symmetry S S Article Bootstrapped Motion of an Agent on an Adaptive Resource Landscape Trung V. Phan 1,† , Gao Wang 2,† , Liyu Liu 2 and Robert H. Austin 1, *   Citation: Phan, T.V.; Wang, G.; Liu, L.; Austin, R.H. Bootstrapped Motion of an Agent on an Adaptive Resource Landscape. Symmetry 2021, 13, 225. https://doi.org/10.3390/ sym13020225 Academic Editors: Theo Odijk and Paul van der Schoot Received: 14 January 2021 Accepted: 27 January 2021 Published: 29 January 2021 Publisher’s Note: MDPI stays neu- tral with regard to jurisdictional clai- ms in published maps and institutio- nal affiliations. Copyright: © 2021 by the authors. Li- censee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and con- ditions of the Creative Commons At- tribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Physics, Princeton University, Princeton, NJ 08544, USA; tvphan@princeton.edu 2 Chongqing Key Laboratory of Soft Condensed Matter Physics and Smart Materials, College of Physics, Chongqing University, Chongqing 400054, China; gaowang@cqu.edu.cn (G.W.); lyliu@cqu.edu.cn (L.L.) * Correspondence: austin@princeton.edu † These authors contributed equally to this work. Abstract: We theoretically show that isolated agents that locally and symmetrically consume re- sources and sense positive resource gradients can generate constant motion via bootstrapped resource gradients in the absence of any externally imposed gradients, and we show a realization of this motion using robots. This self-generated agent motion can be coupled with neighboring agents to act as a spontaneously broken symmetry seed for emergent collective dynamics. We also show that in a sufficiently weak externally imposed gradient, it is possible for an agent to move against an external resource gradient due to the local resource depression on the landscape created by an agent. This counter-intuitive boot-strapped motion against an external gradient is demonstrated with a simple robot system on an light-emitting diode (LED) light-board. Keywords: adaptive interactions; symmetry breaking; boot-strap; mathematical biology; robotics 1. Introduction In biology, it is often assumed that agents do not actively influence their environment, but rather passively respond to the environment. For example, at the simplest level of chemotaxis [1–4], it is assumed that agents sense gradients and move along the gradients, but do not generate or influence the gradients themselves. While the agents’ movements are changed by chemotaxis, in the absence of any gradients, it is assumed that the agents still have a zero-gradient inherent speed. However, chemotactic agents, such as Escherichia coli and Dictyostelium, can explore complex topologies, such as mazes and fractals, much more rapidly and efficiently than a response to an imposed gradient alone would predict [5,6]. This occurs if agents do not just follow an imposed gradient passively, but also influence the surrounding environment by consuming and producing chemicals [7–9]. This emergent self-generated gradient creates a back-reaction, which helps the agents navigate through non-trivial complex geometries. This paper presents a fundamental mechanism involving spontaneous symmetry breaking by which individual agents can self-generate motion to solve the complex topolo- gies that nature presents and even show the ability to back out of a dead-end path that would otherwise trap an agent. The next step will be to let the agents interact with each other and look for emergent large-scale symmetry breaking. 2. Approach We present a scenario where an agent consumes local resources symmetrically and responds to the self-generated resource depression by moving up an emergent self-generated gradient. This emergent self-generated gradient creates a back-reaction, which the agents then use to move away from their self-generated resource depression. In effect, they run from their shadow, but the shadow follows them. The analysis predicts that an agent initially at rest with no inherent motion can bootstrap to a constant motion under certain conditions Symmetry 2021, 13, 225. https://doi.org/10.3390/sym13020225 https://www.mdpi.com/journal/symmetry