive and propagative behavior on fractals$

Description of diffusive and propagative behavior on fractals

Physical Review E, 2004

The known properties of diffusion on fractals are reviewed in order to give a general outlook of ... more The known properties of diffusion on fractals are reviewed in order to give a general outlook of these dynamic processes. After that, we propose a description developed in the context of the intrinsic metric of fractals, which leads us to a differential equation able to describe diffusion in real fractals in the asymptotic regime. We show that our approach has a stronger physical justification than previous works on this field. The most important result we present is the introduction of a dependence on time and space for the conductivity in fractals, which is deduced by scaling arguments and supported by computer simulations. Finally, the diffusion equation is used to introduce the possibility of reaction-diffusion processes on fractals and analyze their properties. Specifically, an analytic expression for the speed of the corresponding travelling fronts, which can be of great interest for application purposes, is derived.

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Front propagation in population dynamics with dispersive variability and delayed growth

Physica A: Statistical Mechanics and its Applications, 2006

On the basis of the Cook model, we propose a delayed-growth reaction-diffusion model with an age-... more On the basis of the Cook model, we propose a delayed-growth reaction-diffusion model with an age-dependent disperser-nondisperser transition. We compare the speed of migration fronts between our model and the hyperbolic generalization of the Cook model. In particular, we study for both models the dependence of the migratory fronts speed on the shape of probability distribution function of adult ages for the Neolithic transition in Europe. The migratory fronts speed in both models behaves in a very similar qualitative way, exhibiting both a good agreement with observational data.

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Extinction conditions for isolated populations with Allee effect

Mathematical Biosciences, 2011

One of the main ecological phenomenons is the Allee effect [1-3], in which a positive benefit fro... more 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.

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Effect of environmental fluctuations on invasion fronts

Journal of Theoretical Biology, 2011

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service... more This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Persistent random motion: Uncovering cell migration dynamics

Journal of Theoretical Biology, 2010

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The effects of distributed life cycles on the dynamics of viral infections

Journal of Theoretical Biology, 2008

We explore the role of cellular life cycles for viruses and host cells in an infection process. F... more We explore the role of cellular life cycles for viruses and host cells in an infection process. For this purpose, we derive a generalized version of the basic model of virus dynamics (Nowak, M.A., Bangham, C.R.M., 1996. Population dynamics of immune responses to persistent viruses. Science 272, 74-79) from a mesoscopic description. In its final form the model can be written as a set of Volterra integrodifferential equations. We consider the role of age-distributed delays for death times and the intracellular (eclipse) phase. These processes are implemented by means of probability distribution functions. The basic reproductive ratio R 0 of the infection is properly defined in terms of such distributions by using an analysis of the equilibrium states and their stability. It is concluded that the introduction of distributed delays can strongly modify both the value of R 0 and the predictions for the virus loads, so the effects on the infection dynamics are of major importance. We also show how the model presented here can be applied to some simple situations where direct comparison with experiments is possible. Specifically, phage-bacteria interactions are analysed. The dynamics of the eclipse phase for phages is characterized analytically, which allows us to compare the performance of three different fittings proposed before for the one-step growth curve.

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