Space use by foragers consuming renewable resources
Guillermo Abramson, Marcelo N Kuperman, Juan M Morales, Joel C Miller
aa r X i v : . [ n li n . AO ] J a n EPJ manuscript No. (will be inserted by the editor)
Space use by foragers consuming renewable resources
Guillermo Abramson , Marcelo N Kuperman , Juan M Morales , and Joel C Miller Centro At´omico Bariloche, CONICET and Instituto Balseiro, S. C. de Bariloche, Argentina Laboratorio Ecotono, INIBIOMA, CONICET and Universidad Nacional del Comahue, S. C. de Bariloche, Argentina Department of Biology, Pennsylvania State University, Pennsylvania, USAReceived: date / Revised version: date
Abstract.
We study a simple model of a forager as a walk that modifies a relaxing substrate. Within itsimplicity, this provides an insight on a number of relevant and non-intuitive facts. Even without memoryof the good places to feed and no explicit cost of moving, we observe the emergence of a finite home range.We characterize the walks and the use of resources in several statistical ways, involving the behavior of theaverage used fraction of the system, the length of the cycles followed by the walkers, and the frequencyof visits to plants. Preliminary results on population effects are explored by means of a system of twonon directly interacting animals. Properties of the overlap of home ranges show the existence of a set ofparameters that provides the best utilization of the shared resource.
PACS.
Animals usually exhibit complex patterns of movementwhich arise arise from the interaction between the indi-vidual and the environment [1]. The motivations for move-ment depend on the animal’s internal state (satiation, re-serves, etc.), on the interactions with members of theirown or other species and on previous experiences [2,3].Importantly, the way animals move affects how individ-uals redistribute themselves over space and thus has thepotential to affect many ecological processes [4,3].A broad group of animals move around in order tocollect food from patches of renewable resources such asfruits, nectar, pollen, seeds, etc. For these animals we ex-pect that their movement trajectories will depend stronglyon the spatial arrangement of such patches [5,6]. Oftenthese animals play an important ecological role as part ofmutualistic interactions, as they pollinate or disperse theseeds of the plants they visit. For seed dispersal in par-ticular, empirical and theoretical studies show that thespatial distribution of plants contributes to the seed de-position patterns through its effect on animal movement[7,8,9,10,11,12,13,14,15]. Understanding the emergenceof space use of animals foraging for renewable resources,besides being an interesting theoretical topic, can allow usto build better studies of animal-plant interactions. a e-mail: [email protected] b e-mail: [email protected] c e-mail: [email protected] d e-mail: [email protected] Previous studies considering systems of animals forag-ing on renewable resources have focused on finding optimalsearch strategies under different assumptions of animalperception and memory [16,17,18]. It is clear that someanimals are capable of finding profitable routes without alot of computational power [19]. Also, much discussion hasbeen devoted to animals’ search paths and whether L´evywalks or flights are predominant in nature [20,21,22,23,24].However, less attention has been paid to the emer-gence of space use patterns as the result of an interactionbetween the behavioral rules used by an organism andthe spatial structure of the environment. In this work weuse a simple model of foraging animals while traversinga territory populated by their source of sustenance. It iseffectively a walker on a dissordered substrate, which isan interesting mathematical problem even in simple for-mulations, whose statistical properties has only recentlybeen subject of investigation [25,26,27]. Our main goalis to understand the emergent properties of simple move-ment rules of animals foraging for patchy and renewableresources.
We consider a finite spatial domain, within which thereare N patches of vegetation, that the animal can visit toget food. These patches, which we may refer to informally Guillermo Abramson et al.: Space use by foragers consuming renewable resources as “plants” below, do not overlap and each one is endowedwith a load of fruit f i ( t ), which is the only resource forthe animals to consume in the system. Initially the plantshave f i (0) = k i ∈ (0 , – The space is the unit square, and the plants are set atuniformly distributed points on the square. – The space is of undefined size, on which a first plantis set at random. The rest of the plants are set se-quentially, at a distance from the previous one whichis randomly drawn from a lognormal distribution (anduniformly distributed in azimuth).The lognormal distribution produces hierarchically clus-tered substrates, compared to the Poissonian distance be-tween plants resulting from the uniform one. For the sakeof brevity we will show mainly the results corresponding touniform substrates, with a few mentions to the differencesarising in the lognormal ones.The only dynamics of the vegetation substrate is acontinuous replenishment of the fruitload of each plant,according to an exponential growth that saturates to theinitial value k i . This simple relaxation dynamics can rep-resent a ripening process, for example, in such a way thatthe fruit load available to the animals is only the ripe fruit.Since we are, at the present stage, only interested in theshort time scale of the dynamics, we consider no season-ality nor density dependence in this dynamics. If the timeof the most recent visit of an animal to plant i is t v , thefruit at any time t > t v (before a new visit) grows as: f i ( t ) = k i + ( f i ( t v ) − k i ) e − ( t − t v ) /τ . (1)All plants relax with the same ripening time τ , which isone of the major control parameters of the system. An animal behaves as a foraging walker on the vegetationsubstrate which, being closed and finite, effectively worksas an enclosure. It browses from plant to plant eating fruit.This dynamics is kept as simple as possible, without anytransit time between plants nor perching time while feed-ing. When the animal gets to a plant it reduces the fruit-load an amount of b . Immediately after eating it choosesthe next plant and goes there. Furthermore, we considerno satiation and no rest. The next plant is chosen ac-cording to a stochastic rule of proximity and fruitload, asdescribed below.Our main assumption about the movement is that themain factor of the movement is proximity of the food. Thisis in fact the case in many foraging species, in particularwhen the distribution of the food resource is not hetero-geneous in the extreme. It is, however, unreasonable tosuppose that the animal will visit any site only because itis nearest, disregarding its food availability. We assume, then, that it will take that step only if it perceives that theplant has enough food. Also, we assume that the walkerdoes not exhaust the available resource at a site. This isalso reasonable because animals have a limited gut capac-ity, they might not want to stay at a site too long forfear of predation, etc. Rather, after taking a fraction ofthe available fruitload, it moves on to the next chosen site[11,13].These ingredients take the following mathematical formin the model. The animal checks the distances from theplant it’s occupying, and chooses where to go next withan exponentially decaying probability depending on theirorder of distance. If the probability to visit the nearestplant is P (1) = r , then the animal chooses the n -th plantin order of distance with probability P ( n ) = r (1 − r ) n − .Observe that if r = 1 the probability of visiting the near-est plant is 1, and the rule becomes deterministic. In thiscase, the animal always chooses the nearest plant. If r < any plant in the sys-tem, with closer ones being favorites. Note, also, that theprobability does not depend on the distance , but on the order of distances. The rationale behind this is that theanimal needs to go somewhere to get its food, disregard-ing the distance. So, it doesn’t matter if the n -th patchis close or far away: the animal will choose it with thesame probability. A probability distribution that depends on the distance (associated with a cost of reaching it, forexample) could give a different phenomenology. This willbe explored elsewhere.After the animal chooses a plant according to the geo-metric distribution just explained, it checks whether thisplant has enough food. If the current fruitload of the plantexceeds a threshold u , the animal moves to it. If the fruit-load is less than u it regards the plant as worthless, andchooses the next in the order of distance, checking againwhether this next plant has enough fruit.In summary, the random walk rules are:1. Choose the next plant according to the geometric dis-tribution of their order of distance.2. Check the fruitload of the chosen plant and:(a) If the load is greater than u , keep this choice.(b) If the load is smaller than u , choose the next plantin order of distance. Go to 2.3. Go to the chosen plant and reduce its load by b .The parameters b and u affect weakly the propertiesof the walk (results not shown), and have been kept con-stant in our simulations. Considering that the maximumfruitload is f = 1, which can be interpreted as a clusterof fruit, we have used b = 0 . u = 0 . r affects the walk in a stronger way, andwe have used it as the second (besides the relaxation time τ ) relevant control parameter of our analysis. Let us consider first a single animal in the system. In thedeterministic version of the rule, we observe that a pe-riodic cycle arises (an example is shown in Fig. 1, where uillermo Abramson et al.: Space use by foragers consuming renewable resources 3 fruitload walk
Fig. 1.
Stationary track (i. e., discarding the transient) of asingle animal on a uniform substrate. The stationary patternis a cycle with period 553 steps. the transient part of the walk has been discarded). Bear inmind that the walker has no memory, at variance with re-lated models whith periodic behavior, such as the TouristWalker of Ref. [26]. If stochasticity is allowed in the rule(so the animal can explore farther plants) the strict pe-riodicity is broken, but some aspects of it persist, as willbe discussed below. Let us exemplify these behaviors indifferent situations.Consider the uniform random substrate shown in Fig. 1.The size of the circles corresponds to the fruitload ofeach plant (in the relaxed state). A deterministic walker( r = 1), after a short transient sets on a complex but cyclictrack with a well defined home range. The black lines inthe same panel show the steps of the walk, which has aperiod of 553 steps (of course, much less than 553 plantsare visited during a cycle). Additionally, Fig. 2 shows theplants visited by the walker, indicating with the size ofthe circles the frequency of visit. It is clear that the useof the resource is very heterogeneous, both in space andin time. As this example shows, the frequency of visits isnot simply correlated with the fruiload. It arises not onlyfrom the fruitload but also from the spatial context of thevisited patch.Having a single animal in the system provides a probeto reveal topological properties of the substrate. It is re-markable that the simple rules of this model, which donot consider any memory nor explicit cost in moving, areenough to guarantee the emergence of a finite home range.Since the memory of the walk is effectively stored inthe landscape, the relaxation time τ affects the length ofthe cycles and the fraction of used space. A longer ripening Fig. 2.
Probability of occupation of space. The footprint ofthe home range occupies 12% of the habitat plants. Circle sizeis proportional to the frequency of visits. time makes fruit more difficult to find, and the home rangeexpands considerably with correspondingly longer excur-sions. The track is still cyclic and periodic but the homerange is traced in a very complex manner. A faster relax-ation, in turn, has the effect of shrinking the home rangeand reducing the period of the cycle. In the case of log-normal substrates, the clustered organization of patcheshas the effect of confining the animal’s home rage. In suchcases most of the space use is restricted to one subcluster.
Even when the attractor for a particular set of parame-ters is periodic there is a great deal of variability in theduration of the period due to the strong dependence onthe random substrate. Since both the period and the frac-tion of plants visited by the animal are measures of theuse of the system we analyze here both magnitudes in aseries of simulation runs. After discarding the transientwe measure the period and the fraction of visited plants.The procedure was repeated for 1000 realizations, eachone with a different substrate of N plants and a differentinitial condition, and for a range of ripening times.Figure 3 (left) shows the average length of the cycle, h T i , as a function of the ripening time τ . After the tran-sient, the walk was tested for periodic behavior of period T . With our method, the longest observable period canbe detected as a single repetition of a pattern. As a con-sequence, the detection of longer periods can be affectedby the maximum running time. Within this limitation, we Guillermo Abramson et al.: Space use by foragers consuming renewable resources < T > T max N N N N < S > T max N N N N Fig. 3.
Cycle length and fraction of visited plants as a functionof ripening time. N = 200, maximum observing times shown inthe legend as multiples of N . Averages and standard deviationsof 1000 realizations. Longest period detected: 1 / T max . have adjusted the observed time to avoid such an artifactas best as possible.As anticipated above, for a given value of τ there isgreat variability in the distribution of periods. The shadedareas shown above and below the mean values should beinterpreted as an inherent property of the distributions ofcycle lengths. The size of these fluctuations depends onthe system size and, consequently, may play a noticeablerole in small systems. Since these distributions are broad,and our measurements correspond to a finite observationtime, the averages are biased towards smaller values. Forthis reason we probed the paths with progressively longerobservation times (using lengths of aN with a = 10, 20,40 and 80, as indicated in the figure). These explorationsshow that there exist longer and longer periods.The average length of the cycles grows slowly with τ ,and also with the size of the system (not shown). In somesense, the length of the cycle is combinatorial and it isnot surprising that, as more space or time is available,longer cycles are possible. Interestingly, though, the num-ber of visited plants does not share this property. Figure 3(right) shows this home range size S as a function of τ forthe same set of runs shown in the left panel. We see thatthe fraction of space used by the animal grows with τ asdoes the average cycle length. But it saturates for longerobservation times (it’s already asymptotic after the short-est observation of 10 N time steps). We believe that this isan important result of our findings. Its robustness with re-spect to many details of a wide family of models suggeststhat it may lie at the core of the origin of the existenceof home ranges in animals that forage on renewable re-sources that are relatively fixed in space (i. e. with a slowresource dynamics).The periodicity of the deterministic version of the walkcan be understood as deriving from a finite set of avail-able states in a closely related model, as follows. Assumethat b = u for simplicity, and that all plants are at theirasymptotic levels k i at t = 0. Let p ( t ) denote the positionof the walker at (discrete) time t , and χ i ∈ { , } be thepresence (1) or absence (0) at plant i , when the walkeris at p . Then the amount of fruit at plant i depends on previous visits and can be written as: f i ( t ) = k i − b t − X t ′ =0 χ i ( p ( t ′ )) e − ( t − t ′ ) /τ . (2)Because of the exponential relaxation the main contribu-tion in this sum corresponds to recent times. If t − t ′ islarge, b e − ( t − t ′ ) /τ is small. In particular, for a given ǫ > T such that k i e − T/τ < ǫ (for all i ). This puts abound on the contribution of the history previous than T steps into the past: b t − T X t ′ =0 e − ( t − t ′ ) /τ = b e − T/τ N X t ′ =0 e − ( t − T − t ′ ) /τ , (3)which can be made arbitrarily small by the choice of ǫ .Since what has be substracted from plant i at time t − T is at most k i (its asymptotic value), the contribution ofthe history previous to t − T is at time t at most ǫ : f i ( t ) − ǫ < k i − b t − X t ′ = t − N χ i ( p ( t ′ )) e − ( t − t ′ ) /τ ≤ f i ( t ) . (4)Then we can approximate: f i ( t ) ≈ k i − b t − X t ′ = t − T χ i ( p ( t ′ )) e − ( t − t ′ ) /τ , (5)which tells that all that matters is the history going back T units of time. With N plants, there are N T possible his-tories, so the configuration space is finite, and periodicityfollows.As can be seen, for this proof to be rigorous it is nec-essary that ǫ is small enough that there is no differencein what choice the walker makes at each step. Given thatthe choice is made based on the rank of distances, thiscan always be ensured. If the choice were, instead, a con-tinuous probability based on the distance, the argumentwould be weaker. (Still, it would be approximate and witha valid regime of applicability.) It is also apparent that theshape of the relaxation plays a role in this phenomenon:on the one hand, a longer τ increases the periods—aswas already observed in the simulations above. On theother hand, a relaxation with a functional form slowerthan exponential—algebraic, for example—may hamperthe existence of periodic trajectories. A little randomness can help the animal browse its rangemore thoroughly, as shown in the example of Fig. 4. Herethe same landscape and initial condition were used as inFigs. 1 and 2, but reducing the probability of choosing thenearest plant to r = 0 .
8. The path is now non periodic .Nevertheless, the space usage is still contained within ahome range. The fraction of used plants is now 43% of uillermo Abramson et al.: Space use by foragers consuming renewable resources 5
Fig. 4.
Probability of occupation of space with r = 0 .
8. Thehome range occupies now 43% of the plants. Same substrateand other parameters as in Figs. 1 and 2. the total. The details are very dependent on the fluctua-tions of the substrate (which, with N = 200 plants, arerather strong), but statistical conclusions can be derivednevertheless.An important difference between these noisy walkersand the deterministic ones shown in the previous sectionis that the home range is not as well defined. The reason isthat there is a small but finite probability of migrating to any plant in the system. As a consequence of this, there isa slow drift away from the main track that may eventuallycover the whole system. Preliminary observations showthat the tracks behave as quasi-cycles that slowly drift inthe substrate, but the whole phenomenology of this hasnot been completely explored at the present stage, andwill be addressed elsewhere.The results presented in Fig. 5 are suggestive of theobserved behavior. We show the average home range size h S i (measured as fraction of plants used) as a function ofthe randomness parameter r . For most situations of inter-est in the foraging of animals such as D. gliroides it is tobe expected that this parameter stays near r = 1, indicat-ing a strong preference of near plants. But since nothingprevents the consideration of less discerning species, wehave explored the whole range from r = 0 (complete ran-domness) to r = 1. Figure 5 shows a decaying behavior ofthe home range as the rule approaches determinism. This,in addition, is strongly affected by the relaxation time,the fructification rate τ . When τ is large the resource isdepleted more effectively, and the animal needs to covermore territory to feed before the maturation of new fruitsrelaxes the resource to its stationary value. For this reason < S > r Uniform substrate = 50 = 100 = 500 = 1000
Fig. 5.
Average home range size (measured as fraction ofplants used) as a function of the noise parameter r ( r = 1 corre-sponds to deterministic walks, with the walker always choosingthe nearest plant; when r = 0 it goes to any plant at randomwith uniform probability). Substrates of 400 plants, and runsof 10 N time steps, measured after a transient of 10 N/ r is the average of 1000 realizations (eachwith a different substrate). the curves are higher for the longer values of τ . For thesewe observe an abrupt shrinking of the home range size as r →
1. Slower relaxation times display a very different be-havior, with a faster transition to a confined home rangeat a much smaller value of r . In other words, a fast relax-ation time favors a small or confined home range even fora very non-discriminating behavior in terms of the stepsize of the walk.In the case of a lognormal substrate, as expected, thereis a confinement of the movement of the animal to a smallerrange within its naturally occurring patches. The transi-tion to confined walks occurs at a small value of r for allrelaxation times, even for very slow ones. If more than one animal share the same substrate, a pos-sible mathematical formulation of the dynamics is of thekind: du i dt = F ( u i , v ) , (6) dvdt = G ( u ) , (7)where u = ( u , . . . ) is the distribution of the animals and v that of the resource. Both densities evolve in time ac-cording to appropriate evolution operators F and G . Bysolving formally Eq. (7) as v ( t ) = v ( t, u ( t )) one can reducethe system to an effectively interacting dynamics withinthe animal populations, even in the absence of an explicit Guillermo Abramson et al.: Space use by foragers consuming renewable resources coupling between the equations for the u i ’s: du i dt = F ( u i , v ( t, u ( t ))) , (8)a situation usually called exploitation competition [28].The situation is similar to a multispecies dynamics with ashared common resource, even in a mean-field formulation.In our present study the walks are self and mutually re-pulsive through the interaction with the substrate, sincean animal avoids the plants where the resource has al-ready been depleted. As a consequence, while not directlyinteracting, the animals feel the presence of one anotherthrough the interference mediated by the common re-source. This produces both a repulsion and a growth ofthe home ranges. Two questions are of interest as a firststep to analyze the use of a shared space under this cir-cumstances. Firstly, how big does a system need to be toaccommodate a population with minimum overlap? Be-sides, does a little randomness in the step rule (whichgives animals the ability to move further) help them inkeeping their home ranges apart?To analyze the interference in a population we makeuse of the distribution of space usage of each animal. Thisis defined as the normalized frequency of visits to everyplant in the system. By definition, this is a vector in themultidimensional space R N : f ∈ S N ⊂ R N , (9)that is, P Nk ( f ) k = 1, which places f in the simplex S N .The plots shown in Figs. 2 and 4, for example, are rep-resentations of these vectors, with the size of the blobsproportional to the components of f . The home rangesof two animals, then, are characterized by two of thesevectors, f i and f j (note that these subindices denote ani-mals, while the subindices accompanied with parenthesesindicate components in R N ).A good measure of the overlap of two home ranges isprovided by the standard scalar product in R N : O ( f i , f j ) = N X k =1 ( f i ) k ( f j ) k . (10)Besides all the good properties of a scalar product theoverlap O has the following property, which is of particularinterest in the present context: two vectors with the samesupport (the same plants visited) can have different scalarproduct. In other words: two home ranges with the sameplant support can have different overlaps. This is betterunderstood with a simple example. Suppose that we havetwo animals on a substrate consisting of three plants. So f = ( x , x , x ) and f = ( y , y , y ). If the two animalsshare the same plant, for example plant f = (1 , , , (11) f = (1 , , , (12)then: O ( f , f ) = 1 , (13) which is the maximum possible overlap. But if the animalsshare two equally frequent plants (say plants f = (0 . , . , , (14) f = (0 . , . , , (15)then: O ( f , f ) = 0 . , (16)which is smaller than 1, even though the “footprint” ofthe two animals on the substrate is the same. The usualinterpretation for this is a dynamical one, rather than ageometric one. Since they have two plants to browse, they can take turns between the plants, thus reducing the inter-ference with respect to the case where they share just one plant. For this reason we will refer to O as the dynamicaloverlap below. To complement this feature, and to be able to discernwhen the two home ranges are effectively disjoint (that is,whether the supports of f and f are disjoint: sup( f ) ∩ sup( f ) = ∅ ), we can use another measure of the overlap,namely the same scalar product divided by the Euclideannorms of f i and f j : P ( f i , f j ) = P Nk =1 ( f i ) k ( f j ) k k f i k k f j k . (17)In the usual geometric interpretation of the scalar prod-uct, P ( f i , f j ) is the cosine of the angle between f i and f j .This measure of the overlap gives the same value ( P = 1)whenever the footprints of the home ranges coincide. Assuch, smaller values of P measure an effective spatial sep-aration of the home ranges. Both measures are comple-mentary, as can be seen from the discussion above, so wechose to analyze both, referring to P as the geometrical overlap.To begin the analysis of these matters we have studiedsystems composed of the minimal non-trivial population:two animals. Figure 6 shows the overlaps O and P mea-sured for a range of the probability of choosing the nearestplant, r , and for a set of values of the relaxation time τ .Each curve is the result of the average of 1000 realizationsper value of r , with randomly chosen uniform substratesand initial conditions. The top panel shows O ( f , f ) andthe bottom one P ( f , f ).Observe that the effect of randomness in the step ruleis not the same for the two overlaps. It stands out that alittle chance ( r .
1) increases the overlaps (because thetwo home ranges expand due to the smaller r ). But pro-gressively smaller values of r have different effects on O and P . Observe first the dynamical overlap O : there is amaximum at an intermediate value of r , and a further in-crease in randomness actually reduces the overlap. It caneven become smaller than the value it has for the deter-ministic rule, r = 1. The same behavior has been observed The interpretation of taking turns to share the two plantsis not the only possible one. Observe that we say that they can take turns. Other dynamical possibilities exist for the same setof f i and O . Still we prefer to refer to O as a dynamical overlapto distinguish it from the one defined below.uillermo Abramson et al.: Space use by foragers consuming renewable resources 7 O = 100 = 200 = 500 = 1000 P r Fig. 6.
Overlaps of the home ranges of two animals as a func-tion of r . N = 400 (uniform substrate). Runs of 10 N timesteps, average of 1000 realizations per value of r (each with adifferent substrate). for all values of τ , but is more striking the slower the ripen-ing, in the sense that a very little randomness in the choiceof the step (which is to be expected in most animals) putsthe system in the regime of decreasing dynamical overlap.While this happens to O , observe that the purely ge-ometric overlap P grows monotonically when reducing r .In other words, the two animals are sharing a commonspace (high P ) but taking turns in their use (low O ). Thisis a rather surprising behavior in such a simple model.Moreover, it show that there is good reason to keep bothdefinitions of the overlap as complementary descriptionsof the use of space.What can be said regarding the first question posed atthe beginning of this section, regarding the interference ofanimals occupying progressively bigger areas? We have an-alyzed the dependence of the overlaps of the home rangesof two animals in a wide range of systems sizes (defined bythe number of plants N ). The results, corresponding to aset of values of the main parameters of the model, r and τ ,are shown in Fig. 7. The left panel displays the dynamicaloverlap P , while the right one shows the geometric one O .Observe firstly that, as expected, in all cases a bigger sys-tem accommodates better than a smaller one our minimalpopulation of two animals.We also observe that both measures of the overlapdecay almost algebraically, as a power law of N . In the
10 100 10001E-41E-30.010.1 10 100 10001E-30.010.11 O N r = 0.9 = 100 r = 1.0 = 100 r = 0.9 = 500 r = 1.0 = 500 ~ N -1 ~ N -1 P N Fig. 7.
Overlaps of two animals as a function of the substratesize, N . Runs of 10 N time steps, average of 100 realizationsper value of r (each with a different substrate). Parameters r and τ as shown in the legend. graphics we have added a dashed line that decays as N − .A simple calculation shows that this is the law expected ifthe home ranges where placed at random without correla-tion. It is slower than an exponential one for large systems,but shows a faster drop for rather small systems. Our re-sults show that the deterministic walks ( r = 1) follow the N − behavior for a wide range of system sizes. The depar-ture obeys mainly to a saturation effect in small systems,and to subsampling of the basins of attraction in the biggerones. It is clear, though, that more realistic walks (slightlyrandom with r = 0 .
9) have a very different behavior. Theyalso depend algebraically on the system size, but with aslower decay. For these animals the interference in theirhome ranges is stronger, compared to non-interacting ani-mals. In other words, two animals interacting in the man-ner modelled here need a much bigger system than whatcould be expected from the random overlap of their homeranges.It is also apparent in our results, and completely rea-sonable, that a model with r = 1 (deterministic step)shows smaller overlaps than one with some randomness( r = 0 . N > τ isless obvious. Compared to the system with τ = 100, theplants with τ = 500 replenish the resource so slowly thatthe animals need to browse a bigger fraction of the sys-tem. Indeed, the geometric overlap ( O , right panel) of theslow relaxation system is much greater than the fast onefor both the deterministic and the random steps. This in-dicates that the footprints of their home ranges overlap.Nevertheless, for the same systems the dynamical over-lap remains small, of the same order of magnitude thatthe corresponding systems with the faster τ = 100. Thisshows that, even though the animals need to share a ter-ritory because of the depletion of the resource, they cando it with little interference by taking turns in differentparts of the corresponding home ranges. We have analyzed a simple model of animal foraging inan heterogeneous habitat. The movement rules have been
Guillermo Abramson et al.: Space use by foragers consuming renewable resources kept intentionally simple in order to understand the con-sequences, in the space use, of a minimal model of foragersexploiting renewable and patchy resources. In this spirit,our model incorporates one essential ingredient in the ani-mal movement (the rule of proximity, complemented witha weaker one of abundance), and one in the resource dy-namics (the exponential relaxation).The properties of the walk defined according to theserules are complex enough for a non-intuitive result. Evenif they provide no means for the animal to remember thegood places to feed, and that there is no explicit costin moving, the combination of closeness and fruitload isenough to guarantee the emergence of a finite home range.The reason for this is that memory of the usage of the land-scape is kept in the landscape itself. The walk rule ensuresthat the animal tends to go away from the current loca-tion: it is consuming the resources, so it searches nearbyplants for fruit. Yet, the relaxation of the fruitload en-sures that the resource is eventually replenished. This is atvariance with models of destructive foraging such as [24],for example, where the visited sites are removed from thesystem and never revisited, which display non-localizedtrajectories similar to L´evy flights. When this happens,the proximity rule (that previously allowed the animal tomove away) allows it to come back to places where it hasfed before. The peculiarities of the random distribution ofpatches and the initial location of the walker determinethe sequence in which the plants are visited.It is particularly notable that, even when the proxim-ity rule is weakened with the inclusion of noise, (the r < r <
1. In a sense, this correspondsto a better utilization of the resource, through sharing.The evolutionary effects of this pattern cannot be assessedat the present stage of the model, but it certainly will beone of our interests in future developments. In particular,it will be of interest to complement this resource-mediatedinterference with other known dynamical mechanisms ofthe origin of territoriality, such as scent deposition [29].In the second place, the overlap displays a simple de-pendence on the system size, namely a power law witha smaller exponent than the one corresponding to non-interacting animals. The other relevant parameters ( r and τ ) appear involved in this behavior in a no trivial way.In any case, this result points also in the direction of theprevious one: two animals are more likely to share the re-source (if there is some randomness in the movement rule)than what could be expected from the random overlap oftheir ranges.The emergent properties of our simple approach pro-vides a baseline for more realistic models of animals forag-ing on patchy and renewable resources. An instance wherethe interaction between animal movement and a dynam-ical resource plays an important role is the mutualismbetween plants and their seed dispersers. In the Patago-nian temperate forest, for example, there exists a particu-larly interesting example: the quintral ( Tristerix corymbo-sus ) and the monito del monte (
Dromiciops gliroides ). T.corymbosus , an hemiparasite, depends on agents to dis-perse their seeds to the branches of potential hosts. Itis a keystone species of the temperate forests of south-ern South America, because during the winter is the onlyresource for the hummingbird
Sephanoides sephanoides ,which is one of the most important pollinators in thisecosystem [30]. Furthermore, the fruits of quintral repre-sent an important food source for the
D. gliroides , a mar-supial endemic to the region and the only current represen-tative of the
Microbiotheria order. In turn,
D. gliroides isthe only seeds disperser for
T. corymbosus [31]. The studyof the relevance of the present findings in such systems iscurrently under way and will be reported elsewhere.
This work received support from the Consejo Nacional de In-vestigaciones Cient´ıficas y T´ecnicas (PIP 112-200801-00076),Universidad Nacional de Cuyo (06/C304), and Agencia Na-cional de Promoci´on Cient´ıfica y T´ecnica (PICT-2011-0790).uillermo Abramson et al.: Space use by foragers consuming renewable resources 9
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