Self-organised criticality in the evolution of a thermodynamic model of rodent thermoregulatory huddling
[['In cold environments , huddling together is a ‘good trick’ that is exploited by many endothermic species in order to keep warm .', 'Huddling in many species of mammals and birds has been described as a self-organising system [1–4] , whereby simple local interactions between individual animals collectively give rise to a complex group-level behaviour .', 'The huddle constitutes an adaptive thermoregulatory system in which the interaction between self-organisation and natural selection , two forces that shape all biological systems , may be expressed in terms of a common ( metabolic ) currency .', 'Here we investigate the interaction between self-organisation and selection by first developing a thermodynamic model of huddling , and then evolving the thermal physiologies of individuals in the model under pressure to thermoregulate at low individual metabolic cost .', 'The model reveals how a balance between cooperative and competitive interactions between animals may constrain the dynamics of natural selection .', 'Converging evidence suggests that thermogenesis plays a central role in balancing competition and cooperation in rodent litters .', 'Non-shivering thermogenesis via brown adipose fat tissue ( BAT ) is functional in juvenile rats , and in the cold they huddle together .', 'Rats seek cold when BAT is pharmaceutically increased [5] , and huddling ceases when BAT is pharmaceutically blocked [6] .', 'Hamsters , who develop functional BAT later and do not normally huddle start to huddle when fostered into litters of rats .', 'Thus heat generated by some individuals can trigger huddling in others [7] .', 'Huddling patterns in rat litters can be understood by considering females ( who have more BAT ) as heat sources and males as heat sinks [8] .', 'Computational modelling suggests that BAT-thermogenesis breaks symmetries between body temperatures in the group , which is necessary for the emergence of realistic huddling patterns [9] .', 'The metabolic costs of BAT-thermogenesis to the individual may be offset by the thermoregulatory benefits of huddling to the group .', 'In cold environments individuals must cooperate by contributing heat in order for the huddle to self-organise , but generating too much heat is energetically costly .', 'The price for burning oxygen must ultimately be paid in the consumption of food , e . g . , the milk of the dam , and pressure to thermoregulate thus puts individuals in competition for food when energy resources are limited .', 'Understanding the thermodynamics of huddling may shed light on how biological systems are shaped by the tension between cooperation and competition , and thus by the interaction between self-organisation and selection in more general terms .', 'Thermoregulatory huddling behaviours displayed by litters of juvenile rodents give rise to what has been described as a second-order critical phase transition , i . e . , a continuous but abrupt decrease in the degree of aggregation as the temperature of the environment is increased [2] .', 'In the language of thermodynamics ( i . e . , statistical physics [10] ) , a second-order critical phase transition ( henceforth simply a phase transition ) occurs only in systems whose component interactions express long-range correlations .', 'In such terms the phase transition describes the change in the macrostate of a system of particles , for example the change that occurs when ice turns into water when it is heated .', 'Analogously , the phase transition in rodent behaviours describes a change in state from a stable aggregation at low ambient temperatures ( huddling ) , to a dispersion of animals at higher temperatures .', 'The classic system for investigating phase transitions is the Ising spin-glass model [10 , 11] , according to which magnetic spins arranged on a lattice dynamically re-orient , directed by the orientation of neighbouring spins .', 'Differences in the relative orientations of neighbouring spins yield differences in ‘energy’ across the lattice , leading to dynamics that may either settle into a stable configuration in which spins remain continually aligned , or the states of the system may cycle a dynamic attractor .', 'The attractor into which a particular system will settle is governed by the temperature of the system , which in theoretical formulations translates to the probability that the orientation of a given spin will be perturbed from its current alignment .', 'At extreme temperatures , dynamics tend to stabilise , whereas at intermediate temperatures complex dynamics may persist .', 'In spin-glass models , a phase transition reveals itself as a peak in the fluctuations of the total energy ( the heat capacity ) of the system , around what is described as its critical temperature .', 'The peak in heat capacity is a defining feature of a phase transition .', 'Rodent huddles display a group-level behaviour known as ‘pup flow’ , whereby individuals continually cycle between the cool periphery of the huddle and its warm core [12] .', 'Huddling can be quantified in terms of the exposed surface area of the group , and pup flow can be quantified in terms of the rate of change of huddling .', 'By these metrics , a recent self-organising model of rodent huddling predicts a peak in pup flow around the critical temperature of the phase transition [9] .', 'If we assume that huddling corresponds to the energy in an Ising model , then pup flow reflects the heat capacity of the system , and hence a peak in pup flow is consistent with the predictions from classical thermodynamics .', 'A thermodynamic description of a complex adaptive animal group behaviour in which metabolic cost can be defined explicitly , could open the door for a thermodynamic theory of the interaction between self-organisation and selection .', 'To this end , in the following section we derive a description of self-organising thermoregulatory huddling in thermodynamic terms .', 'Thermal physiologies in the model are then evolved under pressure to reduce metabolic cost , and the model is shown to display a phenomenon called self-organised criticality [13] .', 'The emergence of criticality in the model suggests that a tension between competition and co-operation creates complex evolutionary dynamics that are stable across a wide range of thermal environments .', 'Finally , application of the same evolutionary pressure to a full agent-based simulation of huddling [9] demonstrates that the evolutionary dynamics are maintained when physical interactions between animals in a closed arena are simulated explicitly , validating the thermodynamic description of huddling , and suggesting that criticality may be a more general property of evolving self-organising systems .'], ['The approach here is to construct the simplest possible description of the huddle as a self-organising thermoregulatory system .', 'The main criteria for a successful description of self-organising huddling are', 'i ) that across a range of intermediate ambient temperatures , the average body temperature , computed over all individuals , should remain approximately constant at a preferred body temperature of 37°C ,', 'ii ) that a metric of huddling should display a second-order critical phase transition as the ambient temperature is manipulated , and', 'iii ) that an appropriate metric of ‘pup flow’ should display a peak at the critical ambient temperature of the phase transition .', 'The simplest model that can generate a second-order phase transition is known to be the Ising spin-glass model , and the dynamics of an Ising model can be simulated using a Monte Carlo algorithm ( see e . g . , [11] ) .', 'Our aim here is therefore to develop a Monte Carlo algorithm whose behaviour satisfies the criteria for a successful huddling model , which is defined in terms that are directly interpretable as factors affecting rodent huddling interactions .', 'The Monte Carlo algorithm presented here uses the standard method of Gibbs sampling , which is described next in terms of huddling interactions .', 'The description of huddling is based only on a description of the litter in terms of the sizes of its component groups ( or ‘aggregons’ [4 , 14 , 15] ) , and abstracts the dynamics of real huddling , however it captures the idea that when two pups come into contact they will either stay in contact ( if doing so is energetically favourable ) , or one will avoid the other and so be displaced from its group .', 'Central to the Gibbs sampling method is to determine the probability of making a decision between two outcomes .', 'Consider the situation where two independent groups of aggregated individuals collide .', 'As a result of a contact between two individuals ( from different groups ) , two outcomes are possible .', 'Outcome 1 is that the individuals may remain in contact , and thus the two groups to which they belong will have merged together into one larger group .', 'Outcome 2 is that an individual may become displaced from its group to form a new group of size 1 . Note that all other possibilities that might be observed , for example the possibility of both individuals becoming simultaneously displaced from their groups , of both individuals returning to their original pre-contact groups , or of one individual leaving its group and joining the other , are all possible as consequences of further iterations of the decision between outcomes 1 and 2 . Hence , our algorithm is based on iteratively deciding between outcomes 1 and 2 , and updating the sizes of the groups after each iteration accordingly .', 'On each iteration , the probability of outcome 1 , p1 , is', 'p 1 = 1 + e - T - 1 , ( 1 )', 'where T is referred to as the ‘temperature parameter’ .', 'Outcome 1 is more likely to occur when the temperature parameter is higher .', 'Gibbs sampling involves generating a random number from a uniform distribution r ∈ [0 , 1] , and on each iteration selecting outcome 1 if r < p1 and selecting outcome 2 otherwise .', 'The probability of outcome 2 is therefore p2 = 1 − p1 .', 'The algorithm of the huddling model proceeds as follows .', 'Each of N individuals are assigned to a group .', 'The number of individuals in a particular group is n .', 'For individuals indexed by i ∈ [1 , N] , the number of individuals in the group to which individual i belongs is defined to be ni .', 'On each iteration , two individuals , labelled a and b , are selected at random from two different groups .', 'Accordingly , the probability that individual i is selected to be individual a is pa = ni/N , and for the remaining N − 1 individuals the probability of being selected as individual b is therefore pb = ni/ ( N − na ) .', 'Note that individuals from larger groups are more likely to be selected .', 'If r < p1 , then the groups to which a and b belong are joined together to create a single group of size na + nb ( outcome 1 ) , otherwise a is removed from its group to form a new group of size 1 ( outcome 2 ) .', 'The temperature parameter T is defined in terms of the body temperature , TB , of individuals a and b ,', 'T = 2 T P - T B a - T B b , ( 2 )', 'where TP = 37°C is the preferred ( or target ) body temperature , and the factor 2 accounts for the combination of temperatures measured from two individuals .', 'According to Eq 2 , the further the body temperatures in a group drop below the preferred body temperature , the more likely it is that the group will merge with another , which is consistent with our intuitions about the consequence of individual thermoregulatory huddling behaviours .', 'The body temperature of each individual is computed using a well-established model of endothermy derived from Newton’s law of cooling [16–18];', 'T B i = T A + G i A i k 1 , ( 3 )', 'where TA is the ambient temperature , G is the rate of heat production by thermogenesis , A ∈ [0 , 1] is the proportion of the surface area that is exposed to the ambient temperature , and k1 is the area-specific wet thermal conductance .', 'Note that the area-specific wet thermal conductance is dependent on many factors intrinsic and extrinsic to the body , including emittance , thermal conductivity , thickness of the fur , speed of the wind , viscosity and density of the air , and the geometry of the body [18 , 19] .', 'This model captures the intuition that a highly exposed body that conducts heat more rapidly to a cold environment must generate more heat to maintain a high body temperature .', 'Geometrical considerations and controlled experiments with small rodents [2 , 20 , 21] show that the average exposed surface area that can be achieved by individuals in a group decays exponentially with the size of the group n , according to', 'A i = n i - 1 / 4 .', '( 4 ) Note that the behaviour of the model developed here is not sensitive to other sensible choices of the exponent , e . g . , −1/3 .', 'By combining Eqs 1–4 the Monte Carlo model can be summarised as follows .', 'At each iteration of the algorithm the groups to which randomly chosen individuals a and b belong are joined together if', 'r < 1 + e ( G a n a 1 / 4 + G b n b 1 / 4 ) / k 1 - 2 ( T P - T A ) - 1 , ( 5 )', 'else individual a is isolated to form a new group of size 1 . In the special case where the number of groups is 1 , p1 = 0 , and in the special case where the number of groups is N , p2 = 0 .', 'Note that by this description of huddling , smaller groups are less likely to be selected to potentially join with other groups , but when selected they are ( in cold environments ) more likely to join with other groups .', 'This tension underlies the dynamics of the model that are revealed in Fig 1 to satisfy our criteria for a successful description of self-organising thermoregulatory huddling interactions .', 'All figures in this paper can be recreated using the source code available in the supplementary materials as S1 Code .'], ['The Monte Carlo huddling algorithm was initialized with a single group of size n = N = 12 , and iterated for t = 5000 iterations .', 'In independent simulations , the model was simulated through a range of ambient temperatures ( TA ∈ [0°C , 50°C] ) .', 'For correspondence with later sections , individual rates of thermogenesis , G , were drawn at random from a uniform distribution , G ∈ [0 , 5] .', 'For Fig 1 the parameter k1 was chosen to be k1 = 0 . 3 , which effectively sets the slope of the phase transition , and was found by experiment to provide a qualitative match to similar results derived from a related model [9] .', 'The metric of huddling is defined to be h = 1 - 1 N ∑ i A i , which is shown in Fig 1 as an average over iterations , and the metric of pup flow was defined as the standard deviation of h over iterations .', 'By this metric , pup flow approximates the heat capacity of the system .', 'The model displays a second-order critical phase transition as the ( ambient ) temperature is manipulated , which is revealed in terms of both the average group size ( Fig 1A ) and the average exposed surface area ( Fig 1B ) , as well as a peak in heat capacity ( Fig 1C ) , which is a hallmark of a true phase transition .', 'These are general thermodynamic properties that are inhereted from the Ising spin model from which the huddling model is derived .', 'Confidence that the particular form of the model is a valid description of rodent huddling interactions , and thus confidence in the analogy between huddling pups and magnetic spins , comes from two observations .', 'First , the temperature parameter T is represented here in terms of temperatures that describe factors that influence huddling , i . e . , a combination of body , ambient , and preferred temperatures .', 'Second , Fig 1D shows that the average body temperature remains approximately constant around TP = 37°C over a range of intermediate ambient temperatures , i . e . , huddling aids thermoregulation .', 'The huddling model is particularly interesting from a theoretical perspective because unlike in a particle system in which temperature is a parameter under influence only by factors external to the system ( e . g . , the temperature of a heat bath in which the system is assumed to be situated ) , temperatures in the huddling model are a property that is internal to the components of the system , i . e . , a property that may be influenced by the rate of thermogenesis of the individuals .', 'Moreover , the system is interesting from a biological perspective if we assume that optimisation of the rate of thermogenesis , G , a key determinant of the body temperature , is under the influence of natural selection .', 'The rate of heat production of an animal is directly related to its metabolic rate .', 'Hence optimisation of G can be thought of as optimisation of the metabolic costs of thermoregulation .', 'To investigate how huddling might interact with natural selection we may borrow from another influential model with origins in statistical physics , which has important implications for evolutionary theory , and allows us to capture the intuition that selection within the huddle constitutes a tension between co-operation and competition .', 'The original model of sandpile formation by Bak and colleagues [22] explains self-organised criticality in the distribution of avalanche events that emerges when grains of sand are iteratively added to a pile .', 'A hallmark of such critical systems is that they exhibit scale-invariance , as evidenced in the sandpile model by a power-law distribution in the sizes of avalanches .', 'Bak and Sneppen later proposed a simpler model that exhibits criticality [23 , 24]; iteratively randomizing the fitness ( a number between zero and one ) of the least fit in a population , and simultaneously randomizing the fitnesses of two neighbours , generates avalanches in fitness whose distribution conforms to a power law .', 'Power-law distributions found in complex systems varying from earthquakes to economics have been described in terms of such underlying dynamics [13] .', 'Bak suggests that the underlying long-range correlations ( i . e . , as introduced by mutating also the neighbours of the least fit species ) may account also for punctuated equilibria; step-wise changes in complexity evidenced by fossil records .', 'The implication of this idea is that fitness ( adaptation to the environment ) evolves by a natural selection driven primarily by mutation of the weakest , rather than by selection of the fittest per se .', 'Simply put , if fitnesses are correlated , then a domino effect can emerge , such that removing the weakest individual has a knock-on effect for other inividuals whose fitnesses are higher due to interactions with the weakest .', 'If co-operative huddling behaviours introduce long-range correlations in fitness between interacting rodents that depend on the exchange of thermal energy ( Fig 1 ) , and competition amongst rodents to metabolise at minimal energetic cost ( and thus contribute less heat ) is enforced by selection pressure operating on the weakest huddler , might the resulting tension between competition and competition also lead to self-organised criticality ?', 'If so , huddling would represent an ideal biological system in which laws governing the interaction between self-organisation ( group-level huddling behaviours ) and natural selection ( for individually efficient metabolism ) might reveal themselves .', 'The aim here is therefore to determine whether self-organised criticality might emerge from selection based on huddling efficiency .', 'By direct analogy with the model of Bak and Sneppen [23] , our approach is to iteratively substitute the weakest individual in the simulation for another with a random thermal physiology .', 'Crucially , we will substitute only one individual per generation , such that any evidence that long-range correlations in the huddling model influence the evolution of the group must be attributable to couplings in thermal physiology that are expressed through self-organising huddling interactions .', 'To this end , we next subject the model to a simple evolutionary procedure , starting with the following definition of fitness;', 'F i = - T B ¯ i - T P + α G i , ( 6 )', 'where T B ¯ i is the average body temperature over time .', 'According to the first term in the brackets of Eq 6 , an individual is considered to be better adapted to its thermal environment if it maintains an average body temperature that is closer to the preferred temperature .', 'This term corresponds to the cost of poor thermoregulation , and is influenced by huddling interactions .', 'According to the second term , an individual is considered to be fitter if it has a lower metabolic rate .', 'This term corresponds to the cost of thermogenesis , and is not influenced by huddling interactions .', 'The constant α is a free parameter that is introduced to allow the relative costs of thermoregulation and thermogenesis to be balanced .', 'Thus α can be tuned to place the evolving system in a regime where the costs of thermoregulation and thermogenesis are comparable .', 'By experimentation , a value α = 6 . 06 was chosen as the smallest value that stops metabolic rates evolving immediately towards G = 0 .', 'Note that the appropriate value of α depends on the choice of the thermal conductance k1 and the size of the group N , but importantly once k1 , N , and α are set , the resulting evolutionary dynamics are robust to variations in the control parameter of interest , TA , as explained shortly .', 'On each of 10 , 000 generations of the evolutionary algorithm , the thermodynamic huddling model was simulated through t = 1000 iterations , the average fitness F was computed for each individual at each generation , and the individual with the lowest fitness was replaced in the next generation by an individual with a metabolic rate drawn randomly from a uniform dstribution G ∈ [0 , 5] .', 'The validity of this evolutionary procedure is considered in full in Discussion .', 'However , note that this algorithm effectively poses the following question of the system; what configuration of metabolic rates in the group is optimal when selection pressure on individual physiological thermoregulation ( pressure to reduce G ) is balanced with pressure on behavioural thermoregulation ( pressure to reduce |TB − TP| ) ?', 'Evolution of the distribution of thermal physiologies in the group resulted in extended periods of stasis , sometimes lasting for hundreds of generations , in which the metabolic rates of all in the group remained at constant low values .', 'However , these periods of stasis were interrupted by prolonged bursts in which the fitnesses of several in the group suddenly reached significantly lower values before returning to stasis .', 'In deference to descriptions of self-organised criticality in models of sandpile formation [13] , these sudden transitions are referred to as the onsets of new avalanches .', 'To understand this observation , it is useful to inspect the interaction between the weakest group member , i . e . , the individual that will be replaced in the next generation , and the individual from the remainder of the group that has the highest metabolism , i . e . , that which we might expect to be the second-weakest individual .', 'The emergence and evolution of a representative avalanche is shown in Fig 2 , in terms of the interaction between the metabolic rates of these two individuals at each generation .', 'Fig 2 reveals that the onset of an avalanche occurs when the least fit is substituted for another with a random metabolic rate whose extra heat renders it an attractive target for huddling to the rest of the group .', 'During the period of stasis prior to an avalanche , the metabolic rates of the group are maintained at low values because the fitness function penalises high values of G . The individual with the highest metabolic rate is likely to consistently be the least fit , and hence during periods of stasis the metabolic rate of the least fit varies randomly because it is randomly substituted .', 'However in the generation at which an avalanche is triggered , the random metabolic rate of the individual that has been substituted into the group was observed to be higher than that of the individual subsequently evaluated to be the least fit .', 'This means that the new individual is fitter not because it has a lower metabolic rate , but because it benefits more than the least fit from compensatory savings in thermoregulation .', 'The high metabolic rate G of the new individual is compensated for a small |TB − TP| , which must occur due to effective huddling interactions with the rest of the group .', 'At the onset of an avalanche , the individual that will be replaced in the next generation is therefore not the individual that is contributing the most heat to the group by thermogenesis , hence the total heat available to the group in the next generation increases .', 'In general , the fitness function penalises high metabolic rates and thus the heat from individual thermogenesis tends to reduce as the system evolves , but at the onset of an avalanche the total heat is instead transiently increased .', 'Thus the evolving configuration of the group becomes critically stable , such that the total heat in the group gradually reduces until the overall reduction of heat triggers the next avalanche .', 'To reveal how the critically stable dynamics revealed by inspection of a single avalanche unfold over the full evolutionary history , Fig 3 shows the distribution of avalanche events across all 10000 generations .', 'Fig 3A shows the full distribution of fitnesses , F , at each generation for all individuals except for the least fit , which as we have seen in Fig 2 has an essentially random distribution that would otherwise mask the interesting group dynamics .', 'Fig 3A confirms that long periods of stasis in fitness are punctuated by avalanches that persist for several generations and involve multiple group members .', 'This example simulation was run at an ambient temperature of TA = 10°C , hence we expect pressure to reduce G towards 0 to maintain fitnesses at around |TA − TP| = −27 ( by assuming TB = TA ) .', 'Values of F that exceed this baseline indicate where huddling interactions have increased the body temperature and values below the baseline indicate where additional costs of sustaining a high metabolic rate have been incurred .', 'Fig 3B shows the metabolic rate of the second-least fit individual at each generation , which is referred to as Gmin , and the close temporal correspondence between Fig 3A and 3B confirms that the avalanches in fitness are caused by transient increases in Gmin , by the mechanism explained in relation to Fig 2 .', 'A number of avalanches are shown in greater detail in Fig 3C .', 'A histogram of the distribution of metabolic rates in the second least fit , H ( Gmin ) , reveals a fall-off in frequency indicative of a power-law distribution .', 'A fitted curve obtained by linear regression of the histogram after a log-log transformation , i . e . , a fit to the model log H ( Gmin ) = m log Gmin + c , is shown in Fig 3D , which reveals a good fit to a negative power-law relationship .', 'Hence the distribution conforms to H ( G min ) = e m G min + c .', 'A power-law relationship is a strong indicator of self-organised criticality , ( see e . g . , [13] ) .', 'A second hallmark of self-organised criticality is that the emergence of the power law distribution is robust to manipulation of the external influences on the system .', 'The external thermal environment of the huddling model is represented by the ambient temperature TA , hence obtaining a power law fit to the distribution of metabolic rates from simulations conducted through a range of ambient temperatures would be stronger evidence of self-organised criticality .', 'Twenty-four evolutionary simulations were run , each at a constant ambient temperature Ta ∈ [0°C , 50°C] .', 'For all simulations at TA < TP , where huddling is expected to improve fitness , the distribution conformed well to a power-law model .', 'The exponent m of the power-law fit obtained at each ambient temperature is shown in Fig 4 .', 'This plot confirms that m remains approximately constant over a wide range of lower ambient temperatures , and the steepness of the curve increases as the ambient temperature approaches 37°C .', 'The emergence of self-organised criticality in the thermodynamic huddling model reveals how the tension between cooperative and competitive interactions in animal groups may constrain evolutionary dynamics .', 'Next we ask whether the effects are a specific feature of the abstract thermodynamic description of huddling developed here , or whether self-organised criticality might also emerge in the evolution of a physically realisable model of self-organising huddling behaviours , i . e . , one that describes also the movements of the individuals .', 'The thermodynamic model of huddling was developed on the basis of an analogy between interacting rodents and the magnetic spins of an Ising model .', 'In an Ising model , spins are fixed in position on a lattice .', 'However , the characteristic thermodynamics of Ising models are preserved in what is known as a Vicsek model [25] .', 'A Vicsek model is a system of continually moving particles whose directions of travel are affected by the directions of proximal particles [25–27] .', 'The Vicsek model provides a theoretical bridge between the thermodynamics of stationary spins and of moving particles , and has successfully been applied to explain the emergent properties of self-organising flocking behaviours of animals interacting in large groups , where the trajectories of individuals ( e . g . , flocking birds or shoaling fish ) are influenced by the trajectories of nearby animals [28 , 29] ( see [30] for a review ) .', 'A Vicsek model is formulated as an agent-based simulation in which each individual particle moves according to', 'd x i d t = v 1 cos θ i ( t ) sin θ i ( t ) , ( 7 )', 'where x is the 2D position of the individual , v1 is its velocity , and θ ( t ) is its direction of travel at time t .', 'The change in the direction of travel is a function of its interactions with proximal particles .', 'A recent agent-based model of rodent thermoregulatory huddling describes the huddle as a self-organising system , generating a second-order phase transition in huddling and a peak in pup flow at the critical temperature [9] .', 'Interestingly , this model can be formulated as a Vicsek model , where dθ/dt is a function of the difference between the body temperature and the preferred temperature .', 'According to the model of [9] , rat pups are simulated as circles that move continually forwards in a two-dimensional arena with a circular boundary .', 'The orientations and body temperatures of the simulated pups are continually updated so that they move to minimise the discrepancy between the current body temperature and a preferred temperature , and when they make contact they exchange body heat .', 'According to the agent-based huddling model , pups turn using a strategy of ‘homeothermotaxis’ , defined as follows .', 'The temperature on the left and right of the body , TLi and TRi , are computed by averaging over many thermal sensors around its circumference , each registering either the ambient temperature or the body temperature of another pup where a contact occurs .', 'The resulting temperatures are mapped to two ‘sensor’ values that incorporate the difference between the preferred temperature and the body temperature , sLi = ( 1 + exp ( −σ ( TP − TBi ) TLi ) ) −1 and sRi = ( 1 + exp ( −σ ( TP − TBi ) TRi ) ) −1 , which are combined so as to turn the body in the direction that will reduce this difference ,', 'd θ i d t = arctan v 2 ( s L i - s R i ) / ( s L i + s R i ) , ( 8 )', 'where v2 sets the turning rate .', 'The body temperature of pup i changes according to', 'd T B i d t = G i - k 1 A i ( T A - T B i ) - k 2 ( 1 - A i ) ( T B i - T C i ) , ( 9 )', 'where TCi is the average temperature of the thermal sensors of pup i that are in contact with other pups ( who are thus reducing its exposed area ) .', 'The parameter k2 is another thermal conductance constant , which determines the rate at which heat is exchanged between pups that are in contact .', 'The three terms on the right of Eq 9 correspond to', 'i ) thermogenesis ,', 'ii ) heat exchange with the environment , and', 'iii ) heat exchange between contacting individuals , respectively .', 'It is important to note that when A = 1 , i . e . , when an ind
['Introduction', 'Models', 'Results', 'Discussion']
['A thermodynamic model of thermoregulatory huddling interactions between endotherms is developed .', 'The model is presented as a Monte Carlo algorithm in which animals are iteratively exchanged between groups , with a probability of exchanging groups defined in terms of the temperature of the environment and the body temperatures of the animals .', 'The temperature-dependent exchange of animals between groups is shown to reproduce a second-order critical phase transition , i . e . , a smooth switch to huddling when the environment gets colder , as measured in recent experiments .', 'A peak in the rate at which group sizes change , referred to as pup flow , is predicted at the critical temperature of the phase transition , consistent with a thermodynamic description of huddling , and with a description of the huddle as a self-organising system .', 'The model was subjected to a simple evolutionary procedure , by iteratively substituting the physiologies of individuals that fail to balance the costs of thermoregulation ( by huddling in groups ) with the costs of thermogenesis ( by contributing heat ) .', 'The resulting tension between cooperative and competitive interactions was found to generate a phenomenon called self-organised criticality , as evidenced by the emergence of avalanches in fitness that propagate across many generations .', 'The emergence of avalanches reveals how huddling can introduce correlations in fitness between individuals and thereby constrain evolutionary dynamics .', 'Finally , a full agent-based model of huddling interactions is also shown to generate criticality when subjected to the same evolutionary pressures .', 'The agent-based model is related to the Monte Carlo model in the way that a Vicsek model is related to an Ising model in statistical physics .', 'Huddling therefore presents an opportunity to use thermodynamic theory to study an emergent adaptive animal behaviour .', 'In more general terms , huddling is proposed as an ideal system for investigating the interaction between self-organisation and natural selection empirically .']
['Huddling is an adaptive behavior that emerges from simple interactions between animals .', 'Huddling is a particularly important self-organising system because the behavior that emerges at the level of the group directly impacts the fitness of the individual .', 'The huddle insulates the group , allowing pups to thermoregulate at a reduced metabolic cost , however a huddle can only self-organise if pups in turn contribute heat .', 'Contributing too much heat is costly but contributing too little compromises the ability of the huddle to self-organise .', 'To investigate how the resulting tension between co-operation and competition in the huddle might affect natural selection , litters of simulated rodents were subjected to a simple evolutionary process .', 'After interacting with its littermates , the individual that incurred the greatest metabolic cost for thermoregulation was iteratively replaced by another with random thermal properties .', 'Simulations resulted in the emergence of a phenomenon called self-organised criticality .', 'Criticality is a hallmark of complex systems , and is evidenced here by the emergence of a power-law distribution of thermal properties in the evolving composition of the group .', 'The model therefore reveals how complexity can emerge in a well-defined biological system ( thermoregulation ) , where experiments can be designed to investigate the interaction between self-organisation and natural selection .']
['organismal evolution', 'medicine and health sciences', 'vertebrates', 'animals', 'mammals', 'simulation and modeling', 'physiological processes', 'systems science', 'physiological parameters', 'mathematics', 'body temperature', 'thermodynamics', 'research and analysis methods', 'computer and information sciences', 'agent-based modeling', 'physics', 'thermogenesis', 'rodents', 'natural selection', 'physiology', 'evolutionary physiology', 'animal evolution', 'biology and life sciences', 'physical sciences', 'evolutionary biology', 'amniotes', 'evolutionary processes', 'organisms']