CAM Final Report John Scheele Advisor: Paul Ohmann I. Introduction Herds are a classic complex system found in nature. From interactions amongst individual animals, group behavior emerges. Historically the interactions of predator and prey have been modeled using systems of differential equations. Recently though it has become possible to model the behavior of individual animals. In doing so, it is possible to find the optimal movement model for a herd of animal that will increase its survivability. In nature many animals herd together to increase their protection, which is encapsulated in the Selfish Herd theory (Hamilton 1971). Previous papers have looked at how the application of random motion (noise) to a movement model helps herd survivability against a predator (Ose and Ohmann 2017) and optimizes a herd s ability to forage (Kay 2018). Our current project is meant to combine these separate ideas into the same system and evaluate which movement rules best optimize herd survivability as they traverse a barrier full of predators. It is hoped that in doing so the results will shed light on how foraging herds interact with predators. II. Overview In Nick Ose and Paul Ohmanns paper The selfish herd: Noise effects in Local Crowded Horizon and Voronoi Models, they examined how noise effects the survivability of a herd. They used two different tests based on ideas from Hamilton and from Vine to evaluate the effect of noise on a herd s survivability. Hamilton connected predation risk with the area around an animal that is exclusively closest to it, or Domain of Danger (DOD) (Hamilton 1971). While Vine pointed out the need to consider predators from outside of the herd (Vine 1971). To find the DOD for each animal in the herd, Ose and Ohmann used Voronoi tessellations. A Voronoi tessellation is comprised of individual voronoi cells, which encapsulate the area around an animal that is exclusively closest to it.
(Figure -1: A Voronoi tessellation for 20 animals. The area inside of the borders for each blue dot is the animals DOD.) To find the change in DOD for each animal in Ose and Ohmann, they calculated the area of the Voronoi cells at the beginning and end of a simulation. The percent change in the Voronoi area for the herd was used to evaluate how effective a movement rule was. Ose and Ohmanns paper examined how different movement rules, with and without noise, effected the DOD and predation risk of herd members. For modeling herd behavior, two movement models, the Local Crowded Horizon (LCH), and Voronoi (V) movement models were evaluated. These two models have differing approaches to modeling the motions of animals in a herd. The LCH model is a weighted distance function that creates a movement vector for a herd member based on a weighted distance to every other animal in the herd. The LCH model comes from observations of fiddler crabs (Viscido et al, 2002).
(Figure 2: A diagram illustrating the weighted vectors to other herd members and the combined movement vector for the LCH rule) For a herd member following the LCH movement model a weighted distance vector is created using the equation f(x) = 1 1+kx where x is the distance from the ith animal to its neighbor for each animal. This process is done for each animal in the herd so that for n animals there are n 1 movement vectors. These vectors are combined to make the movement vector for each animal. Rather than look at every other herd members position, as in the LCH model, the V model focuses only on the animals voronoi neighbors. A herd members voronoi neighbors are the herd members that are in adjacent Voronoi cells to the animal. The voronoi tessellations used are the same as those used to calculate a herd memebrs DOD. An example tessellation can be seen below.
(Figure 3: An example of the voronoi movement rule. The red area is the current animal and the animals in green are its voronoi neighbors.) In their paper, Ose and Ohmann (2017) evaluated the LCH and V movement models effectiveness w/ w/out noise using two tests, Domain of Danger (DOD) and Mixed Herd. In DOD tests, they looked at how the DOD of one hundred animals improved for both movement rule with and without noise. These simulations highlighted the importance of noise in promoting a cohesive herd and minimizing the herds overall DOD. For Mixed Herd simulations, two herds were spawned which would follow separate movement rules, while still interacting between them. They moved following their respective movement rules for a set amount of time steps before a predator was spawned outside of the herd. This predator than moved towards the nearest herd member until it caught it. In mixed herd runs, the movement rule with the most deaths was considered the less successful of the two. In both simulations, Ose and Ohmann (2017) found that noise improved the herd s survivability for both movement rules. In the recent paper A Matlab user-interface tool for modeling herds by Scheele, Ohmann and Green (2018), this idea is taken a step further by building a User Interface (UI) that
facilitates the running of different simulations with ease. This UI allows a person to set up a simulation without having to change the underlying code, and simultaneously makes it easier to collect and interpret data. (Figure 4: User Interface from Scheele et al using the UI makes it possible to run the code and collect data much more efficiently then without.) The other addition to Ose and Ohmann that comes from the UI is the ability to vary the noise weight for different simulations. This allows the evaluation of how much noise is needed to optimize the movement rules. Kay and Ohmann (2018) looks at how herds move and forage with three influences acting on each individual: Voronoi neighbors, a destination and random motion. From their simulations they found that a noise weight greater than the other influences optimized a herd s movement speed and ability to forage while traveling towards a destination. III. Methods To build upon the previous work we will continue to use MATLAB to code our herding models for complex simulations. We are using MATLAB because the previous work on herd behavior is coded in MATLAB and because of our familiarity with the coding language. MATLAB also has built in functions for calculating and plotting voronoi tessellations that can be used instead of coding them from scratch. We are also using MATLAB to process and save data
from the runs with built in functions when possible and by coding our own analysis code when nessacery. In order to facilitate multiple simulations without changing the underlying code we will also be rebuilding the old UI for Scheele et al for use with new code. We are also perusing journals online to find new ideas and/or examples of the system we want to model. This allows us to tie in our research with existing observations and ideas on animal behaviors. In doing so we can test our models on the computer versus events that actually happen in the wild. IV. Current Work For our research project, we started by using the EJP code as a starting point and incorporated Kay and Ohmann s usage of a destination for the herd. This destination is the foraging zone for an animal herd that they will try to reach by crossing a barrier full of predators. (Figure 5: Plot of a herd crossing a barrier that contains one predator while there is food on the other side.) By having, the herd spawn on one side of a barrier and travel across it to eat from these food sources using different strategies; we will try to find which strategies optimize a herd s ability to cross a barrier with predator to reach food sources on the other side.
We began with an overhaul of the UI from Scheele et al to incorporate new features for modeling more herd behaviors. These new features allows us to control settings for the predator, herd, foraging and barrier crossing strategies. By running simulations, using various settings it is then possible for us to look at which strategies are effective in promoting herd survivability. We also changed the UI from two to three columns to have room for all of the new controls that are needed. (Figure 6: Revised UI with controls for each aspect of the simulation. The revised UI still allows us to view data as it is collected from the simulations. ) Along with updating the UI, we had to change the underlying code to allow new runs to be carried out. This meant that everything needed for our new herd model has to have a corresponding function in the code; other functions previously used also had to be modified or removed. The major changes in new the code from the one in Scheele et al is that mixed herd testing is no longer supported. This allows us to focus on the optimal strategy for a single herd s survivability. For the new simulations, we added a food budget for each herd member. This food budget is used in combination with a foraging threshold to determine when a herd member will switch from herding to foraging behavior. Every time step the animals food budget will decrease based on what it did, moving costs an animal more food than if it stood still, and if a herd
member runs out of food it will die. We use this decreasing food budget only as a way to get the animals to begin crossing the barrier to eat food and replenish their reserves. As the food budget decreases, an animal is more inclined to travel to a food source than to herd. A scattered food spawn is used to spread the specified amount of food chosen in the UI around the foraging zone for an animal to eat. Each food source is worth 10% of an animal s food budget. When a food source is eaten, a new food source s then randomly spawned so that there is always the same amount of food in the foraging zone. While there are herd members in the barrier, each predator will move towards the nearest animal to it inside the barrier. During their time inside the barrier, there are different strategies that a herd member can follow to improve its survivability. Each of these different strategies are controlled in the UI and change which components of the movement vector a herd will follow inside the barrier between the span and the food. It is also possible for the herd member to move directly across the barrier ignoring any other movement vector. V. Results Currently, we have coded most of the simulation tools we need to examine which of the various strategies for herds crossing a barrier are optimal. The UI and code are currently able to carry out a simulation that has a herd cross the central barrier to get food with different settings. However, there are a few things left to do before we can begin collecting data with the UI. There are still a few simulation tools that need to be added to the code and UI. We still have to fine tune the simulation and set up test parameters for the runs that reflect a general situation. Even though the UI is incomplete, we are still able to use it to get a better feel for the research problem as a whole. By doing simulations with the current features we can see if there are any patterns or flaws in the strategies used by the herd. This allows us to better focus our literature search to find information necessary realistically model natural systems. In doing so we are able to see how our model can be improved or simplified to better match the real world. VI. Future Ideas While we have a good start on the research task, there is still a large amount of work to be done on the research project before we can begin collecting data to search for optimal
movement strategies. The current simulation types need to be expanded in order to reflect different foraging strategies. For example, we may incorporate a food density based foraging method where a herd member will travel to the nearest patch of food above a set density. We are also looking how to run simulations where the predator, prey and food are all present in the same region. Instead of looking for optimal barrier, crossing strategies we would be looking for which movement strategies optimized the survival of the herd. This will be accomplished by one of two methods: having a single spawn region that they all spawn in or by allowing predators to spawn in the foraging and prey starting zones. Even though the predators spawn locations is changed, we will not change its movement or eating functionality so that we can continue to focus on optimal herd strategies. VII. Bibliography Hamilton, W. D. (1971).Geometry for the selfish herd. J. Theor. Biol. 31, 295 311 John Scheele et al 2018 Eur. J. Phys. N.J. Ose and P.R. Ohmann (2017), Journal of Theoretical Biology 424, 84-90 T.M. Kay, P.R. Ohmann, Effects of random motion in traveling and grazing herds, Journal of Theoretical Biology (2018) Vine, I. (1971). Risk of Visual Detection and Pursuit by a Predator and the Selective Advantage of Flocking Behavior. J. Theor. Biol. 30, 405 422. Viscido, S. V., Miller, M. & Wethey, D. S. (2002). The Dilemma of the Selfish Herd: The Search for a Realistic Movement Rule. J. Theor. Biol. 217, 183 194.