Rate limitation due to encounters is fundamental to many ecological interactions. Since encounter rate governs reaction rates, and thus, dynamics of systems, it deserves systematic study. In classical population biology, ecological dynamics rely on the assumption of perfectly mixed interacting entities (e.g., individuals, populations, etc.) in a spaceless world. The so-called mean field assumption assumes that encounter rates are driven exclusively by changes in the density of the interacting entities and not on how they are distributed or move in space. Therefore, the mean field assumption does not give any insight into relevant spatiotemporal statistical properties produced by the trajectories of moving entities through space. In the present study, we develop spatially explicit simulations of random walking particles (i.e., Lévy walkers) to evaluate encounter rate constraints beyond the mean field assumption. We show that encounter rate fluctuations are driven not only by physical aspects such as the size or the velocity of the interacting particles, but also by different motion patterns. In particular, superdiffusion phenomena might be relevant at low densities and/or low spatial dimensionality. Finally, we discuss potential adaptive responses of living organisms that may allow individuals to control how they diffuse through space and/or the spatial dimensions employed in the exploration process.

Complex networks with high numbers of nodes or links are often difficult to analyse. However, not all elements contribute equally to their structural patterns. A small number of elements (the hubs) seem to play a particularly relevant role in organizing the overall structure around them. But other parts of the architecture (such as hub-hub connecting elements) are also important. In this letter we present a new type of substructure, to be named the $K$-scaffold subgraph, able to capture all the essential network components. Their key features, including the so called critical scaffold graph, are analytically derived.

In this paper, we conducted a comparative study of how social organization takes place in a wasp colony and OSS developer communities. Both these systems display similar global organization patterns, such as hierarchies and clear labor divisions. As our analysis shows, both systems also define interacting agent networks with similar common features that reflect limited information sharing among agents. As far as we know, this is the first research study analyzing the patterns and functional significance of these systems' weighted-interaction networks. By illuminating the extent to which self-organization is responsible for patterns such as hierarchical structure, we can gain insight into the origins of organization in OSS communities.

Many urban settlements result from a spatially distributed, decentralized building process. Here we analyze the topological patterns of organization of a large collection of such settlements using the approach of complex networks. The global efficiency (based on the inverse of shortest-path lengths), robustness to disconnections and cost (in terms of length) of these graphs is studied and their possible origins analyzed. A wide range of patterns is found, from tree-like settlements (highly vulnerable to random failures) to meshed urban patterns. The latter are shown to be more robust and efficient.

In a recent paper, Krapivsky and Redner (Phys. Rev. E, 71 (2005) 036118) proposed a new growing network model with new nodes being attached to a randomly selected node, as well to all ancestors of the target node. The model leads to a sparse graph with an average degree growing logarithmically with the system size. Here we present compeling evidence for software networks being the result of a similar class of growing dynamics. The predicted pattern of network growth, as well as the stationary in- and out-degree distributions are consistent with the model. Our results confirm the view of large-scale software topology being generated through duplication-rewiring mechanisms. Implications of these findings are outlined.

Complex networks in both nature and technology have been shown to display characteristic, small subgraphs (so-called motifs) which appear to be related to their underlying functionality. All these networks share a common trait: they manipulate information at different scales in order to perform some kind of computation. Here we analyze a large set of software class diagrams and show that several highly frequent network motifs appear to be a consequence of network heterogeneity and size, thus suggesting a somewhat less relevant role of functionality. However, by using a simple model of network growth by duplication and rewiring, it is shown the rules of graph evolution seem to be largely responsible for the observed motif distribution.

Patterns of protein interactions are organized around complex heterogeneous networks. Their architecture has been suggested to be of relevance in understanding the interactome and its functional organization, which pervades cellular robustness. Transcription factors are particularly relevant in this context, given their central role in gene regulation. Here we present the first topological study of the human protein–protein interacting transcription factor network built using the TRANSFAC database. We show that the network exhibits scale‐free and small‐world properties with a hierarchical and modular structure, which is built around a small number of key proteins. Most of these proteins are associated with proliferative diseases and are typically not linked to each other, thus reducing the propagation of failures through compartmentalization. Network modularity is consistent with common structural and functional features and the features are generated by two distinct evolutionary strategies: amplification and shuffling of interacting domains through tinkering and acquisition of specific interacting regions. The function of the regulatory complexes may have played an active role in choosing one of them.

Rainforests biodiversity is sustained by the three‐dimensional structure of their canopy which provides a wide range of physical microenvironments. Given the dynamic nature of the forest, the recognition of stable vertical layers or strata in the canopy is controversial. The spatial characterisation of potential habitats of understory species is not straightforward due to the complex structure of rainforest canopies and the wide ecological variability to which rainforest species can be adapted. Here we present a new description of potential understory habitats that give rise to a well‐defined characteristic vertical scale of forest organization hc≈13 m. Species living in microenvironments occurring at canopy heights below this critical height hccan only experience landscapes with disconnected habitat patches (i.e. fragmented habitat landscapes), while those species capable of living also above hcwill experience a fully connected landscape of suitable microenvironmental conditions. The possible implications for plant dispersal and animal colonisation strategies living at the understory or close‐to‐floor are discussed in relation to rainforest gap‐dynamics, habitat loss and habitat fragmentation processes. Long‐range and directed dispersal strategies (e.g. plant seed dispersal by animals) are optimal for those species living below hc, providing the best exploration of scarce habitats and a major robustness to habitat changes. On the other hand, dispersal strategies of those species capable to exploit habitats above hcneed not to be based on directed long‐range mechanisms. Different dispersal strategies may in turn imply different sensitiveness of species to habitat loss and habitat fragmentation processes in the rainforest.

Internet is known to display a highly heterogeneous structure and complex fluctuations in its traffic dynamics. Congestion seems to be an inevitable result of user's behavior coupled to the network dynamics and it effects should be minimized by choosing appropriate routing strategies. But what are the requirements of routing depth in order to optimize the traffic flow? In this paper we analyse the behavior of Internet traffic with a topologically realistic spatial structure as described in a previous study [S.-H. Yook et al. , Proc. Natl Acad. Sci. USA 99, 13382 (2002)]. The model involves self-regulation of packet generation and different levels of routing depth. It is shown that it reproduces the relevant key, statistical features of Internet's traffic. Moreover, we also report the existence of a critical path horizon defining a transition from low-efficient traffic to highly efficient flow. This transition is actually a direct consequence of the web's small world architecture exploited by the routing algorithm. Once routing tables reach the network diameter, the traffic experiences a sudden transition from a low-efficient to a highly-efficient behavior. It is conjectured that routing policies might have spontaneously reached such a compromise in a distributed manner. Internet would thus be operating close to such critical path horizon.

Recent theoretical and empirical studies have focused on the topology of large networks of communication/interactions in biological, social and technological systems. Most of them have been studied in the scope of the small-world and scale-free networks’ theory. Here we analyze the characteristics of ant networks of galleries produced in a 2-D experimental setup. These networks are neither small-worlds nor scale-free networks and belong to a particular class of network, i.e. embedded planar graphs emerging from a distributed growth mechanism. We compare the networks of galleries with both minimal spanning trees and greedy triangulations. We show that the networks of galleries have a path system efficiency and robustness to disconnections closer to the one observed in triangulated networks though their cost is closer to the one of a tree. These networks may have been prevented to evolve toward the classes of small-world and scale-free networks because of the strong spatial constraints under which they grow, but they may share with many real networks a similar trend to result from a balance of constraints leading them to achieve both path system efficiency and robustness at low cost.

Models of habitat fragmentation have mainly explored the effects on a few-species ecologies or on a hierarchical community of competitors. These models reveal that, under different conditions, ecosystem response can involve sharp changes when some given thresholds are reached. However, perturbations, recruitment limitation and other causes may prevent competitive hierarchies from actually operating in natural conditions: the process of competitive exclusion underlying hierarchies could not be a determinant factor structuring communities. Here we explore both spatially-implicit and spatially-explicit metapopulation models for a competitive community, where the colonization-extinction dynamics takes place through neutral interactions. Here species interactions are not hierarchical at all but are somehow ecologically equivalent and just compete for space and resources through recruitment limitation. Our analysis shows the existence of a common destruction threshold for all species: whenever habitat loss reaches certain value a sudden biodiversity collapse takes place. Furthermore, the model is able to reproduce species-rank distributions and its spatially explicit counterpart predicts also species–area laws obtained from recent studies on rainforest plots. It is also discussed the relevance of percolation thresholds in limiting diversity once the landscape is broken into many patches.

A comparison between the evolution of cancer cell populations and RNA viruses reveals a number of remarkable similarities. Both display high levels of plasticity and adaptability as a consequence of high degrees of genetic variation. It has been suggested that, as it occurs with RNA viruses, there is a threshold in the levels of genetic instability affordable by cancer cells in order to be able to overcome selection barriers (Trends Genet. 15 (1999) M57). Here we explore this concept by means of a simple mathematical model. It is shown that an error threshold exists in this model, which investigates both competition between cancer cell populations and its impact on overall tumor growth dynamics. Once the threshold is reached, the highly unstable tumor cell populations, which were sustaining malignant growth, become unable to maintain their genetic information, which in turn triggers a slowed down overall tumor growth regime.

Plankton is the productive base of aquatic ecosystems and plays a major role in the global control of atmospheric carbon dioxide. Nevertheless, after intensive study, the factors that drive its spatial distribution are still far from being clear. The models proposed so far show very limited agreement with actual data as many of their results are not consistent with field observations. Here we show that fluctuations and turbulent diffusion in standard prey-predator models are able to accurately and consistently explain plankton field observations at mesoscales (1-100 km). This includes not only the spatial pattern but also its temporal evolution. We explicitly elucidate the interplay between physical and biological factors, suggesting that the form in which small scale biotic fluctuations are transferred to larger scales may constitute one of the key elements in determining the spatial distribution of plankton in the sea.

. The dynamics of cancer evolution is studied by means of a simple quasispecies model involving cells displaying high levels of genetic instability. Both continuous, mean-field and discrete, bit-string models are analysed. The string model is simulated on a single-peak landscape. It is shown that a phase transition exists at high levels of genetic instability, thus separating two phases of slow and rapid growth. The results suggest that, under a conserved level of genetic instability the cancer cell population will be close to the threshold level. Implications for therapy are outlined.

Many complex systems can be described in terms of networks of interacting units. Recent studies have shown that a wide class of both natural and artificial nets display a surprisingly widespread feature: the presence of highly heterogeneous distributions of links, providing an extraordinary source of robustness against perturbations. Although most theories concerning the origin of these topologies use growing graphs, here we show that a simple optimization process can also account for the observed regularities displayed by most complex nets. Using an evolutionary algorithm involving minimization of link density and average distance, four major types of networks are encountered: (a) sparse exponential-like networks, (b) sparse scale-free networks, (c) star networks and (d) highly dense networks, apparently defining three major phases. These constraints provide a new explanation for scaling of exponent about -3. The evolutionary consequences of these results are outlined.

In this paper, we present a general selection–mutation model of evolution on a one-dimensional continuous fitness space. The formulation of our model includes both the classical diffusion approach to mutation process as well as an alternative approach based on an integral operator with a mutation kernel. We show that both approaches produce fundamentally equivalent results. To illustrate the suitability of our model, we focus its analytical study into its application to recent experimental studies of in vitro viral evolution. More specifically, these experiments were designed to test previous theoretical predictions regarding the effects of multiple infection dynamics (i.e., coinfection and superinfection) on the virulence of evolving viral populations. The results of these experiments, however, did not match with previous theory. By contrast, the model we present here helps to understand the underlying viral dynamics on these experiments and makes new testable predictions about the role of parameters such the time between successive infections and the growth rates of resident and invading populations.

The emergence of a complex language is one of the fundamental events of human evolution, and several remarkable features suggest the presence of fundamental principles of organization. These principles seem to be common to all languages. The best known is the so-called Zipf's law, which states that the frequency of a word decays as a (universal) power law of its rank. The possible origins of this law have been controversial, and its meaningfulness is still an open question. In this article, the early hypothesis of Zipf of a principle of least effort for explaining the law is shown to be sound. Simultaneous minimization in the effort of both hearer and speaker is formalized with a simple optimization process operating on a binary matrix of signal–object associations. Zipf's law is found in the transition between referentially useless systems and indexical reference systems. Our finding strongly suggests that Zipf's law is a hallmark of symbolic reference and not a meaningless feature. The implications for the evolution of language are discussed. We explain how language evolution can take advantage of a communicative phase transition.

We introduce a simple search algorithm that explores the parameter of periodically perturbed discrete maps in order to find desired orbits through chaos control. The method has been applied to one-dimensional maps but is easily extendable to higher-dimensional systems. Here, we consider two types of chaos control involving proportional pulses in the system variables [Phys. Rev. Lett. 72, 1455 (1994)] and constant feedback [Phys. Rev. E 51, 6239 (1995)], the first case being presented in detail. It is shown that our method allows a rapid exploration of parameter space and the finding of high-fitness (i.e., controlled) solutions close to the target orbits, even when high periodicities are required.

Why are some ecosystems so rich, yet contain so many rare species? High species diversity, together with rarity, is a general trend in neotropical forests and coral reefs. However, the origin of such diversity and the consequences of food web complexity in both species abundances and temporal fluctuations are not well understood. Several regularities are observed in complex, multispecies ecosystems that suggest that these ecologies might be organized close to points of instability. We explore, in greater depth, a recent stochastic model of population dynamics that is shown to reproduce: (i) the scaling law linking species number and connectivity; (ii) the observed distributions of species abundance reported from field studies (showing long tails and thus a predominance of rare species); (iii) the complex fluctuations displayed by natural communities (including chaotic dynamics); and (iv) the species-area relations displayed by rainforest plots. It is conjectured that the conflict between the natural tendency towards higher diversity due to immigration, and the ecosystem level constraints derived from an increasing number of links, leaves the system poised at a critical boundary separating stable from unstable communities, where large fluctuations are expected to occur. We suggest that the patterns displayed by species-rich communities, including rarity, would result from such a spontaneous tendency towards instability.