Adapting to changing environments

Unless stated otherwise, the presented results were obtained from studies of the behaviour of different-sized colonies N ∈ {6, 10, 100, 1000} using a maximal threshold θmax= 100 in an environment exhibiting oscillation-like demand variation with t^∆ = 400. Variation of demands is between low values a standard value of D₁^s = D^s₂ = 0.3 and demand change value of D^c₁ = D₂^c = 0.7 (as in Fig. 3.1). A simulation run lasted for 770000 simulation steps leading to 1800 demand variations per simulation.

These experiments were conducted to test how fast a colony is able to adapt to task-related changes in the environment. In order to investigate how the behaviour of different-sized colonies differs when confronted with environmental changes, colonies of different sizes were studied using the same set of parameters.

A good indicator of how fast a colony can adapt to a change in the environment are the task-related stimuli. A colony that has successfully adapted to environmental changes should be able to fulfil the colony’s needs. This means that its individuals should neither work too little nor too much for the tasks present in the system. Such a behaviour should

3. Division of labour in dynamic environments

Figure 3.3.: Change of the stimulus associated with task T₁ in different-sized colonies dur-ing a whole oscillation-like demand variation interval t^∆= 400.

result in stable stimuli. On the other hand, stimuli should change if the colony has not yet adapted, depending on whether too much or too little work is being done on the given tasks.

Figure 3.3 depicts the evolution of the average stimuli values of task T1 for colonies of size 6,10,100, and 1000. Remember that for task T₁ the demand is D₁ = 0.7 (while D2 = 0.3) in the first 200 steps and D1 = 0.3 (while D2 = 0.7) in the remaining steps.

As can be seen in Figure 3.3, stimulus behaviour differs strongly between different-sized colonies. While large colonies (N = {100, 1000}) are able to adapt fast to the change in task-demand, this does not hold for colonies with few individuals. Adaptation speed seems to scale with colony-size, meaning the larger the colony the faster the adaptation to changes in task demand.

How well a colony is able to adapt to environmental changes should also be reflected in the activity of the colony during a demand variation interval. The expected ideal amount of work W_j^ideal which should be done by a colony for a given task T_j (having a demand

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3.3. Experiments

Table 3.1.: Work statistics for the first t^∆/2 steps in an oscillation-like demand variation interval t^∆ = 400; N : colony size, W_j^ideal: expected ideal amount of work for task T_j, W_j^ave: work for task T_j in the first 200 steps of a demand variation phase (averaged over all demand variation intervals); standard deviation is given in parentheses; r_j := W_j^ave/W_j^ideal: relative amount of ideal work done.

N W₁^ideal W₁^ave r1 W₂^ideal W₂^ave r2

6 35.0 24.7 (6.7) 70.6% 15.0 25.3 (5.6) 168.7%

10 58.3 45.6 (6.6) 78.2% 25.0 37.9 (6.4) 151.6%

100 583.3 577.2 (4.7) 99.0% 250.0 257.9 (4.8) 103.2%

1000 5833.3 5842.6 (3.6) 100.1% 2500.0 2524.0 (6.1) 101.0%

parameter D_j) in t^v time steps can be easily calculated via (for details of this formula please refer to Chapter 2).

W_j^ideal = t^v· δj = t^v· Dj ·N m· α

1 + p (3.1)

Here, we are interested in t^v = 200, which corresponds to half of the demand variation interval t^∆, whether we look at the first of second half of t^∆ is irrelevant, as the task demands are symmetric.

Table 3.1 contains the amount of work, the expected ideal amount of work and the fraction of the ideal work that has been done for both tasks and colonies of sizes 6, 10, 100, and 1000, each calculated for the first 200 steps in a demand variation interval (i.e., D1 = 0.7, D2= 0.3). All presented values are averaged over all demand variation intervals in one simulation run.

From the table it is clear that colonies of all sizes fulfil or exceed the ideal amount of work in the demand variation interval, for the task with the unchanged demand (i.e., task T₂). However, small colonies exceed the ideal amount of work far more than large colonies (e.g., the colony of size 6 works around 68.7% more than necessary while a colony of size 1000 exceeds the ideal amount of work only around 1% ).

The ideal amount of work for the task with the increased demand is only fulfilled / exceeded by the largest colony (i.e., N = 1000). Smaller colonies are not able to accomplish the necessary workload. Furthermore, the ability to deal with a demand increase seems to be positively correlated with colony size.

Our suggestion is that the phenomenon seen here (i.e., the difference in the work perfor-mance during an demand variation interval) is due to the colony-size dependent stimuli.

Stimuli trigger the awareness of a colony’s individuals for a task. While stimuli in large colonies are very flexible (i.e., their level can increase or decrease significantly in one time

3. Division of labour in dynamic environments

Figure 3.4.: Change of the stimulus associated with task 1 in a model with colony-size independent stimuli, for colonies of size 6 and 1000 during a whole oscillation-like demand variation interval t^∆= 400.

step, see Eq. 2.5 in Chapter 2 for more detail), this does not hold for small colonies. This contrasts with the individual’s threshold update – all individuals, no matter what colony size, exhibit the same threshold learning and forgetting rate. It seems therefore that indi-viduals in small colonies are simply not as quickly aware of the workload dimensions they have to deal with, unlike the individuals in large colonies.

To check whether the observed differences between different-sized colonies in the amount of work, which is done during a demand variation interval, is a consequence of the colony-size dependency of the stimuli, we studied a modified model which uses colony colony-size inde-pendent stimuli.

Within the model, colony-size independent stimuli can be achieved easily by the follow-ing modification of the stimulus update formula:

S_j = S_j+ (δ_j− Ej· α) · 1/N (3.2)

Table 3.2.: Work statistics for the first t^∆/2 steps in an oscillation-like demand variation interval t^∆= 400 in a model with colony-size independent stimuli update; for parameters see Table 3.1.

N W₁^ideal W₁^ave r1 W₂^ideal W₂^ave r2

6 35.0 24.7 (8.9) 70.6% 15.0 25.4 (9.4) 169.3%

10 58.3 41.3 (12.9) 70.8% 25.0 42.0 (11.5) 168.0%

100 583.3 410.4 (39.6) 70.4% 250.0 422.6 (38.7) 169.0%

1000 5833.3 4105.7 (161.0) 70.6% 2500.0 4230.8 (190.8) 169.2%

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3.3. Experiments

Figure 3.5.: Difference of amount of work done by a colony of size N and the ex-pected ideal amount of work W_j^ideal; results are depicted for task j = 1, colony sizes N ∈ {10, 100, 1000, 10000} and maximal threshold values θmax∈ {10, 50, 100, 300, 500, 1000, 1500}; boxplots show the difference between the amount if work done and the expected ideal amount of work W_j^ideal to be done for all demand variation phases in all test runs; dotted line in the sub-figures shows the value W₁^ideal.

This modification ensures that the stimuli have a growth rate that is independent of the colony size. Stimulus development and the work statistics of different-sized colonies in the modified model confirm our hypothesis. The stimulus evolution of colonies of size 6 and 1000 are depicted in Figure 3.4 and the work statistics for N ∈ {6, 10, 100, 1000} are presented in Table 3.2. Within the modified model, the stimulus development and work statistics of different-sized colonies are relatively similar, which is reflected in the similar values of rj for task j ∈ {1, 2}.