Resource and population dynamics in an agent-environment interaction model

TL;DR

This work addresses how mobile agents forage in a dynamically regenerating environment by coupling an energy depot–based agent model to a patchy, logistic nutrient field. It combines an agent-based implementation with analytical limiting-case calculations to reveal two viable movement strategies—static and high-motility—alongside an absorbing phase resulting from finite-size oscillations. A key finding is that population size is often inversely related to the mean energy per agent, yet resource scarcity or higher metabolic expenditure can, under certain conditions, sustain larger populations through dispersive exploration. The framework provides a principled link between active-matter dynamics and movement ecology, offering insights for predicting habitat thresholds, assessing conservation strategies, and exploring resource-rationing as a collective strategy under environmental constraints.

Abstract

In any ecosystem, the conditions of the environment and the characteristics of the species that inhabit it are entangled, co-evolving in space and time. We introduce a model that couples active agents with a dynamic environment, interpreted as a nutrient source. Agents are persistent random walkers that gather food from the environment and store it in an inner energy depot. This energy is used for self-propulsion, metabolic expenses, and reproduction. The environment is a two-dimensional surface divided into patches, each of them producing food. Thus, population size and resource distribution become emergent properties of the system. Combining simulations and analytical framework to analyze limiting cases, we show that the system exhibits distinct phases separating quasi-static and highly motile regimes. We observe that, in general, population sizes are inversely proportional to the average energy per agent. Furthermore, we find that, counter-intuitively, reduced access to resources or increased metabolic expenditure can lead to a larger population size. The proposed theoretical framework provides a link between active matter and movement ecology, allowing to investigate short vs long-term strategies to resource exploitation and rationing, as well as sedentary vs wandering strategy. The introduced approach may serve as a tool to describe real-world ecological systems and to test environmental strategies to prevent species extinction.

Figures (4)

• Figure 1: Phase diagramas for: Left column: Constant population, $N_0=253$, $\kappa =0$. (a) Average energy per agent $\left\langle e \right\rangle$. White diagonal arrow indicates the $D$ increasing direction (same for all graphs), while dashed lines correspond to $D=10^{-5}$, $10^{-3}$ and $5$. Agent energy remains constant at both low and high $D$ ($\left\langle e \right\rangle \approx 0.2$ for $D<10^{-5}$ and $\left\langle e \right\rangle \approx 0.5$ for $D>5$), reaching its minimum at $D\approx10^{-3}$ ($\left\langle e \right\rangle \approx 0.13$). (b) Average energy per patch $\left\langle f \right\rangle$. Diagonal dashed lines correspond to $D=10^{-5}$ and $2\cdot10^{-2}$. Patch energy remains constant at both low and high $D$ ($\left\langle f \right\rangle \approx 0.45$ for $D<10^{-5}$ and $\left\langle f \right\rangle \approx 0.25$ for $D>2\cdot10^{-2}$), and decreases monotonically with $D$ for intermediate values. Center and right columns: Births and deaths, $N(t)$. (c) Average inner energy per agent (diagonal line $D=10^{-3}$), (d) Average inner energy per patch (diagonal lines $D=10^{-5}$ and $D=2\cdot10^{-2}$), (e) Average population per patch (diagonal lines $D=2\cdot10^{-4}$, $D=2\cdot10^{-3}$ and $D=1$). The graphs are divided by a central diagonal band corresponding to an absorbing phase, i.e., extinction of the agent population, where high fluctuations on $\langle e \rangle$ can be observed. This band separates the static regime (low $v_0$, high $D_\theta$), where $\langle e \rangle$ is high, $\langle f \rangle$ is medium and $\langle n \rangle$ is medium, from the high motility regime (high $v_0$, low $D_\theta$), with medium $\langle e \rangle$, low $\langle f \rangle$ and high $\langle n \rangle$. The emergence of a new absorbing phase for $v_{0} \geq v_{0,max}$ (vertical solid line, Eq. \ref{['eq:KR_vmax']}) can be observed.
• Figure 2: Upper row: Time evolution of the agent populations for different initial populations and diffusion coefficients (one realization per line). Bottom row: Equilibrium population as a function of the initial population for different diffusion coefficients. Red dashed lines correspond to $n_r$ and $n_s$. For $D\leq 10$, the equilibrium population does not depend on the initial conditions. For $D>10$, the equilibrium population depends on the initial population, presenting a linear relationship in the region $n_r<n_i<n_s$ and large fluctuations for $n>n_s$. However, the equilibrium population is always bounded within the range $[n_r,n_s]$, when it does not become extinct.
• Figure 3: Heat map of agents equilibrium population $\langle n \rangle$ as a function of diffusion coefficient $D$ and (a) kinetic rate $\kappa$, (b) metabolic rate $m$, (c) growth rate $r$, (d) patch capacity $c$, (e) intake slope $I_s$, (f) intake capacity $I_c$. (a)kinetic rate. White dashed vertical line corresponds to Eq. \ref{['eq:EP_D']}, white dashed diagonal line corresponds to Eq. \ref{['eq:KR_k']}. The figure shows two absorbing phases corresponding to these Eqs. (b)metabolic rate. Main white dashed line corresponds to Eq. \ref{['eq:MR_m1']}, while top and bottom horizontal dashed white lines correspond to Eq. \ref{['eq:MR_m']}. These lines delimit the absorbing phases of the system. (c)growth rate. White dashed line corresponds to Eq. \ref{['eq:GR_r']}, indicating the phase transition. (d)patch capacity. Main white dashed line corresponds to Eq. \ref{['eq:PC_c']}, while top left white dashed line corresponds to $n_r=1$. Both lines delimit the boundary between the active phase and the absorbing phases. (e)intake slope. White dashed line corresponds to Eq. \ref{['eq:Is']}, separating the active phase of the system from the absorbing phases. (f)intake capacity. White dashed line corresponds to Eq. \ref{['eq:Ic_Ic']}.
• Figure 4: Time series for the agent population size $N$ in systems with an unstable non-trivial equilibrium point ($D=10^2$, $m=10^{-4}$). For $M=225$, the amplitude grows until $N$ reaches zero and the population becomes extinct. For $M=14400$, the amplitude stabilizes after a transient period, resulting in a periodic population.