Aggregation-diffusion in heterogeneous environments

TL;DR

This work develops a 1D aggregation-diffusion framework in heterogeneous environments, coupling diffusion, nonlocal self-attraction, and environmental gradients. By focusing on quadratic diffusion and tractable reductions (Laplace kernel or second-moment expansion), it classifies steady states and couples them with an energy minimisation principle to predict emergent space use. In a single clump landscape, the authors reveal a non-monotonic relationship between clump width and aggregation width and, under strong resource attraction, a counterintuitive widening of the aggregation as self-attraction strengthens, with numerical verification. Overall, the study provides a rigorous link between environment and collective movement and demonstrates rapid, low-dimensional prediction of pattern formation, while exploring robustness to model variants and sensitivity to initial conditions.

Abstract

Aggregation-diffusion equations are foundational tools for modelling biological aggregations. Their principal use is to link the collective movement mechanisms of organisms to their emergent space use patterns in a concrete mathematical way. However, most existing studies do not account for the effect of the underlying environment on organism movement. In reality, the environment is often a key determinant of emergent space use patterns, albeit in combination with collective aspects of motion. This work studies aggregation-diffusion equations in a heterogeneous environment in one spatial dimension. Under certain assumptions, it is possible to find exact analytic expressions for the steady-state solutions when diffusion is quadratic. Minimising the associated energy functional across these solutions provides a rapid way of determining the likely emergent space use pattern, which can be verified via numerical simulations. This energy-minimisation procedure is applied to a simple test case, where the environment consists of a single clump of attractive resources. Here, self-attraction and resource-attraction combine to shape the emergent aggregation. Two counter-intuitive findings emerge from these analytic results: (a) a non-monotonic dependence of clump width on the aggregation width, (b) a positive correlation between self-attraction strength and aggregation width when the resource attraction is strong. These are verified through numerical simulations. Overall, the study shows rigorously how environment and collective behaviour combine to shape organism space use, sometimes in counter-intuitive ways.

Figures (7)

• Figure 1: Aggregation in a single-clumped landscape: numerics. Panel (a) shows $A(x)=a_1[1+\cos(\pi x)]$, a single clump of attractive resources centred on $x=0$. With this functional form of $A(x)$ in place, Panels (b)-(d) show initial ('Start') and final ('End') numerical solutions for Equation (\ref{['eq:adfo']}), for example values of $a_1$. The initial condition is the minimum energy solution in the case where $A(x)=0$ (a homogeneous landscape), so we can see how the introduction of landscape heterogeneity affects the shape of the aggregation.
• Figure 2: Aggregation in a single-clump landscape: minimum energy solutions. Panels (a), (b), and (c) show the minimum energy solution of the form given by Equations (\ref{['eq:ustarx_1hump']})-(\ref{['eq:1hump_Qcts']}) (solid curves), for $n=1$, alongside the numerical steady-state solution (dashed curves) for different values of $a_1$ given in the plots. Panels (d), (e), and (f) show the energy as a function of $r$ for the values of $a_1$ given in Panels (a), (b), and (c) respectively. The respective minimum energy $r$-values are $r=0.289, r=0.248,$ and $r=0.175$.
• Figure 3: Dependence of aggregation size on model parameters. Aggregation sizes refer to minimum energy steady-state solutions to Equation (\ref{['eq:adfond']}) with $A(x)$ as given in Equation (\ref{['eq:ax_1hump']}), and calculated using the energy-minimisation procedure from Section \ref{['sec:scl']}. Unless otherwise stated, $n=1$, $a_n=1$, $\gamma=2$, and $\sigma=0.1$.
• Figure 4: Numerical verification of analytic insights. Panel (a) shows a zoomed-in version of the case $\sigma=0.1$ from Figure \ref{['fig:adh_plot_r_vs_params']}e. Panel (b) shows numerical steady state-solutions to Equation (\ref{['eq:adfond']}) corresponding to the parameters from Panel (a) (namely, $\gamma=2$, $a_n=1$, $\sigma=0.1$). Panel (c) shows the case $a_1=1000$ from Figure \ref{['fig:adh_plot_r_vs_params']}f and Panel (d) gives the corresponding numerical steady-state solutions.
• Figure 5: Numerical investigation. Panel (a) shows numerical steady state solutions of Equation (\ref{['eq:aggdiffnd']}) with a Laplace kernel (Equation \ref{['eq:lap_loc']}), for various $n$ and $m=10$. Panel (b) shows numerical steady state solutions of Equation (\ref{['eq:aggdiffnd']}) with $K_m$ replaced with a top hat kernel (Equation \ref{['eq:th']}), for various $n$ and $\delta=0.1$. Panel (c) shows numerical steady state solutions of Equation (\ref{['eq:aggdiff_lin']}) with a top hat kernel with $\delta=0.1$. Panels (d-f) show the aggregation half-size as a function of the resource hump width, corresponding to the numerics shown in Panels (a-c) respectively. In Panels (d-e), the aggregation sizes are calculated from the support of the numerical steady state. For Panel (f), with linear diffusion, we have $u_\ast(x)>0$ across the whole interval $[-1,1]$, so the aggregation sizes are calculated at height $u_\ast(x)=0.1$. In all panels, $\gamma=2$, $a_n=1$, $p=1$, and $A(x)$ is as in Equation (\ref{['eq:ax_1hump']}).

Theorems & Definitions (3)

• Proposition 1
• Proposition 2
• Proposition 3