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Heterogeneous diffusion with recruitment and mortality

This paper introduces a generalized framework for heterogeneous diffusion with position-dependent recruitment and mortality, revealing that such systems reach a non-equilibrium stationary state with a recruitment-independent saturation limit and exhibit a counter-intuitive "mortality-assisted search" effect where an optimal removal rate minimizes search times and accelerates epidemic spread.

Original authors: Esther D. Gutiérrez, Alexander López, R. K. Singh, Sébastien Fumeron, Malte Henkel, Trifce Sandev

Published 2026-10-07

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CC BY 4.0

Original authors: Esther D. Gutiérrez, Alexander López, R. K. Singh, Sébastien Fumeron, Malte Henkel, Trifce Sandev

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the crowded, complex world of biological cells or the shifting landscapes of porous rocks, particles do not move with the steady, predictable rhythm of a clock. Instead, their motion is often erratic and uneven, speeding up in some regions and slowing down in others depending on exactly where they are. Scientists call this heterogeneous diffusion. It is a fundamental way to describe how things spread in environments that are not uniform, a concept crucial for understanding everything from how nutrients travel inside a living cell to how pollutants disperse through groundwater. For decades, researchers have studied these movements assuming the number of particles remains constant, like a fixed group of hikers exploring a forest. However, many real-world systems are far more dynamic: individuals are constantly being born or infected, while others die or recover, causing the total number of active travelers to fluctuate wildly. Understanding how these changing populations move through uneven terrain is essential for predicting everything from the spread of a virus to the growth of bacterial colonies.

A team of physicists has now built a mathematical model to describe this chaotic, open system, where particles are simultaneously created and destroyed as they wander through a space where their speed changes with location. They discovered that over long periods, the group of particles settles into a stable, unchanging pattern, but the size of this pattern is determined entirely by how quickly particles are removed, not by how fast new ones are added. Perhaps more surprisingly, they found that a moderate amount of removal can actually help the group find a specific target faster than if no one were removed at all. This counter-intuitive effect, which the researchers call "mortality-assisted search," suggests that in a system where new travelers are constantly arriving, getting rid of the ones who have wandered too far away allows the fresh arrivals to dominate the search, effectively pruning the unsuccessful paths and accelerating the arrival of the first successful traveler.

The researchers began by defining a process where the ability of a particle to move depends on its position. In some areas, the environment is so restrictive that movement is sluggish, while in others, it is so open that particles zip along quickly. They introduced two opposing forces: recruitment, which adds new particles at a specific starting point, and mortality, which removes particles at random moments. This setup generalizes a previously studied concept known as stochastic resetting, where a particle is suddenly sent back to its starting point. In the new model, the particle is not just reset but replaced by a new one, and the rate of replacement does not have to match the rate of removal. By solving the equations governing this system, the team showed that as time goes on, the cloud of particles stops expanding indefinitely. Instead, it reaches a fixed size. This limit is determined solely by the mortality rate and the nature of the terrain, completely independent of how many new particles are being injected. The recruitment rate simply scales the total number of particles in the cloud, like turning up the volume on a radio, but it does not change the shape or the reach of the signal.

The most striking discovery emerged when the team asked how long it would take for the first particle to reach a specific target location. They calculated the average time required for this first arrival under different conditions. When they kept the ratio of new arrivals to deaths constant but varied the speed at which particles died, they found a distinct sweet spot. If the mortality rate was too low, the search was slowed down by particles that had wandered far off course and were unlikely to ever return. If the mortality rate was too high, particles were removed so quickly that there were not enough of them left to find the target. However, at a specific, intermediate rate of removal, the search time was minimized. This happens because the removal process acts as a filter, eliminating the "unsuccessful" searchers who have strayed too far, while the constant stream of new recruits ensures that there are always fresh attempts starting from the source. The result is a system that finds its target faster than it would if every single attempt were allowed to continue indefinitely.

This phenomenon holds true regardless of whether the environment makes movement generally slow or fast, though the exact optimal rate of removal shifts depending on the terrain. The researchers confirmed these findings through detailed numerical simulations, showing that the effect is robust across a wide range of conditions. They also demonstrated that the speed at which the system reaches a steady state and the likelihood of a single particle reaching the target before dying are directly linked to the "effective distance" to the goal, a measure that accounts for how the changing speed of the environment stretches or compresses the journey.

To illustrate the real-world relevance of these findings, the authors applied their model to the spread of infectious diseases. In this context, the "particles" are infected individuals, "recruitment" represents new infections arising from contact, and "mortality" represents the removal of infected individuals through death or recovery. The target is a new region or city that has not yet been infected. The model suggests a startling possibility: a moderate rate of removal, such as through effective quarantine or treatment, could paradoxically speed up the arrival of the infection to a new area. This is not because the disease becomes more contagious, but because removing infected individuals who are far from the new region prevents them from wasting time on long, fruitless journeys, allowing the newly infected individuals closer to the target to reach it sooner. While this sounds dangerous, it highlights a complex dynamic where interventions that reduce the total number of cases might not linearly delay the spread to new areas. The study offers a new lens for understanding how populations of moving, changing entities navigate complex landscapes, with implications for managing epidemics, optimizing search strategies in robotics, and understanding cellular dynamics in crowded biological environments.

Technical Summary: Heterogeneous Diffusion with Recruitment and Mortality

Problem Statement
This work addresses the dynamics of heterogeneous diffusion processes (HDPs) in open stochastic systems where the number of particles is not conserved. While HDPs with space-dependent diffusion coefficients have been extensively studied for conserved particle numbers, many physical and biological phenomena—such as epidemic spreading, bacterial growth, or tracer diffusion in reactive fluids—involve concurrent particle creation (recruitment) and destruction (mortality). The paper seeks to generalize the framework of HDPs to include independent recruitment and mortality rates, $\lambda$ and $\mu$ , respectively. This framework encompasses stochastic resetting as a special case where $\lambda = \mu$ . The central questions concern the long-time stationary state of such systems and the first-passage properties (search efficiency) when the population of searchers fluctuates.

Methodology
The study employs a one-dimensional Langevin equation with a multiplicative noise term, defined by a position-dependent diffusion coefficient $D(x) = D_\alpha |x|^\alpha$ (with $\alpha < 2$ ). The authors adopt the Stratonovich interpretation for the stochastic integrals.

Propagator Derivation: The evolution of the probability density function (PDF), $P(x,t)$ , is governed by a Fokker-Planck equation modified by a loss term $-\mu P(x,t)$ and a source term $+\lambda \delta(x-x_0)$ . Using Laplace transform techniques, the authors derive an exact solution for $P(x,t)$ by mapping the recruitment-mortality problem onto the known solution of the free HDP ( $P_0(x,t)$ ). The solution is expressed as a renewal equation:
$P(x, t) = e^{-\mu t}P_0(x, t) + \lambda \int_0^t e^{-\mu t'} P_0(x, t') dt'$
Moment Analysis: The mean squared displacement (MSD) is calculated by integrating the derived PDF, utilizing the properties of the lower incomplete gamma function to handle the time integrals.
First-Passage Analysis: To analyze search efficiency, the authors define a target at $x=L$ with an absorbing boundary condition. They derive the survival probability $S(t)$ for the full recruitment-mortality process by combining the survival probability of a single mortal trajectory, $S_\mu(t)$ , with a "dynamic redundancy" framework. This accounts for the fact that new trajectories are continuously injected, increasing the probability of a successful arrival over time.
Numerical Evaluation: Due to the complexity of the integrals involving complementary error functions, the authors perform numerical evaluations of the survival probability and mean first-passage time (MFPT) to characterize the system's behavior across different parameter regimes.

Key Results

Non-Equilibrium Stationary State (NESS): At long times, the system approaches a NESS. Crucially, the saturated value of the MSD, $\langle x^2 \rangle_{\text{sat}}$ , is found to be independent of the recruitment rate $\lambda$ . Instead, it depends solely on the mortality rate $\mu$ and the heterogeneity exponent $\alpha$ :
$\langle x^2 \rangle_{\text{sat}} \sim \frac{\Gamma(1+2p)}{p^{2p}} \frac{D_\alpha^p}{\mu^p}$
where $p = 2/(2-\alpha)$ . This indicates that while recruitment determines the total population density, the spatial extent of the cloud is governed by the balance between transport efficiency and particle lifespan.
Mortality-Assisted Search: The analysis of the mean first-passage time (MFPT) to a target reveals a non-monotonic dependence on the mortality rate $\mu$ $μ$ for a fixed recruitment-to-mortality ratio $\chi = \lambda/\mu$ $χ = λ / μ$ .
- At low $\mu$ , increasing mortality improves search efficiency by removing trajectories that have wandered far from the target (pruning unproductive searchers), allowing recently recruited particles closer to the target to dominate.
- At high $\mu$ , mortality removes particles faster than they can be replaced, hindering the search.
- Consequently, an optimal mortality rate $\mu^*$ exists that minimizes the MFPT. This "mortality-assisted search" effect is robust across different heterogeneity exponents $\alpha$ .
Role of Heterogeneity: The heterogeneity exponent $\alpha$ influences the search dynamics by altering the "effective distance" $\Delta y$ between the source and the target. As $\alpha$ increases (super-diffusive regimes), the effective distance shrinks, making the target dynamically closer. This shifts the optimal mortality rate $\mu^*$ to higher values, as the system requires stronger pruning to be effective when the target is already easily accessible.
Epidemic Spreading Application: Reinterpreting recruitment as infection and mortality as removal (death or recovery), the model suggests that a moderate removal rate can counter-intuitively accelerate the arrival of an infection at a new spatial region (target). This occurs because removal eliminates "unsuccessful" infection chains that are far from the target, allowing new infections to start fresh from the source.

Significance and Claims
The paper claims to provide a generalized framework for heterogeneous diffusion in open systems, bridging the gap between conserved-particle HDPs and stochastic resetting models. Its primary theoretical contribution is the demonstration that the stationary spatial spread of a population is determined by the removal rate rather than the injection rate, a result that directly encodes the underlying transport regime through an effective exponent.

In the context of first-passage problems, the work identifies and characterizes the "mortality-assisted search" phenomenon in a continuum setting with heterogeneous mobility. The authors assert that this effect offers a counter-intuitive insight for epidemic management: interventions that increase the removal rate (e.g., quarantine, treatment) may not only reduce the total number of cases but could, within specific parameter ranges, paradoxically hasten the arrival of the first infection to a new region by optimizing the search dynamics of the remaining infectious population. The study concludes that these findings are relevant to systems involving concurrent creation and destruction, such as biological cell dynamics and tracer diffusion in reactive fluids, and suggests that the optimal removal rate is a tunable parameter dependent on both the recruitment balance and the spatial heterogeneity of the environment.

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