A mathematical approach to territorial pattern formation
AA mathematical approach to territorialpattern formation
Jonathan R. Potts ,a , Mark A. Lewis , ,b Abstract
Territorial behaviour is widespread in the animal kingdom, with creaturesseeking to gain parts of space for their exclusive use. It arises through acomplicated interplay of many different behavioural features. Extractingand quantifying the processes that give rise to territorial patterns requiresboth mathematical models of movement and interaction mechanisms, to-gether with statistical techniques for rigorously extracting parameters fromdata. Here, we give a brisk, pedagogical overview of the techniques so fardeveloped to tackle the problem of territory formation. We give some ex-amples of what has already been achieved using these techniques, togetherwith pointers as to where we believe the future lies in this area of study.This progress is a single example of a major aim for 21st century science: toconstruct quantitatively predictive theory for ecological systems.
Introduction.
The natural world is full of complex systems, where constituent parts in-teract to cause patterns that can be very rich in diversity and unexpectedin form. These range from the detailed termite hills structures that emergefrom the collective actions of individually very simple animals, to oscillatoryand chaotic patterns in predator-prey systems; from spatially heterogeneousterritorial segregations to climactic effects of changing ecosystems [1, 2, 3, 4].To understand how one, more ‘macroscopic’, level of description emergesfrom finer-grained, ‘microscopic’ processes presents a formidable challenge1 a r X i v : . [ q - b i o . P E ] J a n hat can only be tackled with sophisticated mathematical and computationaltools, many of which may need to be created for each new problem.Statistical physics has seen remarkable success at describing emergentphenomena from underlying mechanisms. Properties that were originallyonly phenomenologically understood, such as the relationships between heat,pressure and volume of a gas, have been accurately and analytically derivedfrom the movements of tiny, jiggling particles [5]. During the 20th century,these ideas have been extended into many areas of physics, including optics,fluid dynamics and soft matter studies, to name but a few [6]. However, inthe field of physics, the constituent agents are relatively simple particles ofinorganic matter.By contrast, the organisms constituting an ecosystem are living plantsand animals, with evolutionarily driven goals and complex behavioral traits.Despite this complexity, the last few decades have seen many scientists andmathematicians embarking on a journey to develop an analogue of statisticalmechanics for ecosystems [7]. This has been spurred on by an increasingawareness of the need to develop a predictive ecology, so that we can accu-rately foretell the effects of anthropogenic changes on ecosystems [8].Here, we review a small part of that journey, the quest to understand howanimal populations self-organize into territorial structures from the move-ments and interactions of individual animals. Along the way, we will de-scribe a number of mathematical and statistical techniques that have beenused to help tackle this problem. We follow the philosophy that biologi-cal questions should drive the decisions to use one particular mathematicaltechnique over another, rather than starting with a field of mathematics andtrying to see what it can add to biological understanding. Consequently, ourreview is broad rather than deep, intended to entice the reader into readingmore about the techniques covered than explain them exhaustively. We hopethat this article will help readers quickly understand the problem of territoryformation and introduce them to tools that are helpful in solving problemsregarding animal movement, interactions and space use. Ecological issues.
As well as theoretical curiosity, there are various ecological considerationsthat make it important to build a mathematical theory of territory formation.In conservation biology, understanding territory size is vital for designing2ature reserves to fit a given population [9]. In epidemiology, if a diseaseis spreading through a population of territorial animals then it is crucial tounderstand how the movements and contact rates of the animals relate to thesize and shape of the territories [10]. Territorial interactions can also causespatial structures to arise that can affect predator-prey dynamics [11].Traditionally, territorial structures have been understood by statisticalanalysis of positional data. This can often be as simple as drawing theminimum convex polygon around a set of points [12] or assuming that theanimal’s territory is roughly given by the mean of narrow Gaussian distribu-tions around each observed location [13]. Recently, more involved techniquesthat take into account the animal’s probable movement between successivelocations have been proposed as a more accurate way of determining spatialpatterns [14]. It makes sense, therefore, to build on these ideas by also in-cluding the interactions between animals into our understanding of territorialformation. The techniques described in this review explain the mathematicaldetails behind this theory. Information about the practical lessons providedby such theory is given in our more biologically-oriented companion review[15].
Building a model from the ground up: the ran-dom walk approach.
Though animal movement decisions are complex and multi-faceted, we be-lieve the best approach to model building is to start with as simple a modelas possible, then build up the complexity one facet at a time, rigorously test-ing whether each additional term significantly improves how the model fitsthe available data. This requires a certain amount of imagination, as a verysimple model is unlikely to model well a real animal in an actual environ-ment. So we start by asking ourselves: what would an animal do if it wereplaced alone in a barren, featureless landscape?We might imagine that it moves for a certain time in one direction, toexplore. Then turns at random and moves in another direction for a shortwhile, and so on. These so-called random walk patterns have been successfullyused as the basis for animal movement models for some time [16, 17]. So westart with this idea as the basis for our animal movement models.Mathematically, a random walk can be described by the probability den-3ity p τ ( x | y ) of moving to position x at time τ in the future, given thatthe animal is currently at position y , which can be in one-, two- or three-dimensional space. Animals may move a variety of different distances in thistime period, but are more likely to move a short distance, and extremelyunlikely to move a large distance. Therefore a suitable distribution of theso-called step lengths , i.e. lengths of straight-line movement between randomturns, might be the exponential distribution: p τ ( x | y ) ∝ exp( − δ | x − y | ) , (1)where δ is a free parameter and the constant of proportionality is calculatedby ensuring that the integral of p τ ( x | y ) with respect to x across the domainof study is 1. This is often called the step length distribution of an ani-mal. Placing such steps together in succession gives a hypothesised possiblemovement path, as in figure 1.Our ultimate goal is to move from this ‘microscopic’ description of animalmovement, to a description of the expected space use of the animal, in anon-speculative, mathematical fashion. While this is difficult for complexmovement models, the random walk equation (1) is simple enough that wecan derive an analytic formula for the space use patterns. We show this herein the one-dimensional (1D) case.Let u ( x, t ) be the probability that the animal is in position x at time t .(We no longer use bold font since the positions are 1D.) Then we can writedown the so-called master equation for this system as u ( x, t + τ ) = (cid:90) ∞−∞ p τ ( x | y ) u ( y, t )d y. (2)Given some initial distribution u ( x,
0) = u ( x ), this can be solved numericallyover time. Moreover, by taking the limit as τ →
0, it is possible to derive adiffusion equation, which is a type of partial differential equation (PDE).To do this, we first define z = y − x to give the following master equation u ( x, t + τ ) = δ (cid:90) ∞−∞ exp( − δ | z | ) u ( x + z, t )d z. (3)A Taylor expansion of u ( x + z, t ) gives u ( x, t + τ ) = δ (cid:90) ∞−∞ exp( − δ | z | ) (cid:20) u ( x, t ) + z ∂u∂z ( x, t )+ z ∂ u∂z ( x, t ) + O ( z ) (cid:21) d z. (4)4earranging this, and calculating the integrals of z exp( − δ | z | ) and z exp( − δ | z | )gives u ( x, t + τ ) − u ( x, t ) = 12 δ ∂ u∂x ( x, t ) + δ (cid:90) ∞−∞ exp( − δ | z | ) O ( z )d z. (5)Now we need to take the limit as τ →
0. When doing this, we need to noticethat δ will not remain constant, because the animal’s step length distributionwill narrow as the time intervals become smaller.The only way to obtain a sensible limit for equation (5), i.e. where ∂u/∂t is not zero or infinity, is to insist that τ goes to 0 in such a way that D =1 / δ τ is kept constant. Then u ( x, t + τ ) − u ( x, t ) τ = D ∂ u∂x ( x, t ) + O ( √ τ ) , (6)where the O ( √ τ ) arises from the fact that the n -th order moment of theexponential distribution is propotional to δ − n , which scales as τ − n/ in thelimit. Taking the limit as τ → ∂u∂t = D ∂ u∂x ( x, t ) , (7)which is the classical diffusion equation.Since the animal starts in a given position x , the initial condition isthe Dirac delta function u ( x ) = δ ( x ). The exact solution of equation (7),with this initial condition, is just a normal distribution with variance thatincreases linearly in time u ( x, t ) = exp[ − ( x − x ) / Dt ]2 √ πDt . (8)This result, in fact, generalises to higher dimensions. In the case of animalterritoriality, we are interested in the 2D result u ( x , t ) = exp[ − ( x − x ) / Dt ]4 πDt , (9)where the power of two denotes the scalar product of a vector with itself.This gives our first example of the sort of space use patterns that can beshown to arise from descriptions of animal movement. If the animal moves in5 random fashion, its space use distribution at any point in time is describedby equation (9). Of course, animal movement is typically far from random.We explain how to add realism in the next section.When estimating space use patterns in reality, field biologists have tomeasure positions over a period of time. As such, they cannot derive theprobability distribution at any particular point in time, but instead con-struct a utilisation distribution, sometimes called a home range [18], over agiven time window such as a day, month or season. To compare this simplerandom walk model with the positional data, it is necessary to make use ofthe utilisation distribution derived from equation (9) over the time T duringwhich the data were collected (figure 1) (cid:82) T u ( x , t )d t (cid:82) Ω (cid:104)(cid:82) T u ( x , t )d t (cid:105) d x . (10) Getting the model right: statistical techniques.
Real animals, of course, do not exist alone in featureless environments. Thenatural drive to survive and reproduce means that interactions with the envi-ronment and other animals greatly affect movement decisions. For example,the needs to gather food, avoid predation, maintain territories, or find matesmay all be contributing factors in an animal’s movement choices [19, 20, 21].The challenge is both to disentangle which of these factors make signifi-cant contributions to movement, and to quantify their effects. By doing this,we can construct realistic models of animal movement, from which accuratepredictions of spatial patterns should emerge.There are two main approaches to constructing such models in a rigor-ous, data-driven fashion. One is the hypothesis testing approach, wherebyparameters are added to the model one at a time. Each time a parameter isadded, we test whether it significantly improves the model. If so, we keep it;if not, it is discarded. The second is the model selection approach, whereby anumber of plausible models are constructed and fitted to the data. We thenuse the best one to construct our model of space use.In the context of animal movement, the so-called likelihood ratio test provides a useful means for hypothesis testing. Suppose we have a simplemodel of animal movement, for example the random walk model of equation61). Denote our data on the positions of the animal at times 0 , τ, τ, . . . , N τ by x , x , . . . , x N .Since we have spent many weeks observing the animals, or have spentmany hours listening carefully to someone who has, we have sufficient intu-ition to construct hypotheses about the animals’ movements. For example,we might hypothesise that they have a tendency to move towards areas ofhigh resource biomass. If b ( x ) is the resource biomass at position x then wecould construct the following model to take this into account p τ ( x | y , α ) ∝ exp[ − δ | x − y | + αb ( x )] , (11)which is the probability of moving from y to x in a time-interval τ , analogousto equation (1). This model assumes implicitly that the distribution of steplengths, disregarding the effect of resources, is exponential. It also assumesthat the effect of resources on movement is proportional to the exponential ofthe biomass. Neither of these are necessarily true for a given data set, so it isimportant in practice to try several functional forms for equation (1) and usea model selection method (e.g. the Akaike Information Criterion detailedin the next section) to find the best one. We are just using the model inequation (1) for ease of explanation and mathematical exposition.The null hypothesis H is that α = 0, hypothesising that the best modelis equation (1), whereas the alternative hypothesis H is that α (cid:54) = 0. Testingthis requires constructing the so-called likelihood function , which is the prob-ability of the data given the model and parameters. If we assume that themovement steps are independent from one another (which may not always betrue, but is a surpisingly fruitful assumption [10, 19, 20, 21]), the likelihoodtakes the following form L ( x , . . . , x N | α ) = N − (cid:89) n =0 p τ ( x n +1 | x n , α ) . (12)The method of maximum likelihood chooses the parameters so as to make thelikelihood as large as possible. Let α max be the value of α that maximises ex-pression (12). Then the likelihood ratio test tells us that 2 log[ L ( x , . . . , x N | α max )] − L ( x , . . . , x N | χ -squared distributed with 1 degreeof freedom [22].We can therefore use the χ -squared test to test whether there is sufficientevidence to reject H . If we reject H then we can consider p τ ( x | y , α max )7s an improved model of animal movement, compared to the simple randomwalk model of equation (1). We may therefore use it to compute predictedspace use patterns, via constructing the master equation as per equation(2), and if possible taking a PDE limit (see [23, 24] for an example). Thepredictions can then be compared with those of the diffusion equation (9) tosee if they are more accurate. Proceeding in this way, we can add parametersone at a time to improve the fit of our model to the data until we are satisfiedwith the space use predictions (figure 2).As an alternative to the reductionist approach of hypothesis testing, wemay wish to use the approach of multiple working hypotheses. Here we wouldformulate an array of plausible models, each defined by a different group ofnonzero parameters, and see which of these models is best supported by thedata. This can be more computationally intensive, but it takes into accountthe idea that a blend of different factors may affect movement, and that theeffects might only be observed when all the covariates are included at thesame time.As a simple example, suppose that both resource biomass and the distance | x − x c | from a predator’s home range centre, x c , are hypothesised to influencemovement decisions. Then we can construct four different movement models p τ ( x | y , α ) ∝ exp[ − δ | x − y | ] ,p τ ( x | y , α ) ∝ exp[ − δ | x − y | + αb ( x )] ,p τ ( x | y , α ) ∝ exp[ − δ | x − y | + β | x − x c | ] ,p τ ( x | y , α ) ∝ exp[ − δ | x − y | + αb ( x ) + β | x − x c | ] . (13)The Akaike information criterion (AIC) gives a technique for choosing be-tween these models. The AIC of a model is 2 k − L max ) where L max isthe maximum of the likelihood function and k the number of model param-eters. Intuitively, it measures the relative closeness of models to the data,with some penalisation for models with a larger number of parameters, al-though the exact form of the AIC expression can be derived more deeplyusing information theory [22]. The various models can be used to build amechanistic model of animal movement, by constructing the master equationas in equation (2). The model with the lowest AIC is likely to describe thespace use patterns most accurately.Once we have our best model, though, it is important to ask how muchbetter it is than the rest. In other words, what is the chance that we arewrong and one of the other models is in fact the best? The theory of AIC8ives a nice answer for this. Let AIC min denote the AIC of the best modeland AIC i that of model i . Then each model i is exp[(AIC min − AIC i ) /
2] timesas likely as the model with the minimum AIC to be the ‘best’ model, insofaras it minimises information loss (see [22]).We have given something of a whistle-stop tour of model selection andhypothesis testing, emphasising the relations to building animal movementmodels. There are a number of textbooks that give detailed descriptions ofAIC, likelihood and related topics, e.g. [22], to which we refer the interestedreader for more information. For an example of model selection and hypoth-esis testing in the context of movement models, see [25]. Sometimes authorsuse AICc, Bayesian Informaion Criteriion (BIC) or other related techniques,which have various pros and cons, which are discussed in detail elsewhere(e.g. [22]).
Adding territorial interactions.
Now we have the tools to build up a model of individual animal movement,we come to the main aspect of this paper: accounting for territorial inter-actions. There are two approaches to this in the modelling literature. Thefirst, typified in the book by [26], is to derive space use patterns from aplausible model of interaction mechanisms, then fit these patterns to loca-tion data. One can then test various candidate models against the data, forexample using the AIC techniques described in the previous section, to inferinformation about the drivers of territorial structures.The second is to continue along a similar path as in the previous sec-tion, fitting the individual movement and interaction model to data, thensee whether the spatial patterns that emerge are similar to the territories de-scribed by animal locations. This more conservative approach is newer andless well-developed, but as such raises a number of interesting challenges forfuture developers of territorial models.Perhaps the first use of partial differential equations to capture territoriesemerging from animal movements and interactions was that of [11], in thecontext of modelling wolf pack territoriality. The simplest model in thatpaper was of two packs with densities U ( x, t ) and V ( x, t ) and can be derivedon a 1D lattice with zero-flux boundary conditions [28, section 3.2]. It positedthat as wolves move, they deposit scent which decays at a rate µ . The densityof scent for packs U and V at position x and time t are denoted P ( x, t ) and9 ( x, t ) respectively. The rate of scent deposition of pack U is l + νQ ( x, t ),and is l + νP ( x, t ) for pack V , modelling the fact that there is a baseline rateof scent deposition, but that it also increases in the presence of the otherpack’s scent.The wolves have a constant speed but switch direction at a rate dependenton the presence of conspecific scent. We assume that the den site of pack U is at the left-hand boundary of the lattice, while pack V has den site to theright. Therefore the probability of a member of pack U switching from rightto left (resp. left to right) is λ + σQ ( x, t ) (resp. λ − σQ ( x, t )), whereas theprobability of a member of pack V switching from left to right (resp. rightto left) is λ + σP ( x, t ) (resp. λ − σP ( x, t )). Letting the positions of left-and right-moving members of pack U be denoted by U − ( x, t ) and U + ( x, t )respectively, and similarly for V , we arrive at the following master equationin discrete space and time [28, section 3.2] U − ( x, t + τ ) =[1 − τ { λ − σQ ( x, t ) } ] U − ( x + a, t )+ τ [ λ + σQ ( x, t )] U + ( x − a, t ) ,U + ( x, t + τ ) =[1 − τ { λ + σQ ( x, t ) } ] U + ( x − a, t )+ τ [ λ − σQ ( x, t )] U − ( x + a, t ) ,V − ( x, t + τ ) =[1 − τ { λ − σP ( x, t ) } ] V − ( x + a, t )+ τ [ λ + σP ( x, t )] V + ( x − a, t ) ,V + ( x, t + τ ) =[1 − τ { λ + σP ( x, t ) } ] V + ( x − a, t )+ τ [ λ − σP ( x, t )] V − ( x + a, t ) ,P ( x, t + τ ) =(1 − µτ ) P ( x, t ) + U ( x, t )[ l + νQ ( x, t )] τ,Q ( x, t + τ ) =(1 − µτ ) Q ( x, t ) + V ( x, t )[ l + νP ( x, t )] τ, (14)where a is the lattice spacing and τ the time it takes to move distance a .To derive partial differential equations from this stochastic model, a meanfield approximation is needed that assumes the distributions of scent marksand individuals are uncorrelated. This could be unreasonable on short timescales, but is reasonable on the longer times scales that are relevant to theformation of territorial patterns since the movement of the individuals israpid compared to the change in scent mark density. Again, we need to takea delicate limiting process. Based on our experience with the earlier diffusionlimit taken on equation (5), and observing that the space step a takes theplace of the mean space step δ − in equation (3), we would expect that thespace and time steps, a and τ would approach zero with a scaling so that10 /τ is constant. However, we now have additional parameters, λ and σ .How should they scale? A natural choice is to have λ increase so that λτ approaches a constant as τ approaches zero, and to have σ increase so that σa approaches a constant as a approaches zero. That way, the switchingand scent mark bias terms are incorporated into model as significant factorsduring the limiting process. Denoting by lower case letters the densities thatcorrespond to upper case letters for probabilities in equation (14), we arriveat the following PDEs describing the emergent space use patterns (see [28,section 3.2] for a derivation) ∂u ( x, t ) ∂t − c ∂∂x [ q ( x, t ) u ( x, t )] = d ∂ u ( x, t ) ∂x ,∂v ( x, t ) ∂t + c ∂∂x [ p ( x, t ) v ( x, t )] = d ∂ v ( x, t ) ∂x ,∂p ( x, t ) ∂t =[ l + νq ( x, t )] u ( x, t ) − µp ( x, t ) ,∂q ( x, t ) ∂t =[ l + νp ( x, t )] v ( x, t ) − µq ( x, t ) , (15)where ( σa ) / ( τ λ ) → c and ( a ) / (2 τ λ ) → d as a and τ tend to zero.It is interesting to consider other possible limiting processes. For example,we could argue, quite reasonably, that the speed of movement a/τ shouldremain constant during the limiting process rather than becoming arbitrarilylarge. This limit is mathematically possible, and leads to related hyperbolicmodels for animal movement (see [37] for the general theory). However, asargued in [28], the hyperbolic and parabolic limiting equations have similarbehaviour when evaluating territorial pattern formation over long time scales.Thus we have a system of PDEs describing territorial patterns that isrigorously derived from the underlying movement and interaction processes.These equations generate steady-state solutions that exhibit spatial patternsthat correspond qualitatively to found in territories [27] (figure 3). Herethere is spatial segregation between the the two packs, u and v , and thescent marks, p and q are highest along the boundary between the two packs.To understand this qualitatively we see that segregation arises from advectionterms in equation (15) that drive individuals back towards their den site whenthe encounter foreign scent marks, and heightened scent mark densities atboundaries arise from a positive feedback loop where scent marking fromone pack gives rise to heightened scent marking rates from the other pack.Indeed, it actually possible to choose a feedback loop that is so strong that11cent marks exhibit mathematical “blow up” at territorial boundaries [27].This is intriguing, although biologically nonsensical, if only due to the finitebladder capacity of animals involved.Although interesting to analyse, these equations are only described in 1Dand contain no behavioural features other that scent marking and conspecificavoidance. To add further realism, it is necessary first to extend the resultsinto 2D, then add further plausible drivers of spatial patterns to the PDE,to create a suite of possible models that describe predicted territorial dis-tributions. These distributions can be fitted to data on animal locations todeduce which model is the best at explaining the complex patterns observedin nature.We demonstrate this with an example from [3], where the authors usea mechanistic model of territory formation to determine the movement ten-dencies that underlie coyote territories. They start with a 2D version ofthe model in equations (15) that applies to n packs with position densities u ( x , t ) , . . . , u n ( x , t ) and scent densities p ( x , t ) , . . . , p n ( x , t ) ∂u i ( x , t ) ∂t = d ∇ u i ( x , t ) − c ∇ · (cid:34) x i u i ( x , t ) (cid:88) j (cid:54) = i q j ( x , t ) (cid:35) ,∂p i ( x , t ) ∂t = (cid:34) l + (cid:88) j (cid:54) = i νp j ( x , t ) (cid:35) u j ( x , t ) − µp i ( x , t ) , (16)where x i is the unit vector in the direction from x to the den site for pack i . This equation can be derived from a biased random walk process withscent-marking in two spatial dimensions (see [26] for details). By non-dimensionalising appropriately [26] and assuming that the system is con-tained within a finite domain with zero flux boundary conditions, the systemsteady state is given by0 = ∇ u i ( x ) − β ∇ · (cid:34) x i u i ( x ) (cid:88) j (cid:54) = i q j ( x ) (cid:35) , (cid:34) (cid:88) j (cid:54) = i mp j ( x ) (cid:35) u j ( x , t ) − p i ( x ) . (17)To this system of ordinary differential equations (ODEs), the authors addtwo different terms. The first corresponds to a tendency for coyotes to move12way from steep terrain, where it’s difficult for them to roam, giving thefollowing model0 = ∇ u i ( x ) − β ∇ · (cid:34) x i u i ( x ) (cid:88) j (cid:54) = i q j ( x ) (cid:35) − ∇ [ α z u i ( x ) ∇ z ( x )] , (18)where z ( x ) is the elevation of the landscape and α z the strength of the ten-dency to move away from high ground. The second models the tendency tomove towards areas with higher prey availability, giving the following model0 = ∇ · [e − α r h ( x ) ∇ u i ( x )] − β ∇ · (cid:34) e − α r h ( x ) x i u i ( x ) (cid:88) j (cid:54) = i q j ( x ) (cid:35) −∇ · [e − α r h ( x ) u i ( x ) ∇ h ( x )] , (19)where h ( x ) is the amount of prey available and α r a measure of the strengthof the tendency to move towards areas of higher prey availability. The threemodels in equations (17), (18) and (19) are then fitted to data on coyotelocations to determine which gave the best fit and therefore which gives themost likely explanation as to the causes of territorial patterns.The procedure for fitting to data uses an AIC test, by taking locationssufficiently far apart (in both space and time) so that they can be consideredindependent random variables drawn from the steady state distribution. Foreach pack i , let x , . . . , x N i be the positions of these independent relocations.Then the likelihood of the data given the model is L = n (cid:89) i =1 N i (cid:89) k i =0 u i ( x k i ) , (20)where u i is the solution of one of the three differential equations (17), (18)or (19). Using the AIC model selection techniques outlined in the previousselection, equation (19) turns out to fit the data significantly better thanthe other two models, suggesting that movements towards available prey,together with conspecific avoidance (CA), explain the territorial patternsbetter than either avoidance of steep terrain plus CA or just CA on its own[3] (figure 4). Notice that AIC penalises the number of model parameters.Therefore it is important that we use the non-dimensional forms of the mod-els, which have the minimum number of parameters needed to specify themodel. 13hile such reaction-diffusion approaches have been successful in bothdescribing territorial patterns and inferring behavioural features, the abovemethods do not give sufficient evidence to conclude that the best fit param-eters values of β, m, α z accurately reflect the underlying mechanisms. First,reaction diffusion approximations can sometimes fail to describe the patternsthat arise from underlying individual-level descriptions, especially when in-teractions are rare, as is the case with territorial behaviour. Second, thoughthe models are built from one level of description, the movement and inter-action mechanisms, they are fitted to data on another level, the space usepatterns. From a logical perspective, it is not necessarily true that a goodfit to space use implies an accurate description of the underlying movementand interaction mechanisms.The first issue was addressed recently by building a model of territorialityby directly simulating individual-level movement and interaction processes[29]. Territorial patterns emerged that well-fitted long-term data on foxmovements and could be used to infer information about the longevity ofscent cues that accurately replicated field observations [10] (figure 5). Onestriking difference between this approach and that of [26] is that no anchoringden site is necessary to see territorial patterns emerge. This causes theterritories to continually move, never settling to a non-trivial steady state,which means the approach of analysing steady-state ODEs as in equations(17), (18) and (19) is no longer usable.Instead, the authors developed semi-analytic techniques for describing theanimal’s movements based on trends observed in the movement of simulatedterritories [30, 31, 32]. They noticed that the territories move in a subdiffu-sive fashion, as predicted by the theory of exclusion processes, and that thegeneralised diffusion constant of the territory decays exponentially with thedimensionless product DρT , where D is the intrinsic diffusion constant of theanimal, ρ is the population density and T is the scent-mark longevity. Thisenabled them to build an analytic model of animal movement within terri-tory borders (figure 6) which could be fitted to movement data. However, asyet this technique is unable to infer the territorial interaction processes with-out using simulation analysis, though see [33] for some first steps towardsrectifying this.The second issue can be addressed by extending the program of buildingmovement models described in the previous section ‘Getting the model right’to include territorial, as well as environmental, interactions. Rather thanconstructing one function for all animals, as in equations (13), this approach14equires constructing different functions for different animals or packs, thencoupling them together via the territorial interactions (figure 2). The genericform of such a model is p t,τi ( x | y ) ∝ φ i ( x | y ) W i ( x , y , E ) C i ( x , y , P ti ) , (21)where φ i ( x | y ) denotes the intrinsic movement of a single animal i on its own,e.g. equation (1), W i ( x , y , E ) represents interactions with the environment E , as in equations (13), and C i ( x , y , P ti ) is a coupling term representing theinteractions between the various animals, such as territorial avoidance. Theterm P ti contains the information about the population required to describethese interactions [34].By using methods identical to those in the previous section, one cantest candidate models of territorial interaction processes and parameterisean individual-level movement and interaction model. In the same way aswe moved from equation (1) to (2), we can use equation (21) to constructa master equation to derive predicted spatial patterns. This can either besolved numerically, or a continuous-time PDE limit may be found, whichmight give some analytic insight.Before we draw general conclusions, it is interesting to note that modelsof the sort we describe here also have been applied to human populations.In ground-breaking work, Andrea Bertozzi and colleagues have started tounderstand the mathematics of crime [35]. Their approach to mapping LosAngeles gang territories was to fit a modified version of equation terrain-taxismodel 18, that includes structures impeding gang movement, such as freewaysand rivers, instead of terrain elevation. This was fitted to an extensive LosAngeles database on gang reports, and the analysis provided new insightsregarding gang interactions [36]. Conclusion and future directions.
The focus of this paper has been to demonstrate how to derive movement andinteraction mechanisms from animal location data, and use these to constructmodels of territorial patterns. We have given a brief, pedagogical overview ofthe various techniques used so far to attack this problem, which we hope willleave the reader in a position to begin using them, together with informationabout what to read to obtain a deeper and more thorough understanding.15hile most current approaches build models of territory formation fromplausible movement and interaction mechanisms, then validate the modelby fitting it to data, the recent attempts to derive spatial patterns fromready-parametrised movement-and-interaction models give a more conserva-tive approach, which is likely to be more accurate at uncovering the actualmechanisms used by animals. This is vitally important in predicting theeffect of future environmental changes on animal populations in a quantita-tively as well as qualitatively accurate way.Currently this approach is in its infancy. The challenge for the future is tobuild mathematical theory that details the best ways to use equations suchas (21) to derive spatial patterns. The simplest way is numerical derivation.However, it is more mathematically pleasing, and may save computationaleffort, to derive a theory of PDEs for the so-called coupled step selectionfunctions of equation (21) [34].There are various approaches to deriving PDEs from individual-level de-scriptions, reviewed in [37] in a biological context. Different limits of themaster equation may uncover different biological aspects of the patterns. Itwould be an important future advancement to see which limits give rise toaccurate territorial structures.Another approach, used more in the physics literature, are van Kampenapproximations of Markovian processes. Recent work by Alan McKane andothers has shown that, when biological models exhibit behaviours quite dif-ferent from those of mean field models, van Kampen approximations oftendo a better job [38, 39, 40]. Since these approximations result in the meanfield description in certain limits, they can often be used to tease out thereasons why and how mean field approaches may fail.It is natural to ask whether territorial animals might try to modify theirbehaviour so as to gain an advantage over their neighbours. Here neighbour-ing packs could effectively play a spatial game where each tries to maximizeits fitness via increased resource consumption arising from territorial expan-sion while, at the same time, attempting to minimize losses incurred throughterritorial altercations. To play the game, the packs should thus be able tomodulate their spatial movement behaviours, described in the partial dif-ferential equation models, in pursuit of enhanced fitness. Preliminary workapplying the theory of differential games to one-dimensional territorial pat-tern formation has shown how certain movement behaviours are stable froman evolutionary perspective while others are not [41]. One fascinating aspectof this analysis has been it’s ability to explain the spontaneous emergence of16 uffer zones , where neither pack is found, between wolf territories as the out-come of an evolutionarily stable strategy. Such buffer zones have observed innature, and have been studied in detail for wolf populations in northeasternMinnesota [42].In conclusion, while we have gone a significant way to understanding themathematics behind territory formation, much work needs to be done. Inthis era of rapid ecological change, predictive ecology is becoming an increas-ingly important subject. With ever-changing ecosystems, understanding themechanisms behind observed spatial patterns is vital for such predictions tobe possible. We are on the first steps of a journey towards making ecologyquantitatively predictive. But it is one that cannot be tackled by a smallnumber of scientists. We hope that this paper has helped you understandthis area and its importance, and perhaps encouraged you to join us in thisendeavour.
Acknowledgment.
This study was partly funded by NSERC Discovery and Acceleration grants(MAL, JRP). MAL also gratefully acknowledges a Canada Research Chairand a Killam Research Fellowship. We are grateful to members of the LewisResearch Group for helpful discussions.
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Expected spatial patterns from a random walk.
Three pathsof 100-step random walks with exponential step-length distribution are shown, asdescribed by equation (1) with δ = 1. Each path has a different shape, cross,dot or triangle, at the start/end of its steps. The contours denote the utilisationdistribution described by mathematical analysis of the random walk, from equation(10). This displays a very simple example of the predicted spatial patterns thatcan arise by mathematical analysis of an animal movement model. Where next?
This example demonstrates the probability of ananimal’s next move, dependent on two factors: (a) collective interactionsvia the strength of conspecific territorial marks and (b) environmental in-teractions via resource quality. Both aspects can be modelled separately ortogether, then the models can be tested against the data using hypothesistesting or model prediction to find out if either significantly affect movementprocesses. The strength of territory marks in this example does not changein the Y -direction, so that animal 1 has territory on the left and animal 2on the right. The probability of animal 1’s (resp. animal 2’s) next positionafter some time interval τ , given that it’s current position is in the middle ofthe landscape (black dot), is shown in panel (c) (resp, panel d). As each ani-mals moves, it marks the terrain causing the territorial profile to change overtime, which in turn influences the other animal’s movements. This feedbackmechanism can cause territorial confinement to emerge (reproduced from[34]). 22 istance (x) W o l f P a ck / S c en t D en s i t y F un c t i on s pack 1 (u)pack 2 (v)Scent 1 (p)Scent 2 (q)den 1 den 2 Figure 3:
One-dimensional model results
Sample solutions for the one-dimensional partial differential equation model equation (15). Note the seg-regation of u and v and the bowl-shaped scent densities for p and q (basedon [27]). 23igure 4: Mechanistic models capturing coyote territorial patterns.
Thetop panel shows the best fit of equation (18) to data on coyotes in Lamar Valley,Yellowstone National Park and the bottom panel shows the same for equation (19).Contour lines show the space use distributions of the best-fit model, whereas dotsgive relocation fixes for coyotes. Different colours represent positions of differentpacks. The coordinates are measure in UTM. Reproduced with permission from[3].
Output from an individual-level model of territory formation.
Contours show the utilisation distribution of various animal positions from anindividual based model of territory formation with periodic boundary conditions.Each colour denotes a different animal’s territory (based on [29]).
Schematic of an analytic model of animal movement withina dynamic territory.