Bartemeus et al., 2005 Animal Search Strategies

Ref: Bartumeus, F., da Luz, M. G. E., Viswanathan, G. M., & Catalan, J. (2005). Animal search strategies: A quantitative random-walk analysis. Ecology, 86, 3078-3087.

This paper takes a look at two random-walk strategies often used in foraging simulations. Before this, no one allegedly analyzed the two of them directly. Random walks used in foraging need to account for the tendency of animals to keep moving in the same direction - directional persistence. They analyze 3 types of walks: correlated random walks (CRWs), Levy walks (LWs), and a new combination, Levy-modulated correlated random walks (LMCRWs).

Random walks are discretized into

• Displacement event (movement length)
• Orientation event (turning angle)

CRW models. Use a Gaussian (or similar) distribution of move lengths combined with a nonuniform angular distribution of turning angles. They used a wrapped Cauchy distribution (WCD), which is basically a normal cuve with tails at plus/minus 180 degrees. The parameter control the extremeness of the curve's peak about 0 degrees (see figure a).
LW models. Use a uniform distribution for turning angles, and a power law distribution for the move lengths. The exponent is the Levy index and controls the range of the correlations (see figure c).

LMCRW models. Uses (i) WCD for turning angles within a flight, (ii) a Gaussian distribution of move steps within a flight, (iii), a uniform distribution of turning angles between flights, and (iv) a power law distribution of flight lengths. The examination of this strategy will allow for the analysis of whether it is the power law distribution of move lengths of the WCD of the turning angle that is the method of directional persistence that controls the optimization of random search.

Simulations looked at a few things with a few conditions.

• Mean-square displacement (msd): The squared distance moved from the start point to another location during a iven time, averaged over many trials.
• Search efficiencies (). is a ratio of the number of target sites visited to the total distance traveled. is the average distance between two target sites (inverse of the target density). Their product is a measure of search efficiency that is independent on target site density.
  

Nondestructive searches - the same target can be revisited many times.

Destructive searches. Targets disappear after first encounter, replaced with a new target at a random location.

Results

CRWs cannot generate long-range correlations in movement. Thus, the msd can depart from the linear increase with time only over a particular range of temporal and spatial scales, but in the long-term limit, CRWs are like uncorrelated random walks: they can only give rise to Brownian motion.

However, a gradual change in the Levy exponent () corresponds to a gradual change in the diffusivity () that does not vanish at the long-term limit. Changing the Levy exponent implies a qualitative change in the macroscopic and long-term properties of the movement as a whole.
On the search efficiencies of the random-walk models.

In all cases, the LWs are more efficient than the CRWs (see figure).
The higher values for LWs in the nondestructive searches may be due to the fact that LWs (and not CRWs) are scale invariant (due to the power law structure).

The runs with LMCRWs for nondestructive search showed that reorienting the movement at power-law time intervals have more influence in the search efficiencies than small direction deviations during flight. Thus, LWs appear robust in their efficiencies even with sinuous flights due to the embedded CRWs.

Discussion

Random search strategies can only exist where there is uncertainty in the behavior of targets. So far, stochastic animal search rules are not generally considered in behavioral evolution (but see Hills, 2006). However, it should be considered that (i) in some search processes, a high degress of uncertainty is unavoidable, and (ii) in such scenarios, the success can be improved by optimizing random search strategies.

Mechanisms to optimize the "chance of finding" something are not necessarily the same as the "detection" of such items. The combination of both mechanisms probably evolved together.

The suggestion is that scale-free puntuations in animal movement (i.e., stops, strong reorientations, distinctive interruptions in the walk, etc.) could be the basis for a stochastic organization of the search at the landscape level. Levy-walk patterns have been identified for a variety of organisms.
Two remaining questions: (1) Are these oatterns caused by a random search strategy or have they emerged from complex behavioral processes, external drivers, etc.? (2) Can we identify reorientation mechanisms within animal behavioral traits? Investigations into how and when organisms actively discretize their movements will facilitate the finding of adaptive mechanisms capable of optimizing random search.