Modeling the aggregative behavior of ants of the species Polyergus rufescens

Boi, Silvia; Capasso, Vincenzo; Morale, Daniela

doi:10.1016/s0362-546x(99)00399-5

Cited by 95 publications

(83 citation statements)

References 6 publications

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“…Mathematical modelling of the collective behavior of individuals, such as swarming, has also been treated by continuum models steaming from discrete particle models [30,14,38,31,39,15,33,13,21,22,16,17]. Typical examples of interaction potentials appearing in these works are the attractive Morse potential W (x) = −e −|x| , attractiverepulsive Morse potentials W (x) = −C a e −|x|/ a + C r e −|x|/ r , W (x) = −e −|x| 2 , W (x) = −C a e −|x| 2 / a + C r e −|x| 2 / r , or W being the characteristic function of a set in R d .…”

Section: Introductionmentioning

confidence: 99%

Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations

et al. 2011

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Abstract. In this paper, we provide a well-posedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The main phenomenon of interest is that, even with smooth initial data, the solutions can concentrate mass in finite time. We develop an existence theory that enables one to go beyond the blow-up time in classical norms and allows for solutions to form atomic parts of the measure in finite time. The weak measure solutions are shown to be unique and exist globally in time. Moreover, in the case of sufficiently attractive potentials, we show the finite time total collapse of the solution onto a single point, for compactly supported initial measures. Finally, we give conditions on compensation between the attraction at large distances and local repulsion of the potentials to have global-in-time confined systems for compactly supported initial data. Our approach is based on the theory of gradient flows in the space of probability measures endowed with the Wasserstein metric. In addition to classical tools, we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations.

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Section: Introductionmentioning

confidence: 99%

Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations

et al. 2011

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“…The purely gradient flow case has been studied for self-interacting individuals via pairwise potentials arising in the modelling of animal collective behavior: flocks, schools or swarms formed by insects, fishes and birds. The simplest models based on ODEs systems [15,24,29,43,44] led to continuum descriptions [19,18,14,37,42,47,48] for the evolution of densities of individuals. It is this class of models that we focus on here, although we will draw parallels to well-known problems and results from the incompressible flow literature.…”

Section: Introductionmentioning

confidence: 99%

The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels

Bertozzi

Laurent

2009

Chin. Ann. Math. Ser. B

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We consider the multidimensional aggregation equation ∂ t ρ−div(ρ∇K * ρ) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better). We review recent results on this problem concerning well-posedness of nonnegative solutions and finite time blowup in multiple space dimensions depending on the behavior of the kernel at the origin. We consider the problem with bounded initial data, data in L p ∩ L 1 , and measure-valued solutions.

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“…These models are ubiquitous in mathematical biology where they have been used as macroscopic descriptions for collective behavior or swarming of animal species, see [53,13,54,55,67,16] for instance, or more classically in chemotaxis-type models, see [60,44,40,39,11,10,20] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Уравнение Агрегации С Анизотропной Диффузией

Вильданова¹

2017

Trudy IMM UrO RAN

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Abstract. We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrization techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations as t → ∞.

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Modeling the aggregative behavior of ants of the species Polyergus rufescens

Cited by 95 publications

References 6 publications

Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations

Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations

The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels

Уравнение Агрегации С Анизотропной Диффузией

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