Collective behaviour of active particles with mean-field interaction
This thesis is concerned with providing mathematical foundations for the emergent collective behaviour of ants. That is, we rigorously derive a macroscopic partial differential equation (PDE) from a nonlinear interacting stochastic particle system used to model ants and use the PDE to provide a phase diagram for the collective behaviour of the particle model. The PDE model sustains ants forming bidirectional lanes and ants forming spherical aggregates, and the behaviour depends critically on the ant antenna. This thesis contributes to the nonlinear analysis of pattern formations and emergent collective behaviour by showing rigorously how a relatively simple phase-space extended non-gradient, nonlocal and nonlinear single species model can create multiple metastable higher-dimensional patterns. In the second chapter, we introduce a microscopic agent-based model for a collective of interacting ants. We model the ants as a stochastic interacting particle system, where each particle has a position and orientation, as an active Brownian particle. The interaction of the particles couples the spatial distribution of the particles by a torque that acts on the orientations, in contrast with the chemotactic drift in the parabolic-elliptic Keller--Segel model. The particles also have a look-ahead mechanism, which models the antenna of ants, and allows the particles to anticipate the chemical field. We show typical particle trajectories for the linear model with several background chemical fields. The behaviour includes (damped) oscillations that are predicted by the inviscid linearised theory. We also show the typical trajectories for the interacting nonlinear model for bounded radially symmetric interaction kernels. This behaviour features aggregation and the formation of travelling clusters. We also introduce a singular interaction kernel, such that the chemical field is a distributional solution of an elliptic equation. Motivated by the behaviour of the nonlinear microscopic model, in the third chapter, we derive a mean-field limit PDE. We pass to the large-particle limit for the microscopic model and rigorously derive a mean-field limit PDE for both bounded interaction kernels using classical Lipschitz stochastic theory and for the singular kernel via recent results for the singular stochastic particle Keller--Segel model. The mean-field limit for the singular kernel is obtained via a compactness method. Next, in the fourth chapter, we provide a well-posedness theory for the parabolic-elliptic mean-field limit PDE using the parabolic regularity of the PDE for the spatially-integrated phase space density. We also show that the solutions of the mean-field PDE model exist globally in time and do not blow up at infinite time, which makes use of the $L^p$ Alikakos iteration method. We show a nonlinear stability result for the time-dependent problem for small interaction strength and small initial perturbation. We also show a uniqueness result for the stationary PDE for small interaction strength. We then show the behaviour of time-dependent solutions for the nonlinear PDE via a finite volume scheme. To quantitatively distinguish the lane formation from the aggregation behaviour, we introduce a new metric. This shows the presence of a phase transition and a bistability region in the parameter space. We also show evidence for the presence of quasi-steady states that are smaller copies of the stationary states, repeated over the domain. In the fifth chapter, we show the convergence of an implicit-in-time finite volume scheme for the nonlinear ant mean-field PDE, which provides a rigorous foundation for the numerical results from the previous chapter. We also establish higher regularity estimates, which are discrete analogues of parabolic PDE higher regularity results, and make use of a recently established finite volume Morrey inequality to establish the discrete analogue of the no blow-up result. In the last chapter, we show an asymptotic analysis of the linear model in the high interaction strength limit (the overdamped limit). We show explicit asymptotic solutions in the outer regions for the $x$ variable for a $y$-invariant harmonic well. We also show the inner problem in $x$ and provide adequate boundary conditions. We check the asymptotic analysis using the numerical scheme from the previous chapter, which also provides evidence that the scheme is preserving the asymptotic limit. We also verify the scaling of the inner region. This shows evidence for how our ant model relates to a recently studied Keller--Segel model with discontinuous advection.
active matter, PDE, SDE, analysis, numerical analysis, pattern formation, singular, phase transition, well-posedness, Alikakos, Sobolev, overdamped, asymptotic
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