Finding analytical approximations for discrete, stochastic, individual-based models of ecology.

Discrete time, spatially extended models play an important role in ecology, modelling population dynamics of species ranging from micro-organisms to birds. An important question is how 'bottom up', individual-based models can be approximated by 'top down' models of dynamics. Here, we study a class of spatially explicit individual-based models with contest competition: where species compete for space in local cells and then disperse to nearby cells. We start by describing simulations of the model, which exhibit large-scale discrete oscillations and characterize these oscillations by measuring spatial correlations. We then develop two new approximate descriptions of the resulting spatial population dynamics. The first is based on local interactions of the individuals and allows us to give a difference equation approximation of the system over small dispersal distances. The second approximates the long-range interactions of the individual-based model. These approximations capture demographic stochasticity from the individual-based model and show that dispersal stabilizes population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the non-spatial global approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Our results provide new approximate analytical descriptions of a complex bottom-up model and deepen understanding of spatial population dynamics.

Using neuronal models to capture burst-and-glide motion and leadership in fish.

While mathematical models, in particular self-propelled particle models, capture many properties of large fish schools, they do not always capture the interactions of smaller shoals. Nor do these models tend to account for the use of intermittent locomotion, often referred to as burst-and-glide, by many species. In this paper, we propose a model of social burst-and-glide motion by combining a well-studied model of neuronal dynamics, the FitzHugh-Nagumo model, with a model of fish motion. We first show that our model can capture the motion of a single fish swimming down a channel. Extending to a two-fish model, where visual stimulus of a neighbour affects the internal burst or glide state of the fish, we observe a rich set of dynamics found in many species. These include: leader-follower behaviour; periodic changes in leadership; apparently random (i.e. chaotic) leadership change; and tit-for-tat turn taking. Moreover, unlike previous studies where a randomness is required for leadership switching to occur, we show that this can instead be the result of deterministic interactions. We give several empirically testable predictions for how bursting fish interact and discuss our results in light of recently established correlations between fish locomotion and brain activity.

The lost art of mathematical modelling.

We provide a critique of mathematical biology in light of rapid developments in modern machine learning. We argue that out of the three modelling activities - (1) formulating models; (2) analysing models; and (3) fitting or comparing models to data - inherent to mathematical biology, researchers currently focus too much on activity (2) at the cost of (1). This trend, we propose, can be reversed by realising that any given biological phenomenon can be modelled in an infinite number of different ways, through the adoption of a pluralistic approach, where we view a system from multiple, different points of view. We explain this pluralistic approach using fish locomotion as a case study and illustrate some of the pitfalls - universalism, creating models of models, etc. - that hinder mathematical biology. We then ask how we might rediscover a lost art: that of creative mathematical modelling.

Friend of a friend models of network growth.

One of the best-known models in network science is preferential attachment. In this model, the probability of attaching to a node depends on the degree of all nodes in the population, and thus depends on global information. In many biological, physical and social systems, however, interactions between individuals depend only on local information. Here, we investigate a truly local model of network formation-based on the idea of a friend of a friend-with the following rule: individuals choose one node at random and link to it with probability p, then they choose a neighbour of that node and link with probability q. Our model produces power-laws with empirical exponents ranging from 1.5 upwards and clustering coefficients ranging from 0 up to 0.5 (consistent with many real networks). For small p and q = 1, the model produces super-hub networks, and we prove that for p = 0 and q = 1, the proportion of non-hubs tends to 1 as the network grows. We show that power-law degree distributions, small world clustering and super-hub networks are all outcomes of this, more general, yet conceptually simple model.

Rapid evolution of coordinated and collective movement in response to artificial selection.

Collective motion occurs when individuals use social interaction rules to respond to the movements and positions of their neighbors. How readily these social decisions are shaped by selection remains unknown. Through artificial selection on fish (guppies, Poecilia reticulata) for increased group polarization, we demonstrate rapid evolution in how individuals use social interaction rules. Within only three generations, groups of polarization-selected females showed a 15% increase in polarization, coupled with increased cohesiveness, compared to fish from control lines. Although lines did not differ in their physical swimming ability or exploratory behavior, polarization-selected fish adopted faster speeds, particularly in social contexts, and showed stronger alignment and attraction responses to multiple neighbors. Our results reveal the social interaction rules that change when collective behavior evolves.

Inferring the dynamics of rising radical right-wing party support using Gaussian processes.

The use of classical regression techniques in social science can prevent the discovery of complex, nonlinear mechanisms and often relies too heavily on both the expertise and prior expectations of the data analyst. In this paper, we present a regression methodology that combines the interpretability of traditional, well used, statistical methods with the full predictability and flexibility of Bayesian statistics techniques. Our modelling approach allows us to find and explain the mechanisms behind the rise of Radical Right-wing Populist parties (RRPs) that we would have been unable to find using traditional methods. Using Swedish municipality-level data (2002-2018), we find no evidence that the proportion of foreign-born residents is predictive of increases in RRP support. Instead, education levels and population density are the significant variables that impact the change in support for the RRP, in addition to spatial and temporal control variables. We argue that our methodology, which produces models with considerably better fit of the complexity and nonlinearities often found in social systems, provides a better tool for hypothesis testing and exploration of theories about RRPs and other social movements. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Phenotypic variability predicts decision accuracy in unicellular organisms.

When deciding between different options, animals including humans face the dilemma that fast decisions tend to be erroneous, whereas accurate decisions tend to be relatively slow. Recently, it has been suggested that differences in the efficacy with which animals make a decision relate closely to individual behavioural differences. In this paper, we tested this hypothesis in a unique unicellular organism, the slime mould Physarum polycephalum. We first confirmed that slime moulds differed consistently in their exploratory behaviour from 'fast' to 'slow' explorers. Second, we showed that slow explorers made more accurate decisions than fast explorers. Third, we demonstrated that slime moulds integrated food cues in time and achieved higher accuracy when sampling time was longer. Lastly, we showed that in a competition context, fast explorers excelled when a single food source was offered, while slow explorers excelled when two food sources varying in quality were offered. Our results revealed that individual differences in accuracy were partly driven by differences in exploratory behaviour. These findings support the hypothesis that decision-making abilities are associated with behavioural types, even in unicellular organisms.

Choice modelling with Gaussian processes in the social sciences: A case study of neighbourhood choice in Stockholm.

We present a non-parametric extension of the conditional logit model, using Gaussian process priors. The conditional logit model is used in quantitative social science for inferring interaction effects between personal features and choice characteristics from observations of individual multinomial decisions, such as where to live, which car to buy or which school to choose. The classic, parametric model presupposes a latent utility function that is a linear combination of choice characteristics and their interactions with personal features. This imposes strong and unrealistic constraints on the form of individuals' preferences. Extensions using non-linear basis functions derived from the original features can ameliorate this problem but at the cost of high model complexity and increased reliance on the user in model specification. In this paper we develop a non-parametric conditional logit model based on Gaussian process logit models. We demonstrate its application on housing choice data from over 50,000 moving households from the Stockholm area over a two year period to reveal complex homophilic patterns in income, ethnicity and parental status.

Mathematical modeling reveals spontaneous emergence of self-replication in chemical reaction systems.

Explaining the origin of life requires us to elucidate how self-replication arises. To be specific, how can a self-replicating entity develop spontaneously from a chemical reaction system in which no reaction is self-replicating? Previously proposed mathematical models either supply an explicit framework for a minimal living system or consider only catalyzed reactions, and thus fail to provide a comprehensive theory. Here, we set up a general mathematical model for chemical reaction systems that properly accounts for energetics, kinetics, and the conservation law. We found that 1) some systems are collectively catalytic, a mode whereby reactants are transformed into end products with the assistance of intermediates (as in the citric acid cycle), whereas some others are self-replicating, that is, different parts replicate each other and the system self-replicates as a whole (as in the formose reaction, in which sugar is replicated from formaldehyde); 2) side reactions do not always inhibit such systems; 3) randomly chosen chemical universes (namely random artificial chemistries) often contain one or more such systems; 4) it is possible to construct a self-replicating system in which the entropy of some parts spontaneously decreases, in a manner similar to that discussed by Schrödinger; and 5) complex self-replicating molecules can emerge spontaneously and relatively easily from simple chemical reaction systems through a sequence of transitions. Together, these results start to explain the origins of prebiotic evolution.

Is the golden ratio a universal constant for self-replication?

The golden ratio, ϕ = 1.61803…, has often been found in connection with biological phenomena, ranging from spirals in sunflowers to gene frequency. One example where the golden ratio often arises is in self-replication, having its mathematical origins in Fibonacci's sequence for "rabbit reproduction". Recently, it has been claimed that ϕ determines the ratio between the number of different nucleobases in human genome. Such empirical examples continue to give credence to the idea that the golden ratio is a universal constant, not only in mathematics but also for biology. In this paper, we employ a general framework for chemically realistic self-replicating reaction systems and investigate whether the ratio of chemical species population follows "universal constants". We find that many self-replicating systems can be characterised by an algebraic number, which, in some cases, is the golden ratio. However, many other algebraic numbers arise from these systems, and some of them-such as [Formula: see text] and 1.22074… which is also known as the 3rd lower golden ratio-arise more frequently in self-replicating systems than the golden ratio. The "universal constants" in these systems arise as roots of a limited number of distinct characteristic equations. In addition, these "universal constants" are transient behaviours of self-replicating systems, corresponding to the scenario that the resource inside the system is infinite, which is not always the case in practice. Therefore, we argue that the golden ratio should not be considered as a special universal constant in self-replicating systems, and that the ratios between different chemical species only go to certain numbers under some idealised scenarios.

Using Bayesian dynamical systems, model averaging and neural networks to determine interactions between socio-economic indicators.

Social and economic systems produce complex and nonlinear relationships in the indicator variables that describe them. We present a Bayesian methodology to analyze the dynamical relationships between indicator variables by identifying the nonlinear functions that best describe their interactions. We search for the 'best' explicit functions by fitting data using Bayesian linear regression on a vast number of models and then comparing their Bayes factors. The model with the highest Bayes factor, having the best trade-off between explanatory power and interpretability, is chosen as the 'best' model. To be able to compare a vast number of models, we use conjugate priors, resulting in fast computation times. We check the robustness of our approach by comparison with more prediction oriented approaches such as model averaging and neural networks. Our modelling approach is illustrated using the classical example of how democracy and economic growth relate to each other. We find that the best dynamical model for democracy suggests that long term democratic increase is only possible if the economic situation gets better. No robust model explaining economic development using these two variables was found.

Using activity and sociability to characterize collective motion.

A wide range of measurements can be made on the collective motion of groups, and the movement of individuals within them. These include, but are not limited to: group size, polarization, speed, turning speed, speed or directional correlations, and distances to near neighbours. From an ecological and evolutionary perspective, we would like to know which of these measurements capture biologically meaningful aspects of an animal's behaviour and contribute to its survival chances. Previous simulation studies have emphasized two main factors shaping individuals' behaviour in groups; attraction and alignment. Alignment responses appear to be important in transferring information between group members and providing synergistic benefits to group members. Likewise, attraction to conspecifics is thought to provide benefits through, for example, selfish herding. Here, we use a factor analysis on a wide range of simple measurements to identify two main axes of collective motion in guppies (Poecilia reticulata): (i) sociability, which corresponds to attraction (and to a lesser degree alignment) to neighbours, and (ii) activity, which combines alignment with directed movement. We show that for guppies, predation in a natural environment produces higher degrees of sociability and (in females) lower degrees of activity, while female guppies sorted for higher degrees of collective alignment have higher degrees of both sociability and activity. We suggest that the activity and sociability axes provide a useful framework for measuring the behaviour of animals in groups, allowing the comparison of individual and collective behaviours within and between species.This article is part of the theme issue 'Collective movement ecology'.

Brain size does not impact shoaling dynamics in unfamiliar groups of guppies (Poecilia reticulata).

Collective movement is achieved when individuals adopt local rules to interact with their neighbours. How the brain processes information about neighbours' positions and movements may affect how individuals interact in groups. As brain size can determine such information processing it should impact collective animal movement. Here we investigate whether brain size affects the structure and organisation of newly forming fish shoals by quantifying the collective movement of guppies (Poecilia reticulata) from large- and small-brained selection lines, with known differences in learning and memory. We used automated tracking software to determine shoaling behaviour of single-sex groups of eight or two fish and found no evidence that brain size affected the speed, group size, or spatial and directional organisation of fish shoals. Our results suggest that brain size does not play an important role in how fish interact with each other in these types of moving groups of unfamiliar individuals. Based on these results, we propose that shoal dynamics are likely to be governed by relatively basic cognitive processes that do not differ in these brain size selected lines of guppies.

How predation shapes the social interaction rules of shoaling fish.

Predation is thought to shape the macroscopic properties of animal groups, making moving groups more cohesive and coordinated. Precisely how predation has shaped individuals' fine-scale social interactions in natural populations, however, is unknown. Using high-resolution tracking data of shoaling fish (Poecilia reticulata) from populations differing in natural predation pressure, we show how predation adapts individuals' social interaction rules. Fish originating from high predation environments formed larger, more cohesive, but not more polarized groups than fish from low predation environments. Using a new approach to detect the discrete points in time when individuals decide to update their movements based on the available social cues, we determine how these collective properties emerge from individuals' microscopic social interactions. We first confirm predictions that predation shapes the attraction-repulsion dynamic of these fish, reducing the critical distance at which neighbours move apart, or come back together. While we find strong evidence that fish align with their near neighbours, we do not find that predation shapes the strength or likelihood of these alignment tendencies. We also find that predation sharpens individuals' acceleration and deceleration responses, implying key perceptual and energetic differences associated with how individuals move in different predation regimes. Our results reveal how predation can shape the social interactions of individuals in groups, ultimately driving differences in groups' collective behaviour.

Local interactions and global properties of wild, free-ranging stickleback shoals.

Collective motion describes the global properties of moving groups of animals and the self-organized, coordinated patterns of individual behaviour that produce them. We examined the group-level patterns and local interactions between individuals in wild, free-ranging shoals of three-spine sticklebacks, Gasterosteus aculeatus. Our data reveal that the highest frequencies of near-neighbour encounters occur at between one and two body lengths from a focal fish, with the peak frequency alongside a focal individual. Fish also show the highest alignment with these laterally placed individuals, and generally with animals in front of themselves. Furthermore, fish are more closely matched in size, speed and orientation to their near neighbours than to more distant neighbours, indicating local organization within groups. Among the group-level properties reported here, we find that polarization is strongly influenced by group speed, but also the variation in speed among individuals and the nearest neighbour distances of group members. While we find no relationship between group order and group size, we do find that larger groups tend to have lower nearest neighbour distances, which in turn may be important in maintaining group order.

Body size affects the strength of social interactions and spatial organization of a schooling fish (Pseudomugil signifer).

While a rich variety of self-propelled particle models propose to explain the collective motion of fish and other animals, rigorous statistical comparison between models and data remains a challenge. Plausible models should be flexible enough to capture changes in the collective behaviour of animal groups at their different developmental stages and group sizes. Here, we analyse the statistical properties of schooling fish (Pseudomugil signifer) through a combination of experiments and simulations. We make novel use of a Boltzmann inversion method, usually applied in molecular dynamics, to identify the effective potential of the mean force of fish interactions. Specifically, we show that larger fish have a larger repulsion zone, but stronger attraction, resulting in greater alignment in their collective motion. We model the collective dynamics of schools using a self-propelled particle model, modified to include varying particle speed and a local repulsion rule. We demonstrate that the statistical properties of the fish schools are reproduced by our model, thereby capturing a number of features of the behaviour and development of schooling fish.

Modelling optimal allocation of resources in the context of an incurable disease.

Nodding syndrome has affected and led to the deaths of children between the ages of 5 and 15 in Northern Uganda since 2009. There is no reliable explanation of the disease, and currently the only treatment is through a nutritional programme of vitamins, combined with medication to prevent symptoms. In the absence of a proper medical treatment, we develop a dynamic compartmental model to plan the management of the syndrome and to curb its effects. We use incidence data from 2012 and 2013 from Pader, Lamwo and Kitgum regions of Uganda to parameterize the model. The model is then used to look at how to best plan the nutritional programme in terms of first getting children on to the programme through outreach, and then making sure they remain on the programme, through follow-up. For the current outbreak of nodding disease, we estimate that about half of available resources should be put into outreach. We show how to optimize the balance between outreach and follow-up in this particular example, and provide a general methodology for allocating resources in similar situations. Given the uncertainty of parameter estimates in such situations, we perform a robustness analysis to identify the best investment strategy. Our analysis offers a way of using available data to determine the best investment strategy of controlling nodding syndrome.

Setting development goals using stochastic dynamical system models.

The Millennium Development Goals (MDG) programme was an ambitious attempt to encourage a globalised solution to important but often-overlooked development problems. The programme led to wide-ranging development but it has also been criticised for unrealistic and arbitrary targets. In this paper, we show how country-specific development targets can be set using stochastic, dynamical system models built from historical data. In particular, we show that the MDG target of two-thirds reduction of child mortality from 1990 levels was infeasible for most countries, especially in sub-Saharan Africa. At the same time, the MDG targets were not ambitious enough for fast-developing countries such as Brazil and China. We suggest that model-based setting of country-specific targets is essential for the success of global development programmes such as the Sustainable Development Goals (SDG). This approach should provide clear, quantifiable targets for policymakers.

Insights into resource consumption, cross-feeding, system collapse, stability and biodiversity from an artificial ecosystem.

Community ecosystems at very different levels of biological organization often have similar properties. Coexistence of multiple species, cross-feeding, biodiversity and fluctuating population dynamics are just a few of the properties that arise in a range of ecological settings. Here we develop a bottom-up model of consumer-resource interactions, in the form of an artificial ecosystem 'number soup', which reflects basic properties of many bacterial and other community ecologies. We demonstrate four key properties of the number soup model: (i) communities self-organize so that all available resources are fully consumed; (ii) reciprocal cross-feeding is a common evolutionary outcome, which evolves in a number of stages, and many transitional species are involved; (iii) the evolved ecosystems are often 'robust yet fragile', with keystone species required to prevent the whole system from collapsing; (iv) non-equilibrium dynamics and chaotic patterns are general properties, readily generating rich biodiversity. These properties have been observed in empirical ecosystems, ranging from bacteria to rainforests. Establishing similar properties in an evolutionary model as simple as the number soup suggests that these four properties are ubiquitous features of all community ecosystems, and raises questions about how we interpret ecosystem structure in the context of natural selection.

Emergent Structural Mechanisms for High-Density Collective Motion Inspired by Human Crowds.

Collective motion of large human crowds often depends on their density. In extreme cases like heavy metal concerts and black Friday sales events, motion is dominated by physical interactions instead of conventional social norms. Here, we study an active matter model inspired by situations when large groups of people gather at a point of common interest. Our analysis takes an approach developed for jammed granular media and identifies Goldstone modes, soft spots, and stochastic resonance as structurally driven mechanisms for potentially dangerous emergent collective motion.