A study of general reaction-advection-diffusion equations describing dynamics between target, partaker, and guardian

TL;DR

The paper develops a general reaction–advection–diffusion model for three sociological actors—target $u$, partaker $v$, and guardian $w$— grounded in routine activity theory and applied to crime, protest dynamics, and bullying. It proves local and global well-posedness in two spatial dimensions under structural hypotheses, and performs linear stability analysis of the homogeneous steady state to derive explicit stability conditions. Through two sociological applications—protest propensity and school bullying—it demonstrates, via 2D numerical simulations, that the RAD system can produce hotspots, uniform states, or oscillatory patterns depending on interaction terms and management strategies. The work highlights how control policies influence spatial-temporal patterns and provides a framework for further theoretical analysis, model refinement, and data-driven intervention design.

Abstract

This paper introduces a reaction-advection-diffusion system that models interactions among three actors: a target, a partaker, and a guardian. The framework is versatile, capturing phenomena ranging from the emergence and movement of crime hotspots in urban areas to shifts in public attitudes during critical events as individuals and control units move through space. We prove local and global existence of solutions under realistic assumptions and showcase the model through two applications: protest dynamics driven by new demonstrators joining and escalating hostility in enclosed environments such as classrooms or offices. Numerical simulations highlight the resulting complex spatial patterns and temporal dynamics.

Figures (6)

• Figure 1: Root mean square of the solutions to system \ref{['eq:protest_peace']}. The two subfigures show how a change in the single parameter $\Psi$ affects the solution behavior. In both (a) and (b), the amplitudes become constant. Another common feature is that $A_{Amp}$ and $P_{Amp}$ rise and stabilize at a value higher than the initial. However, the spatial landscape is different: in (a), it is spatially heterogeneous, while in (b) it is uniform. The simulations are performed on a square domain with a side length of $\pi$ with parameters: $D_A=0.1$, $D_P=0.1$, $D_M=0.1$, $\chi_P=2$, $\chi_M=1$, $\Phi_A=1$, $\Phi_P=2$, $\psi=0.1$. $\Psi$ differs between the two subfigures. The initial conditions are chosen as $[A_0,P_0,M_0]+\varepsilon_0e^{-x-y}$, where $A_0=1.0$, $P_0=0.0$, $M_0=1.0$, and $\varepsilon_0=0.01$.
• Figure 2: The final time frames of the solutions to system \ref{['eq:protest_peace']}. (a) Shows a grid-like pattern of circular clusters at the end of the simulation in Figure \ref{['fig:negot_amp_hetero']}. (b) Shows the corresponding simulation in Figure \ref{['fig:negot_amp_const']}, where decreasing $\Psi$ leads to a homogeneous landscape. The set-up and parameters are the same as in Figure \ref{['fig:negot_amp']}.
• Figure 3: Root mean square of the solutions to system \ref{['eq:protest_elevated']}. The two subfigures illustrate the effect of decreasing $\Psi$ on the system’s dynamics. In (a), a higher $\Psi$ leads the system to a constant steady state, while in (b), lowering $\Psi$ results in high-frequency periodic oscillations, demonstrating a transition from predictable to more complex behavior. The simulations are performed in the same setting as in Figure \ref{['fig:negot_amp']}.
• Figure 4: Final frames of the solutions to system \ref{['eq:protest_elevated']}. (a) Symmetric pattern formation where $A$ and $P$ are concentrated at the center and $M$ at the corners. (b) A similar pattern with the clustering reversed. All components oscillate at the same frequency, but $M$ lags behind $A$ in its motion. The setup and parameters are the same as in Figure \ref{['fig:enhan_amp_per']}.
• Figure 5: Root mean square of the solutions to system \ref{['eq:protest_elevated']}. The three subfigures illustrate how changes in the initial density of the guardian agents affect solution behavior, particularly the bullies' steady-state density. Consecutive decrease in $G_0$ and corresponding increase in $V_0$ changes the steady state solution behavior from (a) trivial to (b) non-trivial constant to (c) periodic. $B_0=0$ in every settings, yet average $B$ increases from (a), where it is zero to (c), where it fluctuates around $0.05$. The domain of the simulation is a square with a side length of $\pi$. The parameters: $D_V=0.05$, $D_B=0.05$, $D_G=0.05$, $\chi_B=2$, $\chi_G=2$, $\Phi_G=0.5$, $\Phi_B=1$, $\Psi=10$. The initial conditions are chosen as $[V_0,B_0,G_0]+\varepsilon_0e^{-x-y}$, where $B_0=0.0$, $\varepsilon_0=0.01$, and $V_0, G_0$ varies between subfigures.

Theorems & Definitions (26)

• Remark 2.1
• Theorem 2.1: Local existence
• proof
• Remark 2.2
• Theorem 2.2: Global existence
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3