Sensory Information and Encounter Rates of Interacting Species

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 power (where is the dimension of the environment) when prey density is low. Similar anomalous encounter rate functions emerge even when predators pursue prey using only noisy, directionless signals. Thus, in both the high-information extreme of long-distance directional sensing, and the low-information extreme of noisy non-directional sensing, encounter rate kinetics differ qualitatively from those derived by classic theory of species interactions. Using a standard model of predator-prey population dynamics, we show that the new encounter rate kinetics derived here can change the outcome of species interactions. Our results demonstrate how the use of sensory information can alter the rates and outcomes of physical interactions in biological systems.


Scent propagation
We model scent propagation in turbulence as packets that appear at the prey position x 0 according to a Poisson arrival process and move as a Brownian motion. From the predator's perspective, this is equivalent to encountering a random number of units of scent, H ∼ Pois(t o R(|x − x o |)), at its location x during a scanning phase of length t o , where R is the rate of scent arrival. Denoting = |x−x 0 |, under these assumptions, the likelihood of h encounters is To derive R( ), let u(x) represent the mean concentration of scent at predator position x emitted by a prey item located at position x 0 . The steady-state diffusion process without advection is described by where D represents the combined molecular and turbulent diffusivity (m 2 s −1 ), µ represents the rate of dissolution of scent patches (s −1 ), and λ represents the rate of scent emission at the prey (s −1 ). In two dimensions, the rate of scent patch encounters by a predator of linear size a located at x is given by where K 0 represents a modified Bessel function of the second kind. Two terms are sufficient to characterize the scent environment: the typical propagation length r o , which corresponds to the distance at which a predator will register on average one unit of scent per scanning period, and the expected number of encounters per unit t o at a distance of one body length from the prey.
2 Dependence of regime break on scent signal propagation length, and dependence of results on properties of the intrinsic movement distribution To determine whether the prey density at which linear regimes in the encounter rate function transitioned to nonlinear regimes depended on the length scale of predator scent detection, we repeated simulations to compute Γ(ρ) over a range of values of the olfaction radius r o . Figure S1 shows that the prey density at which the linear regime transitions to a sublinear regime decreases as r o increases. Thus, when prey scent propagates over a longer distance, the sublinear scaling of encounter rate persists to lower prey density.
As described in the main text, many studies have disputed whether organisms use search strategies that can be described as random walks in which the lengths of movements are drawn from statistical distributions with heavy tails, resulting in so called "Lévy walk" behavior. In the main text, we adopt a distribution for the intrinsic movements γ(θ, ), that has a power law tail with the exponent α = 2, which will lead to superdiffusive Lévy walk behavior. However, as we show in the main text and as has been shown in past work [1], such behavior does not lead to nonlinear scaling of the encounter rate function with prey density in the absence of sensory signals. When signals are incorporated, the effect of signal data can dominate the choice of intrinsic movement strategy such that, regardless of whether a predator uses a heavy tailed intrinsic movement distribution, or one that decays more quickly, the realized movement behavior is very similar [2]. Still, to ensure that our results were not determined by the use of a superdiffusive intrinsic movement strategy, we repeated all search simulations after changing the value of the Pareto exponent α to 3.5. For values of α above 3, the variance of the Pareto distribution is finite and the long-term behavior is diffusive rather than superdiffusive. Figure  3 Encounter rate of a predator with perfect sensing and response, and non-zero encounter radius Suppose that a predator is located at the origin of an n-dimensional environment containing prey distributed according to a Poisson spatial process with intensity ρ. We calculate the expected distance to the nearest prey to reveal the general relationship that the expected encounter rate scales with prey density ρ as ρ 1/n . Let np denote this distance to the nearest prey. Because we have assumed the prey are distributed according to a spatial Poisson process, the probability that there are no prey within a radius r of the predator is given by r | is the volume of an n-dimensional ball of radius r. Defining is the gamma function, we note that |B (n) r | = C n r n . Now, let e := max(0, np − r e ) denote the distance the omniscient predator has to travel to reach the encounter radius of the nearest prey. Using the formula for expectation and then integrating by parts, we can write Under the substitution y = ρC n r n , this integral becomes .

(S4)
where K is a constant that depends only on N . In the specific case of two dimensions, the random distance between the predator and the nearest prey is given by the Rayleigh distribution, which has density p( ) = 2ρπ e −ρπ 2 . We then observe that To observe the square root scaling, simply note that erf(x) → 0 as x → 0. It follows that Γ(ρ) ∼ 2 √ ρ/v in this regime. For larger ρ, the error function behaves like 1 − erf(x) = e −x 2 x √ π + O x −3 e −x 2 so that, to leading order, Because r e and ρ are small in the parameter regime of interest, there is a range of ρ, roughly from 10 to 100, for which encounter rate scales roughly linearly with ρ (i.e. e r 2 e πρ ≈ 1). This is seen in Figure 1 in the main text. As ρ becomes large, the scaling is exponential; however, for the cases of interest here (i.e. relatively low prey density), the exponential regime is not relevant.

Encounter probabilities in the sparse regime
When prey density is very sparse, each prey exists essentially in isolation. This is why the empirically observed probability of encounter with nearby prey stabilizes for low prey density (see Fig. 5, main text) In this section, we aim to estimate this probability in the sparse prey regime. As described in the main text, a proximity event begins when the predator comes within a radius r o of the prey. We pick this length because the expected signal size is one unit and the probability of the signal being nonzero is nontrivial (0.63). If the predator happens to take steps away from the prey it may reach a distance where it is exceedingly unlikely that another signal will be received from that prey. At such a distance, we consider the interaction to have ended without an encounter (and hence subsequent capture). To find an analytical estimate for this sparse regime scaling, we propose the following problem from classical probability theory. We approximate predator motion by a Brownian motion that has diffusivity D. The prey is located at the origin and the predator is located uniformly at random among all points that are a distance distance r o from the origin. We compute the probability that the predator hits a circle of radius r e before exiting a concentric circle of radius zr o . This is an exactly solvable problem. The predator's radial distance from the origin evolves according to a Bessel process R(t) that satisfies following Itô form stochastic differential equation [3] The probability that this process hits the level r e before zr o is given by the solution to the ODE Dp .

(S5)
The approximation is successful because in the presence of signal, the likelihood function in the Bayesian update, Equation (1), truncates the power law tail of the default Pareto distribution. Random walks with exponential jump tails are diffusive in character, meaning that Brownian motion can give a somewhat authentic scaling in r o and r e . Furthermore, note that the hitting probability for Brownian motion is insensitive to its diffusivity, meaning we do not have to attempt to tune the Brownian motion to match the imperfectly sensing predator. On the other hand, the effective diffusivity of the imperfectly sensing predator is certainly state dependent because larger signal magnitudes lead to shorter jump lengths. A further defect of the Brownian approximation is that it will always overestimate the encounter probability because the imperfect predator will occasionally experience zero signal when somewhat distant from the prey. This means imperfectly sensing predators will occasionally sample from the jump distribution with heavy tail and increase its chance of escape before reaching the prey.

Stability Analysis for the Predator-Prey model
In order to determine the local stability of the coexistence fixed point, we compute the Jacobian and evaluate at (R * , P * ), In the above formulation we have used the fact that, when written in terms of R * , the predator fixed point value is P * = f (R * ) ϕ(R * ) . The stability of this system depends on whether the trace occur when the trace T (R * ) = f (R * ) − f (R * ) ϕ (R * ) ϕ(R * ) is positive (unstable) or negative (stable). As we will see, for relevant choices of f and ϕ, there is a critical prey density R c that satisfies To understand the bifurcation more clearly, we consider the special choices f (x) = rx(1−x/K) and ϕ(x) = µ S µ T Γ(x) 1+µ T Γ(x) . We further suppose that Γ(R) = γR ν for some γ > 0 and ν ∈ (0, 1] in a neighborhood of the prey density fixed point R * . Checking whether the trace is positive reduces to checking whether A quick calculation shows that there is a unique R c and for all R * > R c , the coexistence steady state is stable. For all R * < R c , the fixed point is unstable; however numerical studies demonstrate there is stable limit cycle that contains the unstable coexistence equilibrium. For the form of functional response considered here (which is a Holling type II functional response when encounter rate is linear in prey density), all encounter rate models yield dynamics with a region of instability. Indeed, when the encounter rate is linear, Γ(x) = γx for some γ, which has the units of number of prey encountered per hour, R c = 1 2 K − 1 2γµ T revealing that the region of instability exists even for predators with linear encounter rate functions.