Spatial patterns of tree yield explained by endogenous forces through a correspondence between the Ising model and ecology

Generalized synchronization in a tritrophic food web metacommunity

Complete synchronization among the metacommunity is known to elevate the risk of their extinction due to stochasticity and other environmental perturbations. Owing to the inherent heterogeneous nature of the metacommunity, we demonstrate the emergence of generalized synchronization among the patches of dispersally connected tritrophic food web using the framework of an auxiliary system approach and the mutual false nearest neighbor. We find that the critical value of the dispersal rate increases significantly with the size of the metacommunity for both unidirectional and bidirectional dispersals, which in turn corroborates that larger metacommunities are more stable than smaller ones. Further, we find that the critical value of the dispersal for the onset of generalized synchronization is smaller(larger) for bidirectional dispersal than that for unidirectional dispersal for smaller(larger) metacommunities. Most importantly, complete synchronization error remains finite even after the onset of generalized synchronization in a wider range of dispersal rate elucidating that the latter can serve as an early warning signal for the extinction of the metacommunity.

Prediction of depinning transitions in interface models using Gini and Kolkata indices

The intermittent dynamics of driven interfaces through disordered media and its subsequent depinning for large enough driving force is a common feature for a myriad of diverse systems, starting from mode-I fracture, vortex lines in superconductors, and magnetic domain walls to invading fluid in a porous medium, to name a few. In this work, we outline a framework that can give a precursory signal of the imminent depinning transition by monitoring the variations in sizes or the inequality of the intermittent responses of a system that are seen prior to the depinning point. In particular, we use measures traditionally used to quantify economic inequality, i.e., the Gini index and the Kolkata index, for the case of the unequal responses of precritical systems. The crossing point of these two indices serves as a precursor to imminent depinning. Given a scale-free size distribution of the responses, we calculate the expressions for these indices, evaluate their crossing points, and give a recipe for forecasting depinning transitions. We apply this method to the Edwards-Wilkinson, Kardar-Parisi-Zhang, and fiber bundle model interface with variable interaction strengths and quenched disorder. The results are applicable for any interface dynamics undergoing a depinning transition. The results also explain previously observed near-universal values of Gini and Kolkata indices in self-organized critical systems.

Finding critical points and correlation length exponents using finite size scaling of Gini index

The order parameter for a continuous transition shows diverging fluctuation near the critical point. Here we show, through numerical simulations and scaling arguments, that the inequality (or variability) between the values of an order parameter, measured near a critical point, is independent of the system size. Quantification of such variability through the Gini index (g) therefore leads to a scaling form g=G[|F-F_{c}|N^{1/dν}], where F denotes the driving parameter for the transition (e.g., temperature T for ferromagnetic to paramagnetic transition, or lattice occupation probability p in percolation), N is the system size, d is the spatial dimension and ν is the correlation length exponent. We demonstrate the scaling for the Ising model in two and three dimensions, site percolation on square lattice, and the fiber bundle model of fracture.

Modular tipping points: How local network structure impacts critical transitions in networked spin systems

Critical transitions describe a phenomenon where a system abruptly shifts from one stable state to an alternative, often detrimental, stable state. Understanding and possibly preventing the occurrence of a critical transition is thus highly relevant to many ecological, sociological, and physical systems. In this context, it has been shown that the underlying network structure of a system heavily impacts the transition behavior of that system. In this paper, we study a crucial but often overlooked aspect in critical transitions: the modularity of the system's underlying network topology. In particular, we investigate how the transition behavior of a networked system changes as we alter the local network structure of the system through controlled changes of the degree assortativity. We observe that systems with high modularity undergo cascading transitions, while systems with low modularity undergo more unified transitions. We also observe that networked systems that consist of nodes with varying degrees of connectivity tend to transition earlier in response to changes in a control parameter than one would anticipate based solely on the average degree of that network. However, in rare cases, such as when there is both low modularity and high degree disassortativity, the transition behavior aligns with what we would expected given the network's average degree. Results are confirmed for a diverse set of degree distributions including stylized two-degree networks, uniform, Poisson, and power-law degree distributions. On the basis of these results, we argue that to understand critical transitions in networked systems, they must be understood in terms of individual system components and their roles within the network structure.

Critical Scaling through Gini Index

In the systems showing critical behavior, various response functions have a singularity at the critical point. Therefore, as the driving field is tuned toward its critical value, the response functions change drastically, typically diverging with universal critical exponents. In this Letter, we quantify the inequality of response functions with measures traditionally used in economics, namely by constructing a Lorenz curve and calculating the corresponding Gini index. The scaling of such a response function, when written in terms of the Gini index, shows singularity at a point that is at least as universal as the corresponding critical exponent. The critical scaling, therefore, becomes a single parameter fit, which is a considerable simplification from the usual form where the critical point and critical exponents are independent. We also show that another measure of inequality, the Kolkata index, crosses the Gini index at a point just prior to the critical point. Therefore, monitoring these two inequality indices for a system where the critical point is not known can produce a precursory signal for the imminent criticality. This could be useful in many systems, including that in condensed matter, bio- and geophysics to atmospheric physics. The generality and numerical validity of the calculations are shown with the Monte Carlo simulations of the two dimensional Ising model, site percolation on square lattice, and the fiber bundle model of fracture.

The kinetic Ising model encapsulates essential dynamics of land pattern change

A land pattern change represents a globally significant trend with implications for the environment, climate and societal well-being. While various methods have been developed to predict land change, our understanding of the underlying change processes remains inadequate. To address this issue, we investigate the suitability of the two-dimensional kinetic Ising model (IM), an idealized model from statistical mechanics, for simulating land change dynamics. We test the IM on a variety of patterns, each with different focus land type. Specifically, we investigate four sites characterized by distinct patterns, presumably driven by different physical processes. Each site is observed on eight occasions between 2001 and 2019. Given the observed pattern at the time ti we find two parameters of the IM such that the model-evolved land pattern at ti+1 resembles the observed land pattern at that time. The data support simulating seven such transitions per site. Our findings indicate that the IM produces approximate matches to the observed patterns in terms of layout, composition, texture and patch size distributions. Notably, the IM simulations even achieve a high degree of cell-scale pattern accuracy in two of the sites. Nevertheless, the IM has certain limitations, including its inability to model linear features, account for the formation of new large patches and handle pattern shifts.

Spatial biology of Ising-like synthetic genetic networks

BACKGROUND

Understanding how spatial patterns of gene expression emerge from the interaction of individual gene networks is a fundamental challenge in biology. Developing a synthetic experimental system with a common theoretical framework that captures the emergence of short- and long-range spatial correlations (and anti-correlations) from interacting gene networks could serve to uncover generic scaling properties of these ubiquitous phenomena.

RESULTS

Here, we combine synthetic biology, statistical mechanics models, and computational simulations to study the spatial behavior of synthetic gene networks (SGNs) in Escherichia coli quasi-2D colonies growing on hard agar surfaces. Guided by the combined mechanisms of the contact process lattice simulation and two-dimensional Ising model (CPIM), we describe the spatial behavior of bi-stable and chemically coupled SGNs that self-organize into patterns of long-range correlations with power-law scaling or short-range anti-correlations. These patterns, resembling ferromagnetic and anti-ferromagnetic configurations of the Ising model near critical points, maintain their scaling properties upon changes in growth rate and cell shape.

CONCLUSIONS

Our findings shed light on the spatial biology of coupled and bistable gene networks in growing cell populations. This emergent spatial behavior could provide insights into the study and engineering of self-organizing gene patterns in eukaryotic tissues and bacterial consortia.

Critical transitions in degree mixed networks: A discovery of forbidden tipping regions in networked spin systems

Critical transitions can be conceptualized as abrupt shifts in the state of a system typically induced by changes in the system's critical parameter. They have been observed in a variety of systems across many scientific disciplines including physics, ecology, and social science. Because critical transitions are important to such a diverse set of systems it is crucial to understand what parts of a system drive and shape the transition. The underlying network structure plays an important role in this regard. In this paper, we investigate how changes in a network's degree sequence impact the resilience of a networked system. We find that critical transitions in degree mixed networks occur in general sooner than in their degree homogeneous counterparts of equal average degree. This relationship can be expressed with parabolic curves that describe how the tipping point changes when the nodes of an initially homogeneous degree network composed only of nodes with degree k1 are replaced by nodes of a different degree k2. These curves mark clear tipping boundaries for a given degree mixed network and thus allow the identification of possible tipping intersections and forbidden tipping regions when comparing networks with different degree sequences.

Noise-induced versus intrinsic oscillation in ecological systems

Studies of oscillatory populations have a long history in ecology. A first-principles understanding of these dynamics can provide insights into causes of population regulation and help with selecting detailed predictive models. A particularly difficult challenge is determining the relative role of deterministic versus stochastic forces in producing oscillations. We employ statistical physics concepts, including measures of spatial synchrony, that incorporate patterns at all scales and are novel to ecology, to show that spatial patterns can, under broad and well-defined circumstances, elucidate drivers of population dynamics. We find that when neighbours are coupled (e.g. by dispersal), noisy intrinsic oscillations become distinguishable from noise-induced oscillations at a transition point related to synchronisation that is distinct from the deterministic bifurcation point. We derive this transition point and show that it diverges from the deterministic bifurcation point as stochasticity increases. The concept of universality suggests that the results are robust and widely applicable.

By wind or wing: pollination syndromes and alternate bearing in horticultural systems

Cyclical fluctuations in reproductive output are widespread among perennial plants, from multi-year masting cycles in forest trees to alternate bearing in horticultural crops. In natural systems, ecological drivers such as climate and pollen limitation can result in synchrony among plants. Agricultural practices are generally assumed to outweigh ecological drivers that might synchronize alternate-bearing individuals, but this assumption has not been rigorously assessed and little is known about the role of pollen limitation as a driver of synchrony in alternate-bearing crops. We tested whether alternate-bearing perennial crops show signs of alternate bearing at a national scale and whether the magnitude of national-scale alternate bearing differs across pollination syndromes. We analysed the Food and Agriculture Organization of the United Nations time series (1961-2018) of national crop yields across the top-producing countries of 27 alternate-bearing taxa, 6 wind-pollinated and 21 insect-pollinated. Alternate bearing was common in these national data and more pronounced in wind-pollinated taxa, which exhibited a more negative lag-1 autocorrelation and a higher coefficient of variation (CV). We highlight the mutual benefits of integrating ecological theory and agricultural data for (i) advancing our understanding of perennial plant reproduction across time, space and taxa, and (ii) promoting stable farmer livelihoods and global food supply. This article is part of the theme issue 'The ecology and evolution of synchronized seed production in plants'.

Transition prediction in the Ising-model

Dynamical systems can be subject to critical transitions where a system’s state abruptly shifts from one stable equilibrium to another. To a certain extent such transitions can be predicted with a set of methods known as early warning signals. These methods are often developed and tested on systems simulated with equation-based approaches that focus on the aggregate dynamics of a system. Many ecological phenomena however seem to necessitate the consideration of a system’s micro-level interactions since only there the actual reasons for sudden state transitions become apparent. Agent-based approaches that simulate systems from the bottom up by explicitly focusing on these micro-level interactions have only rarely been used in such investigations. This study compares the performance of a bifurcation estimation method for predicting state transitions when applied to data from an equation-based and an agent-based version of the Ising-model. The results show that the method can be applied to agent-based models and, despite its greater stochasticity, can provide useful predictions about state changes in complex systems.

Early warning signals from the periphery: A model suggestion for the study of critical transitions

Studies on the possibility of predicting critical transitions with statistical methods known as early warning signals (EWS) are often conducted on data generated with equation-based models (EBMs). These models base on difference or differential equations, which aggregate a system's components in a mathematical term and therefore do not allow for a detailed analysis of interactions on micro-level. As an alternative, we suggest a simple, but highly flexible agent-based model (ABM), which, when applying EWS-analysis, gives reason to (a) consider social interaction, in particular negative feedback effects, as an essential trigger of critical transitions, and (b) to differentiate social interactions, for example in network representations, into a core and a periphery of agents and focus attention on the periphery. Results are tested against time series from a networked version of the Ising-model, which is often used as example for generating hysteretic critical transitions.

Environmental variation drives continental-scale synchrony of European beech reproduction

Spatial synchrony is the tendency of spatially separated populations to display similar temporal fluctuations. Synchrony affects regional ecosystem functioning, but it remains difficult to disentangle its underlying mechanisms. We leveraged regression on distance matrices and geography of synchrony to understand the processes driving synchrony of European beech masting over the European continent. Masting in beech shows distance-decay, but significant synchrony is maintained at spatial scales of up to 1,500 km. The spatial synchrony of the weather cues that drive interannual variation in reproduction also explains the regional spatial synchrony of masting. Proximity played no apparent role in influencing beech masting synchrony after controlling for synchrony in environmental variation. Synchrony of beech reproduction shows a clear biogeographical pattern, decreasing from the northwest to southeast Europe. Synchrony networks for weather cues resemble networks for beech masting, indicating that the geographical structure of weather synchrony underlies the biogeography of masting synchrony. Our results support the hypothesis that environmental factors, the Moran effect, are key drivers of spatial synchrony in beech seed production at regional scales. The geographical patterns of regional synchronization of masting have implications for regional forest production, gene flow, carbon cycling, disease dynamics, biodiversity, and conservation.

Cooperative root graft networks benefit mangrove trees under stress

The occurrence of natural root grafts, the union of roots of the same or different trees, is common and shared across tree species. However, their significance for forest ecology remains little understood. While early research suggested negative effects of root grafting with the risk of pathogen transmission, recent evidence supports the hypothesis that it is an adaptive strategy that reduces stress by facilitating resource exchange. Here, by analysing mangrove root graft networks in a non-destructive way at stand level, we show further evidence of cooperation-associated benefits of root grafting. Grafted trees were found to dominate the upper canopy of the forest, and as the probability of grafting and the frequency of grafted groups increased with a higher environmental stress, the mean number of trees within grafted groups decreased. While trees do not actively 'choose' neighbours to graft to, our findings point to the existence of underlying mechanisms that regulate 'optimal group size' selection related to resource use within cooperating networks. This work calls for further studies to better understand tree interactions (i.e. network hydraulic redistribution) and their consequences for individual tree and forest stand resilience.

Growth, death, and resource competition in sessile organisms

Population-level scaling in ecological systems arises from individual growth and death with competitive constraints. We build on a minimal dynamical model of metabolic growth where the tension between individual growth and mortality determines population size distribution. We then separately include resource competition based on shared capture area. By varying rates of growth, death, and competitive attrition, we connect regular and random spatial patterns across sessile organisms from forests to ants, termites, and fairy circles. Then, we consider transient temporal dynamics in the context of asymmetric competition, such as canopy shading or large colony dominance, whose effects primarily weaken the smaller of two competitors. When such competition couples slow timescales of growth to fast competitive death, it generates population shocks and demographic oscillations similar to those observed in forest data. Our minimal quantitative theory unifies spatiotemporal patterns across sessile organisms through local competition mediated by the laws of metabolic growth, which in turn, are the result of long-term evolutionary dynamics.

Joint assessment of density correlations and fluctuations for analysing spatial tree patterns

Inferring the processes underlying the emergence of observed patterns is a key challenge in theoretical ecology. Much effort has been made in the past decades to collect extensive and detailed information about the spatial distribution of tropical rainforests, as demonstrated, e.g. in the 50 ha tropical forest plot on Barro Colorado Island, Panama. These kinds of plots have been crucial to shed light on diverse qualitative features, emerging both at the single-species or the community level, like the spatial aggregation or clustering at short scales. Here, we build on the progress made in the study of the density correlation functions applied to biological systems, focusing on the importance of accurately defining the borders of the set of trees, and removing the induced biases. We also pinpoint the importance of combining the study of correlations with the scale dependence of fluctuations in density, which are linked to the well-known empirical Taylor's power law. Density correlations and fluctuations, in conjunction, provide a unique opportunity to interpret the behaviours and, possibly, to allow comparisons between data and models. We also study such quantities in models of spatial patterns and, in particular, we find that a spatially explicit neutral model generates patterns with many qualitative features in common with the empirical ones.

Synchronization within synchronization: transients and intermittency in ecological networks

Transients are fundamental to ecological systems with significant implications to management, conservation and biological control. We uncover a type of transient synchronization behavior in spatial ecological networks whose local dynamics are of the chaotic, predator–prey type. In the parameter regime where there is phase synchronization among all the patches, complete synchronization (i.e. synchronization in both phase and amplitude) can arise in certain pairs of patches as determined by the network symmetry—henceforth the phenomenon of  ‘synchronization within synchronization.’ Distinct patterns of complete synchronization coexist but, due to intrinsic instability or noise, each pattern is a transient and there is random, intermittent switching among the patterns in the course of time evolution. The probability distribution of the transient time is found to follow an algebraic scaling law with a divergent average transient lifetime. Based on symmetry considerations, we develop a stability analysis to understand these phenomena. The general principle of symmetry can also be exploited to explain previously discovered, counterintuitive synchronization behaviors in ecological networks.

Dynamical Ising model of spatially coupled ecological oscillators

Long-range synchrony from short-range interactions is a familiar pattern in biological and physical systems, many of which share a common set of 'universal' properties at the point of synchronization. Common biological systems of coupled oscillators have been shown to be members of the Ising universality class, meaning that the very simple Ising model replicates certain spatial statistics of these systems at stationarity. This observation is useful because it reveals which aspects of spatial pattern arise independently of the details governing local dynamics, resulting in both deeper understanding of and a simpler baseline model for biological synchrony. However, in many situations a system's dynamics are of greater interest than their static spatial properties. Here, we ask whether a dynamical Ising model can replicate universal and non-universal features of ecological systems, using noisy coupled metapopulation models with two-cycle dynamics as a case study. The standard Ising model makes unrealistic dynamical predictions, but the Ising model with memory corrects this by using an additional parameter to reflect the tendency for local dynamics to maintain their phase of oscillation. By fitting the two parameters of the Ising model with memory to simulated ecological dynamics, we assess the correspondence between the Ising and ecological models in several of their features (location of the critical boundary in parameter space between synchronous and asynchronous dynamics, probability of local phase changes and ability to predict future dynamics). We find that the Ising model with memory is reasonably good at representing these properties of ecological metapopulations. The correspondence between these models creates the potential for the simple and well-known Ising class of models to become a valuable tool for understanding complex biological systems.

Flowering synchrony drives reproductive success in a wind-pollinated tree

Synchronised and quasi-periodic production of seeds by plant populations, known as masting, is implicated in many ecological processes, but how it arises remains poorly understood. Flowering and pollination dynamics are hypothesised to provide the mechanistic link for the observed relationship between weather and population-level seed production. We report the first experimental test of the phenological synchrony hypotheses as a driver of pollen limitation in mast seeding oaks (Quercus ilex). Higher flowering synchrony yielded greater pollination efficiency, which resulted in 2-fold greater seed set in highly synchronised oaks compared to asynchronous individuals. Pollen addition removed the negative effect of asynchronous flowering on seed set. Because phenological synchrony operates through environmental variation, this result suggests that oak masting is synchronised by exogenous rather than endogenous factors. It also points to a mechanism by which changes in flowering phenology can affect plant reproduction of mast-seeding plants, with subsequent implications for community dynamics.

Density dependent Resource Budget Model for alternate bearing

Alternate bearing, seen in many types of plants, is the variable yield with a strongly biennial pattern. In this paper, we introduce a new model for alternate bearing behavior. Similar to the well-known Resource Budget Model, our model is based on the balance between photosynthesis or other limiting resource accumulation and reproduction processes. We consider two novel features with our model, 1) the existence of a finite capacity in the tree's resource reservoir and 2) the possibility of having low (but non-zero) yield when the tree's resource level is low. We achieve the former using a density dependent resource accumulation function, and the latter by removing the concept of the well-defined threshold used in the Resource Budget Model. At the level of an individual tree, our model has a stable two-cycle solution, which is suitable to model plants in which the alternate bearing behavior is pronounced. We incorporate environmental stochasticity by adding two uncorrelated noise terms to the parameters of the model associated with the nutrient accumulation and reproduction processes. Furthermore, we examine the model's behavior on a system of two coupled trees with direct coupling. Unlike the coupled Resource Budget Model, for which the only stable solution is the out-of-phase solution, our model with direct coupling has stable in-phase period-2 solutions. This suggests that our model might serve to explain spatial synchrony on a larger scale.

Robust design from systems physics

A crucial challenge in engineering modern, integrated systems is to produce robust designs. However, quantifying the robustness of a design is less straightforward than quantifying the robustness of products. For products, in particular engineering materials, intuitive, plain language terms of strong versus weak and brittle versus ductile take on precise, quantitative meaning in terms of stress–strain relationships. Here, we show that a “systems physics” framing of integrated system design produces stress–strain relationships in design space. From these stress–strain relationships, we find that both the mathematical and intuitive notions of strong versus weak and brittle versus directly characterize the robustness of designs. We use this to show that the relative robustness of designs against changes in problem objectives has a simple graphical representation. This graphical representation, and its underlying stress–strain foundation, provide new metrics that can be applied to classes of designs to assess robustness from feature- to system-level.

Environmental Veto Synchronizes Mast Seeding in Four Contrasting Tree Species

Synchronized and variable reproduction by perennial plants, called mast seeding, is a major reproductive strategy of trees. The need to accumulate sufficient resources after depletion following fruiting (resource budget), the efficiency of mass flowering for outcross pollination (pollen coupling), or the external factors preventing reproduction (environmental veto) could all synchronize masting. We used seed production data for four species (Quercus ilex, Quercus humilis, Sorbus aucuparia, and Pinus albicaulis) to parametrize resource budget models of masting. Based on species life-history characteristics, we hypothesized that pollen coupling should synchronize reproduction in S. aucuparia and P. albicaulis, while in Q. ilex and Q. humilis, environmental veto should be a major factor. Pollen coupling was stronger in S. aucuparia and P. albicaulis than in oaks, while veto was more frequent in the latter. Yet in all species, costs of reproduction were too small to impose a replenishment period. A synchronous environmental veto, in the presence of environmental stochasticity, was sufficient to produce observed variability and synchrony in reproduction. In the past, vetoes like frost events that prevent reproduction have been perceived as negative for plants. In fact, they could be selectively favored as a way to create mast seeding.