## 1 Introduction  
Controlling the collective behavior of a large number of robots is a complex task. However, large natural multi-agent systems are known to work very well, such as bird flocks [1, 2, 3], fish schools [2, 4, 5], ants using pheromones [2, 6], or aggregations of bacteria [2, 7, 8]. These self-organized systems have served as a source of inspiration to control large formations of robots to prevent collisions between these robots [9]. The emergence principle where complex collective behaviors arise from simple, elementary rules governing individuals is the main topic of interest for artificial swarming systems. However, the emergence of swarm behavior requires some constraints on the number of agents, the environment, and so forth. Also, it is difficult to design the elementary rules for a specific desired collective behavior to emerge. One behavior in particular, flocking or schooling, enjoys a growing interest in the scientific domain. [PERSON]. [10] established a first landmark, giving the three rules of flocking (alignment, cohesion, separation). Then [PERSON], [PERSON] and [PERSON], in a two-part article [11, 12], made a fundamental mathematical analysis of the Reynolds rules using the Artificial Potential Field (APF) approach. They proved that using potentials both attractive and repulsive, the flock becomes homogeneous with equal inter-agent distance and equal speed vectors at equilibrium. This behavior is resilient to external stimuli as long as the agents are within communication range. By using these methods, and despite the remaining challenges to designing the desired group behavior, the robotics community has used swarm-based approaches for applications such as target tracking [13; 14; 15; 16; 17], Search And Rescue (SAR) [18], or Odor Source Localization (OSL) [19; 20; 21; 22; 23; 24; 25; 26], among others. Ant Colony Optimization (ACO) [6] or Particle Swarm Optimization (PSO) [27] are swarm intelligence algorithms used in the mathematical community of optimization. The strength of these approaches lies in using agents distributed in the workspace, sharing information to search for the optimum of a fitness function. In particular, this community strongly uses the PSO algorithm [13; 14; 15; 18; 19; 20; 21; 28]. It is important to stress that optimization algorithms like PSO, can also drive a swarm of actual robots [29]. This bridge between the two communities of robotics and optimization shows that one common way to solve their respective problems can be found through swarming. In the present paper, we focus on tracking a mobile target leaving a semi-permanent trail, such as a chemical scent or radioactive particles carried by the environment. This problem could be described as Mobile Odor Source Localization (MOSL), a generalization of OSL. We assume the information to be \"scalar\", in that its value is a real, one-dimensional number. As Section 2 will show, this problem is equivalent to maximizing a fitness function varying with space and time. The plume model is simplified compared to the state-of-the-art [30; 31] to fasten the simulation results and obtain preliminary results but those models will be upgraded in a future work. In a terrestrial environment, instances of this problem arise for example, when tracking a moving source of nuclear radiation with possible counter-terrorist applications [32; 33], or finding the origin of polluted areas [34]. Applications in the marine field include securing a specified area to detect intruders; to prevent illegal oil discharges by ships using so-called \"magic pipes\"; localizing a pipeline leak [35; 36], or various other similar uses. A variant of PSO called Charged Particle Swarm Optimization (CPSO) was proposed by [20; 21] to solve OSL and head for a static odor source, with interesting results [37]. CPSO differs from PSO in that repulsive components are added to avoid collisions between agents in the swarm.