Persistence of motion is the tendency of an object to maintain motion in a direction for short time scales without necessarily being biased in any direction in the long term. One of the most appropriate mathematical tools to study this behaviour and its implications at multiple scales in a multiple-agent system is the velocity-jump process. In the absence of agent-agent interaction, the system can be described at the population level using the hyperbolic telegraph equation. When agent-agent interaction is included  , a strictly advective system of partial differential equations (PDEs) has been derived . However, no diffusive limit  has been obtained from such a model. A diffusive macroscopic description is desirable, since it allows the exploration of a wider range of scenarios and establishes a direct connection with commonly used statistical tools of movement analysis. We study a generalisation of the  velocity-jump process on a two-dimensional lattice with three forms of agent interaction. This generalisation allows us to take a diffusive limit and obtain a faithful macroscopic description. We investigate the properties of the model  and we elucidate some of the  key characteristic features of interacting persistent agents . In particular, we show an intrinsic anisotropy of the model and we find evidence of a spontaneous form of aggregation at both the micro- and macro-scales.
Persistence of motion is the tendency of an object to maintain motion in a direction for short time scales without necessarily being biased in any direction in the long term. One of the most appropriate mathematical tools to study this behaviour is an agent-based velocity-jump process. In the absence of agent-agent interaction, the mean-field continuum limit of the agent-based model (ABM) gives rise to the well known hyperbolic telegraph equation. When agent-agent interaction is included in the ABM , a strictly advective system of partial differential equations (PDEs) can be derived at the population-level . However, no diffusive limit of the ABM has been obtained from such a model. Connecting the microscopic behaviour of the ABM to a diffusive macroscopic description is desirable, since it allows the exploration of a wider range of scenarios and establishes a direct connection with commonly used statistical tools of movement analysis. In order to connect the ABM at the population-level to a diffusive PDE at the population-level, we consider a generalisation of the agent-based velocity-jump process on a two-dimensional lattice with three forms of agent interaction. This generalisation allows us to take a diffusive limit and obtain a faithful population-level description. We investigate the properties of the model at both the individual and population-level and we elucidate some of the models' key characteristic features . In particular, we show an intrinsic anisotropy inherent to the models and we find evidence of a spontaneous form of aggregation at both the micro- and macro-scales.
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