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.

中文翻译：

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使用内部导航控制系统探索复杂环境的活性粒子的动力学和宏观模型

大量的生物系统——从细菌到绵羊——可以被描述为具有复杂的内部动力学的自推进剂（活性粒子）的集合，控制着代理的行为：休息、缓慢移动、快速移动、进食等。在这项研究中，我们假设这样一个复杂的内部动态可以用马尔可夫链来描述，它控制着智能体的移动方向、速度和内部状态。我们将此马尔可夫链称为导航控制系统（NCS）。此外，我们通过考虑 NCS 的转换率取决于环境的局部（标量）测量值（例如化学浓度、光强度或温度）来模拟代理感知环境。这里，我们研究了在哪些条件下，代理的（渐近）行为可以简化为代理密度的有效对流-扩散方程，为漂移和扩散项提供有效的表达式。我们将开发的通用框架应用于一系列具体示例，以表明为了获得漂移项，应满足三个必要条件：（i）NCS 应具有两个或多个内部状态，（ii）NCS 转换率应取决于代理人的位置，并且（iii）转换率应该是不对称的。此外，我们指出漂移项的符号——即代理是否产生正或负趋化反应——可以通过修改 NCS 的不对称性或通过交换与内部状态相关的速度来改变。