Collective Irrationality and Positive Feedback

Recent experiments on ants and slime moulds have assessed the degree to which they make rational decisions when presented with a number of alternative food sources or shelter. Ants and slime moulds are just two examples of a wide range of species and biological processes that use positive feedback mechanisms to reach decisions. Here we use a generic, experimentally validated model of positive feedback between group members to show that the probability of taking the best of options depends crucially on the strength of feedback. We show how the probability of choosing the best option can be maximized by applying an optimal feedback strength. Importantly, this optimal value depends on the number of options, so that when we change the number of options the preference of the group changes, producing apparent “irrationalities”. We thus reinterpret the idea that collectives show "rational" or "irrational" preferences as being a necessary consequence of the use of positive feedback. We argue that positive feedback is a heuristic which often produces fast and accurate group decision-making, but is always susceptible to apparent irrationality when studied under particular experimental conditions

Symmetry restoring bifurcation in collective decision-making.

How social groups and organisms decide between alternative feeding sites or shelters has been extensively studied both experimentally and theoretically. One key result is the existence of a symmetry-breaking bifurcation at a critical system size, where there is a switch from evenly distributed exploitation of all options to a focussed exploitation of just one. Here we present a decision-making model in which symmetry-breaking is followed by a symmetry restoring bifurcation, whereby very large systems return to an even distribution of exploitation amongst options. The model assumes local positive feedback, coupled with a negative feedback regulating the flow toward the feeding sites. We show that the model is consistent with three different strains of the slime mold Physarum polycephalum, choosing between two feeding sites. We argue that this combination of feedbacks could allow collective foraging organisms to react flexibly in a dynamic environment

Verklighetsanknuten matematikundetvisning

Det står i läroplanen och styrdokument att olika typer av undervisningsmetoder och uppgifter som kan användas vid matematikundervisning för att öka intresset för matematikämnet och elevernas inlärning. Syftet med denna uppsats är därför att undersöka lärares erfarenhet av verklighetsanknuten matematikundervisning och lärarnas åsikter om elevernas attityd och inlärning. Studien genomfördes genom intervjuer med lärare och analyserades utifrån en teori om tillämpningar av samspelet mellan matematik och vardagslivet. Resultaten visar att verklighetsanknutna uppgifter och laborativ samt interaktiv undervisning väcker elevernas intresse för matematik och kan hjälpa eleverna att förstå matematiska begrepp och använda detta i vardagslivet

Mathematical modelling approach to collective decision-making

In everyday situations individuals make decisions. For example, a tourist usually chooses a crowded or recommended restaurant to have dinner. Perhaps it is an individual decision, but the observed pattern of decision-making is a collective phenomenon. Collective behaviour emerges from the local interactions that give rise to a complex pattern at the group level. In our example, the recommendations or simple copying the choices of others make a crowded restaurant even more crowded. The rules of interaction between individuals are important to study. Such studies should be complemented by biological experiments. Recent studies of collective phenomena in animal groups help us to understand these rules and develop mathematical models of collective behaviour. The most important communication mechanism is positive feedback between group members, which we observe in our example. In this thesis, we use a generic experimentally validated model of positive feedback to study collective decision-making. The first part of the thesis is based on the modelling of decision-making associated to the selection of feeding sites. This has been extensively studied for ants and slime moulds. The main contribution of our research is to demonstrate how such aspects as "irrationality", speed and quality of decisions can be modelled using differential equations. We study bifurcation phenomena and describe collective patterns above critical values of a bifurcation points in mathematical and biological terms. In the second part, we demonstrate how the primitive unicellular slime mould Physarum Polycephalum provides an easy test-bed for theoretical assumptions and model predictions about decision-making. We study its searching strategies and model decision-making associated to the selection of food options. We also consider the aggregation model to investigate the fractal structure of Physarum Polycephalum plasmodia.Fel serie i tryckt bok /Wrong series in the printed book</p

Decision quality as a function of the flow rate of individuals and of the quality of the better option () as obtained by Monte Carlo simulations.

<p>(a) case , (b) and (c) . Other parameter values as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0018901#pone-0018901-g001" target="_blank">Fig. 1</a>, number of realizations is 5000.</p

Decision quality as a function of the flow rate of individuals and of the number of options as obtained by Monte Carlo simulations.

<p>(a) case , (b) and (c) . Other parameter values as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0018901#pone-0018901-g001" target="_blank">Fig. 1</a>, number of realizations is 5000.</p

Bifurcation diagrams of corresponding to the steady state level of commitment for the better option (eqs. (3) – (6))with respect to the flow rate .

<p>(a) case , (b) and (c) . Full and dashed lines correspond to stable and unstable solutions respectively. The stability has been checked numerically by integrating the full eqs. (1) – (2). The arrows indicate the evolution of initial conditions on the two sides of a threshold value corresponding to the intermediate unstable state. Parameter values are , , and .</p

Bifurcation diagram corresponding to the steady state solutions of <b>equation (3</b>) with respect to the parameter .

<p>Full and dashed lines correspond to stable and unstable solutions respectively. The black circle shows the first bifurcation, the white circle corresponds to the second bifurcation and the black square labels the third bifurcation. Parameter values are , and .</p

Conditions for existence of the bifurcation points displayed in <b>Fig. 2.</b>

<p>Parameter as a function of for fixed mass , other parameter as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003960#pcbi-1003960-g002" target="_blank">Fig. 2</a>. Bold solid, solid and dashed lines correspond to the condition for existence of the first, second and third bifurcation, respectively.</p