MATLAB HS13 – Research Plan

Fishy03

Naterop Luca, Prasad Nishant, Petrides Ioannis

Testing swarm models for different population sizes and influential neighbours

General Introduction

In a very general sense, the term swarm is usually meant to signify a group of objects (agents) that interact with one another and have a collective goal. It has long been recognized that such interactive, cooperative behaviour within biological swarms is advantageous in avoiding predators or, vice versa, in capturing prey.

A lot of researchers have dabbled with modelling swarm behaviour in the past, not only to attempt to truly reflect actual swarm behaviour, but also to look into the applicability of this concept towards innovations in the real world that involve decentralized organisation, like cooperative robotics, vehicle navigation, optimization techniques, telecommunication network design, and the likes.

While going through different research articles, we found that a lot of them assumed that swarm population size and number of influential neighbours were relatively insignificant parameters contributing to overall swarm dynamics – which may be true for small swarms, but we want to test this notion for relatively larger population sizes, since we intuitively feel that they could heavily influence swarm properties like overall velocity, response time towards external forces and swarm splitting characteristics.

We will therefore use a purely mechanistic mathematical model (ignoring other custom swarm “rules”), and run the same over wide ranges of population, and neighbourhood distances, to gain a broader understanding on the significance of such factors towards overall swarm behaviour.

The Model

In our model, a swarm consists of a group of objects (agents), interacting with each other using forces. One force will act between members of the swarm, repelling agents at short distances and attracting at large distances. A swarm member won't be influenced by each member, but only by those who do not exceed a given neighbourhood distance. A second force attracts the agents to a source of food; and a third, repelling only on short distances, makes sure the agents do not collide with objects in the environment. Since we consider fish swarms, we have to include water resistance as an additional force.

Agents are considered as point masses in a two-dimensional, continuous world with open boundary. In every cycle of computation, positions, velocities and accelerations are changed due to the forces mentioned above.

Variables we want to study:

Independent Variables:

• Population size (number of agents)
• Neighbourhood distance: maximum distance of an influental neighbour

Dependent Variables:

• Interaction forces: Member-member-forces and member-environment-forces
• Acceleration, velocity and position of agents
• Cohesiveness: (1/A, where A is the area of the smallest circle around all members of a swarm)
• Splitting number: Amount of independent subgroups
• Average velocity of agents
• Density: population size x cohesiveness

Fundamental Questions

• How does the population dynamics change with varying neighbourhood distance?

  • Is there a change in the behaviour of the swarm (i.e. when persuing food)?
  • Do individual agents get isolated by the rest?
  • Does the average velocity decrease with increasing neighbourhood distance?

• How is the spread function modified by population size?

  • How does the cohesiveness change with increasing population?
  • What is the evolution of the swarm in an environment with objects (i.e food & obstacles)?
    • Does it have a characteristic or chaotic evolution?
    • Are there any particular formations prefered?

• How does the density vary with respect to changes in population size and neighbourhood distances?

Expected Results

• Even though the number of influential neighbours may be insignificant at low population numbers, we expect a high dependence for larger numbers.

• The cohesiveness should initially decrease drastically with growing population number, and eventually approach a constant value. In addition, formation characteristics (such as splitting into subgroups) are anticipated for large populations.

• By increasing the number of influential neighbours we assume stronger bondings, thus we expect a lower probability of certain characteristics such as splitting into subgroups and secluded agents.

• Increase of cohesiveness for increase neibhbourhood.

• Increased chance of splitting up in presence of objects with decrease of neighbourhood.

• For different population sizes, the product of cohesiveness and population (densitiy) should be constant for constant neighbourhood.

• For decreasing neighbourhood distance, we expect an increasing average velocity.

References

• T.I. Zohdi, “Mechanistic Modeling of Swarms”, Comput. Methods Appl. Mech. Engrg. 198 (2009) 2039–2051
• Viscidoa, Parrisha, Grünbaum, "The effect of population size and number of influential neighbors on the emergent properties of fish schools", Ecological Modelling 183 (2005) 347–363

Research Methods

We intend to implement an agent-based model in MATLAB using discrete time intervals to simulate our model.