Search

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 ωm. 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 ωm-dependent optimal frequency ω = ωoptl 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. [ABSTRACT FROM AUTHOR]

We explore the case of a group of random walkers looking for a target randomly located in space, such that the number of walkers is not constant but new ones can join the search, or those that are active can abandon it, with constant rates r_{b} and r_{d}, respectively. Exact analytical solutions are provided both for the fastest-first-passage time and for the collective time cost required to reach the target, for the exemplifying case of Brownian walkers with r_{d}=0. We prove that even for such a simple situation there exists an optimal rate r_{b} at which walkers should join the search to minimize the collective search costs. We discuss how these results open a new line to understand the optimal regulation in searches conducted through multiparticle random walks, e.g., in chemical or biological processes.

We study the effect of a resetting point randomly distributed around the origin on the mean first-passage time of a Brownian searcher moving in one dimension. We compare the search efficiency with that corresponding to reset to the origin and find that the mean first-passage time of the latter can be larger or smaller than the distributed case, depending on whether the resetting points are symmetrically or asymmetrically distributed. In particular, we prove the existence of an optimal reset rate that minimizes the mean first-passage time for distributed resetting to a finite interval if the target is located outside this interval. When the target position belongs to the resetting interval or it is infinite then no optimal reset rate exists, but there is an optimal resetting interval width or resetting characteristic scale which minimizes the mean first-passage time. We also show that the first-passage density averaged over the resetting points depends on its first moment only. As a consequence, there is an equivalent point such that the first-passage problem with resetting to that point is statistically equivalent to the case of distributed resetting. We end our study by analyzing the fluctuations of the first-passage times for these cases. All our analytical results are verified through numerical simulations.

We analyze a one-dimensional intermittent random walk on an unbounded domain in the presence of stochastic resetting. In this process, the walker alternates between local intensive search, diffusion, and rapid ballistic relocations in which it does not react to the target. We demonstrate that Poissonian resetting leads to the existence of a non-equilibrium steady state. We calculate the distribution of the first arrival time to a target along with its mean and show the existence of an optimal reset rate. In particular, we prove that the initial condition of the walker, i.e., either starting diffusely or relocating, can significantly affect the long-time properties of the search process. Moreover, we demonstrate the presence of distinct parameter regimes for the global optimization of the mean first arrival time when ballistic and diffusive movements are in direct competition.

Stochastic resetting can be naturally understood as a renewal process governing the evolution of an underlying stochastic process. In this work, we formally derive well-known results of diffusion with resets from a renewal theory perspective. Parallel to the concepts from renewal theory, we introduce the conditioned backward B and forward F times being the times since the last and until the next reset, respectively, given that the current state of the system X(t) is known. These magnitudes are introduced with the paradigmatic case of diffusion under resetting, for which the backward and forward times are conditioned to the position of the walker. We find analytical expressions for the conditioned backward and forward time probability density functions (PDFs), and we compare them with numerical simulations. The general expressions allow us to study particular scenarios. For instance, for power-law reset time PDFs such that φ(t)∼t^{-1-α}, significant changes in the properties of the conditioned backward and forward times happen at half-integer values of α due to the composition between the long-time scaling of diffusion P(x,t)∼1/sqrt[t] and the reset time PDF.

We consider a walker moving in a one-dimensional interval with absorbing boundaries under the effect of Markovian resettings to the initial position. The walker's motion follows a random walk characterized by a general waiting time distribution between consecutive short jumps. We investigate the existence of an optimal reset rate, which minimizes the mean exit passage time, in terms of the statistical properties of the waiting time probability. Generalizing previous results, we find that when the waiting time probability has first and second finite moments, resetting can be either (i) never beneficial, (ii) beneficial depending on the distance of the reset point to the boundary, or (iii) always beneficial. Instead, when the waiting time probability has the first or the two first moments diverging we find that resetting is always beneficial. Finally, we have also found that the optimal strategy to exit the domain depends on the reset rate. For low reset rates, walkers with exponential waiting times are found to be optimal, while for high reset rate, anomalous waiting times optimize the search process.

We study a non-Markovian and nonstationary model of animal mobility incorporating both exploration and memory in the form of preferential returns. Exact results for the probability of visiting a given number of sites are derived and a practical WKB approximation to treat the nonstationary problem is developed. A mean-field version of this model, first suggested by Song et al., [Modelling the scaling properties of human mobility, Nat. Phys. 6, 818 (2010)NPAHAX1745-247310.1038/nphys1760] was shown to well describe human movement data. We show that our generalized model adequately describes empirical movement data of Egyptian fruit bats (Rousettus aegyptiacus) when accounting for interindividual variation in the population. We also study the probability of visiting any site a given number of times and derive a mean-field equation. Our analysis yields a remarkable phase transition occurring at preferential returns which scale linearly with past visits. Following empirical evidence, we suggest that this phase transition reflects a trade-off between extensive and intensive foraging modes.

Excited random walks represent a convenient model to study food intake in a media which is progressively depleted by the walker. Trajectories in the model alternate between (i) feeding and (ii) escape (when food is missed and so it must be found again) periods, each governed by different movement rules. Here, we explore the case where the escape dynamics is adaptive, so at short times an area-restricted search is carried out, and a switch to extensive or ballistic motion occurs later if necessary. We derive for this case explicit analytical expressions of the mean escape time and the asymptotic growth of the depleted region in one dimension. These, together with numerical results in two dimensions, provide surprising evidence that ballistic searches are detrimental in such scenarios, a result which could explain why ballistic movement is barely observed in animal searches at microscopic and millimetric scales, therefore providing significant implications for biological foraging.

We investigate the effects of Markovian resetting events on continuous time random walks where the waiting times and the jump lengths are random variables distributed according to power-law probability density functions. We prove the existence of a nonequilibrium stationary state and finite mean first arrival time. However, the existence of an optimum reset rate is conditioned to a specific relationship between the exponents of both power-law tails. We also investigate the search efficiency by finding the optimal random walk which minimizes the mean first arrival time in terms of the reset rate, the distance of the initial position to the target, and the characteristic transport exponents.

Information on the relevant global scales of the search space, even if partial, should conceivably enhance the performance of random searches. Here we show numerically and analytically that the paradigmatic uninformed optimal Lévy searches can be outperformed by informed multiple-scale random searches in one (1D) and two (2D) dimensions, even when the knowledge about the relevant landscape scales is incomplete. We show in the low-density nondestructive regime that the optimal efficiency of biexponential searches that incorporate all key scales of the 1D landscape of size L decays asymptotically as η_{opt}∼1/sqrt[L], overcoming the result η_{opt}∼1/(sqrt[L]lnL) of optimal Lévy searches. We further characterize the level of limited information the searcher can have on these scales. We obtain the phase diagram of bi- and triexponential searches in 1D and 2D. Remarkably, even for a certain degree of lack of information, partially informed searches can still outperform optimal Lévy searches. We discuss our results in connection with the foraging problem.

Random walks with stochastic resetting provides a treatable framework to study interesting features about central-place motion. In this work, we introduce noninstantaneous resetting as a two-state model being a combination of an exploring state where the walker moves randomly according to a propagator and a returning state where the walker performs a ballistic motion with constant velocity towards the origin. We study the emerging transport properties for two types of reset time probability density functions (PDFs): exponential and Pareto. In the first case, we find the stationary distribution and a general expression for the stationary mean-square displacement (MSD) in terms of the propagator. We find that the stationary MSD may increase, decrease or remain constant with the returning velocity. This depends on the moments of the propagator. Regarding the Pareto resetting PDF we also study the stationary distribution and the asymptotic scaling of the MSD for diffusive motion. In this case, we see that the resetting modifies the transport regime, making the overall transport subdiffusive and even reaching a stationary MSD, i.e., a stochastic localization. This phenomena is also observed in diffusion under instantaneous Pareto resetting. We check the main results with stochastic simulations of the process.

Random walks with memory usually involve rules where a preference for either revisiting or avoiding those sites visited in the past are introduced somehow. Such effects have a direct consequence on the transport properties as well as on the statistics of first-passage and subsequent recurrence times through a site. A preference for revisiting sites is thus expected to result in a positive correlation between consecutive recurrence times. Here we derive a continuous-time generalization of the random walk model with preferential relocation to visited sites proposed in Phys. Rev. Lett. 112, 240601 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.240601 to explore this effect, together with the main transport properties induced by the long-range memory. Despite the long-range memory effects governing the process, our analytical treatment allows us to (i) observe the existence of an asymptotic logarithmic (ultraslow) growth for the mean square displacement, in accordance to the results found for the original discrete-time model, and (ii) confirm the existence of positive correlations between first-passage and subsequent recurrence times. This analysis is completed with a comprehensive numerical study which reveals, among other results, that these correlations between first-passage and recurrence times also exhibit clear signatures of this ultraslow relaxation dynamics.

We study simple stochastic scenarios, based on birth-and-death Markovian processes, that describe populations with the Allee effect, to account for the role of demographic stochasticity. In the mean-field deterministic limit we recover well-known deterministic evolution equations widely employed in population ecology. The mean time to extinction is in general obtained by the Wentzel-Kramers-Brillouin (WKB) approximation for populations with the strong and weak Allee effects. An exact solution for the mean time to extinction can be found via a recursive equation for special cases of the stochastic dynamics. We study the conditions for the validity of the WKB solution and analyze the boundary between the weak and strong Allee effect by comparing exact solutions with numerical simulations.

Stochastic resets have lately emerged as a mechanism able to generate finite equilibrium mean-square displacement (MSD) when they are applied to diffusive motion. Furthermore, walkers with an infinite mean first-arrival time (MFAT) to a given position x may reach it in a finite time when they reset their position. In this work we study these emerging phenomena from a unified perspective. On one hand, we study the existence of a finite equilibrium MSD when resets are applied to random motion with 〈x^{2}(t)〉_{m}∼t^{p} for 0

Inspired by recent experiments on the organism Caenorhabditis elegans we present a stochastic problem to capture the adaptive dynamics of search in living beings, which involves the exploration-exploitation dilemma between remaining in a previously preferred area and relocating to new places. We assess the question of search efficiency by introducing a new magnitude, the mean valuable territory covered by a Browinan searcher, for the case where each site in the domain becomes valuable only after a random time controlled by a nonhomogeneous rate which expands from the origin outwards. We explore analytically this magnitude for domains of dimensions 1, 2, and 3 and discuss the theoretical and applied (biological) interest of our approach. As the main results here, we (i) report the existence of some universal scaling properties for the mean valuable territory covered as a function of time and (ii) reveal the emergence of an optimal diffusivity which appears only for domains in two and higher dimensions.

In this work we construct individual-based models that give rise to the generalized logistic model at the mean-field deterministic level and that allow us to interpret the parameters of these models in terms of individual interactions. We also study the effect of internal fluctuations on the long-time dynamics for the different models that have been widely used in the literature, such as the theta-logistic and Savageau models. In particular, we determine the conditions for population extinction and calculate the mean time to extinction. If the population does not become extinct, we obtain analytical expressions for the population abundance distribution. Our theoretical results are based on WKB theory and the probability generating function formalism and are verified by numerical simulations.

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.

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, and compare our analytical results with stochastic simulations.

Recent works have explored the properties of Lévy flights with resetting in one-dimensional domains and have reported the existence of phase transitions in the phase space of parameters which minimizes the mean first passage time (MFPT) through the origin [L. Kusmierz et al., Phys. Rev. Lett. 113, 220602 (2014)]. Here, we show how actually an interesting dynamics, including also phase transitions for the minimization of the MFPT, can also be obtained without invoking the use of Lévy statistics but for the simpler case of random walks with exponentially distributed flights of constant speed. We explore this dynamics both in the case of finite and infinite domains, and for different implementations of the resetting mechanism to show that different ways to introduce resetting consistently lead to a quite similar dynamics. The use of exponential flights has the strong advantage that exact solutions can be obtained easily for the MFPT through the origin, so a complete analytical characterization of the system dynamics can be provided. Furthermore, we discuss in detail how the phase transitions observed in random walks with resetting are closely related to several ideas recurrently used in the field of random search theory, in particular, to other mechanisms proposed to understand random search in space as mortal random walks or multiscale random walks. As a whole, we corroborate that one of the essential ingredients behind MFPT minimization lies in the combination of multiple movement scales (regardless of their specific origin).

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 encounters. So, this optimization necessarily requires some prior information (albeit crude) about target distances or distributions. Furthermore, we found that the incorporation of additional (three, four, …) movement scales and its adjustment to target distances does not improve further the search efficiency. This allows us to claim that optimal random search strategies arise through the informed combination of only two walk scales (related to the exploitative and the explorative scales, respectively), expanding on the well-known result that optimal strategies in strictly uninformed scenarios are achieved through Lévy paths (or, equivalently, through a hierarchical combination of multiple scales).

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 approximation to determine the probability distribution function and the mean time to extinction of the population. Our analytical results agree well with numerical simulations.

We investigate the behavior of five hyperbolic reaction-diffusion equations most commonly employed to describe systems of interacting organisms or reacting particles where dispersal displays inertia. We first discuss the macroscopic or mesoscopic foundation, or lack thereof, of these reaction-transport equations. This is followed by an analysis of the temporal evolution of spatially uniform states. In particular, we determine the uniform steady states of the reaction-transport systems and their stability properties. We then address the spatiotemporal behavior of pure death processes. We end with a unified treatment of the front speed for hyperbolic reaction-diffusion equations with Kolmogorov-Petrosvskii-Piskunov kinetics. In particular, we obtain an exact expression for the front speed of a general class of reaction correlated random walk systems. Our results establish that three out of the five hyperbolic reaction-transport equations provide physically acceptable models of biological and chemical systems.

We derive the Fokker-Planck equation for multivariable Langevin equations with cross-correlated Gaussian white noises for an arbitrary interpretation of the stochastic differential equation. We formulate the conditions when the solution of the Fokker-Planck equation does not depend on which stochastic calculus is adopted. Further, we derive an equivalent multivariable Ito stochastic differential equation for each possible interpretation of the multivariable Langevin equation. To demonstrate the usefulness and significance of these general results, we consider the motion of Brownian particles. We study in detail the stability conditions for harmonic oscillators with two white noises, one of which is additive, random forcing, and the other, which accounts for fluctuations of either the damping or the spring coefficient, is multiplicative. We analyze the role of cross correlation in terms of the different noise interpretations and confirm the theoretical predictions via numerical simulations. We stress the interest of our results for numerical simulations of stochastic differential equations with an arbitrary interpretation of the stochastic integrals.

Random encounters in space are central to describing diffusion-limited reactions, animal foraging, search processes, and many other situations in nature. These encounters, however, are often constrained by the capacity of the searcher to detect and/or recognize its target. This can be due to limited binding and perception abilities of the searcher or hiding and avoiding mechanisms used by the target. Hence detection failure upon passage over the target location turns the process into an n-passage problem, with n being random. Here we provide a general description of this detection problem for arbitrary dimensions and arbitrary detection constraints. The mean detection time (MDT) for a random searcher embedded in a sea of homogeneously distributed targets is obtained as a function of the target density ρ, the size domain L, and the effective detection distance a. While the scaling with ρ and L is found to be universal and equivalent to that found for the corresponding first-passage problem, the scaling of the MDT on a depends on the specific detection mechanism considered.

We investigate the performance of a linear electromechanical oscillator as an energy harvester of finite-bandwidth random vibrations. We derive exact analytical expressions for the net electrical power and the efficiency of the conversion of the power supplied by the noise into electrical power for arbitrary colored noise. We apply our results to the important case of exponentially correlated noise and discuss the tuning of parameters to achieve good performance of the device.

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.

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.

Random search theory has been previously explored for both continuous and intermittent scanning modes with full target detection capacity. Here we present a new class of random search problems in which a single searcher performs flights of random velocities, the detection probability when it passes over a target location being conditioned to the searcher speed. As a result, target detection involves an N-passage process for which the mean search time is here analytically obtained through a renewal approximation. We apply the idea of speed-selective detection to random animal foraging since a fast movement is known to significantly degrade perception abilities in many animals. We show that speed-selective detection naturally introduces an optimal level of behavioral intermittence in order to solve the compromise between fast relocations and target detection capability.

We investigate the instabilities of a linear damped oscillator due to fluctuations of the damping parameter. The fluctuations are driven either by Gaussian white noise or Poisson white noise (white shot noise). We consider three notions of stability. The first two are the well-known notions of stability in the mean and stability in the mean square. We introduce the concept of thermodynamic stability, corresponding to a nonpositive rate of energy dissipation at all times. We derive analytical results for the various instability thresholds, confirm the validity of our approach for white shot noise by numerical simulations, and obtain the unexpected result that mean-square and thermodynamic stability coincide for the two types of white noise.

We explore a reaction-dispersal mechanism for the generation of wave fronts which consists of a set of particles traveling with random velocities (chosen from arbitrary distributions) which experience an autocatalytic reaction. The differences found between this mechanism and approaches based on the continuous-time random walks, where the particles are assumed to perform discrete jumps from one position to another, are analyzed and discussed. A complete analytical treatment of our velocity model is achieved, which allows us to predict the constant speed of traveling fronts or their time dependence in case they are accelerated. Also, a general criterion to distinguish the situations of fronts with constant speed from those accelerated is provided. From it, we conclude that accelerated fronts should be expected in different situations easily achievable in nature.

The problem of nonlinear Hamiltonian oscillators is one of the classical questions in physics. When an analytic solution is not possible, one can resort to obtaining a numerical solution or using perturbation theory around the linear problem. We apply the Fourier series expansion to find approximate solutions to the oscillator position as a function of time as well as the period-amplitude relationship. We compare our results with other recent approaches such as variational methods or heuristic approximations, in particular the Ren-He's method. Based on its application to the Duffing oscillator, the nonlinear pendulum and the eardrum equation, it is shown that the Fourier series expansion method is the most accurate.

Recently there has been a certain controversy about the scaling properties of reaction-subdiffusion fronts. Some works seem to suggest that these fronts should move with constant speed, as do classical reaction-diffusion fronts, while other authors have predicted propagation failure, i.e., that the front speed tends asymptotically to zero. In the present work we confirm by Monte Carlo experiments that the two situations can actually occur depending on the way the reaction process is implemented. Also, we present a general analytical model that includes these two different behaviors as particular cases. From our analysis, we reach two main conclusions. First, the differences found in the scaling properties show the lack of universality of reaction-subdiffusion fronts. Second, we prove that, contrary to the widespread belief, the tail of the waiting time distributions is not always decisive to determine the speed of these fronts, but sometimes it plays just a marginal role in the front dynamics.

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.

In this paper we reconsider the mass action law (MAL) 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-tailed) 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.

We study the conditions for extinction and survival of populations living in a patch surrounded by a hostile environment. We find analytic expressions for the steady states when population dynamics is described by diffusion and reaction is driven by compensation, depensation, or critical depensation growths. The role of initial population density is studied, and the complete bifurcation diagrams are constructed and validated numerically for the three cases studied.

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.

We apply variational principles to reaction-diffusion equations in order to predict, for a general reaction term, the sign of the shift in the front speed due to a cutoff. We develop an improved variational principle to obtain the shift in the front speed for a wide range of reaction terms, and the theoretical results so obtained are in excellent agreement with numerical solutions. This work proves that variational principles are an optimal framework to deal with fronts propagating into unstable and metastable states under cutoff.

From the Hamilton-Jacobi formalism, an explicit expression for the speed of wave front propagation along the backbone of comblike structures is obtained. This expression, through the waiting-time distribution function, takes into account the number of sites and their distribution in the secondary branches. Our theoretical results are supported by numerical simulations of the reaction random-walk process on the structure. Finally, a more complex situation such as the Peano basin structure is also considered, both theoretically and numerically, exhibiting a good agreement too.

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.

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.

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.

Recently, it has been shown that the speed of virus infections can be explained by time-delayed reaction-diffusion [J. Fort and V. Méndez, Phys. Rev. Lett. 89, 178101 (2002)], but no analytical solutions were found. Here we derive formulas for the front speed, valid in appropriate limits. We also integrate numerically the evolution equations of the system. There is good agreement with both numerical and experimental speeds.

The speed of front propagation in fractals is studied by using (i) the reduction of the reaction-transport equation into a Hamilton-Jacobi equation and (ii) the local-equilibrium approach. Different equations proposed for describing transport in fractal media, together with logistic reaction kinetics, are considered. Finally, we analyze the main features of wave fronts resulting from this dynamic process, i.e., why they are accelerated and what is the exact form of this acceleration.

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.