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Talk Abstract
From the Individual Behaviour Subject to Stochastic Fluctuations to the PIDE Describing the Global Behaviour of an Aggregating Population

Vincenzo Capasso
University of Milan


Starting from field experiments, we consider animal grouping due to the interplay of interaction between individuals which works in the direction of the formation of spatially concentrated patterns and the dispersion caused by population pressure in order to avoid overcrowding effects. As a specific example the social behavior of slave-maker ant of the species Polyergus Rufescens taken.

The colonies of these species are characterized by the absence of polyethism in the worker cast which is composed only by soldiers, unable to attend any task (e.g. brood tending or nest maintenance) other than raiding activity. These indispensable tasks are performed by individuals belonging to few specific species which have been kidnapped by Polyergus soldiers so that newborn or pupae grew up in the Polyergus nest. As detected in field experiments carried out in [1], during their raids to the nests of other species of ants, the Amazons tend to aggregate in a transversally organized army (transversal with respect to the main direction of motion); they do not exhibit overcrowding effects. The army looks strikingly different than the foraging ant column we use to see in our houses. This aggregation has a sharp front 10-40 cm wide, composed by 10 or more individuals walking side by side. During the return journey the ants walk single, with a variable distance between successive individuals. So in the same species we observe a different aggregative behavior in different phases of the raid. We interpret this phenomenon as different magnitude of the social response of each individual strictly related to the aim of the group. During the raids they seem to need a big force to overwhelm the defenders of the target nest. As a consequence they assume the form of a compact army: the magnitude of the social response is high. On the other end on their way back they have already reached their aim and can go back by themselves: the magnitude of the social response is very low.

So in the aggregation phases it is clear how it is strong the element of choice in location. How to interpret the social rules? A relevant phenomenon is the spatial dependence of the social parameters on different environmental conditions, such as regularity of the terrain, reciprocal visibility, etc. which may impose restrictions on the sensory range among individuals.

We describe the individual dynamics of social individuals in a population of size N by means of a system of N stochastic differential equations where the drift describes interaction forces among the particles, i.e. forces of aggregation and forces of repulsion. In addition each particle is subject to random dispersal modelled by a family of independent standard Brownian motions.

A rigorous derivation based on a "law of large numbers" shows that the average behaviour of the total population is described by a deterministic advection-diffusion-reaction integro-differential equation for N tending to infinity [5], [4], [6]

The specific aim of our reserch has been to apply the above methods to an aggregation model which may be seen as an extension of a model due to Okubo and Grünbaum [2], [3], [7].

References

[1] Boi, S., Capasso. V, Morale, D. "Modeling the aggregative behavior of ants of the species Polyergus rufescens." Quaderno del Dipartimento di Matematica-Universitá di Milano n. 33/1998.

[2] Grünbaum, D. "Translating stochastic density-dependent individual behaviour with sensory constraints to an Eulerian model of animal swarming." J. Math. Biology, 33(1994), 139-161.

[3] Grünbaum, D., Okubo, A. "Modelling social animal aggregations" In "Frontiers of Theoretical Biology" (S. Levin Ed.), Lectures Notes in Biomathematics, 100, Springer Verlag, New York, 1994

[4] Morale, D., Capasso, V. Oelschläger K. "A rigorous derivation of the mean-field nonlinear integro-differential equation for a population of aggregating individuals subject to stochastic fluctuations," Preprint 98-38 (SFB 359), IWR, Universität Heidelberg, Juni 1998.

[5] Morale, D., Capasso, V. Oelschläger K. "On the derivation of the mean-field nonlinear integro-differential equation for a population of aggregating individuals subject to stochastic fluctuations," Preprint 98-39 (SFB 359), IWR, Universität Heidelberg, Juni 1998.

[6] Morale, D. "Cellular automata and many-particles systems modeling aggregation behaviour among populations," 1998. In preparation.

[7] Okubo, A. "Dynamical aspects of animal grouping: swarms, school, flocks and herds." Adv. BioPhys., 22, (1986), 1-94.



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1998-1999 Mathematics in Biology

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