Agent-based models, multi-agent systems

Fish school

Some of the most beautiful, fascinating or strange phenomena of nature can be understood as emerging from the behaviors of interacting agents. Widely acknowledged examples include the murmuration of birds (or swarming insects, schools of fish), and the societal 'superorganisms' of ant colonies. We have come to understand that despite the obvious organization that we see at the macro-scale, there is no hierarchical center of coordination, but rather the whole emerges from simple interactions at local levels. These have been suggested as examples of emergence, or of self-organization. Which is to say, the whole is greater than a naive sum of its parts.

Agent-based models, or multi-agent systems, attempt to understand how non-trivial organized macro-behavior emerges from typically mobile individuals that interact primarily at a local-level. These individuals are called agents. Besides modeling flocks and colonies, agent-based modeling has applications from microbiology to sociology, as well as video games and computer graphics. As a biological approximation, an agent could refer to anything from an individual protein, virus, cell, bacterium, organism, or deme (a population group). Agent systems also share many features with particle systems, and the two are sometimes conflated.

What is an agent?

The agent abstraction has arisen somewhat independently in different fields, thus the definition of an agent can vary widely. However, in most cases, an autonomous agent interacts within an environment populated by other agents, but behaves independently without taking direct commands from other agents nor a global planner or leader. Agent-based models consist of populations that typically operate in parallel within a spatial environment. Interactions are usually limited to local distances, rather like cellular automata. But unlike CA, which roots a program in a particular spatial location (the cell), an agent-based program is typically mobile. The following components are usually present:

Just like CA, at times, the self-organizing behavior of systems of even relatively simple agents can be unpredictable, complex, and generate new emergent structures of order.

Tortoises and Vehicles

Machina Speculatrix

In the late 1940's and early 1950's, Cyberneticist Grey Walter pioneered the engineering of autonomous agents, as early examples of self-directed robots, with his series of "tortoises".

Remarkably, this direction of research was largely forgotten as efforts in artificial intelligence concentrated on symbolic thinking. A brief history can be read here -- but the story of Walter and other British Cyberneticians is truly fascinating; I highly recommend Andrew Pickering's account in The Cybernetic Brain. Walter's tortoises have inspired many great research products of the last century, including the turtle graphics of Logo, the situated robotics of Rodney Brooks, the flocking behaviors of Craig Reynolds, and Valentino Braitenberg's Vehicles.



A Braitenberg vehicle is an agent that can autonomously move around. It has primitive sensors (measuring some stimulus at a point) and wheels (each driven by its own motor) that function as actuators or effectors. A sensor, in the simplest configuration, is directly connected to an effector, so that a sensed signal immediately produces a movement of the wheel. Depending on how sensors and wheels are connected, the vehicle exhibits different behaviors (which can be goal-oriented). wikipedia

Braitenberg, V. (1984). Vehicles: Experiments in synthetic psychology. Cambridge, MA: MIT Press.

Cyberneticist Valentino Braitenberg argues that his extraordinarily simple mechanical vehicles manifest behaviors that appear identifiable as fear, aggression, love, foresight, and optimism. The vehicle idea was a thought experiment conceived to show that complex, apparently purposive behaviour did not need to depend on complex representations of the environment inside a creature or agents brain. In fact simply by reacting to the environment in a consistent manner was more than enough to explain the low level reactive behaviours exhibited by many animals.

Casey Reas (co-author of the Processing creative coding environment), Yanni Loukissas, and many others have used populations of Braitenberg-inspired vehicles to create artworks based on their combined paths.

Reas' Tissue

Vehicles have also been constructed in hardware of course // see examples here, here, here.

Steering Behaviors

Craig Reynolds' work with robotics is strongly inspired by Braitenberg's and Walter's vehicles. In the late 1980s Reynolds proposed a model of animal motion to model flocks, herds and schools, which he named boids. Reynolds' paper Steering Behaviors for Autonomous Characters covers boids as an aggregate of other steering behaviours. The paper has been very influential in robotics, game design, special effects and simulation, and remains well worth exploring as a collection of patterns for autonomous agent movements, so we will work through it in more detail.

His paper breaks agent movement in general into three layers:

His paper concentrates on several different steering algorithms. Locomotion is extremely simple -- equivalent to a single force vector on a point mass -- but a more complex (and thus constrained) locomotion system could be swapped in without changing the steering model significantly. Action selection is not dealt with, but also arguably independent.

Here for example is the "seek" and "flee" steering methods, which derive desired velocity from a the relative vector toward (or away from) a target point. Given a desired velocity, the steering force is obtained by subtracting the current velocity.

Here for "seek" extended with obstacle avoidance. The basic concept is to project forward a rectangle of the same width as the agent, up to a limit of visible range, and see if it intersects with the obstacle. If it does, a lateral force opposite in magnitude to the expected overlap is added to prevent a collision happening.

Random walks in nature

One of the most useful strategies, which he calls wander, is based on the random walk. Random walks are a well-established model in mathematics, with a physical interpretation as Brownian motion. Essentially, for an agent a random walk involves small random deviations to steering. This form of movement is widely utilized by nature, whether purposefully or simply through environmental interactions.

See the random walker example here. Rather than simple deviations to the current heading, this follows Reynold's method to generate headings based on deviations of an ideal point just ahead of the agent, which is more parametrically defined.

And here are many walkers -- to help demonstrate how to support populations, and how quickly random walkers can cover a terrain. This is on reason why random walks feature in many foraging behaviours in nature.

Boids, flocking, swarms


In the late 1980s Reynolds proposed a model of animal motion to model flocks, herds and schools, which he named boids. Each boid follows a set of rules based on simple principles:

In addition, if none of the conditions above apply, i.e. the boid cannot perceive any others, it may move by random walk.

To make this more realistic, we can consider that each boid can only perceive other boids within a certain distance and viewing angle. We should also restrict how quickly boids can change direction and speed (to account for momentum). Additionally, the avoidance rule may carry greater weight or take precedence over the other rules. Gary Flake also recommends adding an influence for View: to move laterally away from any boid blocking the view.

Evidently the properties of a boid (apart from location) include direction and speed. It could be assumed that viewing angle and range are shared by all boids, or these could also vary per individual. The sensors of a boid include an ability to detect the density of boids in different directions (to detect the center of the flock), as well as the average speed and direction of boids, within a viewable area. The actions of a boid principally are to alter the direction and speed of flight.


The behavior of an agent depends on the other agents that it can perceive (the neighborhood). The simplest way to detect nearby agents is to simply iterate all agents and apply a distance condition (being careful to exclude the agent itself!). We may also include a view angle condition (easily calculated using vector dot product).

This isn't especially efficient, but for small populations it is quite reasonable.

Once a set of visible neighbors is calculated, it can be used to derive the steering influences of the agent. The center force depends on the average location of neighbors, relative to the agent. The copy force depends on the average velocity of neighbors. The avoidance force applies if a neighbor is too close.

Note that since the agents are dependent on each other, it also makes sense to perform movements and information processing in separate steps. Otherwise, the order in which the agent list is iterated may cause unwanted side-effects on the behavior. (This multi-pass approach is similar in motivation to the double buffering required in many cellular automata).

Several choices have to be made to balance the forces well -- how far agents can see, over what radius, what kind of clipping on the strength of forces, or clipping on the resulting velocities, what range and sensitivity does the avoidance force have, what relative weight does the center force have, etc. Varying these can produce quite different behaviours. Do some forces (e.g. avoidance) completely override others? Do some forces apply only intermittently (by probability)? Etc.

See one example of flocking boids here.

Environmental interaction

One of the attractive features of the Boids model is how they behave when attempting to avoid immobile obstacles. For a small number of obstacles, we can simply use a list with a similar avoidance force routine.

Several pretty flockers with obstacles


Chemotaxis is the phenomenon whereby somatic cells, bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemicals in their environment. This is important for bacteria to find food (for example, glucose) by swimming towards the highest concentration of food molecules, or to flee from poisons (for example, phenol). In multicellular organisms, chemotaxis is critical to early development (e.g. movement of sperm towards the egg during fertilization) and subsequent phases of development (e.g. migration of neurons or lymphocytes) as well as in normal function. wikipedia

Note that here the environmental features being interacted with are not spatial objects, but spatial fields: things that occupy all space but in different intensities.

When we look at microbiology we can find some remarkably simple mechanisms to achieve chemotaxis. For example, the E. Coli bacterium's goal is to find the highest sugar concentration. But it has no eyes or ears to sense at a distance, it can only sense the local sugar concentration at its current location. Moreover it has no sense of direction, nor any internal "map".

Here, E. coli uses its flagella to move in just two modes of locomotion:

How does this solve the problem? It uses a very simple chemical memory to detect whether the concentration at the current moment is better, worse, or not much different to how it was a few moments ago. That is, it detects the differential experienced, or gradient traversed. Knowing whether things are getting better or worse is used to select between the swimming or tumbling patterns. With just a few tuning parameters, the method can lead to a very rapid success.

Note that this method works when the variations of sugar concentration in the environment are fairly smooth, which is generally true for an environment in which concentrations diffuse.


The first thing we need is an environment of varying sugar concentrations for the agents to explore. We can use field2D again for this purpose. The behavior of the agents will depend on the spatial distribution of sugar in the field; a totally random space is both unrealistic and will defeat chemotactic strategies; a smoothly distributed landscape is needed. For example, we can use the distance from the center:

var dim = 256;
var sugar = new field2D(dim);
var center = new vec2(0.5, 0.5);

sugar.set(function(x, y) {
    // convert x, y in to 0..1 range:
    var p = new vec2(x / dim, y / dim);
    // get distance from center:
    var d = p.distance(center);
    // make concentration high at center, lower with increasing distance:
    return 1 - d;

Or, for a more interesting and variegated landscape, we can start from uniform noise, and then smoothen it out:

// fill with uniform noise
sugar.set(function(x, y) { return random(); });
// smoothen it out by long-range diffusion:
sugar.diffuse(sugar.clone(), sugar.width, 100);
// make it vary between 0 and 1:

We'll need two different steering behaviours, for normal swimming & for tumbling. These should not be completely independent, i.e. swimming should also include a small amount of turning (to look realistic!), and tumbling should include a small amount of forward movement (otherwise tumbling won't be effective). For example:

// tumbling behaviour:

// swimming behaviour:

Agents can then sample the local field during their update routine as follows:

// in per agent update routine:
var sensed_sugar_concentration = sugar.sample(a.pos);

Then all we need is to compare the sugar concentration with the agent's memory (a stored member variable) of the concentration on the last time step, and choose the behaviour accordingly (and of course, store our sensed value in memory for the next update).

Here's an implementation of Chemotaxis in the editor.

A variety of other taxes worth exploring can be found on the wikipedia page. Note how chemotaxis (and other taxes) can be divided into positive (attractive) and negative (repulsive) characters, just like forces (directly seen in steering forces). This is closely related to the concepts of positive and negative feedback and the explorations of cybernetics.


So far, our sugar field is static. We could regenerate the field in response to user interaction (e.g. click to regenerate), but it would be nice to see what a continuously dynamic field offers. For example, we could let the mouse add more sugar to the space, and watch it gradually diffuse away.

Adding sugar in response to the mouse is fairly easy, by making use of the deposit method of field2D. This method accumulates into the field at a point coordinate, adding to its existing values. Note that the coordinate is in the normalized 0..1 range, and that it will spread the value over the nearest four cells:

function mouse(e, pt) {
    // add one unit of sugar at the location of the mouse:
    sugar.deposit(1, pt);

Another way is to have a separate agent (here called "source") randomly walking and adding sugar, from inside the update() method:

    // apply source agent to sugar:
    sugar.deposit(2, source.pos);
    // and then move it (random walk):

So far this isn't completely effective to attract agents, as there can be large spatial discontinuities. If agents happen to be on part of a drawn path, they might follow it, but otherwise they are unlikely to find it.

To create a smoother gradient, we must diffuse the field continuously. To add continuous dynamics, we need to add field processing to our update() routine. And since the diffuse() method requires a distinct source field, we'll need to double buffer like before:

var sugar_old = sugar.clone();

function update() {
    // update field:
    var tmp = sugar_old;
    sugar_old = sugar;
    sugar = tmp;
    // diffuse it
    sugar.diffuse(sugar_old, 0.1);

    ... update agents as before

Note that since we are applying this process 60 times a second, we set the diffusion rate to be low (0.1 in the above code fragment). Nevertheless, it rapidly diffuses to a lower concentration that is hard to see, so we should probably also increase the quantity deposited each time (e.g. to 10 units). Now the agents are able to find the areas of sugar more easily -- and even show something like trail-following.

But we also notice that even when the sugar is quite dissipated, the agents can still find the strongest concentrations. If we don't find this realistic, one thing we might consider adding is a low level of background randomness. The rationale is that even if sensing is perfectly accurate, small fluctuations in the world (such as due to the Brownian motion of water and sugar molecules) make it impossible to discern very small differences. The randomness added must be signed noise, balanced around zero such that overall the total intensity is statistically preserved. Note that the amplitude of this noise will have to be extremely high if it is applied before the diffusion, or very low if applied after diffusion. The effective results are also quite different.

    // applied after sugar.diffuse:
    // use to modify a cell value
    // adding a small random deviation
    // whose average is zero { 
        return v + 0.01*(random() - 0.5);

Still, the sugar just seems to accumulate over time, and the more we draw, the more the screen tends to grey. This is because our diffuse() method spreads intensity out, but does not change the total quantity. We need some other process to reduce this total quantity to make a balance. One way to do that is to add a very weak overall decay to the field -- as if some particles of sugar occasionally evaporate:

// in update():

Of course, our agents aren't just looking for sugar to show it to us -- they want to eat it! We can also add the effect of this on the environment by removing intensity from the field by each agent. We can do this by calling the field2D.deposit() method with a negative argument (i.e. a debit!):

    // get the sugar level at this location:
    var sense = Math.max(sugar.sample(a.pos), 0);
    // update the field to show that we removed sugar here:
    sugar.deposit(-sense, agent.pos);

Note that we add a Math.max(0, sense) component here, just as a sanity check. There is no such thing as a negative concentration of sugar, but it might happen that our field has negative values. We wouldn't want those to make our agents create food and put it back! Of course this also suggests that we think about how to actually start using the acquired energy for metabolism in the agent...

See the example here

Aside: A very different alternative to diffusion, scaling, and added noise is to use something closer to an asynchronous CA. The basic concept is to emulate Brownian diffusion by occasionally swapping to neighbouring particles. This can be made into a smoother gradient by interpolating the swap. And removal can be simulated as a small portion of randomly chosen cells being set to zero. This is a more challenging dynamic landscape for the agents to navigate:

  // some swaps:
  for (var i=0; i<10000; i++) {
    // pick a point at random
    var x = random(sugar.width);
    var y = random(sugar.height);
    // pick a neighbour
    var x1 = x + random(3)-1;
    var y1 = y + random(3)-1;
    // get the values
    var a = sugar.get(x, y);
    var b = sugar.get(x1, y1);
    // pick a random interpolation factor
    var t = 0.5*random();
    // blend a and b into each other accordingly
    var a1 = a + t*(b-a);
    var b1 = b + t*(a-b);
    // write these values back to the field
    sugar.set(a1, x, y);
    sugar.set(b1, x1, y1);
  // some removals:
  for (var i=0; i<100; i++) {
    var x = random(sugar.width);
    var y = random(sugar.height);
    sugar.set(0, x, y);

Obviously there's a hint here of how it might be interesting to pair our agents with some other CAs for field processes...

Going back to the sketch of Vehicles, now we can implement a two-eyed agent following light as a diffuse field:


How could it be extended to include collision avoidance? Or to have different vehicle types, some attracted to others, some repelled by others, etc.?


Stigmergy is a mechanism of indirect coordination between agents by leaving traces in the environment as a mode of stimulating future action by agents in the same location. For example, ants (and some other social insects) lay down a trace of pheromones when returning to the nest while carrying food. Future ants are attracted to follow these trails, increasing the likelihood of encountering food. This environmental marking constitutes a shared external memory (without needing a map). However if the food source is exhausted, the pheromone trails will gradually fade away, leading to new foraging behavior.

Traces evidently lead to self-reinforcement and self-organization: complex and seeminly intelligent structures without global planning or control. Since the term stigmergy focuses on self-reinforcing, task-oriented signaling, E. O. Wilson suggested a more general term sematectonic communication for environmental communication that is not necessarily task-oriented.

Stigmergy has become a key concept in the field of swarm intelligence, and the method of ant colony optimization in particular. In ACO, the landscape is a parameter space (possibly much larger than two or three dimensions) in which populations of virtual agents leave pheromone trails to high-scoring solutions.

Related environmental communication strategies include social nest construction (e.g. termites) and territory marking.


Being able to leave pheromones behind depends on having a specific signature marker in space, such as a particular smell. A simple way to emulate this is to have a field for each pheromone. For example, we may want one pheromone to signal "food this way", and another to signal "nest this way". To draw multiple fields, we can turn on blending and apply different colors for each one:

function draw() {

    ... now draw agents

We already saw in the chemotaxis example how to leave trails in a field, as well as how to let these diffuse and dissipate in order to attract more distant agents and make way for new trails to be made, and similar processing will be needed for each pheromone field.

But this time, our sugar field won't be dissipative -- we just need to create some clumps of sugar resources somewhat distant from the nest. And, inspired by ants, we may grant our agents two sensors, spatially separated. By comparing these two sensors (and perhaps our memory for a third) we can estimate the spatial gradient of the field, and turn accordingly.

Here's a start in that direction.

Energy exchange, birth and death

Death Note: adding/removing agents from an array

If we want to extend our model to support death, such as being eaten by a predator, infected by a disease, starvation, poison, fire, etc., we will need a way to remove items from our array of agents. Assuming that we can mark death by some property (such as a.dead = true), you might expect to remove it from the agents array as follows:

if (agents[n].dead) {
    agents.splice(n, 1);

(Note that delete agents[n] won't work at all, it doesn't reshuffle the array to accommodate the gap, it just leaves the gap right there.)

Similarly, if an agent gives birth, you might expect to add the child as follows:

if (child) {

But in general, adding to or removing from an array is not safe while iterating the array, which is almost certainly the time that you would want to call it. The reason is that you may end up skipping an item, or trying to iterate over an item that no longer exists, or visiting an item twice, because the exact positions of items and the array length have changed. This is one of the classic problems commonly encountered in many imperative programming languages (e.g. C++ too).

Here are two suggested solutions:

Iterate backwards

It is safe to splice and push while iterating backwards, because reshuffling an array only affects later items:

var i = agents.length;
while (i--) {
    var a = agents[i];

    //.. do stuff, possibly setting a.dead = true or creating a child

    // remove safely:
    if (a.dead) agents.splice(i, 1);

    // add a child?
    if (child) agents.push(child);

Note that in this case, new births will not be visited until the next frame. This is probably what you want, since otherwise you might end up creating an infinite loop!


Rather than modifying the array in-place, we build a new array on each iteration, and replace the original when done. This has the advantage of working with for-of loops:

// create new array
var newagents = [];
for (var a of agents) {

    //.. do stuff, possibly setting a.dead = true or creating a child

    // preserve only living agents:
    if (!a.dead) newagents.push(a);

    // add a child?
    if (child) newagents.push(child);        
// replace original
agents = newagents;

Note that new children are added to newagents, ensuring they are visited next frame. If we added them to agents, they would also be included in the iterating for loop on the current frame, which could potentially lead to an infinite loop.

Energy dynamics

Agents are also exchanging energy, which may be important to model in the simulation. Possible energetic transfers include:

Each of these energetic costs is likely also multiplied by some scalar constant, such as a metabolism_cost, locomotion_cost, etc.

If we are careful to enumerate all the energetic gains and losses in this way, we can count the total energy lost in the system on each frame, and make that available as an energy input to the environment on the next frame. In this way, the total energy circulating in the system should remain constant. This total energy expenditure may also be a useful indicator of the liveliness of a system.

Another (perhaps less bug-prone) way is to count all the energy in the system and compare the the 'ideal' level of energy. If there's an environmental field, you can use field.sum() to return a total. Then iterate the list of agents and add their energy levels too. Then expenditure is the difference of this total between frames.


And here's that example combined with the chemotaxis model

The population can shrink and grow (safely, because the main update loop iterates backwards). But over time, energy is being lost from the world. We can calculate how much energy is active in the system by adding up the total in each field cell (by field.sum()) as well as the energy stored in each agent. When this total energy level drops below our chosen carrying capacity for the world, we can then introduce energy back in, to ensure that the whole system doesn't wind down to zero.

Still, population control can be essential at both ends. Clearly if the population ever drops to zero, we will never see any organisms again. Methods to overcome this include preventing death, or introducing new randomly seeded organisms (perhaps out of view) at low population sizes. At the other end, a population that grows excessively large can slow down the simulation, perhaps even crash it. In theory the conservation of energy can prevent this, but in practice a hard limit may also be necessary.