Impact of memory on clustering in spontaneous particle aggregation

TL;DR

The paper analyzes how memory, implemented as a chain of $K$ internal variables per agent, shapes spontaneous aggregation in a stochastic particle system. It derives a formal macroscopic Fokker–Planck description in the large-population limit and characterizes steady states that permit nonuniform cluster densities via the condition $G(W*\varrho)^2\,\varrho=C_0$. Extensive 1D and 2D simulations reveal three memory-driven regimes: short/medium memory enhances coarsening into fewer, larger clusters, while long memory suppresses clustering and yields many outliers, with memory also dampening responsiveness to local density. The findings connect microscopic memory dynamics to emergent spatial patterns and offer broader implications for memory evolution in collective systems.

Abstract

The effect of short-term and long-term memory on spontaneous aggregation of organisms is investigated using a stochastic agent-based model. Each individual modulates the amplitude of its random motion according to the perceived local density of neighbors. Memory is introduced via a chain of $K$~internal variables that allow agents to retain information about previously encountered densities. The parameter $K$ controls the effective length of memory. A formal mean-field limit yields a macroscopic Fokker--Planck equation, which provides a continuum description of the system in the large-population limit. Steady states of this equation are characterized to interpret the emergence and morphology of clusters. Systematic stochastic simulations in one- and two-dimensional spatial domains reveal that short- or moderate-term memory promotes coarsening, resulting in a smaller number of larger clusters, whereas long-term memory inhibits aggregation and increases the proportion of isolated individuals. Statistical analysis demonstrates that extended memory reduces the agents' responsiveness to environmental stimuli, explaining the transition from aggregation to dispersion as $K$ increases. These findings identify memory as a key factor controlling the collective organization of self-driven agents and provide a bridge between individual-level dynamics and emergent spatial patterns.

Figures (7)

• Figure 1: Plots of the kernel $\kappa=\kappa(t)$ given by \ref{['eq:kappa']} with $K/\alpha = 1$ (left panel) and with $K/\alpha=2$ (right panel).
• Figure 2: One-dimensional equilibrium profiles satisfying the condition $(\ref{['eq:eqrho']})$, obtained by solving equation \ref{['eq:rho']} in domain $(\ref{['domainomega']})$ for $d=1$, subject to the initial datum $\varrho(t=0,x) = 1+ 10^{-1} \sin(2\pi x(2-x))^2$, until a steady state is reached. We use $G(s)$ and $W(x)$ given by $(\ref{['choiceGandW']})$ with $R=0.1$$($top$)$, $R=0.05$$($middle$)$ and $R=0.025$$($bottom$)$. The left panels depict the steady state density $\varrho=\varrho(x)$, while the right panels visualize the function $G(W\ast\varrho)^2$.
• Figure 3: Simulations of the individual-based model \ref{['rho_i']} and \ref{['SDE1']} for $K\in\{1,\dots,5\}$, and the model \ref{['rho_i']}--\ref{['model1']} without memory ($K=0)$. We used $N=400$ agents moving in the domain $\Omega$ given by $(\ref{['domainomega']})$ with $d=1$. $G(s)$ and $W({\mathbf x})$ are given by $(\ref{['choiceGandW']})$ with $R=0.025$. For each value of $K\in\{0,1,\dots,5\}$, the plots capture the particle positions at the final time $t=10^3$$($horizontal axis$)$ and the perceived density of their neighbours $\vartheta_i$ given by \ref{['rho_i']}$($vertical axis$)$. The clusters, differentiated by colour, are identified using the DBSCAN method with parameters $\mathtt{epsilon}=0.025$ and $\mathrm{MinPoints}=20$. The light green points are outliers, i.e., particles not belonging to any cluster.
• Figure 4: Statistics of the clustering behaviour over $100$ realizations of the individual-based model given by equations \ref{['rho_i']} and \ref{['SDE1']} for dimension $d=1$, $N=400$ agents, $G(s)$ and $W(x)$ given by $(\ref{['choiceGandW']})$ with $R=0.025$ and $K \in \{1, 2, \dots, 6\}$. The case $K=0$ (no memory) refers to stochastic simulation of the system \ref{['rho_i']}--\ref{['model1']}. Other parameters are the same as in Figure \ref{['figure2']}. (a) average $($orange solid line$)$, minimum and maximum $($indicated by blue error bars$)$ number of clusters identified in the final timestep of the simulations at time $t=10^3$, (b) average $($orange solid line$)$, minimum and maximum $($blue error bars$)$ cluster sizes, (c) average $($orange solid line$)$, minimum and maximum $($blue error bars$)$ number of outliers, i.e., particles that do not belong to any cluster, (d) percentage of simulation outcomes $($out of the $100$ runs$)$ that did not produce any clusters.
• Figure 5: Simulations of the individual-based model \ref{['rho_i']} and \ref{['SDE1']} for $K\in\{1,\dots,8\}$, and the model \ref{['rho_i']}--\ref{['model1']} without memory ($K=0)$. We used $N=400$ agents moving in the domain $\Omega$ given by $(\ref{['domainomega']})$ with $d=2$. Functions $G(s)$ and $W({\mathbf x})$ are given by $(\ref{['choiceGandW']})$ with $R=0.05$. For each value of $K\in\{0,2,\dots,8\}$, the plots capture the particle positions at the final time $t=10^4$. The clusters, differentiated by colour, are identified using the DBSCAN method with parameters $\mathtt{epsilon}=0.05$ and $\mathtt{minPts}=12$. The points indexed with $-1$ are outliers, i.e., not belonging to any cluster.