Noisy pursuit and pattern formation of self-steering active particles

Abstract

We consider a moving target and an active pursing agent, modeled as an intelligent active Brownian particle capable of sensing the instantaneous target location and adjusting its direction of motion accordingly. An analytical and simulation study in two spatial dimensions reveals that pursuit performance depends on the interplay between self-propulsion, active reorientation, limited maneuverability, and random noise. Noise is found to have two opposing effects: (i) it is necessary to disturb regular, quasi-elliptical orbits around the target, and (ii) slows down pursuit by increasing the traveled distance of the pursuer. For a stationary target, we predict a universal scaling behavior of the mean pursuer-target distance and of the mean first-passage time as a function of $\mathrm{Pe}^2/\Omega$, where the P'eclet number $\mathrm{Pe}$ characterizes the activity and $\Omega$ the maneuverability. Importantly, the scaling variable $\mathrm{Pe}^2/\Omega$ depends implicitly on the level of thermal or active noise. A similar behavior is found for a moving target, but modified by the velocity ratio $\alpha =u_0/v_0$ of target and pursuer velocities $u_0$ and $v_0$, respectively. We also propose a strategy to sort active pursuers according to their motility by circular target trajectories.

Pursuit problem with a stochastic prey that sees its chasers

“…Therefore, also in this case the CT diverges. Note that the critical condition for divergence (v e = 1) is independent of the parameter S. These qualitative considerations are confirmed by the solution of equation (8), which for D = 0 is an ordinary differential equation:…”

“…In this context, predators and preys have to deal with severe uncertainties in sensing each other's position and cannot reliably 'see' each other as in our model [63][64][65][66]. Such uncertainties would translate into rotational noise acting on the velocity vector of pursuers and evaders, a kind of noise that can be beneficial or detrimental in the pursuit of a non-random target [8,47]. It is not clear, however, how this further stochastic element would affect the present model, in particular if the model is endowed with different evasion strategies.…”

“…Several of these models incorporate also random noise elements, which is particularly relevant if predators and preys are as small as, for instance, microorganisms or the increasingly popular artificial microswimmers capable of target tracking [24][25][26][27][28][29]. Typical stochastic chase-and-pursuit models, however, are either so detailed that they can only be investigated through numerical simulations [8,13,14,16,17,19,20,30,31], or rather abstract [4,7,32]. Analytical results have been mostly found for the case that both predators and prey are random walkers confined to a one-dimensional space [33][34][35], to grids [36], or to graphs [37].…”

“…The seemingly simplest case of this problem, i.e. only one chaser hunting a single target, can already pose challenging mathematical problems [1][2][3][4][5][6][7][8]. Different authors have also investigated more elaborate situations in which a pack of hunters chase a single prey [9,10], a single chaser pursues a herd of prey [11,12], or a pack of hunters chasing multiple preys [13][14][15][16][17][18][19][20][21].…”

Light Controlled Biohybrid Microbots

“…Indicating with r the microbot position vector measured from the target and with $$\widehat {\rm{e}}$$ its orientation, Figure a, we can define the angular distance from the target as the angle θ between directions $\[ - \widehat {\rm{r}}\]$ and $\[\widehat {\rm{e}}\]$. Following, [ 32 ] the time evolution of θ will be given by: $$\[\dot{\theta } = \omega (\theta ) + \frac{{v(\theta )\sin \theta }}{r}\]$$…”