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Fair division is typically framed from a centralized perspective. We study a decentralized variant of fair division inspired by the dynamics observed in community-based targeting, mutual aid networks, and community resource management paradigms. We develop an approach for decentralized fair division and compare it with a centralized approach with respect to fairness and social welfare guarantees. In the context of the existing literature, our decentralized model can be viewed as a relaxation of previous models of sequential exchange in light of impossibility results concerning the inability of those models to achieve desirable outcomes. We find that in settings representative of many real world situations, the two models of resource allocation offer contrasting fairness and social welfare guarantees. In particular, we show that under appropriate conditions, our model of decentralized allocation can ensure high-quality allocative decisions in an efficient fashion.

This study revisits the electroabsorption (EA) spectrum of polyacetylene, as functions of the electric field strength, isomerization degree, and light polarization states. The EA spectrum of $cis$-$(CH)_x$ reveals an oscillatory feature that follows the Stark shift-related first derivative of the materials absorption spectrum that contains v(0-1) and v(0-2) sidebands of the excited $C=C$ stretching vibration that agrees well with the Raman spectrum. EA spectrum of $trans $-$(CH)_x$ does not match the first derivative of the materials absorption spectrum, and the phonon sideband frequency does not agree with the RS spectrum. EA spectrum of $trans $-$(CH)_x$ reveals a band below the first allowed $1B_u$ exciton. We interpret this feature as due to the electric field activated even-parity dark (forbidden) exciton, namely $mA_g$ ($m >1$), showing that the nonluminescent $trans $-$(CH)_x$ is due to the reverse order of the excited states, where a dark $mA_g$ exciton lies below the allowed $1B_u$ exciton. This agrees with the unusual phonon sideband in $trans $-$(CH)_x$ absorption, since the excited state attenuation caused by the fast internal conversion from $1B_u$ to $mA_g$ influences the apparent frequency that determines the phonon sideband. Consequently, from the EA and RS spectra we estimate the $1B_u$ lifetime in $trans $-$(CH)_x$ to be $\sim 30$ fs. Integrated EA spectrum of $trans $-$(CH)_x$ shows a traditional Huang-Rhys type series with a relaxation parameter, $S \sim 0.5$. This indicates that the EA spectrum of the $trans $ isomer is also determined by a Stark shift related to the first derivative of the absorption spectrum, but preferentially for the longest chains in the films chain lengths distribution. This is due to the $N^3$ response of the non-linear susceptibility, $\chi^{(3)}$ ($\sim$EA), dependence on the chain length having $N$ monomers.

Deterministic approximations to stochastic Susceptible-Infectious-Susceptible models typically predict a stable endemic steady-state when above threshold. This can be hard to relate to the underlying stochastic dynamics, which has no endemic steady-state but can exhibit approximately stable behaviour. Here we relate the approximate models to the stochastic dynamics via the definition of the quasi-stationary distribution (QSD), which captures this approximately stable behaviour. We develop a system of ordinary differential equations that approximate the number of infected individuals in the QSD for arbitrary contact networks and parameter values. When the epidemic level is high, these QSD approximations coincide with the existing approximation methods. However, as we approach the epidemic threshold, the models deviate, with these models following the QSD and the existing methods approaching the all susceptible state. Through consistently approximating the QSD, the proposed methods provide a more robust link to the stochastic models.

The basic reproduction number, $R_0$ is an important and widely used concept in the study of infectious diseases. We briefly review the recent trend of calculating the average of various $R_0$ estimates in systematic reviews aimed at estimating the basic reproduction number of SARS-CoV-2, and discuss the drawbacks and implications of using such averaging methods. Additionally, we argue that even a theoretically grounded approach such as next generation matrix could have practical impediments in its use. More generally, the practice of associating an infectious disease with a single value of $R_0$ is problematic, when the disease can, in fact have different reproduction numbers in various populations., Comment: 5 pages, 3 figures

Local variations in the intergalactic medium (IGM) neutral hydrogen fraction will affect the Ly-$\alpha$ absorption signature of protoclusters identified in tomographic surveys. Using the IllustrisTNG simulations, we investigate how the AGN proximity effect and hot, collisionally ionised gas arising from gravitational infall and black hole feedback changes the Ly-$\alpha$ absorption associated with $M_{z=0}\simeq10^{14}\,M_\odot$ protoclusters at $z\simeq2.4$. We find that protocluster galaxy overdensities exhibit a weak anti-correlation with Ly-$\alpha$ transmission in IGM transmission maps, but local HI ionisation enhancements due to hot $T>10^{6}\rm\,K$ gas or nearby AGN can disrupt this relationship within individual protoclusters. On average, however, we find that strong reductions in the IGM neutral fraction are limited to within $\lesssim 5h^{-1}\,\textrm{cMpc}$ of the dark matter haloes. Local ionisation enhancements will therefore have a minimal impact on the completeness of protocluster identification in tomographic surveys if smoothing Ly-$\alpha$ transmission maps over scales of $\sim4 h^{-1}\,\textrm{cMpc}$, as is typically done in observations. However, if calibrating the relationship between the matter density and Ly-$\alpha$ transmission in tomographic maps using simple analytical models for the Ly-$\alpha$ forest opacity, the presence of hot gas around haloes can still result in systematically lower estimates of $M_{z=0}$ for the most massive protoclusters., Comment: 14 pages (including 1 appendix), 5 figures (plus 2 in the appendix), 1 table. Accepted for publication in MNRAS

Diseases spread over temporal networks of interaction events between individuals. Structures of these temporal networks hold the keys to understanding epidemic propagation. One early concept of the literature to aid in discussing these structures is concurrency -- quantifying individuals' tendency to form time-overlapping "partnerships". Although conflicting evaluations and an overabundance of operational definitions have marred the history of concurrency, it remains important, especially in the area of sexually transmitted infections. Today, much of theoretical epidemiology uses more direct models of contact patterns, and there is an emerging body of literature trying to connect methods to the concurrency literature. In this review, we will cover the development of the concept of concurrency and these new approaches., Comment: 5 figures

The contact structure of a population plays an important role in transmission of infection. Many ``structured models'' capture aspects of the contact structure through an underlying network or a mixing matrix. An important observation in such models, is that once a fraction $1-1/\mathcal{R}_0$ has been infected, the residual susceptible population can no longer sustain an epidemic. A recent observation of some structured models is that this threshold can be crossed with a smaller fraction of infected individuals, because the disease acts like a targeted vaccine, preferentially immunizing higher-risk individuals who play a greater role in transmission. Therefore, a limited ``first wave'' may leave behind a residual population that cannot support a second wave once interventions are lifted. In this paper, we systematically analyse a number of mean-field models for networks and other structured populations to address issues relevant to the Covid-19 pandemic. In particular, we consider herd-immunity under several scenarios. We confirm that, in networks with high degree heterogeneity, the first wave confers herd-immunity with significantly fewer infections than equivalent models with lower degree heterogeneity. However, if modelling the intervention as a change in the contact network, then this effect might become more subtle. Indeed, modifying the structure can shield highly connected nodes from becoming infected during the first wave and make the second wave more substantial. We confirm this finding by using an age-structured compartmental model parameterised with real data and comparing lockdown periods implemented either as a global scaling of the mixing matrix or age-specific structural changes. We find that results regarding herd immunity levels are strongly dependent on the model, the duration of lockdown and how lockdown is implemented., Comment: 25 pages, 9 figures, 1 Table

Combinations of intense non-pharmaceutical interventions ('lockdowns') were introduced in countries worldwide to reduce SARS-CoV-2 transmission. Many governments have begun to implement lockdown exit strategies that allow restrictions to be relaxed while attempting to control the risk of a surge in cases. Mathematical modelling has played a central role in guiding interventions, but the challenge of designing optimal exit strategies in the face of ongoing transmission is unprecedented. Here, we report discussions from the Isaac Newton Institute 'Models for an exit strategy' workshop (11-15 May 2020). A diverse community of modellers who are providing evidence to governments worldwide were asked to identify the main questions that, if answered, will allow for more accurate predictions of the effects of different exit strategies. Based on these questions, we propose a roadmap to facilitate the development of reliable models to guide exit strategies. The roadmap requires a global collaborative effort from the scientific community and policy-makers, and is made up of three parts: i) improve estimation of key epidemiological parameters; ii) understand sources of heterogeneity in populations; iii) focus on requirements for data collection, particularly in Low-to-Middle-Income countries. This will provide important information for planning exit strategies that balance socio-economic benefits with public health.

The ongoing Coronavirus Disease 2019 (COVID-19) pandemic threatens the health of humans and causes great economic losses. Predictive modelling and forecasting the epidemic trends are essential for developing countermeasures to mitigate this pandemic. We develop a network model, where each node represents an individual and the edges represent contacts between individuals where the infection can spread. The individuals are classified based on the number of contacts they have each day (their node degrees) and their infection status. The transmission network model was respectively fitted to the reported data for the COVID-19 epidemic in Wuhan (China), Toronto (Canada), and the Italian Republic using a Markov Chain Monte Carlo (MCMC) optimization algorithm. Our model fits all three regions well with narrow confidence intervals and could be adapted to simulate other megacities or regions. The model projections on the role of containment strategies can help inform public health authorities to plan control measures.

SARS-CoV-2 causing COVID-19 disease has moved rapidly around the globe, infecting millions and killing hundreds of thousands. The basic reproduction number, which has been widely used and misused to characterize the transmissibility of the virus, hides the fact that transmission is stochastic, is dominated by a small number of individuals, and is driven by super-spreading events (SSEs). The distinct transmission features, such as high stochasticity under low prevalence, and the central role played by SSEs on transmission dynamics, should not be overlooked. Many explosive SSEs have occurred in indoor settings stoking the pandemic and shaping its spread, such as long-term care facilities, prisons, meat-packing plants, fish factories, cruise ships, family gatherings, parties and night clubs. These SSEs demonstrate the urgent need to understand routes of transmission, while posing an opportunity that outbreak can be effectively contained with targeted interventions to eliminate SSEs. Here, we describe the potential types of SSEs, how they influence transmission, and give recommendations for control of SARS-CoV-2., Comment: 10 pages, 3 figures

We provide a description of the Epidemics on Networks (EoN) python package designed for studying disease spread in static networks. The package consists of over $100$ methods available for users to perform stochastic simulation of a range of different processes including SIS and SIR disease, and generic simple or comlex contagions., Comment: This paper is available in a shorter form without examples at https://joss.theoj.org/papers/10.21105/joss.01731.pdf

We use state of the art hydrodyamical simulations from the Sherwood, EAGLE and Illustris projects to examine the signature of $M_{\rm z=0}\simeq 10^{14}M_{\odot}$ protoclusters observed in Ly-$\alpha$ absorption at $z\simeq 2.4$. We find there is a weak correlation between the mass overdensity, $\delta_{\rm m}$, and the Ly-$\alpha$ effective optical depth relative to the mean, $\delta_{\tau_\textrm{eff}}$, averaged over $15~h^{-1}\rm\,cMpc$ scales, although scatter in the $\delta_{\rm m}$--$\delta_{\tau_\textrm{eff}}$ plane means it is not possible to uniquely identify large scale overdensities with strong Ly-$\alpha$ absorption. Although all protoclusters are associated with large scale mass overdensities, most sight lines through protoclusters in a $\sim 10^{6}$ $\rm cMpc^{3}$ volume probe the low column density Ly-$\alpha$ forest. A small subset of sight lines that pass through protoclusters exhibit coherent, strong Ly-$\alpha$ absorption on $15h^{-1}\rm\,cMpc$ scales, although these correspond to a wide range in mass overdensity. Assuming perfect removal of contamination by Ly-$\alpha$ absorbers with damping wings, more than half of the remaining sight lines with $\delta_{\tau_{\rm eff}}>3.5$ trace protoclusters. It is furthermore possible to identify a model dependent $\delta_{\tau_{\rm eff}}$ threshold that selects only protoclusters. However, such regions are rare: excluding absorption caused by damped systems, less than 0.1 per cent of sight lines that pass through a protocluster have $\delta_{\tau_{\rm eff}}>3.5$, meaning that any protocluster sample selected in this manner will also be highly incomplete. On the other hand, coherent regions of Ly-$\alpha$ absorption also provide a promising route for identifying and studying filamentary environments at high redshift., Comment: 18 pages, 18 figures, 3 tables; accepted for publication in MNRAS

We consider the spread of a Susceptible-Infected-Recovered (SIR) disease through finite populations and derive an expression for the final size distribution. Our derivation allows arbitrary distributions of the number of transmissions caused by an infected individual. We show how this calculation can be used to infer parameters of the infectious disease through observations in multiple small populations. The inference suffers from some identifiability difficulties, and it requires many observations to distinguish between parameter combinations that correspond to the same reproductive number.

Pairwise models are used widely to model epidemic spread on networks. These include the modelling of susceptible-infected-removed (SIR) epidemics on regular networks and extensions to SIS dynamics and contact tracing on more exotic networks exhibiting degree heterogeneity, directed and/or weighted links and clustering. However, extra features of the disease dynamics or of the network lead to an increase in system size and analytical tractability becomes problematic. Various `closures' can be used to keep the system tractable. Focusing on SIR epidemics on regular but clustered networks, we show that even for the most complex closure we can determine the epidemic threshold as an asymptotic expansion in terms of the clustering coefficient.We do this by exploiting the presence of a system of fast variables, specified by the correlation structure of the epidemic, whose steady state determines the epidemic threshold. While we do not find the steady state analytically, we create an elegant asymptotic expansion of it. We validate this new threshold by comparing it to the numerical solution of the full system and find excellent agreement over a wide range of values of the clustering coefficient, transmission rate and average degree of the network. The technique carries over to pairwise models with other closures [1] and we note that the epidemic threshold will be model dependent. This emphasises the importance of model choice when dealing with realistic outbreaks., Comment: 12 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1806.06135

We explore the application of probability generating functions (PGFs) to invasive processes, focusing on infectious disease introduced into large populations. Our goal is to acquaint the reader with applications of PGFs, moreso than to derive new results. PGFs help predict a number of properties about early outbreak behavior while the population is still effectively infinite, including the probability of an epidemic, the size distribution after some number of generations, and the cumulative size distribution of non-epidemic outbreaks. We show how PGFs can be used in both discrete-time and continuous-time settings, and discuss how to use these results to infer disease parameters from observed outbreaks. In the large population limit for susceptible-infected-recovered (SIR) epidemics PGFs lead to survival-function based models that are equivalent the the usual mass-action SIR models but with fewer ODEs. We use these to explore properties such as the final size of epidemics or even the dynamics once stochastic effects are negligible. We target this tutorial to biologists and public health researchers who want to learn how to apply PGFs to invasive diseases, but it could also be used in an introductory mathematics course on PGFs. We include many exercises to help demonstrate concepts and to give practice applying the results. We summarize our main results in a few tables. Additionally we provide a small python package which performs many of the relevant calculations., Comment: Re-uploaded with simplified proof of theorem 2.7

The duration, type and structure of connections between individuals in real-world populations play a crucial role in how diseases invade and spread. Here, we incorporate the aforementioned heterogeneities into a model by considering a dual-layer static-dynamic multiplex network. The static network layer affords tunable clustering and describes an individual's permanent community structure. The dynamic network layer describes the transient connections an individual makes with members of the wider population by imposing constant edge rewiring. We follow the edge-based compartmental modelling approach to derive equations describing the evolution of a susceptible - infected - recovered (SIR) epidemic spreading through this multiplex network of individuals. We derive the basic reproduction number, measuring the expected number of new infectious cases caused by a single infectious individual in an otherwise susceptible population. We validate model equations by showing convergence to pre-existing edge-based compartmental model equations in limiting cases and by comparison with stochastically simulated epidemics. We explore the effects of altering model parameters and multiplex network attributes on resultant epidemic dynamics. We validate the basic reproduction number by plotting its value against associated final epidemic sizes measured from simulation and predicted by model equations for a number of setups. Further, we explore the effect of varying individual model parameters on the basic reproduction number. We conclude with a discussion of the significance and interpretation of the model and its relation to existing research literature. We highlight intrinsic limitations and potential extensions of the present model and outline future research considerations, both experimental and theoretical., Comment: 29 pages, 12 Figures. Submitted to Bulletin of Mathematical Biology on 29/03/18. Version 2 has minor changes to the content: a reduced and edited abstract, the author summary is removed, acknowledgements were edited, references to the matrix G were bolded, some Figure titles and captions have been edited, and a reference to providing Matlab code at the end of the Methods section has been removed

For a recently derived pairwise model of network epidemics with non-Markovian recovery, we prove that under some mild technical conditions on the distribution of the infectious periods, smaller variance in the recovery time leads to higher reproduction number, and consequently to a larger epidemic outbreak, when the mean infectious period is fixed. We discuss how this result is related to various stochastic orderings of the distributions of infectious periods. The results are illustrated by a number of explicit stochastic simulations, suggesting that their validity goes beyond regular networks.

We revisit the size distribution of finite components in infinite Configuration Model networks. We provide an elementary combinatorial proof about the sizes of birth-death trees which is more intuitive than previous proofs. We use this to rederive the component size distribution for Configuration Model networks. Our derivation provides a more intuitive interpretation of the formula as contrasted with the previous derivation based on contour integrations. We demonstrate that the formula performs well, even on networks with heavy tails which violate assumptions of the derivation. We explain why the result should remain robust for these networks., Comment: [just updating a few minor typos and obvious little issues. No change to main argument]

Random network models play a prominent role in modeling, analyzing and understanding complex phenomena on real-life networks. However, a key property of networks is often neglected: many real-world networks exhibit spatial structure, the tendency of a node to select neighbors with a probability depending on physical distance. Here, we introduce a class of random spatial networks (RSNs) which generalizes many existing random network models but adds spatial structure. In these networks, nodes are placed randomly in space and joined in edges with a probability depending on their distance and their individual expected degrees, in a manner that crucially remains analytically tractable. We use this network class to propose a new generalization of small-world networks, where the average shortest path lengths in the graph are small, as in classical Watts-Strogatz small-world networks, but with close spatial proximity of nodes that are neighbors in the network playing the role of large clustering. Small-world effects are demonstrated on these spatial small-world networks without clustering. We are able to derive partial integro-differential equations governing susceptible-infectious-recovered disease spreading through an RSN, and we demonstrate the existence of traveling wave solutions. If the distance kernel governing edge placement decays slower than exponential, the population-scale dynamics are dominated by long-range hops followed by local spread of traveling waves. This provides a theoretical modeling framework for recent observations of how epidemics like Ebola evolve in modern connected societies, with long-range connections seeding new focal points from which the epidemic locally spreads in a wavelike manner.

Sexual partnerships that overlap in time (concurrent relationships) may play a significant role in the HIV epidemic, but the precise effect is unclear. We derive edge-based compartmental models of disease spread in idealized dynamic populations with and without concurrency to allow for an investigation of its effects. Our models assume that partnerships change in time and individuals enter and leave the at-risk population. Infected individuals transmit at a constant per-partnership rate to their susceptible partners. In our idealized populations we find regions of parameter space where the existence of concurrent partnerships leads to substantially faster growth and higher equilibrium levels, but also regions in which the existence of concurrent partnerships has very little impact on the growth or the equilibrium. Additionally we find mixed regimes in which concurrency significantly increases the early growth, but has little effect on the ultimate equilibrium level. Guided by model predictions, we discuss general conditions under which concurrent relationships would be expected to have large or small effects in real-world settings. Our observation that the impact of concurrency saturates suggests that concurrency-reducing interventions may be most effective in populations with low to moderate concurrency.

The emergence of diseases such as Zika and Ebola has highlighted the need to understand the role of sexual transmission in the spread of diseases with a primarily non-sexual transmission route. In this paper we develop a number of low-dimensional models which are appropriate for a range of assumptions for how a disease will spread if it has sexual transmission through a sexual contact network combined with some other transmission mechanism, such as direct contact or vectors. The equations derived provide exact predictions for the dynamics of the corresponding simulations in the large population limit.

Organic semiconductors find increasing importance in spin transport devices due to the modulation and control of their properties through chemical synthetic versatility. The organic materials are used as interlayers between two ferromagnet (FM) electrodes in organic spin valves (OSV), as well as for magnetic spin manipulation of metal-organic complexes at the molecular level. In the latter, specifically, the substrate-induced magnetic switching in a paramagnetic molecule has been evoked extensively, but studied by delicate surface spectroscopies. Here we present evidence of the substantial magnetic switching in a nanosized thin film of the paramagnetic molecule, tris(8-hydroxyquinoline)iron(III) (Feq3) deposited on a FM substrate, using the magnetoresistance response of electrical spin-injection in an OSV structure, and the inverse-spin-Hall effect induced by state-of-art pulsed microwave spin-pumping. We show that interfacial spin control at the molecular level may lead to a macroscopic organic spin transport device, thus, bridging the gap between organic spintronics and molecular spintronics.

In recent years, many variants of percolation have been used to study network structure and the behavior of processes spreading on networks. These include bond percolation, site percolation, $k$-core percolation, bootstrap percolation, the generalized epidemic process, and the Watts Threshold Model (WTM). We show that --- except for bond percolation --- each of these processes arises as a special case of the WTM and bond percolation arises from a small modification. In fact "heterogeneous $k$-core percolation", a corresponding "heterogeneous bootstrap percolation" model, and the generalized epidemic process are completely equivalent to one another and the WTM. We further show that a natural generalization of the WTM in which individuals "transmit" or "send a message" to their neighbors with some probability less than $1$ can be reformulated in terms of the WTM, and so this apparent generalization is in fact not more general. Finally, we show that in bond percolation, finding the set of nodes in the component containing a given node is equivalent to finding the set of nodes activated if that node is initially activated and the node thresholds are chosen from the appropriate distribution. A consequence of these results is that mathematical techniques developed for the WTM apply to these other models as well, and techniques that were developed for some particular case may in fact apply much more generally., Comment: To appear in PRE

A complex contagion is an infectious process in which individuals may require multiple transmissions before changing state. These are used to model behaviors if an individual only adopts a particular behavior after perceiving a consensus among others. We may think of individuals as beginning inactive and becoming active after contact with a sufficient number of active partners. These have been studied in a number of cases, but analytic models for the dynamic spread of complex contagions are typically complex. Here we study the dynamics of the Watts Threshold Model (WTM) assuming transmission occurs in continuous time as a Poisson process, or in discrete time where individuals transmit to all partners in the time step following their activation. We adapt techniques developed for infectious disease modeling to develop and analyze analytic models for the dynamics of the WTM in Configuration Model networks and a class of random clustered (triangle-based) networks. The resulting model is relatively simple and compact. We use it to gain insights into the contagion dynamics. In the infinite population limit, we derive conditions under which cascades happen with an arbitrarily small initial proportion active, confirming a hypothesis of Watts for this case. We also observe hybrid phase transitions when cascades are not possible for small initial conditions, but occur for large enough initial conditions. We derive sufficient conditions for this hybrid phase transition to occur. We show that in many cases, if the hybrid phase transition occurs, all individuals eventually become active. Finally, we discuss the role clustering plays in facilitating or impeding the spread and find that the hypothesis of Watts that was confirmed in Configuration Model networks does not hold in general. This approach allows us to unify many existing disparate observations and derive new results., Comment: key words: Watts Threshold Model, Cascades, Hybrid Bifurcation, Complex Contagion, Dynamic Model

Epidemic spreading phenomena are ubiquitous in nature and society. Examples include the spreading of diseases, information, and computer viruses. Epidemics can spread by local spreading, where infected nodes can only infect a limited set of direct target nodes and global spreading, where an infected node can infect every other node. In reality, many epidemics spread using a hybrid mixture of both types of spreading. In this study we develop a theoretical framework for studying hybrid epidemics, and examine the optimum balance between spreading mechanisms in terms of achieving the maximum outbreak size. We show the existence of critically hybrid epidemics where neither spreading mechanism alone can cause a noticeable spread but a combination of the two spreading mechanisms would produce an enormous outbreak. Our results provide new strategies for maximising beneficial epidemics and estimating the worst outcome of damaging hybrid epidemics.

We consider the spread of infectious disease through contact networks of Configuration Model type. We assume that the disease spreads through contacts and infected individuals recover into an immune state. We discuss a number of existing mathematical models used to investigate this system, and show relations between the underlying assumptions of the models. In the process we offer simplifications of some of the existing models. The distinctions between the underlying assumptions are subtle, and in many if not most cases this subtlety is irrelevant. Indeed, under appropriate conditions the models are equivalent. We compare the benefits and disadvantages of the different models, and discuss their application to other populations (\emph{e.g.,} clustered networks). Finally we discuss ongoing challenges for network-based epidemic modeling.

We study a simple model of a forager as a walk that modifies a relaxing substrate. Within it simplicity, this provides an insight on a number of relevant and non-intuitive facts. Even without memory of the good places to feed and no explicit cost of moving, we observe the emergence of a finite home range. We characterize the walks and the use of resources in several statistical ways, involving the behavior of the average used fraction of the system, the length of the cycles followed by the walkers, and the frequency of visits to plants. Preliminary results on population effects are explored by means of a system of two non directly interacting animals. Properties of the overlap of home ranges show the existence of a set of parameters that provides the best utilization of the shared resource.