Population dynamics with limited perception establish global swarm topology Anna Shcherbacheva 1 , Tuomo Kauranne 1 1 Lappeenranta University of Technology, Finland anna.shcherbacheva@lut.fi tuomo.kauranne@lut.fi Abstract We simulate the swarming behavior of three synthetic animal species that differ only by the degree of perception they have on their fellow animals. The species are called mosquitoes, birds and fish. The swarms that comprise many individuals of each species in turn move randomly in a rugged poten- tial landscape. The mosquitoes pay no heed to one another. The birds follow a bunch of their nearest neighbours in front, based on strictly limited visibility. The fish, in turn, sense also far-away neighbors through their lateral line, as mod- eled by an exponentially decaying perception function. The simulations show that such local differences in perception by swarming individuals have global macroscopic consequences to the geometry of the corresponding swarms. These conse- quences are of persistent nature across many simulations with each species. Introduction Humans, like many other animal species, are social. We are fundamentally geared to living in a herd of some 20 - 100 individuals, from our nearest primate cousins to de- cide. Many other animal species have much more intense social lives, with flocks extending to thousands of individu- als. Large flocks - or swarms - have their own requirements as to the means of imposing collective social control over the individuals involved. The process of social co-ordination is bi-directional: on the one hand, swarm dynamics exerts con- trol over each individual. On the other, swarm dynamics is a direct consequence of the collective motion of all of its individuals. In this article, we compare the consequences on collec- tive dynamics of different degrees and forms of perception of swarming animals through computer simulations. Many studies indicate that the bi-directional flow of information described above has an important defining role in determin- ing the nature of swarm dynamics. The impact of informa- tion flow boils down to the question of how do the individu- als in a swarm perceive the collective dynamics of the swarm - and to the reciprocal question of how does the reaction of individuals influence the collective dynamics of the swarm. We shall study this question with computer simulations of the collective motion of swarms of three different types of animals, all capable of moving in two spatial dimensions. These synthetic species are labeled as mosquitoes, birds and fish. They are set to move in a similar synthetic world with some external forces and constraints. But the way they per- ceive their fellow passengers is different, which has a funda- mental impact on the nature of the corresponding collective motion of the swarm. Swarm dynamics with a difference Collective swarm dynamics can be described in many ways, such as using ordinary or partial, deterministic or stochastic, differential equations. Classical models of Eu- lerian type (e.g., see Milewski and Yang (2008), Mur- ray (2002), Mogilner and Edelstein-Keshet (1999) and Na- gai and M. (1983)) are based on the diffusion-advection- reaction equation, governing the spatio-temporal dynamics of the population density: ∂f ∂t = ∂ ∂x  D(f ) ∂f ∂x  - ∂ ∂x (V (f )f )+ B(f ), (1) where the first term on the right-hand side introduces a Brownian motion with diffusion coefficient D(f ), the sec- ond term stands for advection with density-dependent ve- locity V (f ) and the last reaction term may include birth or death processes. Convection term results in attraction and repulsion ef- fects, reflecting forms of social interaction between popula- tion members which implies that the direction and speed of motion of a particular individual is determined by the popu- lation density of the surrounding environment. One advan- tage of continuous models is the diversity of readily avail- able analytic tools that facilitate their study. Since the sensory systems of animals are limited, it is typically assumed that interactions have finite spatial in- fluence. In most PDE-based models advection velocity is specified as a convolution (Mogilner and Edelstein-Keshet (1999),Edelstein-Keshet et al. (1997)): V (f )= K * f =  R K(x - x  )f (x  ,t)dx  , (2) ECAL - General Track 657 ECAL 2013