Brownian jumps vs. L´evy flights

0 0.5 µ1 1.5 2

1 1.2 1.4 1.6 1.8 2

Q

1-D 2-D

Figure 4.9: Improvement of the searching process because of the commu-nication mechanism. Circles correspond to the 2Dmodel and squares to

1D. Lines are interpolations.

ian walker has no directionality in the movement, so provided with sources of information (communication together with the local quality of the landscape) it can search much more efficiently. This effect is less important for L´evy searchers due to the presence of long, straight-line moves that, by themselves, decrease the number of times that a particular area is revisited. In addition the return probability to a given point is much higher in 1Dthan in 2D. Therefore the di-rectionality introduced by the communication has a stronger effect in the simple 1Dscenario that we have studied. It is also important to remark that in this case the walker only can move either to the right or to the left at each step. This will make the influence of the bias due to communication much stronger in the jumping probabilities.

In summary, the communication mechanism is less important in L´evy strategies, so that its effect is less noticeable as it is shown in Fig. 4.9 both in 1Dand 2D.

However, the value of the optimal interaction range changes with the kind of mo-tion. This is shown in the 2Dmodel by the dependence of the mean search time on the communication range for different L´evy exponents (Fig. 4.3). The value ofσoptincreases with the L´evy exponent, so Brownian searchers (µ→2) need to spread the information farther (a larger value ofσopt) than L´evy (µ=1) walkers to obtain the maximum benefit. In Fig. 4.10 we show the value of the optimal communication range,σopt, as a function of the L´evy exponent. L´evy trajectories show clusters of short displacements with frequent turns occasionally broken up by long linear displacements, which account for most of the target encounters.

However, because these steps are often much longer than the average distance between targets they are not positively influenced by communication, so any benefit a L´evy strategy gains from communication occurs during the series of short displacements. The time that an individual spends doing short movements is limited by the interarrival time of the large steps, so unless an individual is already relatively close to a target, it will not have time to reach a target before the next big step comes and moves it far away from that original target. Therefore

0 0.5

µ

0.02 0.03

σ

Figure 4.10: Optimal communication range as a function of the L´evy ex-ponent.

the optimal communication range decreases with decreasing L´evy exponent,µ, as longer displacements become more frequent at lowerµvalues.

In addition, the value ofσopt depends on both the number of targets and their spatial distribution, as was shown in Sec. 4.4 for a simple 1Dsituation where σopt∼L.

4.7

Summary and conclusions

In this chapter we compared Brownian and L´evy search strategies using a pop-ulation of individuals that exchange information about the location of spatially distributed targets. Using a simple 1Dmodel we have provided analytical re-sults on both cases, concluding that frequent long jumps (µ→0, ballistic limit) minimize the searching times.

However the effect of a communication mechanism is more pronounced in the limit of short jumps i.e., Brownian motion. This means that a population of individuals employing Brownian motion gains proportionally more benefit from communicating and sharing information than does a population of L´evy walkers, where long jumps are more or less frequent depending on the value of the L´evy exponent µ. When messages are exchanged in a range that minimizes search duration, communication is the driving force in the Brownian limit, but occasional long jumps are still responsible for most of the encounters with targets in the case of long-tailed step-length distributions.

The main result of this work is rather general: independently of the kind of communication performed by the population, and of the spatial distribution of the targets, a population of individuals with the ability to communicate will find the targets in a shorter time if the information is spread at intermediate ranges.

Both an excess and a lack of information increase the search time. However, the communication mechanism does not have the same quantitative effect on the different moving strategies (i.e., ballistic, L´evy or Brownian). Uninformed

4.7. SUMMARY AND CONCLUSIONS

Brownian individuals perform a random movement revisiting the same position many times, so having an external source of information introduces directionality on the movement, decreasing the number of times that a point in the space is visited. In the case of L´evy and ballistic strategies (µ→ 0), communication is less noticeable because individuals are able to do long jumps. This is already a source of directionality that prevents individuals from revisiting the same points in space many times, and thus weakening the effect of the directionality introduced by communication.

CHAPTER 5

Foraging in Procapra gutturosa

In this chapter we show an application of the model presented in Chapter 4 to the particular case of acoustic communication among Mongolian gazelles (Procapra gutturosa), for which data are available, searching for good habitat areas. Using Monte Carlo simulations, our results point out that the search is optimal (i.e.

the mean first hitting time among searchers is minimum) at intermediate scales of communication. We also present this result in terms of the frequency of the sounds, showing a good agreement with field measurements of the sounds emitted by these gazelles in the wild. The formation of groups in the populations is also studied.

5.1

Introduction

Many living organisms, including bacteria [Liu and Passino, 2002], insects, and mammals [Zuberb ¨uhler et al., 1997; McComb et al., 2003] communicate for a va-riety of reasons including facilitation of social cohesion [Cap et al., 2008; Pfefferle and Fischer, 2006], defense against predators [Zuberb ¨uhler, 2001], maintenance of territories [Slater et al., 1994; Frey et al., 2007], and to pool information on resource locations when no single individual is sufficiently knowledgeable [von Frisch, 1967; Hoare et al., 2004; Berdahl et al., 2013; Simons, 2004; Torney et al., 2009b]. Communication among individuals frequently leads to group forma-tion [Eftimie et al., 2007], which often has clear direct benefits such as reducing individual vulnerability to predators. Such strategies may, however, also have important incidental benefits. For example, an individual that has found a good foraging patch might try to attract conspecifics to reduce its risk of predation, but also provides its conspecifics with information on the location of good forage, thus increasing the foraging efficiency of those responding to the call.

We apply a specialized version of the model introduced in Chapter 4 to the par-ticular case of acoustic communication among Mongolian gazelle, the dominant wild herbivore in the Eastern steppe of Mongolia (Fig. 5.1a) among 14 species

Figure 5.1: (a) Location of the Eastern Steppe. (b) Construction of roads in the step has caused a habitat fragmentation. (c) Hunting is one of the major threats to Mongolian gazelles. (d) Oil and mineral explorations in

the steppe.

of ungulates. A population of about one milion of animals is estimated, but it is difficult to give a good measurement because of large fluctuations due to the extreme conditions in the steppe that cause periods with important population losses. In addition, the nomadism of the species, travelling long distances during the year makes more difficult a demographic control. In any case, the species is still recognized asone of Asia’s largest remaining wildlife populations[Olson et al., 2005], although it has experienced a major reduction in range during the past century, and is further threatened by excessive hunting and continued habi-tat loss and fragmenhabi-tation (grassland steppes are increasingly being carved up by fences, roads, agriculture, and densely settled areas while oil fields and pipelines are being developed in the region) (Fig. 5.1 b-d). In fact, Mongolian gazelles were formerly distributed across the whole area of the Republic of Mongolia but the range of this species, between the 1940s and 1960s, was reduced by 70% owing to excessive official hunting and poaching. Nowadays, although individuals or small groups are found across a wider geographical range, higher concen-trations of this gazelle species are now limited to the eastern steppe (Fig. 5.1a) where they avoid narrow valleys, forest, sand dunes or cultivated fields unless driven there by exceptional circumstances. The plant cover of the dry steppes of eastern Mongolia is extremely sparse, generally 5–20% (Fig. 5.1, right panel), rarely reaching 30–40% [Frey and Gebler, 2003; Mueller et al., 2008].

In summary, gazelles must find each other to form grazing groups less suscep-tible to preadator’s attacks (Fig. 5.1), and relatively small areas of good forage in a vast landscape where sound can travel substantial distances [Frey et al., 2008](Fig. 5.2a) . We aim to explore whether acoustic communication in the Steppe could lead to the formation of observed large aggregations of animals (Fig. 5.2b) [Olson et al., 2009], and how search efficiency depends on the distances