Edited by: Hava T. Siegelmann, University of Massachusetts Amherst, USA
Reviewed by: Thomas Boraud, Universite de Bordeaux, France; Asa Ben-Hur, Colorado State University, USA
*Correspondence: Joel Zylberberg, Redwood Center for Theoretical Neuroscience, University of California, 575A Evans Hall, MC 3198, Berkeley, CA 94720-3198, USA. e-mail:
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A prey animal surveying its environment must decide whether there is a dangerous predator present or not. If there is, it may flee. Flight has an associated cost, so the animal should not flee if there is no danger. However, the prey animal cannot know the state of its environment with certainty, and is thus bound to make some errors. We formulate a probabilistic automaton model of a prey animal's life and use it to compute the optimal escape decision strategy, subject to the animal's uncertainty. The uncertainty is a major factor in determining the decision strategy: only in the presence of uncertainty do economic factors (like mating opportunities lost due to flight) influence the decision. We performed computer simulations and found that
Prey animals frequently assess their surroundings to identify potential threats to their safety. If an animal does not flee soon enough in the presence of a predator (type I error), it may be injured or killed. If it flees when there is no legitimate threat (type II error), it wastes metabolic energy, and loses mating or foraging opportunities (Nelson et al.,
Previous studies have not investigated how escape decisions might be affected by prey animals’ degree of certainty about their environment. Indeed, the predominant assumption in the field appears to be that this uncertainty is not important, and that so long as the prey animal knows the most likely state of the environment (or the expected value of the state), they can still make economically optimal decisions. We question this assumption.
We explicitly consider the animal's uncertainty in our model and subsequently demonstrate that, when a prey animal knows the environmental state with certainty, the optimal decision strategy is simply to flee whenever a threat is present. This strategy is independent of any “economic” factors – predator lethality, predator frequency, loss of mating opportunities, etc. When the state of the environment is less certain, the animal is bound to make errors, and the optimal balance between type I and II errors is determined by economic factors. This is in contrast with previous theoretical studies (Ydenberg and Dill,
It has been observed that prey animals increase their flight initiation distance (predator–prey distance at which they flee) when intruders begin their approach from farther away (Blumstein,
The current models of escape decisions (Ydenberg and Dill,
Inspired by these observations, we propose a new approach for studying escape decisions in prey animals, namely that they are engaged in a decision-theoretic process, wherein they must decide, with imperfect information, whether the current environment is likely to pose a significant enough threat to their safety that they should flee. This view is supported by observations of active risk assessment behaviors in prey animals (Schaik et al.,
Sih (
We demonstrate through computer simulation that animals subject to predation naturally evolve to display the strategy predicted by our model, confirming our choice of objective function.
As a starting point, we will assume that the prey animal chooses the strategy
We formulate a probabilistic automaton (Rabin,
In each time step, the animal assesses a potential threat. For concreteness, we imagine the animal asking “Is that object likely to try to kill me?” Animals that do not flee from a real threat may be killed by a predator, while those that do flee, escape. Those animals that are not killed by predators may mate, and they may or may not die of causes other than predation.
We group potential threats into discrete “zones” in predator–prey distance; Figure
Qualitatively, our automaton model captures many features pertinent to real prey animals. Effects like periodicity of mating opportunities and threat frequency, maturation periods, learning during the lifetime of the animal (Hemmi and Merkle,
Figure
To simplify our notation, we first define the variables
where
The expectation value of the objective function (
The second line follows from the first since
Since the animals in our model assess one potential threat per unit time, the size of the “time steps” in our model is fairly short (seconds, or possibly minutes). In the real-world, we expect that actual threats are relatively uncommon (for example, the probability of encountering a real threat in any given short time period is small):
The anticipated escape response threshold maximizes the expectation value of the objective function
In the standard fashion (Boas,
The last of these equations enforces the constraint. The first two equations yield
And the solution to our optimization problem is (for
Note that, were all of the threats in one distance zone, Eq.
For an explicit computation of where the decision threshold should lie, we require information about the ROC curve. As an example, we assume that the score is distributed as
Note that, given a Gaussian-distributed variable
We need the derivative
Therefore, the optimal threshold for the
It is clear that as |
The decreasing importance of the “economic” term with increasing |
This can be seen by noting that, if the distribution of scores in the presence of a threat is
Solving for
For even
Thus, we can be assured that the decreasing importance of the economic term with increasing |
We note that, while it simplified our automaton model and our notation, nowhere was it necessary to assume that the danger occurs in discrete zones in distance. One could instead utilize a continuous distance measure by considering an infinitely large set of possible values of
To verify that our objective function is the one selected for by natural evolution, we perform a computer simulation of a population of prey animals subject to predation.
Our simulation contains a population of animals whose life cycles are described by the probabilistic automaton model (Figure
The animal is then presented with a “score” variable, with which it makes its decision. As in our analytic example, the scores are randomly drawn from the 𝒩(0,1) distribution if the threat is real, or from the 𝒩(
Those animals that do mate produce
At the end of every time step, the population is trimmed so that it does not get too large. This is done by killing random individuals, thus inducing no selection pressure. This is represented by the value
We initialize the simulation with a population of animals whose decision thresholds are drawn from a uniform distribution.
The strategy that maximizes the expectation value of
Figure
To investigate the influence of uncertainty on the decision strategy, we consider a specific example: the decision is made based on a single “score” variable
This score may be, for example, the output of a neural network that takes into account all of the information available to the animal, including information about the predator behavior, odor, morphology, etc. The use of a single score for the decision can be understood as a dimensionality reduction step: the high-dimensional sensory data is reduced to a single scalar value, upon which the decision can be based. In the case of an animal with a “command neuron” (e.g., the Mauthner cell; Rock et al.,
Let the distribution of scores in the absence of danger in zone
and
For large |
This result is true for Gaussian-distributed score variables, but is not true for all distributions. However, it is straightforward to prove that the result holds for all unimodal distributions in the exponential family
To verify our choice of objective function, and the approximations made in our calculation, we performed a computer simulation of a population of prey animals subject to predation. Unlike previous work (Floreano and Keller,
Some of the results of this experiment are shown in Figure
Variable | Range |
---|---|
[0.05, 0.6] | |
[0.5,1.0] | |
[0.01, 0.5] | |
[3,8] | |
[0.1,0.7] | |
[−7.0, −0.5] |
The results of the simulation (Figure
We stress that we made a specific choice of objective function for our analytic calculation, but that objective function was not available to the animals in our simulation. Had we made a different choice of objective function, our analytic calculations would have yielded different results, and those would necessarily not have been in agreement with the simulation results.
For example, choosing longevity as an objective function, one would choose the strategy that maximizes lifetime. Given the structure of our automaton model, that strategy is clearly to flee all of the time;
It has previously been conjectured (Cooper and Frederick,
We conclude that, given sufficiently accurate and detailed (probabilistic) information about the life cycle of an animal, it may be possible (although difficult) to make quantitative behavioral predictions.
We have found that a prey animal's uncertainty about threats in its environment has a profound effect on the optimal escape strategy. Moreover, computer simulations of the evolution of populations of animals subject to predation demonstrate that the objective function we assumed for our analytic calculations is, indeed, optimized by selection pressure.
Interesting work has modeled the learning process in the presence of uncertainty, in the context of optimal decision making (Rao,
Whereas much previous work (Ydenberg and Dill,
We expect that nearby threats will be more conspicuous: |
Similarly, when an intruder initiates its approach from further afield, the prey animal has more time to gain information about it. Thus, at a greater distance, the animal can correctly assess the threat, leading to a larger flight initiation distance, as observed in real prey animals (Blumstein,
Finally, when an odor or shape is presented to the animal that is associated with common predators, the score of the intruder will be far from the mean of the “no threat” distribution. Thus, defensive behavior is likely to be trigered.
Our decision-theoretic model for prey escape strategy can thus account for several observed behaviors (Blumstein,
We propose that approaches based on optimal performance in the face of imperfect information are likely to be useful for studying further aspects of escape decisions in prey animals, as they have been in other areas of biology such as mate selection (Benton and Evans,
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The authors are grateful to the William J. Fulbright Foundation and the University of California for funding this work.