Simulations as a Mathematical Tool

1. Simulations as a Mathematical Tool D. J. Kaup Department of Mathematics & IST John E. Fauth & Linda Walters Department of Biology Rex Oleson III Institute for Simulation & Training (IST) Linda Malone Department of Industrial Engineering & Management Systems Tom Clarke Institute for Simulation & Training (IST)

2. Goals & Outline • Goals • Introduce simulation models • Demonstrate utility • Outline • Simulation models • Biological example: niche partitioning in salamanders • Simulation of interacting salamanders (R. Oleson) • Future research

3. Simulation Models • Goals of simulation: Understand and predict behaviors of organisms in complex environments • Models 1. Scripted 2. Agent-based • Genetic Algorithm • Neural Networks 3. Social Potential • Flocking • HMFV • Lakoba-Kaup-Finkelstein

4. 1. Scripted

5. 2A. Genetic Algorithms • Attempt to follow features of natural selection. • Train an agent over many iterations by giving positive and negative feedback. • Each iteration the agents reproduce. • A “mutation” is allowed to occur during reproduction. • Each agent must compete at the end of each iteration; only top 10% survive.

6. 2B. Neural Networks • Agents need to be trained in the environment they will experience. • Very generic and requires little initial setup. • Training is very lengthy and time consuming. • Sometimes the agents do something that seems wrong; the neural network does not give insight into why.

7. 3. Social Potentials • Heavy emphasis on path planning and obstacle avoidance • Used for robot motion • cij is the coefficient • σi,j is the inverse power

8. 3A. Flocking Model • Fluid-like motion built up from a series of independent entities • Each entity’s actions based on its local perception of the world • Based on 3 parameters • collision avoidance • velocity matching • flock centering

9. Flocking Implementation • Parameters • Cohesion • Avoidance • Randomness • Consistency • Execute Simulation

10. Social forces Physical forces (repulsion/attraction) (pushing, friction) + 3B. Helbing, Molnar, Farkas & Vicsek (HMFV) Model • HMFV Model builds on the social potential force model: • Individual behavior leads to collective behavior

11. Social forces Tendency to keep preferred speed Repulsion (tendency to keep distance from others, and from boundaries) Attraction to exit(s) D attr >> D rep (non-infinite D attr plays role when a person decides which exit to head) As panic increases,

12. Physical forces Pushing and Friction (when pedestrians come in contact with each other) • Note: • Physical forces do not depend on • relative orientation of • pedestrians. • - By themselves, pushing forces • do NOT prevent pedestrians • from “walking through” each • other !

13. 3C. Lakoba, Kaup & Finkelstein (LKF) Model • Adjusted values of HMFV model to be physically correct. • Included additional social parameters required for realism. • Ability to learn and forget about location of an exit and walls. • Knowledge of locations is used to determine: • Direction in which a pedestrian is looking. • Attraction force to the exit (similarly, repulsion from walls). • Correctly reproduces realistic collective and individual behaviors.

14. LKF Model • Equations are stiff Code has to resolve two disparate scales: • LARGE: distances about the size of the room ( ~ 10 m). • Small: distance between pedestrians when they come into contact ( ~ 1 cm). • New overlap algorithm developed to eliminate overlap among pedestrians. • Allows stable solutions using the explicit 1st-order Euler method.

15. Room Geometry • Using the LKF Model, we simulated angling the walls toward a doorway to see how it affects pedestrian motion. • Angle ranged from 0 to 90 degrees • Goal: Optimize the number of individuals that can escape from the room.

16. Simulation: 0 degrees See videos section this website to view simulation

17. Simulation: 30 degrees See videos section this website to view simulation

18. Simulation: 90 degrees See videos section this website to view simulation

19. Biological Example: Niche Partitioning in Salamanders • Niche partitioning is a core concept in ecology. • Environmental gradients (ecotones) are common in nature. • Salamanders are an ideal model system. • Long history of research on plethodontid salamanders.

20. Prior field experiments • Fauth manipulated the presence and absence of several species of plethodontid salamanders: • Desmognathus, Plethodon, Eurycea, Gyrinophilus, Pseudotriton • Analyzed survival, growth, and microhabitat use along the aquatic-terrestrial ecotone.

21. Niche Partitioning in Mesocosms

22. Mimics Niche Partitioningin Nature Plethodon Desmognathus

23. Simulated Environment cover objects Aquatic Terrestrial

24. Simulated vs. Mesocosm Environments

25. Social forces physical forces (repulsion/attraction) (pushing, friction) + Salamander Parameters • Regional Affinity (= microhabitat preference) • Cover Attraction • Cover Memory Attraction (~ site fidelity) • Food Attraction • Water Attraction • Salamander to Salamander Repulsion • No physical forces

26. Virtual Simulation

27. } enhances survival } shifts niches Predictions from Simulation • Parameters most important for producing niche partitioning (in order of importance): • Regional Affinity: yields microhabitat preference • Cover Attraction • Cover Memory Attraction • Food Attraction • Water Attraction • Salamander to Salamander Repulsion: • Yields one salamander per cover object • Move to cover when threatened.

28. Summary • LKF Model yields specific predictions about forces driving niche partitioning, microhabitat shifts, and survival. • Input to simulations was only qualitative. • Output of simulations is quantitative. • Model is flexible and can be customized to diverse ecological scenarios.

29. Future Directions • Interactions among different species of salamanders along differing environmental gradients. • Behavior of animals using ecopassages. • Dynamics of oyster recruitment. • Other collaborations???