Collective behavior based on agent-environment interactions

TL;DR

The study shows that collective patterns can emerge in active populations solely from interactions with a dynamically evolving resource landscape, without direct agent-to-agent communication. By embedding chemotaxis into a resource-consumption framework with logistic regrowth, the authors derive IBM and PDE descriptions that capture a spectrum of states—from disordered gas to polar traveling waves and nematic clusters. The phase structure is controlled by the interplay of chemotactic sensitivity $\gamma$ and angular noise $D_\theta$, with transitions at $\gamma_p=2D_\theta$ and $\gamma_n=v_0/2$, and an edge-of-chaos region where population density is optimized. The results bridge active-matter physics and movement ecology, highlighting how environmental feedback alone can organize complex spatiotemporal patterns and efficient resource exploitation.

Abstract

We present a model of active particles interacting through a dynamic, heterogeneous environment, leading to emergent collective behaviors without direct agent-to-agent communication. Expanding the resource-dependent framework introduced in Briozzo et al., 2025, arXiv:2512.08762, agents perform a persistent random walk combined with chemotaxis, directing toward nutrient-rich patches, whose resources are generated by logistic regrowth. We identify distinct phases of collective organization, ranging from disordered gas-like states to polar traveling waves and nematic independent clusters, depending on the interplay between chemotactic sensitivity and angular noise. The system exhibits spontaneous symmetry breaking and density waves driven purely by the coupling between population dynamics (birth-death processes) and environmental feedback. Our results bridge active matter physics and movement ecology, demonstrating that complex spatiotemporal patterns can arise without direct interaction between agents, but solely from the maximization of resource intake in a reactive environment.

Figures (5)

• Figure 1: Emergent collective phases: Comparison between IBM and PDE. (a-j) Individual-Based Model (IBM) results. (k-t) Continuum (PDE) solutions. Top row (a-e): Snapshots of agent spatial distribution and inner energies. Second row (f-j): Kymographs showing the time evolution of agent density. Third row (k-o): Numerical solutions of the fields from Eq. \ref{['eq:M_FP_st3']}. Bottom row (p-t): PDE density kymographs. The columns correspond to different dynamical regimes driven by the alignment force $\gamma$: Left column (Gas phase): $v_0=0.70$, $D_\theta=0.01$, $\gamma=0.01$. For $\gamma < \gamma_p$, angular noise dominates, resulting in a globally homogeneous, disordered state. Second column (Polar order): $\gamma=0.10$. In the regime $\gamma_p < \gamma < \gamma_n$, spontaneous symmetry breaking occurs, leading to periodic traveling waves. Third column (Nematic order): $\gamma=1.00$. For $\gamma > \gamma_n$, the wavefronts fragment into compact, independent clusters with opposing velocities (ripple effect). Fourth & Right columns (Proliferation waves): $v_0=0.05$, $D_\theta=0.05$, $\gamma=0.00$. In the absence of chemotaxis, slow diffusion coupled with birth-death dynamics drives the emergence of planar (fourth column) and circular (right column) density waves. See Supplemental Material for descriptive videos.
• Figure 2: Phase diagram and order parameters. Steady-state averages for population density $\langle n_\infty \rangle$ (1st col), spatial entropy $\langle S_\infty \rangle$ (2nd col), polar order $\langle P_\infty \rangle$ (3rd col), and nematic order $\langle Q_\infty \rangle$ (4th col). Top row (a-d): Dependence on active velocity $v_0$. Survival is limited to a window bounded by critical velocities (Eqs. \ref{['eq:v_min']}, \ref{['eq:v_max']}). Middle row (e-h): Dependence on alignment strength $\gamma$. Two transitions are visible: (i) Disorder-to-Polar at $\gamma/D_\theta \approx 2$, where $P$ rises sharply; (ii) Polar-to-Nematic at $\gamma \approx 0.5$, where $P$ declines while $Q$ remains high, indicating wave breakup. Bottom row (i-l): Dependence on the noise-to-alignment ratio $D_\theta/\gamma$. The collapse of curves for different $\gamma$ values highlights $D_\theta/\gamma$ as the primary control parameter for the order transition. Note that population (1st col) peaks near the transition point before decaying in the highly ordered phase, while entropy (2nd col) is minimized in the ordered regimes.
• Figure 3: (a): Numerical integration of Eqs. \ref{['eq:M_FP_st4']} and \ref{['eq:M_AA_fD']}. (b): Numerical integration of Eq. \ref{['eq:M_AA_fPQ']}. (c): Phase portrait and orbit from Eq. \ref{['eq:M_AA_fD']}.
• Figure 4: Temporal dynamics and stability. (a-b) Time evolution of the mean nematic order $\hat{O}$ and spatial entropy $\hat{S}$ from a disordered initial state ($N_i=200, v_0=2^{-0.5}$). The system relaxes to a steady state determined by $\gamma$ after a transient period. (c) Evolution of population density $\langle n(t) \rangle$ for various initial densities $\langle n(0) \rangle$. Red dashed lines indicate reproduction ($n_r$) and starvation ($n_s$) thresholds. Populations starting within or below the stable range converge to an equilibrium $\langle n_{\text{eq}} \rangle \approx (n_r+n_s)/2$. Those starting above $n_s$ exhibit damped oscillations, while excessive initial densities ($\langle n(0) \rangle > n_c \approx 2.6$) lead to extinction. (d) Equilibrium population as a function of initial density, confirming the extinction threshold. (e) Center-of-mass velocity components vs. noise parameter $\tau$ for $\gamma=0.10$. Below the critical threshold $\tau < \tau_c$, wave motion is stable and unidirectional (one component dominates). Near $\tau_c$, fluctuations destabilize the direction, and for $\tau > \tau_c$, coherent motion vanishes. (f) Ripple effect mechanism ($\gamma=0.5, D_\theta=0.0015$): Spatial histograms of agent density before, during, and after a collision event, showing the interpenetration of counter-propagating clusters (red: right-moving, blue: left-moving).
• Figure 5: System energetics and consumption. Steady-state averages of patch energy $\langle f \rangle$ (1st col), agent energy $\langle e \rangle$ (2nd col), and total consumption rate $\Omega$ (3rd col) plotted against the implicit equilibrium population $\langle n_\infty \rangle$. Rows correspond to variations in active velocity $v_0$ (top), alignment force $\gamma$ (middle), and angular diffusion $D_\theta$ (bottom). The data reveals that: (i) Patch energies (left column) follow the mean-field prediction (dashed lines) robustly. (ii) Agent energies (center column) deviate from mean-field theory at high energies/low populations due to fluctuations. (iii) The total consumption $\Omega$ (right column) exhibits a distinct maximum near $\langle n_\infty \rangle \approx 1$, identifying an optimal density for resource exploitation efficiency.