SC741 W12: Division of Labor Part I: Fixed- and Variable- Threshold Algorithms

SC741 W12: Division of Labor Part I: Fixed- and Variable- Threshold Algorithms

Outline Division of labor in natural systems Ants Bees, wasps Models and mechanisms Fixed-threshold mechanisms Variable-threshold mechanisms Algorithms Single-task, multi-caste Multi-task, multi-caste

Different Forms of Division of Labor in Social Insects

The Division of Labor and its Control The control of task allocation The most obvious sign of the division of labor is the existence of castes. We distinguish between three kinds of castes: physical, behavioral, and temporal (temporal polyethism). The individuals belonging to different castes are usually specialized for the performance of a series of precise tasks.

Physical Castes (Wilson, E. O., 1976) Self-grooming Minor worker Dealat e queen Male Carry or roll egg Carry or roll larva Feed larva solids Carry or roll pupa Assist eclosion of adult Minor worker Dealat e queen Male Forage Lay odor t rail Feed inside nest Agression (drag or at t ack) Carry dead larva or pupa Feed on larva or pupa Lick wall of nest Antennal tipping Guard nest entrance Minor 0 02 0 4 0 6 Major 0 0 2 0 4 0 6 Behavioral repertoires of majors and minors In Pheidole guilelmimuelleri the minors show ten times as many different basic behaviors as the majors Average fraction of time spent in a given activity/behavior

Behavioral Castes Allocation of the daily activities in a colony of desert harverster ants (Portal, AZ) From D. Gordon, Ants at Work, 1999

Temporal Polyethism 40 20 10 Cleaning cells Tending brood Behavioral changes in worker bees as a function of age 10 Tending Queen Percentage of time spent in each activity 10 10 10 10 10 10 80 60 40 20 40 20 Eating pollen Feeding & grooming nestmates Ventilating nest Shaping comb Storing nectar Packing pollen Foraging Patrolling Young individuals work on internal tasks (brood care and nest maintenance). Older individuals forage for food and defend the nest. 40 20 Resting 1 4 7 10 13 16 19 22 25 Age of bee (days)

The Division of Labor and its Control Flexibility of social roles The division of labor in social insects is flexible. The number of individuals belonging to different castes and the nature of the tasks to be done are subject to constant change in the course of the life of a colony. The proportions of workers performing the different tasks varies in response to internal or environmental perturbations. This is true under certain conditions even when hard morphological differences exist or irreversible aging processes take place.

Example of Short-Term Evolution of Division of Labor The changes in the population of a bee colony in the course of a season is influenced by reproduction/extinction process, weather changes, etc. overlapped with the irreversible temporal polyethism due to aging 22127 41297 48253 51988 49971 34062 60 AGE (DAYS) 40 May 27- Jun 7 Jun 20- Jul 1 Jul 14-25 Aug 19-30 Sep 12-23 Oct 6-17 20 0 I NCREASI NG PERI OD MAJOR PERI OD CONTRACTI NG PERI OD Egg Larvae Pupae Nurse bee House bee Field bee

How is flexibility implemented at the level of the individual?

The Division of Labor and the Flexibility of Social Roles The control of task allocation Task 1 Task 2 Task 3? 2 3 1 How is dynamic task allocation achieved?

The Division of Labor and its Control Flexibility of social roles How does the colony control the proportions of individuals assigned to each task, given that no individual possesses any global representation of the needs of the colony? The flexibility of the division of labor depends on the behavioral flexibility of the workers. A mechanism arising from the concept of a response threshold allows the production of this flexibility. Or at least it represents a biologically plausible model for explaining such division of labor facts in social insect colonies.

The Division of Labor and the Flexibility of Social Roles The control of task allocation explained with a fixedthreshold model 1 2 3 σ i1 σ i1 σ i1 σ i1 σ i1 σ i1 σ i1 σ i1 σ i1 The lower the threshold, the lower can be the stimulus for achieving a given response; respectively, the lower the threshold, the higher will be the response of an individual for a given stimulus.

An Example of Control of the Division of Labor Implying the Existence of Fixed Response Thresholds

Guy Theraulaz

The Division of Labor and its Control The idea of a response threshold Note: same species mentioned for illustrating physical castes 9 8 Pheidole pubiventris Social behavior Self-grooming 10 9 P. guilelmimuelleri Social behavior Self-grooming 7 8 Behavioral Acts/Major/Hour 6 5 4 3 2 Behavioral Acts/Major/Hour 7 6 5 4 3 2 1 1 0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Fraction of majors Fraction of majors

The Division of Labor and its Control An example of a response threshold 1 θ i,1 Task 11 Task 11 θ j,1 T(s) Response Probability 0,75 0,5 0,25 n=2 Minors Majors T (s) = θ i s n s n + θ i n s : intensity of the stimulus associated with the task θ i : response threshold n : nonlinearity parameter, e.g. n = 2 0 0,1 1 10 100 Stimulus θ minors θ majors

Properties of Mechanisms Arising from Fixed Response Thresholds

Using a Model for Explaining Wilson s Results and more Macroscopicmodel, continuous time Response to stimuli based on fixed thresholds No detailed mechanisms for stimuli preception incorporated in the model Assumptions: nonspatial model, i.e. equiprobable exposure of individuals to the stimuli associated with the task and stimulus homogeneously distributed over space 1 task (e.g., social behavior), 2 castes (e.g., minor and major)

Fixed Threshold Model with 1 Task and 2 Castes Two castes of individuals in the colony (physical, behavioral, or age-based castes); each individual can perform the task or doing nothing. Note n i and N i swapped in comparison to your book! N 1 = number of individuals of type 1 N 2 = number of individuals of type 2 N 1 + N 2 = N = total number of individuals in the colony n 1 = number of individuals of type 1 engaged in carrying out the task n 2 = number of individuals of type 2 engaged in carrying out the task x 1 = n 1 / N 1 : fraction of individuals of type 1 engaged in carrying out the task x 2 = n 2 / N 2 : fraction of individuals of type 2 engaged in carrying out the task 1-x 1 : fraction of individuals of type 1 inactive (or not performing the task) 1-x 2 : fraction of individuals of type 1 inactive (or not performing the task)

Fixed Threshold Model with 1 Task and 2 Castes Dynamics of the fraction of active individuals in each caste: t x 1 t x 2 s 2 = (1-x 1 ) - r a x 1 s 2 + θ 2 1 s 2 = (1-x 2 ) - r a x 2 s 2 + θ 2 2 Note: t x = x t s : intensity of the stimuli associated with the task θ i : response threshold of workers of type i r a : rate of task abandoning (i.e. probability per unit time that an active individual abandons the task on which it is engaged for moving to idle) 1/r a : average time spent by an individual in working on a task before abandoning it

Fixed Threshold Model with 1 Task and 2 Castes Dynamics of demand associated with the task α t s = δ (n 1 + n 2 ) N δ : rate of stimulus increase α : normalized effectiveness rate for the individual contribution on the task (in this case an active individual of type 1 contribute in the same way as an active individual of type 2 when performing the task)

Fixed Threshold Model with 1 Task and 2 Castes Number of acts per major as a function of the fraction of majors for different sizes of colony Parameters of the simulation 80 N=10 N=100 N = 10, 100, 1000 θ 1 = 8, θ 2 = 1 α = 3 δ = 1 r a = 0.2 Number of acts per major during the simulation 60 40 20 N=1000 0 0 0,25 0,5 0,75 1 Fraction of majors

Fixed Threshold Model with 1 Task and 2 Castes Comparison between the macroscopic model and experimental results Parameters of the simulation N = 10, 100 θ 1 = 8, θ 2 = 1 α = 3 δ = 1 r a = 0.2 Number of acts per major during the simulation 75 50 25 Pheidole guilelmimuelleri Pheidole pubiventris simulation N=10 simulation N=100 10 8 6 4 Number of acts per major during the real experiment 2 0 0 0,25 0,5 0,75 1 0 Fraction of majors

Variable Threshold Model for Controlling the Division of Labor

The Division of Labor and the Flexibility of Social Roles The control of task allocation explained with a variablethreshold model σ i1 σ i1 σ i1 - - - + + + 2 3 σ i1 σ i1 σ i1 - - - + + + 1 σ i1 σ i1 σ i1 - - - + + + The lower the threshold, the lower can be the stimulus for achieving a given response; respectively, the lower the threshold, the higher will be the response of an individual for a given stimulus.

Polist Wasps: again an Interesting Example Guy Theraulaz

Origins of the Division of Labor in the Polist Wasps Polists : primitive eusocial species Colonies usually contain only a small number of individuals (ca 20). These species do not show morphological differences between castes in the adult stages, nor any control or physiological determination of the role an individual will play in the colony as an adult. Individual behavior is very flexible; all individuals are able to perform the whole range of tasks which determine the survival of the colony. The integration and coordination of individual activities is achieved through the interactions which occur between the members of the colony, and between the members of the colony and the local environment.

Origins of the Division of Labor in the Polist Wasps The role of learning in the differentiation of activities The state of the brood triggers the activities of foraging and feeding the larvae. In adults there are different response thresholds and these thresholds vary as a consequence of the individuals activities. Within an individual, the act of setting out on a foraging task has the effect of lowering the response threshold for larval stimulation. There is therefore a self-exciting process (positive reinforcement/feedback mechanims) bringing about the specialization of individuals which have carried out foraging tasks; this process can be described by means of a variable threshold model. Specialists can be created from generalists

Properties of Mechanisms Arising from Adaptive Response Thresholds

Variable Threshold Model with Individual behavioral algorithm 1 Task and m Castes θ i θ i - ξ when i performs the task θ i θ i + ϕ when i does not perform the task T (s) = θ i s 2 s 2 + θ i 2 Parameters of individuals Note: special case m= N but remember that this is a macroscopic model, so a certain number of individuals per caste is required for quantitatively correct predictions i : caste index [1 m] s : intensity of stimuli associated with the task θ i : response threshold of individual i to the task: θ i [θ min,θ max ] ξ : incremental learning parameter (the threshold of an individual carrying out a task is reduced by ξ) ϕ : incremental forgetting parameter (the threshold of an individual not carrying out a task increases by ϕ)

Variable Threshold Model with 1 Task and m Castes Description of the algorithm YES θ i,1 ξ Task θ i,1 Execute task Average duration = 1/r a System of DE: NO θ i,1 +ϕ t x i = T θ i (s)(1-x i ) - r a x i r a : abandoning rate (as before for fixed thresholds)

Variable Threshold Model with 1 Task and m Castes Dynamics of the demand associated with a task m α t s = δ ( n i ) N Σ i= 1 δ : rate of stimulus increase α : normalized effectiveness rate for the individual contribution to the task (in this case all the active individual belonging to different castes contribute in the same way) n : i : number of active individual belonging to caste i

Example with 6 Castes (θ 1 θ 6 ) Model Parameters θ i = [1 1000], initial random distribution α = 3 δ = 1 r a = 0.2 ξ = 10 φ = 1 Red and blue caste lower the threshold -> specialists (note started from low thresholds) Thresholds evolution

Example with 6 Castes (θ 1 θ 6 ) Evolution of the demand associated to the stimulus s

Example with 6 castes (θ 1 θ 6 ) High fraction of active red and blue indidivuals Evolution of the fraction of active individuals in each caste (demand profile overlapped)

Example with 6 Castes and one Caste Removed at t = 150 Light gray specialist removed (corresponds to the red specialist in the previous example)

The Control of the Division of Labor in a variable Threshold Model with 2 Tasks

Individual behavioral algorithm Variable Threshold Model with 2 Tasks and m Castes θ ij θ ij θ ij - ξ when caste i performs task j θ ij + ϕ when caste i does not perform task j T (s j ) = θ ij s j 2 s j2 + θ ij 2 Variable and parameters s j : intensity of the stimulus associated with the task j θ ij : response threshold of the caste i to task j, θ ij [θ min,θ max ] ξ : incremental learning parameter (the threshold of an individual carrying out task is reduced by ξ ) ϕ : incremental forgetting parameter (the threshold of an individual not carrying out a task increases by ϕ)

Variable Threshold Model with 2 Tasks and m Castes Dynamics of the demand associated with a task m α j t s j = δ j ( n ij ) N Σ i = 1 Parameters of the demand associated with a task δ j : rate of stimulus increase associated with a task j α j : normalized effectiveness rate for the individual contribution to the task j (in this case all the active individual belonging to different castes contribute in the same way) m : total number of castes in the colony n ij : number of active individuals belonging to caste i and carrying out the task j

Dynamics of response thresholds Variable Threshold Model with 2 Tasks and m Castes Task 1 Task 2 1000 1000 Response threshold 750 500 250 caste 1 caste 2 caste 3 caste 4 Response threshold 750 500 250 caste 1 caste 2 caste 3 caste 4 caste 5 caste 5 0 0 1000 2000 3000 0 0 1000 2000 3000 time time

Variable Threshold Model with 2 Tasks and m Castes Dynamics of the proportion of active individuals per caste involved in each task Task 1 Task 2 1,00 1,00 Proportion of active individuals 0,75 0,50 0,25 Proportion of active individuals 0,75 0,50 0,25 0,00 0 1000 2000 3000 0,00 0 1000 2000 3000 time time

Additional Literature Week 12 Papers Bonabeau E., Theraulaz G., and Deneubourg J.-L., Fixed Response Thresholds and the Regulation of Division of Labour in Insect Societies. Bulletin of Mathematical Biology, 1998, Vol. 60, pp.753-807. Theraulaz G., Bonabeau E., and Deneubourg J.-L., Response Threshold Reinforcement and Division of Labour in Insect Societies. Proc. of the Royal Society of London Series B, 1998, Vol. 265, pp. 327-332. Pacala S. W., Gordon D. M., and Godfray H. C. J. 1996. Effects of Social Group Size on Information Transfer and Task Allocation, Evolutionary Ecology, 10: 127-165.