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Two Approaches for Representing the Stochastic Dynamics of Foraging Behaviors

By Jillian Kunze

“Foraging is a ubiquitous behavior that all animals do, as they self-sustain by searching for food to acquire energy,” Ahmed El Hady of the Max Planck Institute of Animal Behavior and the Center for Advanced Study of Animal Behavior (formerly of Princeton University) said. Foraging occurs across multiple spatiotemporal scales, providing opportunities for study in the lab or the field. It is also sensitive to environmental structure. There is a long history of quantitative research into foraging behavior, and the field of neuroscience has also recently begun to show more interest in this area.

Figure 1. A rat in an environment with multiple patches of resources. Figure courtesy of Ahmed El Hady.
During a minisymposium presentation at the 2022 SIAM Conference on the Life Sciences, which took place this week concurrently with the 2022 SIAM Annual Meeting, El Hady described two modeling approaches for representing foraging in work done with Zachary Kilpatrick (University of Colorado Boulder) and Jack Davidson (Max Planck Institute of Animal Behavior). He specifically focused on patch-leaving, a canonical foraging behavior (see Figure 1). Since El Hady works mostly with rats, he used them as an example throughout the talk. “You can imagine this behavior as you have a rat in an environment with patches of resources,” El Hady said. “The rat will enter, eat some of the resources in the patch, and then leave at a certain time point — then enter into another patch and so on.” There are several behavioral observables in this case: food intake, residence time spent in a patch, and travel time between patches.

The classical approach to optimal foraging theory is the marginal value theorem, which assumes that resources are clumped into patches like trees, bushes, and herds. An animal will leave a patch when the current patch intake becomes equal to the average environmental intake rate. This theorem makes several very strong assumptions: that the animal knows the food arrival rate and environmental food availability, that food is available continuously, and that the environmental statistics do not change. Since this means that the marginal value theorem is very idealized, El Hady used it mostly as a guiding principle and measure for comparison throughout the talk.

Several mechanistic questions arise out of the marginal value theorem. How does an animal implement behavior according to the marginal value theorem or some similar rule, and how can one account for stochasticity or uncertainty in the foraging process? El Hady touched on two modeling approaches to address these questions: a heuristic accumulator model and a Bayesian approach.

Figure 2. An illustration of foraging drift. The rat has a decision variable in its brain that it updates as it finds food in a patch. Once the decision variable reaches a certain threshold, the rat will move on to a different patch. Figure courtesy of Ahmed El Hady.
Talking about the heuristic approach first, El Hady discussed foraging drift with a diffusion model (see Figure 2). “Let us propose that the rat has a decision variable in its brain that tracks the food encountered,” El Hady said. “Whenever it encounters a piece of food, the decision variable drifts; when it reaches a certain threshold, the animal leaves the patch.” The heuristic approach involves taking into consideration patch parameters like food density and patch size, as well as travel time between patches. El Hady proposed that the animal is keeping track of the available energy in the environment, and calculated the evolution of a decision variable in the animal’s brain. This decision probability density drifts over time until it reaches a threshold to leave the patch.

Optimizing the animal’s energy intake via the marginal value theorem can yield a relationship between the threshold and drift. El Hady described varying the threshold and drift rate to find different strategies for the animal to move between patches. In the increment-decrement strategy, if an animal finds food, it is then more likely to stay in a patch. In the decremental or robust counting strategy, animals that find food are more likely to leave. But when should an animal use each of these strategies? El Hady and his collaborators found that the increment-decrement strategy is best when the amount of food is unknown — if a creature does not know ahead of time how much food will be in a patch and then does find food there, it is more likely to stay. The decrement strategy is then optimal when the amount of food is known, since the animal has a sense of how much it is consuming out of the total patch.

“The question that comes afterwards is, ‘Can we derive foraging decision strategies normatively?’” El Hady said. “This is where we turn to the Bayesian approach for patch foraging.” El Hady and his collaborators modeled foraging behavior as a process in which the animal gathers statistical evidence about the qualities of patches and the environment. However, there are drawbacks to the Bayesian models that researchers currently use for foraging, such as not being extendable to learning environmental statistics. They are also difficult to relate to motion data, body posture, and neural dynamics — since El Hady is interested in applying this work to neuroscience in the future, this was a significant disadvantage.

In the new approach that El Hady described, he treated habitat choice and patch leaving as a statistical inference problem. He considered an animal that is entering a new environment and choosing between different large habitat areas, each of which contain multiple patches of resources. This can be a statistical inference problem in that the animal updates its beliefs about a patch as it finds food. The researchers therefore used Bayes’ rule to calculate the animal’s belief and simulate its change over a long time period.

Figure 3. Three habitats that are respectively very rich (green), less rich (yellow), and poor in resources (red). An animal searches between the patch types over time before eventually remaining in the high-resource patch. Figure courtesy of Ahmed El Hady.
El Hady considered an animal that is selecting between three habitats that are respectively very rich, less rich, and poor in resources (see Figure 3). The animal will search for, then remain in the patch with the highest resources. But what strategy should it adopt to minimize the time required to find the highest-quality patch? 

For an idealized situation with a binary environment—one high-yield and one low-yield habitat—El Hady wrote down the stochastic differential equation for the change in an animal’s belief over time. The animal will switch between patches over time until it ultimately remains in the high-yield patch, and its optimal strategy is to arrive and remain in the best patch as soon as possible via a series of decisions to stay or go from the patch it is currently in. “The most important thing we talked about here is minimizing the time to arrive and remain in the high-yield patch,” El Hady said. 

The collaborators found that the arrival time for staying in the high-resource patch gets longer as that patch becomes less discriminable from the low-resource patch. The more discernible that the high patch is in the environment, however, the lower the arrival time. “Minimizing the time to find the high-quality patch is a very strong deviation from the typical assumptions of the marginal value theorem,” El Hady said. “The discriminability of the high patch strongly shapes the time and strategy to find the high patch.”

  Jillian Kunze is the associate editor of SIAM News