A hybrid model of collective motion of discrete particles under alignment and continuum chemotaxis

Abstract

In this paper we propose and study a hybrid discrete in continuous mathematical model of collective motion under alignment and chemotaxis effect. Starting from the paper by Di Costanzo et al (2015a), in which the Cucker-Smale model (Cucker and Smale, 2007) was coupled with other cell mechanisms, to describe the cell migration and self-organization in the zebrafish lateral line primordium, we introduce a simplified model in which the coupling between an alignment and chemotaxis mechanism acts on a system of interacting particles. In particular we rely on a hybrid description in which the agents are discrete entities, while the chemoattractant is considered as a continuous signal. The proposed model is then studied both from an analytical and a numerical point of view. From the analytic point of view we prove, globally in time, existence and uniqueness of the solution. Then, the asymptotic behaviour of a linearised version of the system is investigated. Through a suitable Lyapunov functional we show that for t → +∞, the migrating aggregate exponentially converges to a state in which all the particles have a same position with zero velocity. Finally, we present a comparison between the analytical findings and some numerical results, concerning the behaviour of the full nonlinear system.

Long-time behaviour of the approximate solution to quasi-convolution Volterra equations

“…In [5] a simplified mathematical model of collective motion, in which the coupling between an alignment and chemotaxis mechanism acts on a system of interacting particles, is described as an hybrid system, where particles are considered discrete entities and the chemical signal is supposed to be continuous. The rate of change in time of the concentration in signal is equal to the sum of a diffusion term, a source term depending on the position of each particle, and a degradation term.…”

“…, 3, given. From the analytic point of view, it has been proved in [5] that the velocity of the centre of mass tends time-asymptotically to zero. Equation (5.4) assumes the form (1.4) (and thus (1.6)) with…”

“…However, some recent applications, as cell migration and collective motion (see e.g. [5][6][7]), are characterized by kernels of quasi-convolution type, namely involving a convolution contribution plus a non-convolution term. The presence of this spurious term prevents from using the Lubich approach for describing the long-term behaviour of the numerical solution.…”

“…, : [0, +∞) → R , : [0, +∞) × [0, +∞) → R × , with ( , ) = 0 when > , and : [0, +∞) → R × , ≥ 1. Equation (1.4) includes the model for the motion of the centre of mass of a system of interacting particles described in [5], and the integral formulation of the cell migration model, which appears for example in [6,7], fits into the form of equation (1.7). Therefore, in Section 5 we describe the application to these specific problems.…”

“…The Paley-Wiener condition is straightforwardly satisfied in some particular cases of linearised models for 1D cell migration, described in [6], where due to the boundedness of the forcing term, the numerical solution remains bounded, thus mimicking the behaviour of analytical one. For the motion of the centre of mass of a system of interacting particles described in [5], the forcing function is zero and the exponential decay of the non convolution terms is readily observed. The difficulty for this problem arises in checking the Paley-Wiener condition which requires a dedicated investigation, directly on the discrete equation obtained from the application of Backward Euler method.…”