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The authors have declared that no competing interests exist.

Conceived and designed the experiments: AMH SAM. Performed the experiments: AMH. Analyzed the data: AMH. Contributed reagents/materials/analysis tools: AMH SAM. Wrote the paper: AMH SAM.

Most motile organisms use sensory cues when searching for resources, mates, or prey. The searcher measures sensory data and adjusts its search behavior based on those data. Yet, classical models of species encounter rates assume that searchers move independently of their targets. This assumption leads to the familiar mass action-like encounter rate kinetics typically used in modeling species interactions. Here we show that this common approach can mischaracterize encounter rate kinetics if searchers use sensory information to search actively for targets. We use the example of predator-prey interactions to illustrate that predators capable of long-distance directional sensing can encounter prey at a rate proportional to prey density to the

Encounters between individual organisms are an essential part of biology; in many sexually reproducing species, males and females must encounter one another in order to mate, pollinators must find flowers, and predators must locate prey before capturing and consuming them. Many species accomplish these tasks by actively searching for their targets using sensory information. Despite this, classical mathematical models used to predict the rate of encounters between searchers and their targets assume that searchers make movement decisions without using sensory information. Here we develop a mathematical framework for incorporating sensory information into searcher movement behavior to study how sensory response changes the relationship between encounter rate and target density. By comparing searchers that use sensory information to those that do not, we show that sensory response not only increases encounter rate, but that it also changes the form of the relationship between encounter rate and target density. By using sensory information, predators encounter prey at a rate that is less sensitive to changes in prey density when prey density is low. Our results demonstrate a strong connection between the usage of sensory information and the encounter rates that are so critical to survival and reproduction in nature.

Classical models of species interactions assume that encounters between individuals are governed by a process akin to mass-action; individuals move along random linear trajectories and encounter one another when they come within a critical distance

Recent work has extended the study of encounter rates to consider searchers that encounter targets probabilistically, destroy targets after encounters, search intermittently, and follow trajectories that are not linear

Of course, the assumption that searchers move independently of targets is often made for mathematical convenience. The question is whether models that rely on this assumption capture the salient features of encounter rate kinetics in nature. Empirical studies have shown that inhibiting particular sensory modalities such as chemosensing or flow sensing can dramatically decrease search performance (e.g.,

Below we adopt the language and intuition associated with a predator searching for prey. We assume that the predator samples its environment for sensory cues passively emitted by prey, and adjusts its movement behavior according to explicit mathematical models presented here. This approach builds on a recently developed framework for modeling search decision-making

Studies of biological search typically describe how the type of movement behavior used by a searching organism affects the time needed to encounter its first target

To illustrate how sensory information can affect encounter rates, we consider an idealized model of a searching predator in a two-dimensional environment (we discuss search in three dimensions in

When prey density is low, a predator will typically detect little or no signal and proceed with the information that the target is not likely to be nearby. When prey density is high, an increasing fraction of the landscape is covered by regions that are within the encounter radius

To model predator search behavior, we generalize a recently developed framework for modeling search decision-making

During the movement phase of the search, predator movements are modeled by drawing from the distribution

Suppose the predator detects sensory observation

A) Predator with perfect sensing and response searching in a two-dimensional environment. After collecting sensory data, the predator moves along a linear trajectory toward the nearest prey and encounters the prey when it comes within a distance of ^{−1}, points show mean of 1000 replicates at each density). Prey distribution is randomly generated from a Poisson point process in each simulation. Solid cyan curve shows theoretical mean encounter rate (see text), which approaches

When the predator moves directly from one prey to the next, it will encounter prey at a mean rate that is inversely proportional to the mean distance between prey, which we denote

Purely random search models have a long history, beginning with the models of Lotka

We note that it is possible to formulate a search behavior that does not rely on sensory data using the Bayesian framework of Equation (1) by assuming that the decoding function

In addition to the predator with access to perfect information, and the predator with access to no information, we consider a predator with access to minimal sensory data and a minimal capacity for decision-making. We do this to demonstrate that even minimal information use can significantly alter the encounter rate function _{0}_{0}

This model of olfactory search behavior has two salient features. The first is that, because there is no directional information inherent in the signal, the predator always draws turn angles from the same distribution (Uniform on

In addition to the analytical results described above, we used search simulations to compare the behavior of a predator that moves according to a purely random strategy to a predator with imperfect sensing and response. In both cases, we assume that the intrinsic movement behavior is described by a symmetric two-dimensional Pareto distribution. Because of the symmetry we can separately draw the turn angle

In each simulation, we placed a single predator in a prey periodic environment (i.e., environment was a torus) and populated the environment with a Poisson number of prey with mean 600. The size of the environment was then scaled to achieve the desired prey density. In each scanning phase,

We performed two sets of simulations. In the first, prey positions were generated using a Poisson point process. We then recorded the time required for the predator to encounter the first prey and used this to compute encounter rate

Our primary goal was to characterize the form of the encounter rate function in the low prey density regime. We simulated predators exploring environments with prey densities ranging from 0.5–100 prey per

As in previous investigations (e.g.,

There is a dramatic difference between movement patterns of predators that use sensory data and those that do not. As is evident from

Prey (red points) and locations where predator scans for prey (blue points) for A) signal-modulated and B) purely-random predators. Scan points are semitransparent so darker color indicates locations where predator has scanned more frequently. Data represent searches in which a predator made 1000 consecutive movements without destroying prey.

Signal-modulated predators perform ARS because they move short distances between scans when they receive strong sensory signals and move long distances when they measure weak signals

A) Purely random (yellow circles) and signal-modulated predators (blue triangles, _{o}

Predators that used a purely random search strategy encountered prey at a rate that was nearly proportional to prey density (

Across all densities studied, predators that use sensory data to make movement decisions encounter prey at a higher rate than predators that do not use sensory cues (

As prey density increases, the encounter rate of signal-modulated predators increases nonlinearly and clearly displays multiple scaling regimes (

Predators encounter and destroy

In addition to engaging in area-retricted search, signal-modulated predators successfully locate nearby prey more frequently than purely random predators (

A) Encounter probability as a function of target density. Points represent the probability that a purely random predator (yellow squares) or a signal-modulated predator (blue circles) will wander away from a nearby target without encountering it. Parameters as in

Our results demonstrate that the use of sensory information alters encounter rate kinetics, both at the extreme of perfect information and decision-making, and at the other extreme of minimal sensing and rudimentary decision-making. In studies of coupled population dynamics, the encounter rate function is a central component of the functional response, the relationship that couples prey and predator populations. Given the anomalous scaling of encounter rate shown by predators that use sensory information to make movement decisions, a natural question is whether such predator search behavior might affect coupled population dynamics. Here, we explore this question.

Predator-prey dynamics can be modeled by the following system of equations:

To relate our encounter rate findings to coupled population dynamics, we must translate the encounter rate

We now proceed to study the effect of the form of

To explore how the form of the encounter rate function affects the outcome of coupled population dynamics, we parameterize the population model described above for the sparse prey regime (^{*} theory posits that a species' competitive ability is determined by its ability to deplete resources and persist when resources are rare

A) Steady state prey density as a function of the ratio of predator conversion efficiency to predator mortality rate (

A second important observation is that, for a given value of the parameter

Our results demonstrate that the use of information about the position of targets fundamentally alters the relationship between encounter rates and target density. Not only do predators that use sensory information encounter prey more often, but they are less sensitive to changes in prey density. This is true even when sensory cues contain a minimal amount of information about target locations, and searchers do not remember past signals. This increased robustness provides an ecological mechanism through which sensory response may allow predators to cope with fluctuations in prey density. Moreover, it can alter coupled population dynamics. These findings are robust to a range of assumptions about target distribution, capture behavior, and the length over which searchers detect scent signals (

Reaching a general understanding of the effect of sensory data on species encounter rates is challenging. Searching organisms collect a wide variety of sensory data and biologists do not know, in general, how they use these data to make decisions

Our work suggests several ways to better integrate experiments with models of encounter rates. For example, we suggest that encounter rate and functional response of predators should be nonlinear at low prey densities. Yet, most experimental studies of encounter rates and functional response measure rates at high prey density, where handling time and predator satiation determine the shape of the rate function (but see

Finally, we note that nonlinearity of the encounter rate function depends on the ratio of the length scale of sensory signal transmission to the length scale at which encounters occur. When predators can only detect prey that are very nearby (i.e. detection distance/encounter distance

Our results show that introducing a response to even relatively information-poor, noisy sensory signals qualitatively alters the relationship between predator-prey encounter rate and prey density in many biologically plausible scenarios. Behaviors such as area-restricted search emerge naturally from our model of search behavior, even in the absence of signal gradients, complex signal processing, and memory of past signal and target encounters

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This work benefited from discussions with B. Holt, J. Principe, J. Casas, and the Biomath group in the Department of Mathematics at the University of Florida.

Conceived and designed the experiments: AMH SAM. Performed the experiments: AMH. Analyzed the data: AMH. Contributed reagents/materials/analysis tools: AMH SAM. Wrote the paper: AMH SAM.