Simon Benhamou

Random walk models in biology

Mathematical modelling of the movement of animals, micro-organisms and cells is of great relevance in the fields of biology, ecology and medicine. Movement models can take many different forms, but the most widely used are based on the extensions of simple random walk processes. In this review paper, our aim is twofold: to introduce the mathematics behind random walks in a straightforward manner and to explain how such models can be used to aid our understanding of biological processes. We introduce the mathematical theory behind the simple random walk and explain how this relates to Brownian motion and diffusive processes in general. We demonstrate how these simple models can be extended to include drift and waiting times or be used to calculate first passage times. We discuss biased random walks and show how hyperbolic models can be used to generate correlated random walks. We cover two main applications of the random walk model. Firstly, we review models and results relating to the movement, dispersal and population redistribution of animals and micro-organisms. This includes direct calculation of mean squared displacement, mean dispersal distance, tortuosity measures, as well as possible limitations of these model approaches. Secondly, oriented movement and chemotaxis models are reviewed. General hyperbolic models based on the linear transport equation are introduced and we show how a reinforced random walk can be used to model movement where the individual changes its environment. We discuss the applications of these models in the context of cell migration leading to blood vessel growth (angiogenesis). Finally, we discuss how the various random walk models and approaches are related and the connections that underpin many of the key processes involved.

Spatial analysis of animals' movements using a correlated random walk model*

A probabilistic model was developed that applies to the analysis of erratic movements made by animals foraging in a stochastic environment. This model is a first order correlated random walk model in which the following two biological constraints are integrated into the Brownian motion model: bilateral symmetry and the cephalocaudal polarization that leads to a tendency to go forward. The main properties of the model were studied by numerical simulations using a pseudo-random generator. It was found that the spatial pattern of search paths could be quantified by a single numerical index of sinuosity. Some advice on the concrete use of this model is given. Correlated random walk diffusion was studied in relation to the sinuosity by analysing the probabilistic distribution of the bee-line distance between the first and last points of a path. Some theoretical applications of this model are developed.

ANIMAL MOVEMENTS IN HETEROGENEOUS LANDSCAPES : IDENTIFYING PROFITABLE PLACES AND HOMOGENEOUS MOVEMENT BOUTS

Because of the heterogeneity of natural landscapes, animals have to move through various types of areas that are more or less suitable with respect to their current needs. The locations of the profitable places actually used, which may be only a subset of the whole set of suitable areas available, are usually unknown, but can be inferred from movement analysis by assuming that these places correspond to the limited areas where the animals spend more time than elsewhere. Identifying these intensively used areas makes it possible, through subsequent analyses, to address both how they are distributed with respect to key habitat features, and the underlying behavioral mechanisms used to find these areas and capitalize on such habitats. We critically reviewed the few previously published methods to detect changes in movement behavior likely to occur when an animal enters a profitable place. As all of them appeared to be too narrowly tuned to specific situations, we designed a new, easy-to-use method based on the time spent in the vicinity of successive path locations. We used computer simulations to show that our method is both quite general and robust to noisy data.

Spatial memory and animal movement

Memory is critical to understanding animal movement but has proven challenging to study. Advances in animal tracking technology, theoretical movement models and cognitive sciences have facilitated research in each of these fields, but also created a need for synthetic examination of the linkages between memory and animal movement. Here, we draw together research from several disciplines to understand the relationship between animal memory and movement processes. First, we frame the problem in terms of the characteristics, costs and benefits of memory as outlined in psychology and neuroscience. Next, we provide an overview of the theories and conceptual frameworks that have emerged from behavioural ecology and animal cognition. Third, we turn to movement ecology and summarise recent, rapid developments in the types and quantities of available movement data, and in the statistical measures applicable to such data. Fourth, we discuss the advantages and interrelationships of diverse modelling approaches that have been used to explore the memory-movement interface. Finally, we outline key research challenges for the memory and movement communities, focusing on data needs and mathematical and computational challenges. We conclude with a roadmap for future work in this area, outlining axes along which focused research should yield rapid progress.

Dynamic Approach to Space and Habitat Use Based on Biased Random Bridges

Background Although habitat use reflects a dynamic process, most studies assess habitat use statically as if an animals successively recorded locations reflected a point rather than a movement process. By relying on the activity time between successive locations instead of the local density of individual locations, movement-based methods can substantially improve the biological relevance of utilization distribution (UD) estimates (i.e. the relative frequencies with which an animal uses the various areas of its home range, HR). One such method rests on Brownian bridges (BBs). Its theoretical foundation (purely and constantly diffusive movements) is paradoxically inconsistent with both HR settlement and habitat selection. An alternative involves movement-based kernel density estimation (MKDE) through location interpolation, which may be applied to various movement behaviours but lacks a sound theoretical basis. Methodology/Principal Findings I introduce the concept of a biased random (advective-diffusive) bridge (BRB) and show that the MKDE method is a practical means to estimate UDs based on simplified (isotropically diffusive) BRBs. The equation governing BRBs is constrained by the maximum delay between successive relocations warranting constant within-bridge advection (allowed to vary between bridges) but remains otherwise similar to the BB equation. Despite its theoretical inconsistencies, the BB method can therefore be applied to animals that regularly reorientate within their HRs and adapt their movements to the habitats crossed, provided that they were relocated with a high enough frequency. Conclusions/Significance Biased random walks can approximate various movement types at short times from a given relocation. Their simplified form constitutes an effective trade-off between too simple, unrealistic movement models, such as Brownian motion, and more sophisticated and realistic ones, such as biased correlated random walks (BCRWs), which are too complex to yield functional bridges. Relying on simplified BRBs proves to be the most reliable and easily usable way to estimate UDs from serially correlated relocations and raw activity information.

Efficiency of area-concentrated searching behaviour in a continuous patchy environment*

In many natural situations involving terrestrial predators, prey items are clustered in patches that have no perceptible boundaries and can only be defined as areas where the local resource density is higher than the mean overall resource density. In this type of continuous patchy environment, an efficient forager has to regulate the sinuosity (klinokinesis) and the speed (orthokinesis) of its random search movements so as to concentrate its search effort in the high resource density areas. This study determined the theoretical efficiency of this area-concentrated behaviour using computer simulations. Two search modes were assumed to be used by the model predator: an intensive search mode involving high sinuosity and low speed within high resource density areas, which is triggered by the detection of a prey item, and an extensive search mode involving low sinuosity and high speed between these areas, which is resumed after an unsuccessful search. Among the four habitat types included in this study, the efficiency of this movement control was higher in coarse-grained than in fine-grained habitats and increased also with the level of heterogeneity. The optimal values of the movement control parameters were similar whatever the habitat type and a large range of values led to nearly maximal efficiency. This suggests that even if they can only poorly estimate the resource distribution, terrestrial predators may ultimately search efficiently for clustered prey items by using elementary proximate mechanisms roughly controlling their random search movements.

Spatial memory in large scale movements: Efficiency and limitation of the egocentric coding process

The orientation abilities exhibited by many species are based on spatial memorization. The location of a place to which an animal intends to return may be memorized either independently of the animals position (location and orientation) by an exocentric coding which consists of processing the location-based (site-dependent) information provided by nearby landmarks perceptible both from the animals location and the place, or in relation to the animals position by an egocentric coding which consists of processing the route-based (site-independent) information provided by the animals movements (rotations and translations). An animal seldom has the opportunity of processing useful location-based information during large scale movements in its home range, except during the terminal adjustment of its path. Consequently, it may memorize the directions and the distances of the preferential places in its home range by processing route-based information. The efficiency of this egocentric coding process was quantified as a function of the path characteristics and the type and magnitude of route-based information estimation errors. This process was found to require a high degree of accuracy in idiothetic (internal) estimations of rotatory movements but not in estimations of translatory movements or in allothetic (compass-based) estimations of rotatory movements. The complementarity of the exocentric and egocentric coding processes during large scale movements is discussed.

How animals use their environment: a new look at kinesis

Abstract The aim of this paper is to show how animals can orient themselves in relation to a stimulation gradient or exploid patchy environments using simple kinetic mechanisms. From a new look at klinokinesis and orthokinesis, the properties of these two mechanisms were determined and their respective contributions to the phenomenon of animal aggregation in the most suitable areas of the environment were specified. When movement regulation is a function of variations in the stimulus intensity, klinokinesis can be seen as an elementary spatial orientation mechanism, whereas orthokinesis seems to have no biological application. In contrast, when movement regulation is a function of the value of stimulus intensity, klinokinesis and orthokinesis can both be seen as elementary space-use mechanisms. Some examples of applications of the models are given. In particular, it is suggested how klinokinetic and orthokinetic models can formalize the ‘area-restricted’ searching behaviour exhibited by many foraging animals. Finally, the place of klinokinetic and orthokinetic mechanisms in the framework of a general theory of spatial orientation and space use in animals is discussed.

Landmark use by navigating rats (Rattus norvegicus) contrasting geometric and featural information.

Place navigation in the rat is based on an allocentric localization process in which places are recognized in reference to the surrounding environment. However, the nature of the spatial information used for such a process remains unclear. In particular, the relative importance of environmental geometry and of individual landmarks is still a matter of debate. This issue was addressed in rats (Rattus norvegicus, Long-Evans strain) by means of a water maze experiment in which the importance of the identity of landmarks and of their geometric arrangement were compared. Results showed that place navigation appears to be primarily based on the geometric arrangement of landmarks. These results are discussed in the more general context of the construction of spatial representations in rodents.

How Predictability of Feeding Patches Affects Home Range and Foraging Habitat Selection in Avian Social Scavengers

Feeding stations are commonly used to sustain conservation programs of scavengers but their impact on behaviour is still debated. They increase the temporal and spatial predictability of food resources while scavengers have supposedly evolved to search for unpredictable resources. In the Grands Causses (France), a reintroduced population of Griffon vultures Gyps fulvus can find carcasses at three types of sites: 1. “light feeding stations”, where farmers can drop carcasses at their farm (spatially predictable), 2. “heavy feeding stations”, where carcasses from nearby farms are concentrated (spatially and temporally predictable) and 3. open grasslands, where resources are randomly distributed (unpredictable). The impact of feeding stations on vulture’s foraging behaviour was investigated using 28 GPS-tracked vultures. The average home range size was maximal in spring (1272±752 km2) and minimal in winter (473±237 km2) and was highly variable among individuals. Analyses of home range characteristics and feeding habitat selection via compositional analysis showed that feeding stations were always preferred compared to the rest of the habitat where vultures can find unpredictable resources. Feeding stations were particularly used when resources were scarce (summer) or when flight conditions were poor (winter), limiting long-ranging movements. However, when flight conditions were optimal, home ranges also encompassed large areas of grassland where vultures could find unpredictable resources, suggesting that vultures did not lose their natural ability to forage on unpredictable resources, even when feeding stations were available. However during seasons when food abundance and flight conditions were not limited, vultures seemed to favour light over heavy feeding stations, probably because of the reduced intraspecific competition and a pattern closer to the natural dispersion of resources in the landscape. Light feeding stations are interesting tools for managing food resources, but don’t prevent vultures to feed at other places with possibly high risk of intoxication (poison).