Abstract
Understanding the synchronization of oscillations across space is fundamentally important to many scientific disciplines. In ecology, longrange synchronization of oscillations in spatial populations may elevate extinction risk and signal an impending catastrophe. The prevailing assumption is that synchronization on distances longer than the dispersal scale can only be due to environmental correlation (the Moran effect). In contrast, we show how longrange synchronization can emerge over distances much longer than the length scales of either dispersal or environmental correlation. In particular, we demonstrate that the transition from incoherence to longrange synchronization of twocycle oscillations in noisy spatial population models is described by the Ising universality class of statistical physics. This result shows, in contrast to all previous work, how the Ising critical transition can emerge directly from the dynamics of ecological populations.
Introduction
Synchrony of dynamics across space has been a subject of intense study across a diverse array of scientific disciplines^{1,2,3,4,5}. The study of synchrony of oscillations in population numbers across space and time has a long history in ecology^{6} and has provided great insights into fundamental issues of population dynamics^{3,5}. The absence of spatial synchrony is generally considered to be the key to persistence in exploiter–victim systems^{5,7,8,9}, and may be vitally important in the conservation of species^{3,5,7} and disease eradication^{10}. There is longstanding recognition of the need for new models and measures of spatial synchrony relevant to ecology^{3,5,10,11} that do not depend upon the simplifying assumptions of the Kuramoto^{1,12} and related models^{2,4}.
Three wellknown causes of spatial synchrony are the longrange correlation of environmental variation (known as the Moran effect^{6}), entrainment to neighbouring populations via trophic interactions and dispersal of individuals among habitat patches^{3,5,13}. In the absence of environmental noise, shortrange dispersal is sufficient to phaselock spatial populations over arbitrarily large distance scales^{7}. In the more realistic case of environmental variation, shortrange dispersal has been shown to synchronize population oscillations only over distance scales comparable to the correlation length of the environmental noise^{5,14,15,16}. The remaining question is whether and how synchrony over distances much longer than the scale of environmental correlations can arise from shortrange dispersal in the presence of strong noise. This paper is the first to demonstrate the emergence of longrange synchronization and modelindependent powerlaw scalings in noisy cyclic populations. Note that our results apply to the onset of spatial synchrony of oscillations in population numbers far from the extinction threshold, and our work is distinct from previous investigations of model independent power laws near the extinction threshold^{17,18,19}.
Many of the most prominent studies of spatial synchrony have focused on examples where the underlying dynamics are two cycles, including the dynamics of childhood diseases^{10} and small mammals^{13}. Because both of these examples have been argued to arise in part from the seasonal forces that impact the vast majority of ecological populations, twocycle behaviour is truly a ubiquitous phenomenon. That two cycles are so common is expected from the structure of overcompensatory models in population biology, such as the logistic map, because of the large range of growth rates for which the asymptotic dynamics lie in an exact twocycle. With the addition of noise, approximate twocycle behaviour can be found over an even larger range of growth rates^{20} along the entire perioddoubling sequence and through the twobanded chaotic region, as can be seen in the bifurcation diagram for any quadratic map^{21}.
In statistical physics, the canonical model for the emergence of longrange order from the collective behaviour of local interactions is the Ising model^{22,23,24,25}. The Ising model describes emergent phenomena at a phase transition where a simple twofold symmetry is broken. Physical systems with unrelated microscopic dynamics, such as magnetic, liquid–vapour and binary alloy systems, exhibit the powerlaw scaling behaviour of the Ising model at a critical phase transition. The Ising model thus defines a ‘universality class’—a set of many disparate systems that are all characterized by identical powerlaw scaling behaviour over large distances despite wide differences in local interactions and shortrange behaviour. Here, for a suite of ecological models, we will show that the powerlaws describing asymptotic behaviour at the emergence of longrange synchronization are in the twodimensional (2D) Ising universality class, providing a powerful tool for understanding the emergence of collective synchronization in spatial populations with local dispersal.
This quantitative correspondence between ecological population models and the Ising model is quite surprising and novel. The Ising universality class describes phase transitions for systems in thermal equilibrium, that is, systems with a welldefined energy and temperature^{22,23,24,25}. On the other hand, ecological population models are dynamical, dissipative systems lying far from thermal equilibrium. Previous attempts to apply the Ising model to population ecology simply assume thermal equilibrium and lack any clear ecological motivation. In this paper, we show how a quantitative correspondence to the Ising model emerges naturally in spatially extended dynamical models of ecological populations, providing new insights for both fields.
Here, we first introduce a suite of discretetime spatial population models and define a statistic for discriminating between incoherent and synchronous behaviour. Next, we demonstrate that the asymptotic behaviour of ecological population dynamics at the emergence of longrange synchronization is described by the Ising universality class. Finally, we define new statistics of spatial synchrony, each corresponding to a statistic of the Ising model, that can rapidly detect sudden changes in the spatial synchrony of ecological populations. We anticipate that our methods will be broadly applicable to the study of transitions in the spatial synchrony of cyclic populations, and we underscore the importance of universality for ecological modelling.
Results
Definition of the synchronization order parameter
We investigate the longrange synchronization of twocycle oscillations in several discretetime spatial population models with local dispersal and uncorrelated environmental noise: a singlespecies Ricker model (Methods equations (4) and (5))^{26,27} with two different dispersal kernels, a singlespecies Logistic model (Methods equations (4) and (6))^{28,29,30,31} and a version of the Nicholson–Bailey Host–Parasitoid model (Methods equations (7) and (8))^{32}. The analogue of the local ‘spin’ variable at a lattice site in the Ising model is a local twocycle amplitude, m_{j,t} (Methods equation (9)), defined at each habitat patch j and generation t in the spatial population models. A spatiotemporal average of the m_{j,t} is the synchronization order parameter, m (Methods equation (11)), which discriminates between phases of incoherence and longrange synchronization: an order parameter value of zero signals complete incoherence (the disordered phase), while nonzero values measure the degree of longrange synchronization (the ordered phase). On a landscape with one or two spatial dimensions (onedimensional (1D) or 2D) and L habitat patches along each side (such that the total number of habitat patches is N=L or N=L^{2}, respectively), finitesize estimates of the synchronization order parameter, m_{L} (Methods equation (13)), can be obtained from numerical simulations.
Ising universality at the critical transition
We present results for 1D and 2D ecological models with various numbers of habitat patches. The synchronization order parameter as a function of uncorrelated environmental noise level is plotted in Fig. 1a,b,d,e and Supplementary Figs 1a,b and 2a,b. On 1D landscapes, for sufficiently large N, the order parameter approaches zero in all the population models, and we find no evidence of longrange synchronization at nonzero noise levels. The same behaviour occurs in the 1D Ising model^{22,23,24,25}. The behaviour on 2D landscapes is quite different. A continuous transition in the synchronization order parameter sharpens with increasing landscape size. Near this ‘critical point’, synchronized habitat clusters appear on the landscape and form fluctuating fractal patterns (Fig. 1c,f; Supplementary Figs 1c and 2c) associated with the emergence of longrange order and critical slowing down (Supplementary Fig. 3). These results demonstrate both that longrange synchronization can be an emergent phenomenon on 2D landscapes even in the presence of noise, and also that the existence of a critical transition from incoherence to synchrony is independent of the details of the local ecological dynamics.
We now quantify the strong correspondence between the critical behaviour of the Ising model and the emergence of longrange synchronization on 2D landscapes. In addition to finitesize estimates of the order parameter, m_{L} (Methods equation (13)), we define two statistics of the local twocycle amplitudes motivated by the Ising correspondence: the susceptibility, χ_{L} (Methods equation (14)), measures the variance of twocycle amplitudes, and the fluctuation correlation length, ξ_{L} (Methods equation (16)), measures the length scale of longrange spatial correlations among twocycle amplitudes. We find ‘finitesize scaling collapses’ in which measurements of m_{L}, χ_{L} and ξ_{L} on landscapes of different sizes and noise levels, when multiplied by appropriate powers of L given by the Ising ‘critical exponents’, lie on universal curves (Methods equation (24); Fig. 2a–c; Supplementary Figs 4–7). We also identify a scaling collapse across models: statistics of the different ecological population models, multiplied by modeldependent prefactors, collapse onto corresponding Ising statistics near the critical transition (Methods equation (24); Fig. 2d–f; Supplementary Figs 8 and 9).
While it is not the main focus of our work, the slow dynamics near the Ising critical point is also ecologically interesting. The Ising model equipped with the relevant nonconservative dynamics is referred to as ‘model A’^{33,34,35}. Critical dynamics described by model A are universal and known to hold across a wide spectrum of physical systems near and far from thermal equilibrium^{36,37,38,39,40,41,42,43,44,45,46}. We expect that the critical dynamics of all of the spatial population models studied here fall into the modelA dynamic universality class.
Ising universality explains the robustness and broad applicability of our results and provides a quantitative description of scaling behaviour at the onset of longrange synchronization that is independent of the details of local population dynamics. In particular, universality allows us to conclude that longrange synchronization among ecological oscillators can emerge from shortrange dispersal even when a detailed description of the local dispersal behaviour and dynamics is unknown. We have explicitly demonstrated identical scaling behaviour in the spatial Ricker model with either the nearestneighbour or Moore neighbourhood dispersal kernel (Fig. 2; Supplementary Figs 4,7 and 9), and universality ensures that the same scaling behaviour would be found for any localized dispersal kernel. Ecological populations exhibit even greater variability in their local connectivity but are nonetheless expected to be in the (random) Ising universality class^{23} as long as dispersal remains shortranged.
We can determine the conditions for longrange synchronization as a function of the parameters in each ecological model. For the spatial Ricker model, this information is summarized in Fig. 3 as a ‘phase diagram’ for incoherence and synchrony (see also Supplementary Fig. 10). We find that longrange synchronization emerging from local dispersal can persist in the presence of strong uncorrelated environmental noise. We conjecture that all critical points on the phase boundary lie in the 2D Ising universality class. The shape of the phase boundary as a function of growth rate, dispersal fraction and noise level will be qualitatively similar in the other population models.
Rapid detection of sudden changes in spatial synchrony
Given that largescale disturbances are not uncommon, the study of transient dynamics is especially important in ecology^{47}. The rapid formation of complex patterns of spatial synchrony can be difficult to infer on timescales short enough to allow for an effective response. Using the relatively short timeseries data typically available, new measures are needed to rapidly detect sudden changes in spatial synchrony. We introduce three such measures, each corresponding to a statistic of the Ising model: the synchronization index, S_{t} (Methods equation (17)), measures instantaneous synchronization as a nonspatial, twogeneration, simple moving average; the susceptibility index, V_{t} (Methods equation (19)), measures the magnitude of fluctuations in synchronization windowed in both space and time; the correlation length index, CL_{t} (Methods equation (23)), measures a distance scale proportional to the size of synchronized clusters and is also windowed in both space and time. The spatial windowing of V_{t} and CL_{t} allows statistical power to be gathered across space in the absence of a long time series^{48}.
We demonstrate the application of these new measures to the rapid detection of sudden changes in spatial synchrony from incoherence to synchronized cluster formation and growth. In Fig. 4, starting with an ensemble of incoherent population configurations at generation zero (t=0), we calculate trajectories of S_{t}, V_{t} and CL_{t} emerging from the dynamics of a Ricker model where the growth rate, dispersal fraction and noise level correspond to either the synchronous phase (the signal, red trajectories) or the incoherent phase (the baseline, blue trajectories). The nonspatial index, S_{t} (Fig. 4a), is outperformed by the spatially windowed indices, V_{t} and CL_{t} (Fig. 4b,c), each of which allows for detection of a sudden change in spatial synchrony within two or three generations.
Synchronized cluster formation and growth in ecological models is analogous to a ‘quench’ in the Ising model with ‘nonconserved’ dynamics, where the size of clusters increases according to a powerlaw in time^{22,23,24,25}. This powerlaw is universal^{22,23,24,25}, and therefore, independent of the details of local dynamics. The same powerlaw behaviour can be found in the Ricker model (Fig. 4c) and in the host–parasitoid model of Rost et al.^{32} These observations suggest that the sudden, spontaneous emergence of sharp ecological boundaries between clusters of habitat patches with oppositesign synchronization, as seen in Fig. 4d, might occur in many types of ecological systems with local dispersal and uncorrelated environmental noise.
Discussion
By demonstrating the existence of Ising critical transitions in the spatial synchrony of noisy spatial populations, our work shows, for the first time, how longrange synchronization in ecological systems can emerge in the absence of external forcing (the Moran effect^{6}). Beyond the important case of transitions between incoherence and twocycle synchrony, our results suggest a general correspondence between emergent longrange synchronization in ecology and the universality classes of critical transitions in statistical physics. The methods developed here can be readily generalized and applied to establish those correspondences. This high degree of model independence implies that the possibility of emergent longrange synchronization must be carefully considered when seeking to understand the causes and consequences of spatial synchrony^{3,5,6,7,8,9,10,11,12,13,14,15,16,27,32,49,50,51,52,53,54,55,56}, including in the dynamics of childhood diseases, such as measles and pertussis^{10}, in the biological control of outbreaks of insects, such as bark beetles and gypsy moths^{5} and in the mitigation of synchronous masting, which threatens food security in agroecosystems^{56}.
Finding appropriate measures of spatial synchrony in 2D ecological systems has been a serious challenge^{5}. We have shown how the Ising correspondence introduces new statistics, both asymptotic measures and transient indices, to the study of spatial synchrony. In contrast to the Kuramoto model^{1} and other spatially implicit approaches to quantifying spatial synchrony, these new statistics of spatial synchrony depend on the full spatiotemporal dynamics of a noisy population. Isinglike transient indices should be especially useful in quantifying spatial synchrony over timescales relevant to ecology and in the rapid detection of sudden transitions^{57}.
This work and others underscore the vital importance of universality for making robust, quantitative predictions about transitions to longrange order in ecological populations where the full complexity of local dynamics could never be modelled explicitly. Connections between spatial population models and the universality classes of statistical physics have previously been demonstrated in phase transitions to flocking^{58}, to extinction^{17,19} and to other ordered phases relevant to ecological systems^{25,59}. Universality of transitions in the spatial synchrony of noisy populations promises to remain a fruitful area of investigation.
Methods
Ising model and Ising universality
In ecology, the modelling of spatial populations begins with a dynamical systems approach, while in statistical physics, the modelling of a spin system in thermal equilibrium begins with a probability distribution called the Gibbs distribution^{22,23,24,25}
where the spin system is in thermal contact with a thermal reservoir that maintains a fixed temperature, T, and the energy of the state is E(state). (Boltzmann’s constant is set to one in equation (1).) The energy function typically models local interactions among nearestneighbour spins. In the Ising model^{22,23,24,25}, lattice sites are indexed by an integer, i, and the local spin, the s_{i}, is either up (+1) or down (−1). The Ising energy function can be written as a sum over products of neighbouring spins
where is the set of indices for the neighbouring spins of lattice site i and J is a coupling constant. The overall negative sign means that the energy is lower (higher) when neighbouring spins are more (less) aligned. Because the probability of any given configuration of local spin states over the lattice is given by the Gibbs distribution, configurations with a high level of local alignment are the most probable at low temperatures. The expected sum of all the spins
defines the order parameter, m. Nonzero values signal an ordered phase in which spins are globally aligned; a zero value signals a disordered phase.
A 1D chain of spins will only order at zero temperature^{22,23,24,25}. For 2D lattices, a phase transition from disorder to order occurs at a nonzero critical temperature, where the order parameter goes continuously to zero. Near the critical point, the fluctuation correlation length diverges: clusters of aligned spins form on all length scales, and clusters of oppositesign spins coexist in a fluctuating fractal pattern. These emergent, longrange correlations render many details of the local interactions unimportant to emergent behaviour near the critical point. Indeed, the critical point of the Ising model is universal: the same powerlaw scaling relationships are common to critical points observed in seemingly unrelated magnetic, liquid–gas and binary alloy systems^{22,23,24,25}, as well as systems far from thermal equilibrium^{36,37,38,39,40,41,42,43,44,45,46}. The numerical values of the exponents in those powerlaws quantitatively define the Ising universality class^{22,23,24,25}.
Ecological population models
We start by investigating longrange synchronization in three models of singlespecies populations with local dispersal and uncorrelated environmental noise. The population density of habitat patch j at generation t is a continuous random variable, X_{j,t}. The geometry of N habitat patches on a spatial landscape is given by a linear or square lattice with periodic boundary conditions and one or two spatial dimensions. (If the linear dimension is L, then the landscape size is N=L^{d} where d=1, 2.) The spatial dynamics are given by noisy coupled maps
In each generation, local populations are regulated by the per capita density dependence of a noisy quadratic map, f(X_{j,t}), in which random environmental variation in population regulation, uncorrelated in both space and time, is modelled as multiplicative white noise. Afterwards, a fraction of each local population, , disperses evenly among z neighbouring habitat patches, as indexed by the elements of , the set of indices for the neighbouring spins of lattice site j. Note that changing the number of spatial dimensions does not change the fraction of a local population that disperses.
For any finitesized landscape and nonzero noise levels, the asymptotic (t→∞) phase of the system is extinction (X_{j}=0 for all j). Nonetheless, typical extinction times rapidly increase with the number of habitat patches. In numerical simulations for finite N, we gather statistics on the single, longlived, metastable state that is conjectured to be fully stable in the N→∞ (or ‘thermodynamic’) limit. In the following, ‘asymptotic’ will refer to times much larger than one generation but much less than the extinction time.
The Ricker model is equation (4) with a nearestneighbour dispersal kernel on a linear or square lattice of habitat patches (z=2d) and local populations regulated by a noisy Ricker map^{20,26}
where r is the growth rate, ξ_{j,t} are standard normal deviates, uncorrelated in both space and time, and λ is the noise level, that is, the s.d. of the environmental noise distribution. Equation (5) is a wellknown extension of the per capita density dependence of the Ricker map^{26} to include the effects of multiplicative uncorrelated environmental noise^{20}. A similar coupled map has previously been used to understand the causes of intraspecific spatial synchrony in largescale moth and aphid populations^{27}.
The Logistic model is equation (4) with a nearestneighbour dispersal kernel on a linear or square lattice of habitat patches (z=2d) and local populations regulated by a noisy logistic map^{30}
The study of noisy logistic maps has a long history^{30}. Pattern formation in coupled logistic maps with local twocycle oscillations has been investigated previously^{31}.
The Ricker–Moore model is the same as the Ricker model but with a broader dispersal kernel. On 1D landscapes, the dispersal fraction is evenly distributed among the nearest and nexttonearest neighbours (z=4). On 2D landscapes, the dispersal fraction is evenly distributed among the habitat patches of the Moore neighbourhood (the z=8 nearest and nexttonearest neighbours on a square lattice).
The host–parasitoid model is a mechanistic twospecies model in which the population dynamics of two noninterbreeding cohorts of hosts are coupled together by a shared parasitoid. The spatial dynamics of the host (H) and parasitoid (P) is taken from equation (8) of Rost et al.^{32}
with local populations regulated by a noisy Nicholson–Bailey coupled map
where r is the growth rate of the host, a is the searching efficiency of the parasitoid, the and the are two sets of standard normal deviates, uncorrelated in space and time and with each other, and λ is the noise level. A nearestneighbour dispersal kernel is assumed, where is the dispersal fraction for both host and parasitoid. Rost et al.^{32} demonstrated the existence of coarsening behaviour and an asymptotic phase of spatial synchrony. Phase separation and coarsening in twovariable, discretetime systems with local interactions have also been studied in the physics literature^{60,61}. Our work focuses on the emergence of longrange synchronization from incoherence at a critical transition and the Ising universality of that transition.
Synchronization order parameter
In the Ricker, Logistic and Ricker–Moore models, we define a twocycle amplitude for each local population given by
The spatial average of the m_{j,t} defines the instantaneous synchronization
In the physics literature, order parameters are defined by an ensemble average denoted by angle brackets. Here, the expectation value of m_{t} taken over the probability distribution of the X_{j,t} at long times defines the synchronization order parameter, m
An order parameter value of zero signals complete incoherence (the disordered phase); nonzero values measure the degree of partial synchronization (the ordered phase). The synchronization order parameter of the host–parasitoid model is the same as in equations (9)–(11), , but with N_{j,t} substituted for X_{j,t} in the definition of the twocycle amplitude. In our numerical estimates, we obtain the same results if P_{j,t} is substituted instead. The onset of longrange synchrony in both the host and parasitoid populations is conjectured to occur at the same critical point.
To gather some intuition for the emergence of an Ising critical transition in the synchronization order parameter, we first consider a population in which density, x_{t}, is regulated by the deterministic Ricker map. This is a limiting case of the Ricker model. The limiting value of the twocycle amplitude, as defined in equation (9), is
and the limiting value of the synchronization order parameter, as defined in equation (11), is the average value of m_{t} in the asymptotic regime. For growth rates in the steadystate range, the asymptotic population density is a constant, x_{t}=x*, and the order parameter vanishes, m=0. The asymptotic system is clearly invariant under time translations, that is, t→t+n for any integer n. For growth rates in the twocycle range, the asymptotic population density oscillates between two distinct values, and . The order parameter takes on one of two possible nonzero values, , depending on the temporal phase of the twocycle oscillation. The asymptotic system remains invariant to evenperiod, but not oddperiod, time translations. Therefore, in the deterministic limit, increasing the growth rate from the steady state to the twocycle range breaks a twofold, or , symmetry of the asymptotic steadystate system. In the full Ricker model, high noise levels wash out any cycling in the underlying density dependence, and the asymptotic spatially averaged population density tends to a constant value as the size of the landscape becomes very large. In this phase, the synchronization order parameter vanishes. Only at sufficiently low noise levels can cycling emerge in asymptotic spatially averaged population density. In this phase, the asymptotic system is no longer invariant to oddperiod time translations. A discrete twofold symmetry is broken in the transition from incoherence to synchrony, and this is the symmetry breaking pattern of the Ising universality class^{22,23,24,25}.
Asymptotic measures of spatial synchrony
We define new measures of spatial synchrony that complement existing approaches^{3,5,6,7,8,9,10,11,12,13,14,15,16,27,32,49,50,51,52,53,54,55,56}. Each asymptotic measure is identical to a statistic in the Ising model if the m_{j,t} are the Monte Carlo trajectories of a local spin rather than a time series of local twocycle amplitudes. Each transient measure corresponds to an asymptotic measure but can be estimated at a single generation of the population model and does not require the existence of a long nor stationary time series.
Integrated autocorrelation time. This statistic measures the characteristic timescale over which correlations in trajectories of the instantaneous synchronization decay. Our estimates and error bars, as plotted in Supplementary Fig. 3, are based on p. 433 of Janke^{62}.
Synchronization order parameter (m_{L}). Because critical transitions occur only in the limit of infinite system size, numerical estimates of the synchronization order parameter on finitesized L × L landscapes are denoted with a subscript, m_{L}, and are based on a spatial average of the m_{j,t,L} (see, for example, equation (3) of Janke^{62}). The estimate of the synchronization order parameter, as plotted in Fig. 1 and elsewhere, is
where N is the number of samples in the spatial averaging (the number of habitat patches on the landscape) and M is the number of samples in the time averaging (the length of the time series). Error bars are estimated based on the integrated autocorrelation time^{63}.
Susceptibility (χ_{L}). Defined as
the susceptibility measures the strength of fluctuations in the synchronization order parameter. Error bars are estimated in a block bootstrap^{63} with 100 blocks.
Fluctuation correlation length (ξ_{L}). This statistic measures the characteristic length scale over which correlations in local twocycle amplitudes decay. On a landscape of infinite size, let the displacement between habitat patches i and j be r_{i,j}. The strength of correlations in the m_{j,t}, as a function of r_{i,j}, is given by the spatially averaged connected correlation function^{63}
Over large distances, we assume that G(r) decays exponentially. The fluctuation correlation length is the characteristic length scale of that exponential decay^{62}
Our estimates of mean values and error bars for the fluctuation correlation length on finitesized landscapes, ξ_{L}, are based on p. 23 of Janke^{64}.
Critical noise levels (λ_{c}). Our estimates of critical noise levels, λ_{c}, at the onset of longrange synchronization are based on crossings of Binder’s cumulant, U^{65}. In the 2D Ising model, the critical value, U*, is universal and lies between −1.830 and −1.835 (ref. 39). We estimate U* in each of the four population models and find a value consistent with the Ising result.
Transient measures of spatial synchrony
Each transient measure of spatial synchrony corresponds to an asymptotic measure but can be estimated at a single generation of the population model and does not require the existence of a long nor stationary time series. Transient measures are defined on a 2D landscape of N=L^{2} habitat patches. We omit an explicit subscript ‘L’ on the symbols below to simplify notation.
Synchronization index (S_{t}). This statistic is an estimate of the synchronization order parameter windowed in time. We define the index as the absolute value of a twogeneration simple moving average of the instantaneous synchronization (equation (10))
Susceptibility index (V_{t}). This statistic is an estimate of susceptibility windowed in both space and time. We define a square spatial window covering N′=L/2 × L/2 habitat patches of the L × L landscape. With periodic boundary conditions, that window can be placed on the landscape in N possible ways. Let the kth placement of the spatial window cover N′ habitat patches with their locations indexed by the elements of . For each placement in each generation, we calculate the average value of the twocycle amplitudes (equation (10)) that fall within the spatial window
Over two generations, we obtain 2N estimates of m_{k,t}. The absolute value of the mean of the estimates is simply the synchronization index defined above. The susceptibility index is given by
and provides a measure of variability in local synchronization.
Correlation length index (CL_{t}). This statistic defines a characteristic length scale for the size of synchronized clusters using the same approach to spatial and temporal windowing as in the definition of the susceptibility index. In the absence of longrange correlations and a long time series, we cannot obtain a good estimate of the longdistance exponential decay of the connected correlation function (equation (15)) that defines the finitesized fluctuation correlation length, ξ_{L} (equation (16)). Instead, we define CL_{t} to be a length scale determined by the shortdistance exponential decay of a spatially windowed correlation function. On the kth spatial window covering N′=L/2 × L/2 habitat patches at generation t (see the discussion of spatial windowing in the definition of V_{t} above), we define the correlation function, G_{k,t}(r), as
This function can be averaged over N possible placements of the spatial window in each of two generations, defining G_{t}(r) as
We hypothesize that G_{t}(r) decays exponentially even over short distances such that
and that CL_{t} can be estimated simply as
The overall scale factor of 4 is chosen such that CL_{t} is roughly equal to the typical diameter of a synchronized cluster. In a sudden transition from incoherence to cluster formation and growth, a typical trajectory of CL_{t} lies on the curve CL_{t}∝t^{1/2}, as shown in Fig. 4c. This is the expected powerlaw scaling for coarsening behaviour in the Ising model with nonconserved dynamics^{22,23,24,25}.
Finitesize scaling and evidence of Ising universality
Strictly speaking, the Ising critical transition occurs only on an infinite lattice of spins. Nonetheless, critical behaviour can be studied in numerical Monte Carlo simulations of finitesized lattices^{22,23,24,25}. Finitesize scaling theory has previously been used to provide strong numerical evidence of Ising (or in some cases, ‘Isinglike’) universality in the critical transitions of probabilistic cellular automata^{36} and chaotic map lattices^{37,38,39,43}. Here, we apply similar methods to study critical transitions in coupled maps of nonchaotic, noisy twocycles with applications to ecology. We first define the analogue of reduced temperature in our noisy population models and then introduce our finitesize scaling hypothesis.
Reduced control parameters. In studies of the Ising model, the reduced temperature (τ_{T}) is defined as (T–T_{c})/T, where T is the temperature and T_{c} is the critical temperature. To investigate the Ising universality of noiseinduced critical transitions in ecological models, we define the reduced noise level (τ_{λ}) as (λ–λ_{c})/λ, where λ is the noise level and λ_{c} is the critical noise level. If we want to refer to either τ_{T} or τ_{λ}, we will simply use the symbol τ and call this the reduced control parameter.
Data collapse for ecological statistics. Near a critical transition in the Ising model and the ecological models, we gather finitesize measurements of the synchronization order parameter, m_{L}, susceptibility, χ_{L}, and fluctuation correlation length, ξ_{L}, for various landscape sizes, L, and reduced control parameter values, τ. For each model, we find that these measurements can be scaled to lie on a function, F_{m}, F_{χ} or F_{ξ}, that is independent of both the model and the landscape size^{22,23,24,25}
The powers of L are determined by the 2D Ising critical exponents (ν=1, β=1/8, γ=7/4) and are model independent. The modeldependent scale factors are a_{model} and b_{model}, where a_{model} is a scaling of the reduced noise level and b_{model} is a scaling of local twocycle amplitudes
The asymptotic measures of spatial synchrony, m_{L}, χ_{L} and ξ_{L}, defined above in terms of the m_{j,t,L}, scale with b_{model} as
We choose a_{Ising}=b_{Ising}=1 without loss of generality. For the ecological models, corrections to scaling and quantitative estimates of critical exponents will be published elsewhere.
Monte Carlo simulations
Our numerical methods for estimating asymptotic statistics characterizing noisy ecological populations are based on wellestablished methods in the statistical physics literature on Monte Carlo simulations^{62,63,64,65}. In each Monte Carlo simulation, we iterate equation (4) or (7), for a burnin period of T_{burnin} generations before running for T_{steps} generations and gathering M samples of each statistic at regular intervals. In all simulations, initial population densities are drawn from a normal distribution with mean μ_{0} and s.d. σ_{0} (negative numbers, if drawn, are thrown out), but results are independent of initial conditions, so long as local population densities are nonzero. Details follow on the Monte Carlo simulations used to estimate the statistics plotted in the main text and Supplementary Information.
Fig. 1: (a,b) On the basis of Monte Carlo simulations, with T_{burnin}=2 × 10^{6}, T_{steps}=2 × 10^{7} and M=1 × 10^{5}, of the Ricker model, with r=2.3, and λ one of 64 evenly spaced values on the interval [0.12,0.18]. Snapshots are shown in c for λ=0.145. Initial population densities are drawn from a normal distribution with μ_{0}=0.50 and σ_{0}=0.1. (d,e) On the basis of Monte Carlo simulations, with T_{burnin}=2 × 10^{6}, T_{steps}=2 × 10^{7} and M=1 × 10^{5}, of the host–parasitoid model, with r=1.5, , a=20 and λ one of 64 evenly spaced values on the interval [0.3,0.5]. Snapshots are shown in f for λ=0.376. Initial population densities are drawn from a normal distribution with μ_{0}=0.50 and σ_{0}=0.1.
Fig. 2: (a–c) On the basis of Monte Carlo simulations, with T_{burnin}=2 × 10^{7}, T_{steps}=2 × 10^{8} and M=1 × 10^{5}, of the Ricker model, with r=2.3, . We scanned 72 evenly spaced values of λ on the interval [0.135,0.150] but plotted statistics over a smaller range closer to the critical transition. Initial population densities are drawn from a normal distribution with μ_{0}=0.50 and σ_{0}=0.1. The critical noise level, λ_{c}=0.14131(4), is estimated from the critical value of Binder’s cumulant and used to calculate the reduced noise level as plotted on the horizontal axis. (d–f) The points and error bars are the N=4,096 series in panels b, d and f of Supplementary Figs 4–8 (see details below), after a modeldependent scaling of the horizontal and vertical axes by a_{model} and b_{model}. Without loss of generality, we define a_{Ising}=b_{Ising}=1. Then we estimate a_{Ricker}=2.60, a_{host–parasitoid}=3.15, a_{Logistic}=2.85, a_{Ricker–Moore}=2.65, and b_{Ricker}=2.028, b_{host–parasitoid}=4.382, b_{Logistic}=7.228, b_{Ricker–Moore}=2.179.
Fig. 3: The phase boundary plotted in this figure is an interpolation of the maximal values of fluctuation correlation length in Supplementary Fig. 10 (see details below).
Fig. 4: On the basis of Monte Carlo simulations, with T_{burnin}=0 and T_{steps}=10, of the Ricker model, with and λ=0.12. For the signal (red lines), r=2.3; for the baseline behaviour (blue lines), r=1.5. In each scenario, trajectories of the synchronization, susceptibility and correlation length indices are plotted for 1,000 initial population configurations. Initial population densities are drawn from a normal distribution with μ_{0}=1.00 and σ_{0}=0.1.
Supplementary Fig. 1: same as Fig. 1a–c but for the Logistic model, with r=3.3, ɛ=0.1 and λ one of 64 evenly spaced values on the interval [0.04,0.07]. Snapshots are shown in panel (C) for λ=0.0519.
Supplementary Fig. 2: (A,B) same as Fig. 1 a,b but for the Ricker–Moore model, with r=3.3, ɛ=0.1 and λ one of 64 evenly spaced values on the interval [0.13,0.19]. Snapshots are shown in panel (C) for λ=0.154.
Supplementary Fig. 3: integrated autocorrelation times of the synchronization order parameter are estimated from the same Monte Carlo simulations used to generate Fig. 1.
Supplementary Fig. 4: panels D–F are the same as Fig. 2 a–c. Panels A–C in this figure plot the same statistics without finitesize scaling.Supplementary Fig. 5: same as Supplementary Fig. 4 but for the Host–Parasitoid model, with r=1.5, ɛ=0.1, α=20 and λ one of 72 evenly spaced values on [0.347,0.397]. Initial population densities, for both host and parasitoid, are drawn from a normal distribution with and . The critical noise level is .
Supplementary Fig. 6: same as Supplementary Fig. 4 but for the Logistic model, with with , and one of 72 evenly spaced values . The critical noise level is .
Supplementary Fig. 7: same as Supplementary Fig. 4 but for the Ricker–Moore model, with , and one of 72 evenly spaced values on . The critical noise level is .
Supplementary Fig. 8: based on simulations of the 2D Ising model using the Wolff algorithm[63], with , and , and (in units where Botzmann’s constant is set to unity) one of 64 evenly spaced values on the interval . Initial spin configurations are fully aligned (all spin values are ). The critical temperature and theoretical predictions for an infinite lattice are well known in the physics literature^{22,23,24,25}.
Supplementary Fig. 9: same as Fig. 2 d–f but for the A–C and D–F series.
Supplementary Fig. 10: based on Monte Carlo simulations, with , and , of the Ricker model where and is one of 26 evenly spaced values on the interval . (A) and is one of 30 evenly spaced values on . (B) and is one of 30 evenly spaced values on .
Additional information
How to cite this article: Noble, A. E. et al. Emergent longrange synchronization of oscillating ecological populations without external forcing described by Ising universality. Nat. Commun. 6:6664 doi: 10.1038/ncomms7664 (2015).
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Acknowledgements
We are grateful to Carl Boettiger, Jeff Gore, Marcel Holyoak and Kirill Korolev for reading and commenting on an early version of this manuscript. We thank Erwin Frey, Nigel Goldenfeld, Ilkka Hanski and Edward Ott for productive conversations. Simulations were performed on a highperformance computing cluster housed within the Computer Science and Engineering department at UC Davis. We are grateful to Bill Broadley and Terri Knight for their excellent administration of that resource. This work is supported by the National Science Foundation under INSPIRE Grant No. 1344187.
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This study has its origins in the lab of A.H. The results are the product of a tightknit interdisciplinary collaboration among the three authors. J.M. and A.H. contributed to the development and analysis of the results, and to the writing and editing of the manuscript. A.E.N. developed analytical results, ran numerical simulations and took primary responsibility for the manuscript.
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Noble, A., Machta, J. & Hastings, A. Emergent longrange synchronization of oscillating ecological populations without external forcing described by Ising universality. Nat Commun 6, 6664 (2015). https://doi.org/10.1038/ncomms7664
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