Phase transitions in biology: from bird flocks to population dynamics

Abstract

Phase transitions are an important and extensively studied concept in physics. The insights derived from understanding phase transitions in physics have recently and successfully been applied to a number of different phenomena in biological systems. Here, we provide a brief review of phase transitions and their role in explaining biological processes ranging from collective behaviour in animal flocks to neuronal firing. We also highlight a new and exciting area where phase transition theory is particularly applicable: population collapse and extinction. We discuss how phase transition theory can give insight into a range of extinction events such as population decline due to climate change or microbial responses to stressors such as antibiotic treatment.

1. Introduction

The study of how systems undergo transitions between states has been an important topic in physics and chemistry for centuries. Some of the first recognized phase transitions involved the thermodynamic properties of transitions between liquid, solid and gaseous states of matter. Much of the terminology used in the study of phase transitions was codified by Paul Ehrenfest's work on statistical mechanics in the early twentieth century [1]. The study of phase transitions has been expanded to other physical phenomena ranging from magnetization, which can be described with the well-studied Ising model [1,2], to liquid crystals [3,4], Bose–Einstein condensates [5] and superconductivity [6]. A phase transition, most simply defined, is a shift of a system from one identifiable state of order to another, in response to the alteration of a control parameter such as temperature or the strength of an applied magnetic field [1].

In recent decades, phase transitions have been identified in biological phenomena [7]. Whereas physical or chemical phase transitions involve populations of inanimate particles transitioning from one well-defined state of organization to another, phase transitions in biology are often far more complex, since they involve active, interacting organisms whose behaviour can be characterized by an immense range of possible parameters. Quantifying an ordered state in a complex biological system is also more challenging than in a physical or chemical system, which further complicates the study of phase transitions in such systems.

While the investigation of phase transitions in a biological system necessitates some simplification in the characterization of the system, since not all parameters can be considered, this approach can still provide profound insights into the underlying biological processes. Biological phase transitions have been identified in neural coding and synchronization [8–11], bird flocking [12], migration of keratocytes [13], rigidity of actin networks [14], response of yeast and bacterial populations to environmental stress [15] and ecological phenomena [16–21]. Phase transitions even occur in the formation of traffic jams [22], and such jamming transitions occur at the cellular level as well [23].

One of the most exciting areas of potential for identifying phase transitions in biology is population dynamics. It is broadly relevant to biological fields such as disease biology and conservation, where it can be used to inform applied methods for disease treatment and population management. In conservation biology, where it may be essential to prevent stress levels from provoking population collapse, predicting tipping points has clear, immediate applications [24,25]. The dynamics of populations are also crucial in clinical settings, where it is important to ensure that the level of an antimicrobial stressor overcomes a threshold to trigger the collapse of a pathogenic population [15]. Understanding the dynamics of collapse could thus lead to the design of interventions in order to rescue endangered populations—an increasingly important task in this age of the ‘sixth extinction’ [26]—as well as to steer clinical initiatives towards controlling pathogenic species.

Here, we briefly review the underlying concepts of phase transitions (§2) and discuss their applications to biological problems (§3), with a particular emphasis on population dynamics (§4). We conclude with a discussion of the implications that modes of population decay may have for understanding underlying ecological processes (§5).

2. Phase transitions in physics

In physics and chemistry, the phase of a system is defined by an order parameter, which quantifies the degree of organization of the system. For instance, the state change of a material from solid to liquid to gas can be quantified by changes in pressure or density. In the magnetization of a material, the phase can be characterized by the net alignment of spins [1]. Phase transitions occur when a system undergoes a change of state, which is characterized by a sharp transition in the value of the order parameter. This occurs at the critical value of a control parameter which influences the state of the system. Classical examples of control parameters are temperature, enthalpy or the strength of an applied external magnetic field [1]. The order parameter operates as a dependent variable, while the control parameter acts as the independent variable.

While order parameters undergo a sharp change during a phase transition, there are various degrees of ‘sharpness’. The behaviour of the order parameter in the critical range is key to understanding the underlying dynamics of the transition process. A first-order phase transition exhibits discontinuity in the order parameter at the critical point. A plot of the order parameter as a function of the control parameter will thus undergo an instantaneous ‘jump’ at the critical value of the control parameter (figure 1a). By contrast, a second-order, critical or continuous phase transition shows a sharp but continuous change, as shown in figure 1b. The ‘first-order’ and ‘second-order’ terminology originates from the description of thermodynamic phase transitions, referring to the lowest derivative of the free energy that shows a discontinuity at the critical point. To avoid confusion, we will refer to second-order transitions simply as critical transitions throughout the rest of the paper.

Figure 1.

Schematic representations of first-order versus critical phase transitions. (a) An example of a first-order phase transition, where the order parameter displays a discontinuous jump at the critical value of the control parameter. (b) An example of a critical phase transition, where the order parameter increases smoothly as the control parameter changes. (c) In critical phase transitions, fluctuations in the order parameter increase in the critical range, with the largest standard deviation at the critical value.

Critical phase transitions are particularly interesting because of the large fluctuations in the value of the order parameter in the critical range (figure 1c). These fluctuations exhibit so-called critical behaviour, characterized by power-law scaling [1]. In the Ising model, for example, which was designed as a simple model for magnetization, the distribution of sizes of clusters of aligned spins exhibits a power-law distribution at the critical point, with many small clusters and a few large ones. Such distributions have a so-called ‘longer tail’ than exponential distributions, meaning that there are more large clusters than would be expected with an exponential distribution [1]. The parameters that describe the way in which this scaling occurs (for example, how certain measures fluctuate as a function of the distance from the critical point) define the universality class of the phase transition. Transitions in the same universality class can characterize a wide variety of systems.

The ferromagnetic transition, as the spins in a piece of iron become aligned in the presence of an external magnetic field, is a particularly well-studied example of a critical transition [1]. In some more complex systems, however, or in systems with an added temporal or spatial disorder, it can be more difficult in practice to identify the type of transition. For example, there has been considerable debate over whether the transition from order to disorder in computational models of self-propelled ‘swarming’ particles is a first-order or a critical transition [27].

In addition to the first-order versus critical classification, phase transitions can be categorized as equilibrium or non-equilibrium. An equilibrium phase transition is reversible, or in equilibrium, between two states. The melting and freezing of water is an example of an equilibrium transition. A non-equilibrium phase transition is irreversible; the state from which the system cannot return is described as the ‘absorbing state’ [28]. In a biological system, a transition from a viable to an inviable state is an example of a transition to an absorbing state.

A canonical example of an absorbing, critical phase transition is the directed percolation transition, named in analogy to water moving downward through coffee grounds under the influence of a gravitational field. In addition to theoretical and computational studies [28], directed percolation has been experimentally demonstrated in a system of turbulent liquid crystals [3,4]. In this case, the transition was induced between two types of turbulence, called dynamic scattering mode 1 (DSM1) and dynamic scattering mode 2 (DSM2); voltage was tuned as the control parameter. While other experimental demonstrations of directed percolation appear more elusive [29], this important type of transition is under active investigation since it is the best-understood example of a non-equilibrium phase transition, and other non-equilibrium transitions can be described using modifications of the directed percolation framework [28,29]. Though many questions remain unanswered, there is a rapidly growing body of literature on phase transitions of all varieties in physical systems, both classical and quantum, including theoretical, computational and experimental studies.

3. Phase transitions in biology

Phase transitions have recently been particularly successful in describing biological systems and have been invoked to describe a range of phenomena from membrane fluidity [30] to neuronal encoding mode [31] and neuronal synchronization [10,32,33]. Some examples of such transitions are summarized in table 1 and are discussed in more detail in this section.

Table 1.

Selected examples of phase transitions.

transition/system	control parameter	order parameter	first-order versus critical	selected references
melting, freezing, sublimation of fluid	temperature, pressure	fluidity	first-order or critical	[1]
melting of plasma membrane	temperature	fluidity	critical	[30]
ferromagnetization	temperature; external magnetic field	magnetization	critical	[1]
neuron rate encoding mode	relative roughness of voltage versus voltage threshold	temporal encoding versus rate encoding	‘abrupt,’ presumably first-order	[31]
neuronal synchronization	synaptic strength; ratio of excitation to inhibition	synchronization of firing rates	first-order	[10,33]
flocking/swarming	density; noise in perception of neighbours' behaviour	alignment of flight patterns	first-order or critical	[12,27,34–42]
actin rigidity	density	rigidity	unspecified	[14,43]
microbial population collapse	stressor level	growth	critical	[15,44]
cultural evolution	landscape changes accompanied by epistasis between cultural traits	cultural paradigm	first-order	[45]
invasive species spread	biodiversity threshold	population size	unspecified (presumably first-order)	[19]
phytoplankton traits	light conditions	chlorophyll content	critical	[21]
megafaunal population collapse	varied	population size	critical, unspecified	[18,46–48]
rigidity percolation in zebrafish embryongenesis	adhesion-dependent cell connectivity	tissue viscosity, cluster size	critical	[49]

At the interface between biochemistry and biophysics, phase transitions are particularly important in the interaction of phospholipid molecules to form biological membranes. Different temperatures and concentrations can result in the formation of micelles, or in the assembly of a lipid bilayer. This system undergoes a critical phase transition as lipid bilayers shift from a crystalline to a liquid crystalline state [30]. Here, temperature serves as a control parameter, and the order parameter is the cell swelling rate, calculated from measurements of absorbance. Importantly, changing the experimental conditions can change the shape of the phase transition. The curve plotting the order parameter against temperature can be shifted, for example, by increasing the cholesterol concentration of the membrane, or by changing its phospholipid composition, both of which alter the membrane fluidity at each given temperature, and hence shift the range of temperatures at which the membrane will pass from a crystalline to a liquid crystalline state [30].

A more abstract biological phase transition has been proposed in a model suggested by Taillefumier and Magnasco to characterize the switching of a neuron from rate encoding to temporal encoding [31]. An individual neuron can transmit information in two different ways: (i) the rate at which it fires (rate encoding) or (ii) the precise timing at which it fires (temporal encoding). In Taillefumier and Magnasco's model, the mode of information transmission is determined by the ‘roughness’ of the action potential voltage threshold as a function of time, compared to the ‘roughness’ of the transmembrane potential. If the former is rougher, the neuron is a temporal encoder; otherwise, the neuron is a rate encoder. The relative roughness thus serves as a control parameter. The phase transition here is equilibrium—it is reversible—and is first-order, as the neuron abruptly switches from one information encoding mode to another. Whether this proposed phase transition occurs experimentally in individual neurons remains to be seen. Other types of phase transitions in neural systems involve the collective behaviour of neural ensembles, as when groups of neurons synchronize their electrical activity [10,32], or when neural activity is tuned by the balance between excitation and inhibition [33].

Many other examples of phase transitions in biological systems involve the collective behaviour of large ensembles of individual organisms. Beekman et al. [34] showed that ant foraging behaviour undergoes a phase transition from a disordered to an ordered state as a function of colony size. Density often serves as an order parameter in such examples of phase transitions in collective behaviour. For example, as the density of locusts is increased, random movement in a population undergoes a critical phase transition to an ordered ‘marching’ state [35].

There is extensive literature on phase transitions in the collective swarming and flocking behaviour of ensembles of simulated ‘active particles’ [27,36–40,50]. Swarming and flocking phase transitions have also been studied experimentally, for example, in swarms of midges [41]. Detailed data collected from starling flocks together with a related computational model [42] suggest that a critical phase transition may occur from a disordered to an ordered state as the control parameter of the noise in birds' perception of their neighbours' behaviour is varied.

Phase transitions in biological systems are also increasingly well-studied at the level of cells and tissues. A critical transition between ordered and disordered states was observed experimentally in groups of migrating goldfish keratocytes, and the results can be well fit with a computational model of flocking [13]. A phase transition model has been applied to the interaction between malignant and stromal cells in cancer cell migration [51] and phase transitions have been identified in a model of cancer cell migration that incorporates both the polarity of migrating cells and cell–cell interactions in densely packed tissue [52]. Flocking transitions have been investigated in detail in a range of confluent tissue types [53], explaining experimental observations of epithelial monolayer migration and having implications for understanding cell migration during the process of wound healing. In these cases, the transition is driven by the alignment of the cell's local migration velocity with its polarity.

A fluid-to-solid jamming transition has been identified as a key step in elongation of the body axis in zebrafish embryos [23,54]. Jamming phase transitions have also been implicated in the pathology of asthma [55]. A jamming transition occurs when a material composed of multiple elements (such as solid inanimate particles in the physics of granular materials or moving cells in a biological system) reaches a critical density at which the system ‘jams’ or enters a disordered packing state in which net flow cannot occur [56]. Transitions that are broadly referred to as ‘jamming’ also occur in many other situations where collective cell motility comes to a stop; as reviewed by Lawton-Keister & Manning [57], these transitions can be driven not only by changes in density, but also by tension-driven rigidity or by a decrease in fluctuations. Cellular jamming transitions can also be modulated by cell speed and by the balance between cell-to-cell adhesion and the tension in the cell's actomyosin cortex [52]. It has been suggested that the mechanical heterogeneity of cancer cells may allow ‘softer’ cells to become unjammed and better able to migrate, thus increasing tumour invasiveness [52].

A related type of phase transition, the rigidity percolation transition, has recently also been observed in embryonic tissues such as the zebrafish blastoderm. This transition can be characterized by a change in tissue viscosity that also constitutes an essential step in embryogenesis [49]. The transition is triggered by a drop in adhesion-dependent cell connectivity. A rigidity percolation transition in the shear properties of articular cartilage [58] has been adapted to model the mineralization of collagen fibrils in bone [59].

At the scale of molecular biophysics, phase transitions have been observed experimentally in the alignment of actin filaments moving along a heavy meromyosin-coated surface, as a function of actin concentration [43]. A subsequent study identified a non-equilibrium transition between homogeneous and aggregated actin networks [60]. Gurmessa et al. [14] used a combination of optical tweezers and microrheology techniques to demonstrate a rigidity phase transition in actin networks during chemically triggered assembly and disassembly.

The capacity of some systems to reach an ordered state may be impacted by factors such as environmental stress, as in the case of swarming bacteria. Grobas et al. [61] recently showed that, in the presence of a gradient of the antibiotic kanamycin, swarming Bacillus subtilis undergo a transition to a biofilm state. In another recent study, B. subtilis aggregation was disrupted by reactive oxygen species in the presence of light, and the experimental results were simulated by a model of interacting self-propelled particles in the presence of different levels of noise (i.e. uncertainty in the motion of the simulated bacteria) [62]. As noted earlier, noise has also been shown to play a role in driving the swarming behaviour of bird flocks [42]; it can also modulate phase transition behaviour in generalized computational models of aggregating biological agents [39], highlighting the common collective behaviour of aggregating organisms over a wide range of size scales. The effect of disorder on phase transitions is also a topic of significant theoretical importance [27].

4. Population collapse and extinction

Population collapse and species extinction due to anthropogenic change of natural environments is one of the most pressing scientific issues faced by both developing and developed countries. Population collapse, as a result of environmental and other factors, has been studied in numerous systems ranging from bird communities [63] to honeybees [64] and other desirable pollinators [65], to pathogenic bacteria [66]. Climate change has driven the collapse of megafaunal communities in past geological eras [67] and is threatening many animal species in the Anthropocene [68], including terrestrial herbivores [69] and marine megafauna [70].

The process of population collapse often displays strikingly similar mathematical properties across organisms when modelled as a phase transition, given the appropriate selection of order and control parameters. Phase transition-like behaviour has been observed in numerous ecological systems, but analyses have not always clarified whether the observed behaviour constitutes a true phase transition, and if so, what type of transition it is. Ecological transitions are also often described as ‘tipping points’ [71] and analysed in terms of transitions between stable dynamical states [16,72,73]. While coming from different areas of physics (tipping points and stable states from nonlinear dynamics, and phase transitions from statistical physics), these approaches are closely linked theoretically, as discussed in [72,73]; it has been suggested that in some cases it may be advantageous for biological systems to be poised at criticality [7].

The tipping point approach allows for the identification of early warning signals that presage the onset of collapse [16]. One such warning signal is critical slowing down, in which a system takes longer and longer to recover from a challenge as it nears a tipping point from one stable state (or from one phase) to another. This was observed by Dai et al. in an experimental S. cerevisiae population [74] and has been investigated in the context of more general models of rescuing populations on the brink of collapse [24,25].

Ecological collapse and tipping points have been investigated in a wide range of contexts, including human societies. One study analysed the collapse of the Rapa Nui society, constructing a population dynamics model based on a wide array of environmental factors, from climate to ecology to demographics [17]. Another study developed a population dynamics model for cultural ‘paradigm shifts’, depending on the cultural landscape and epistasis between cultural traits, noting that ‘phenomenon of paradigm shifts in cultural evolution [is] in the same category as catastrophic shifts in ecology or phase transitions in physics' [45]. Crossover between various types of phase transitions (e.g. from first-order to critical) has been studied in a model of social contagion with a heterogeneous threshold for adoption of new behaviour [75]. Mathematical models of this type of problem have been investigated in the voter model in probability theory, in which phase transitions have also been observed [76].

Other studies have addressed tipping points in the context of population recovery. One study in voles showed phase transition-like collapse of a population followed by sudden recovery driven by immigration of additional animals rather than an increase in reproduction rate [18]. In coral reefs, seaweed chemical cues serve as a control parameter guiding a tipping point in whether juvenile corals and fishes can repopulate an area [46]. This highlights the fact that the collapse or recovery of one population in an ecological community can have a cascading effect on other species [77]. In a mutualistic network, noise (variability) has been shown to enable population recovery [78]. Such studies will likely prove increasingly important in the design of rescue protocols to bring species back from the brink of collapse, as in the management of marine ecosystems [47] and in ocean ecosystems more broadly [48]. However, there are limiting conditions beyond which evolutionary rescue from a tipping point may be impossible [79].

The problem of ecological tipping points can be addressed within the specific framework of phase transition theory. A theoretical study used a Lotka–Volterra model to study a population subject to migration from other species; a phase transition was observed from a state with a single globally attracting fixed point (corresponding to one possible invasion-resistant community composition) to a state in which multiple stable outcomes (i.e. multiple distributions of species abundance and diversity) coexist [20]. The coexistence of stable states led to history-dependent community properties. This is an example of hysteresis, a phenomenon in which a system can fall into one stable state or another, depending on the direction in which control parameters are varied. Another recent study addressed hysteresis in the context of tipping points and environmental feedback in tropical forests, with implications for forest resilience in the coming decades of climate change [80].

The determination of specific phase transition characteristics may be daunting in studies involving megaflora and megafauna, due to the difficulty in collecting sufficiently large datasets from which to draw robust quantitative conclusions. Studies in smaller scale organisms offer the possibility of collecting large datasets which are also subject to direct experimental manipulation. In phytoplankton, for example, critical phase transition behaviour has been observed in cellular chlorophyll content as a function of external light conditions; this transition has been suggested to belong to a new universality class [21].

Microbial populations offer a convenient and important experimental system for the study of phase transitions in biological systems as studies can be conducted over relatively short time periods with large populations, allowing researchers to parse the interplay between biological factors and population dynamics. There is strong evidence that phase transitions can occur in microbial populations grown under stress where the stressor level serves as the control parameter and population growth serves as the order parameter [15]. This breaks from the canonical practice of visualizing growth over time under a given pharmacological condition (kill curve) and provides a broader representation of the system's behaviour under a range of conditions, which can be readily compared to similar representations in other biological systems.

The transition of a population from viable to inviable is, virtually by definition, a non-equilibrium (irreversible) phase transition. In the opposite direction, phase transitions have also been proposed in models for the origin of life [81,82]. Non-equilibrium phase transitions have been investigated in models of the transition from extinction to survival in evolutionary models [83–86]. The transition from extinction to survival in a computational model of evolutionary dynamics on a two-dimensional phenotype space, for example, exhibits characteristics of directed percolation [83–85], as does a two-component model of a species and its mutant [86].

In microbial populations, factors such as temperature, salt concentration or antibiotic concentration can act as control parameters; doubling rate or growth serves as the order parameter. Recent work by Ordway et al. [15] identified phase transition-like behaviour for yeast and bacterial systems exposed to stressors such as high salt concentration, high temperature and antibiotics. Notably, different types of stressors cause different types of phase transition-like responses. For example, for some antibiotics, Escherichia coli exhibits a sharp decline in doubling rate as a function of antibiotic concentration, consistent with a critical phase transition. For other antibiotics, however, E. coli undergoes a very gradual decline in doubling rate, inconsistent with any phase transition model. Interestingly, these diverse patterns of population decline were observed within and across different classes of antibiotics. In a parallel set of experiments in yeast (Sacchromyces cerevisiae), critical phase transition-like behaviour was observed when the cells were subjected to high-temperature stress, while a much more gradual transition was observed in the presence of a different stressor, high NaCl concentration.

In S. cerevisiae, the difference in transition behaviour observed by Ordway et al. [15] presumably occurs because of differences in stress responses to increased temperature compared to increased salinity. As temperature increases, there is a rapid denaturation of the proteins and enzymes required for S. cerevisiae growth, which results in a sharp decline in viability [44,87,88]. Conversely, S. cerevisiae has a number of mechanisms to cope with increasing salt stress, which may ameliorate population decline [89–91].

It is less clear why different antibiotics induce such different phase transition responses in bacterial populations. Even chemically similar compounds (i.e. aminoglycosides) induce drastically different responses, while similar responses can be induced by antibiotics from different functional classes (figure 2). The answer to this question may offer clinically important insights into treatment failure; the identification of factors determining the point at which a bacterium becomes susceptible to a particular antibiotic will have immediate implications for treatment design. From an evolutionary perspective, it is notable that related antibiotics such as spectinomycin and streptomycin cause such disparate phase transition responses (figure 2). This suggests that even closely related antibiotics may impose drastically different selection conditions for resistance evolution and raises the prospect of experimentally testing (at a broader level) what ecological factors, or (at a more proximal level) what specific pressures, alter the dynamics of population collapse [92–94]. Identifying the underlying genetic and metabolic features that affect the dynamics of the transition could in principle be used not only to determine the steepness of response to antibiotic concentration, but also to precisely tune antibiotic concentrations for optimal response in a clinical setting. For example, shifting a population's behaviour away from a linear transition, toward a steeper dose–response curve, could increase the effectiveness of an antibiotic and reduce subsequent evolution of resistance. Can a single mutation change the curve from a sharp phase transition to a gradual decline? Might resistance mutations in proteins at or near antibiotic binding sites, or mutations in proteins that export or break down antibiotics, have such an effect? How does the phase transition curve change when genetic or environmental backgrounds are altered? Studies showing that a gradual increase in antibiotic concentration gives rise to more successful sets of resistance and compensatory mutations than a sharp elevation in antibiotic dose [94,95] may point the way toward further experiments that could resolve these questions.

Figure 2.

E. coli strain MG1655 growth measured using optical density at 600 nm with respect to log antibiotic concentration in μg ml⁻¹. MG1655 were grown for 24 h in 200 µl M9 glucose minimal media from an optical density of 0.005 at 600 nm in various antibiotic concentrations, with six replicates at each concentration. Different antibiotics induce different phase transition behaviour between population success and collapse; high standard deviations of the order parameter near the inflection point of the curve appear for some, but not for all, antibiotics. (Online version in colour.)

5. Outlook and future perspectives

Despite an emerging wealth of current work, there remain a number of gaps in current knowledge of phase transitions in biological systems. A better understanding of these could inform both management of both invasive species and pathogens, as well as management of ecologically beneficial groups such as pollinators, key species in ecosystem networks and probiotic bacteria. From a physics perspective, the problem of noise in modulating transitions is an active area of research that has the potential to greatly inform biological phase transitions. The introduction of disorder into a system can change the underlying nature of the phase transition [96,97] and could lead to rescue in the case of transitions that might at first seem to lead to an absorbing state [78].

A key question in biological phase transitions is to more precisely determine what factors drive ecological systems towards collapse, and how to design interventions to modulate or prevent these processes. This can be addressed in the laboratory, with the use of, among other techniques, experimental evolution [92,98]. Studies of population collapse in microbes have been used to test fundamental ecological theory such as the effect of rapidly changing environments on evolutionary responses [99–101]. Similarly, experimental systems such as E. coli and yeast may also facilitate the identification of factors predictive of a population's approach to a ‘tipping point’ into collapse [74]. In the near term, single-celled organisms will likely provide a test-bed for population-rescue protocols that could be applied to macro-organisms in conservation efforts [99–101].

Additional open questions include the identification of the optimal order parameters to characterize population dynamics and the identification of the primary control parameters that drive population collapse. How robust are results across systems and scales? What factors (genetic, physiological and ecological) affect the shape of phase transitions in population dynamics? Here also, microbial models can provide insights into globally relevant phenomena, which may be applicable to the dynamics of larger ecosystems, including endangered megafauna. Studies of biological phase transitions and their modulation will likely trigger new insights into current issues in the physics of phase transition phenomena as well, such as the role of disorder and the determinants of crossover between types of phase transition.

Collaboration between physicists and biologists will also be critically important for understanding the dynamics of populations at the macro-scale. Large datasets will be essential in order to map the population dynamics of animals such as endangered terrestrial vertebrates. Sources of aggregated data from many studies that are available to the public, like the Living Planet Index, will become increasingly valuable. Studies of population dynamics at the macro-scale will also need to combine temporal analyses of population size with studies of spatial distributions of animals. An extensive literature already exists on the topic of spatial distributions and population dynamics over a range of systems, species and scales ([102–107], among many other examples), and could surely be quickly deployed in fruitful collaborations between physicists, biologists and interdisciplinary scientists in the identification of tipping points and the development of rescue protocols.

Data accessibility

Data are available from the Dryad Digital Repository: https://doi.org/10.5061/dryad.xksn02vg4 [108].

Authors' contributions

E.H., S.B. and R.F.I.: conceived of the review; R.F.I.: designed the supporting experiments; E.H. and H.H.: performed the experiments; R.F.I. performed the data analysis. E.H., H.H., S.B. and R.F.I.: drafted the manuscript. All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Competing interests

We declare we have no competing interests.

Funding

We received no funding for this study.