Kinetic and macroscopic models for active particles exploring complex environments with an internal navigation control system

Abstract

A large number of biological systems — from bacteria to sheep — can be described as ensembles of self-propelled agents (active particles) with a complex internal dynamic that controls the agent’s behavior: resting, moving slow, moving fast, feeding, etc. In this study, we assume that such a complex internal dynamic can be described by a Markov chain, which controls the moving direction, speed, and internal state of the agent. We refer to this Markov chain as the Navigation Control System (NCS). Furthermore, we model that agents sense the environment by considering that the transition rates of the NCS depend on local (scalar) measurements of the environment such as e.g. chemical concentrations, light intensity, or temperature. Here, we investigate under which conditions the (asymptotic) behavior of the agents can be reduced to an effective convection–diffusion equation for the density of the agents, providing effective expressions for the drift and diffusion terms. We apply the developed generic framework to a series of specific examples to show that in order to obtain a drift term three necessary conditions should be fulfilled: (i) the NCS should possess two or more internal states, (ii) the NCS transition rates should depend on the agent’s position, and (iii) transition rates should be asymmetric. In addition, we indicate that the sign of the drift term — i.e. whether agents develop a positive or negative chemotactic response — can be changed by modifying the asymmetry of the NCS or by swapping the speed associated to the internal states. The developed theoretical framework paves the way to model a large variety of biological systems and provides a solid proof that chemotactic responses can be developed, counterintuitively, by agents that cannot measure gradients and lack memory as to store past measurements of the environment.