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Last October, a paper I co-authored hit the arXivotubes (1110.3845, to be specific). This was, on reflection, one of the better things which happened to me last October. (It was, as the song sez, a lonesome month in a rather immemorial year.) Since then, more relevant work from other people has appeared. I’m collecting pointers here, most of them to freely available articles.

I read this one a while ago in non-arXiv preprint form, but now it’s on the arXiv. M. Raghib et al. (2011), “A Multiscale maximum entropy moment closure for locally regulated space-time point process models of population dynamics”, Journal of Mathematical Biology 62, 5: 605–53. arXiv:1202.6092 [q-bio].

Abstract: The pervasive presence spatial and size structure in biological populations challenges fundamental assumptions at the heart of continuum models of population dynamics based on mean densities (local or global) only. Individual-based models (IBM’s) were introduced over the last decade in an attempt to overcome this limitation by following explicitly each individual in the population. Although the IBM approach has been quite insightful, the capability to follow each individual usually comes at the expense of analytical tractability, which limits the generality of the statements that can be made. For the specific case of spatial structure in populations of sessile (and identical) organisms, space-time point processes with local regulation seem to cover the middle ground between analytical tractability and a higher degree of biological realism. Continuum approximations of these stochastic processes distill their fundamental properties, but they often result in infinite hierarchies of moment equations. We use the principle of constrained maximum entropy to derive a closure relationship for one such hierarchy truncated at second order using normalization and the product densities of first and second orders as constraints. The resulting `maxent’ closure is similar to the Kirkwood superposition approximation, but it is complemented with previously unknown correction terms that depend on on the area for which third order correlations are irreducible. This region also serves as a validation check, since it can only be found if the assumptions of the closure are met. Comparisons between simulations of the point process, alternative heuristic closures, and the maxent closure show significant improvements in the ability of the maxent closure to predict equilibrium values for mildly aggregated spatial patterns.


J. A. Bonachela, M. A. Munoz and S. A. Levin (2012). “Patchiness and Demographic Noise in Three Ecological Examples” [arXiv:1205.3389].

Abstract: Understanding the causes and effects of spatial aggregation is one of the most fundamental problems in ecology. Aggregation is an emergent phenomenon arising from the interactions between the individuals of the population, able to sense only—at most—local densities of their cohorts. Thus, taking into account the individual-level interactions and fluctuations is essential to reach a correct description of the population. Classic deterministic equations are suitable to describe some aspects of the population, but leave out features related to the stochasticity inherent to the discreteness of the individuals.Stochastic equations for the population do account for these fluctuation-generated effects by means of demographic noise terms but, owing to their complexity, they can be difficult (or, at times, impossible) to deal with. Even when they can be written in a simple form, they are still difficult to numerically integrate due to the presence of the “square-root” intrinsic noise. In this paper, we discuss a simple way to add the effect of demographic stochasticity to three classic, deterministic ecological examples where aggregation plays an important role. We study the resulting equations using a recently-introduced integration scheme especially devised to integrate numerically stochastic equations with demographic noise. Aimed at scrutinizing the ability of these stochastic examples to show aggregation, we find that the three systems not only show patchy configurations, but also undergo a phase transition belonging to the directed percolation universality class.

A. F. Lütz et al. (2012), “Intransitivity and coexistence in four species cyclic games” [arXiv:1205.6411].

Abstract: Intransitivity is a property of connected, oriented graphs representing species interactions that may drive their coexistence even in the presence of competition, the standard example being the three species Rock-Paper-Scissors game. We consider here a generalization with four species, the minimum number of species that allows other interactions beyond the single loop (one predator, one prey). We show that, contrary to the mean field prediction, on a square lattice the model presents a transition, as the invasion rates change, from a coexistence to a state in which one species gets extinct. Such a dependence on the invasion rates shows that the interaction graph structure alone is not enough to predict the outcome of such models. In addition, different invasion rates permit to tune the level of transitiveness, indicating that for the coexistence of all species to persist, there must be a minimum amount of intransitivity.

Interesting things the authors observe about their model:

1. A phase transition as a function of invasion rate $\chi$ appears in the lattice simulations which the mean-field approximation doesn’t pick up at all.

2. The pair approximation (first-order correction to the mean field) fails too. I’d like to see this treated in more detail, as I’ve grown used to seeing mean-field approximations fail for spatial lattices, so the interesting question is whether the adjustments on top of them do any better.

U. Dobramysl and U. C. Täuber (2012), “Environmental vs. demographic variability in two-species predator-prey models” [arXiv:1206.0973].

Abstract: We investigate the competing effects and relative importance of intrinsic demographic and environmental variability on the evolutionary dynamics of a stochastic two-species Lotka-Volterra model by means of Monte Carlo simulations on a two-dimensional lattice. Individuals are assigned inheritable predation efficiencies; quenched randomness in the spatially varying reaction rates serves as environmental noise. We find that environmental variability enhances the population densities of both predators and prey while demographic variability leads to essentially neutral optimization.

X. Chen et al. (2012), “Impact of generalized benefit functions on the evolution of cooperation in spatial public goods games with continuous strategies” Physical Review E 85: 066133 [arXiv:1206.7119]. Interesting on a first skim (I’ve done some numerical work on a case analogous to their sharp-cutoff limit, in various population structures).

Abstract: Cooperation and defection may be considered as two extreme responses to a social dilemma. Yet the reality is much less clear-cut. Between the two extremes lies an interval of ambivalent choices, which may be captured theoretically by means of continuous strategies defining the extent of the contributions of each individual player to the common pool. If strategies are chosen from the unit interval, where 0 corresponds to pure defection and 1 corresponds to the maximal contribution, the question is what is the characteristic level of individual investments to the common pool that emerges if the evolution is guided by different benefit functions. Here we consider the steepness and the threshold as two parameters defining an array of generalized benefit functions, and we show that in a structured population there exist intermediate values of both at which the collective contributions are maximal. However, as the cost-to-benefit ratio of cooperation increases the characteristic threshold decreases, while the corresponding steepness increases. Our observations remain valid if more complex sigmoid functions are used, thus reenforcing the importance of carefully adjusted benefits for high levels of public cooperation.

L. D. Fernandes and M. A. M. de Aguiar (2012), “Turing patterns and apparent competition in predator-prey food webs on networks” [arXiv:1207.3424].

Abstract: Reaction-diffusion systems may lead to the formation of steady state heterogeneous spatial patterns, known as Turing patterns. Their mathematical formulation is important for the study of pattern formation in general and play central roles in many fields of biology, such as ecology and morphogenesis. In the present study we focus on the role of Turing patterns in describing the abundance distribution of predator and prey species distributed in patches in a scale free network structure. We extend the original model proposed by Nakao and Mikhailov by considering food chains with several interacting pairs of preys and predators. We identify patterns of species distribution displaying high degrees of apparent competition driven by Turing instabilities. Our results provide further indication that differences in abundance distribution among patches may be, at least in part, due to self organized Turing patterns, and not necessarily to intrinsic environmental heterogeneity.

A. Szolnoki et al. (2012), “Defense mechanisms of empathetic players in the spatial ultimatum game” Physical Review Letters 109: 078701 [arXiv:1207.4786].

Abstract: Experiments on the ultimatum game have revealed that humans are remarkably fond of fair play. When asked to share an amount of money, unfair offers are rare and their acceptance rate small. While empathy and spatiality may lead to the evolution of fairness, thus far considered continuous strategies have precluded the observation of solutions that would be driven by pattern formation. Here we introduce a spatial ultimatum game with discrete strategies, and we show that this simple alteration opens the gate to fascinatingly rich dynamical behavior. Besides mixed stationary states, we report the occurrence of traveling waves and cyclic dominance, where one strategy in the cycle can be an alliance of two strategies. The highly webbed phase diagram, entailing continuous and discontinuous phase transitions, reveals hidden complexity in the pursuit of human fair play.

S. M. Messinger and A. Ostling (2012), “The influence of host demography, pathogen virulence, and relationships with pathogen virulence on the evolution of pathogen transmission in a spatial context” Evolutionary Ecology DOI: 10.1007/s10682-012-9594-y.

Abstract: A major challenge in evolutionary ecology is to explain extensive natural variation in transmission rates and virulence across pathogens. Host and pathogen ecology is a potentially important source of that variation. Theory of its effects has been developed through the study of non-spatial models, but host population spatial structure has been shown to influence evolutionary outcomes. To date, the effects of basic host and pathogen demography on pathogen evolution have not been thoroughly explored in a spatial context. Here we use simulations to show that space produces novel predictions of the influence of the shape of the pathogen’s transmission–virulence tradeoff, as well as host reproduction and mortality, on the pathogen’s evolutionary stable transmission rate. Importantly, non-spatial models predict that neither the slope of linear transmission–virulence relationships, nor the host reproduction rate will influence pathogen evolution, and that host mortality will only influence it when there is a transmission–virulence tradeoff. We show that this is not the case in a spatial context, and identify the ecological conditions under which spatial effects are most influential. Thus, these results may help explain observed natural variation among pathogens unexplainable by non-spatial models, and provide guidance about when space should be considered. We additionally evaluate the ability of existing analytical approaches to predict the influence of ecology, namely spatial moment equations closed with an improved pair approximation (IPA). The IPA is known to have limited accuracy, but here we show that in the context of pathogens the limitations are substantial: in many cases, IPA incorrectly predicts evolution to pathogen-driven extinction. Despite these limitations, we suggest that the impact of ecology can still be understood within the conceptual framework arising from spatial moment equations, that of “self-shading’’, whereby the spread of highly transmissible pathogens is impeded by local depletion of susceptible hosts.

I think “descendant-shading” would be a better term than “self-shading” here, since it indicates more clearly the timescales involved.

S. Heilmann, K. Sneppen and S. Krishna (2012), “Coexistence of phage and bacteria on the boundary of self-organized refuges” PNAS 109, 31: 12828–33. DOI: 10.1073/pnas.1200771109.

Abstract: Bacteriophage are voracious predators of bacteria and a major determinant in shaping bacterial life strategies. Many phage species are virulent, meaning that infection leads to certain death of the host and immediate release of a large batch of phage progeny. Despite this apparent voraciousness, bacteria have stably coexisted with virulent phages for eons. Here, using individual-based stochastic spatial models, we study the conditions for achieving coexistence on the edge between two habitats, one of which is a bacterial refuge with conditions hostile to phage whereas the other is phage friendly. We show how bacterial density-dependent, or quorum-sensing, mechanisms such as the formation of biofilm can produce such refuges and edges in a self-organized manner. Coexistence on these edges exhibits the following properties, all of which are observed in real phage–bacteria ecosystems but difficult to achieve together in nonspatial ecosystem models: (i) highly efficient virulent phage with relatively long lifetimes, high infection rates and large burst sizes; (ii) large, stable, and high-density populations of phage and bacteria; (iii) a fast turnover of both phage and bacteria; and (iv) stability over evolutionary timescales despite imbalances in the rates of phage vs. bacterial evolution.

G. Demirel, F. Vazquez, G. A. Böhme and T. Gross (2012), “Moment-Closure Approximations for Discrete Adaptive Networks” [arXiv:1211.0449].

Moment-closure approximations are an important tool in the analysis of the dynamics on both static and adaptive networks. Here, we provide a broad survey over different approximation schemes by applying each of them to the adaptive voter model. While already the simplest schemes provide reasonable qualitative results, even very complex and sophisticated approximations fail to capture the dynamics quantitatively. We then perform a detailed analysis that identifies the emergence of specific correlations as the reason for the failure of established approaches, before presenting a simple approximation scheme that works best in the parameter range where all other approaches fail. By combining a focused review of published results with new analysis and illustrations, we seek to build up an intuition regarding the situations when existing approaches work, when they fail, and how new approaches can be tailored to specific problems.

C. Reigada and M. A. M. de Aguiar (2012), “Host-parasitoid persistence over variable spatio-temporally susceptible habitats: bottom-up effects of ephemeral resourcesOikos 121: 1665–79. DOI:10.1111/j.1600-0706.2011.20259.x.

We experimentally and theoretically investigated the persistence of hosts and parasitoids interacting in a Metapopulation structure consisting of Ephemeral Local Patches (MELPs). We used a host-parasitoid system consisting of necrophagous Diptera species and their pupal parasitoids. The basal resources used by the host species were assumed to be ephemeral, supporting only one generation of individuals before completely disappearing from the environment. We experimentally measured the host-parasitoid persistence and the effects of local demographic processes in two scenarios: (1) constant occurrence of basal resources at a single site (no dispersion or colonization of other sites) and (2) variable occurrence of basal resources between two sites (colonization of a new patch requiring species dispersal). The experimental setup and findings were then formalized into a mathematical model describing the interaction dynamics in a MELP structure. We evaluated the contribution of several factors to the host-parasitoid coexistence, such as resource allocation probability (probability of resource appearance in a site), variation in resource size and number of sites available to receive resources in the MELP. We found that demographic fluctuations and environmental stochasticity affected the density of migrants, patch habitat connectivity, persistence and spatial distribution of interacting species.

They did experiments! On nonsimulated life forms! With rotting meat and everything!

B. Allen, C. E. Tarnita (2012), “Measures of success in a class of evolutionary models with fixed population size and structure” Journal of Mathematical Biology online before print, DOI:10.1007/s00285-012-0622-x.

We investigate a class of evolutionary models, encompassing many established models of well-mixed and spatially structured populations. Models in this class have fixed population size and structure. Evolution proceeds as a Markov chain, with birth and death probabilities dependent on the current population state. Starting from basic assumptions, we show how the asymptotic (long-term) behavior of the evolutionary process can be characterized by probability distributions over the set of possible states. We then define and compare three quantities characterizing evolutionary success: fixation probability, expected frequency, and expected change due to selection. We show that these quantities yield the same conditions for success in the limit of low mutation rate, but may disagree when mutation is present. As part of our analysis, we derive versions of the Price equation and the replicator equation that describe the asymptotic behavior of the entire evolutionary process, rather than the change from a single state. We illustrate our results using the frequency-dependent Moran process and the birth–death process on graphs as examples. Our broader aim is to spearhead a new approach to evolutionary theory, in which general principles of evolution are proven as mathematical theorems from axioms.

(Last update: 2 December 2012.)