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This paper investigates a heterogeneous diffusion process with position-dependent diffusion, incorporating independent recruitment and mortality rates, generalizing stochastic resetting. The system converges to a non-equilibrium stationary state where the mean squared displacement saturates, reflecting the transport regime through an effective exponent of the diffusion coefficient. The study also reveals a "mortality-assisted search" phenomenon, where an optimal mortality rate minimizes mean first-passage time, and demonstrates applications in epidemic spreading and other reactive systems.
Optimal mortality can accelerate search and epidemic spread, challenging intuition about removal processes.
We study a heterogeneous diffusion process with a position-dependent diffusion coefficient in the presence of independent recruitment and mortality. This framework generalises heterogeneous diffusion under stochastic resetting, which is recovered as the special case of equal recruitment and mortality rates. At long times, the system approaches a non-equilibrium stationary state with a mean squared displacement that saturates to a value independent of the recruitment rate. Crucially, this saturation limit directly encodes the underlying transport regime via an effective exponent applied to the diffusion coefficient, and we characterise the non-trivial transition to this stationary regime. We further derive the mean first-passage time to a target and show that, for a fixed recruitment-to-mortality ratio, an optimal mortality rate minimises the search time. This"mortality-assisted search"effect is confirmed by numerical evaluation. Reinterpreting recruitment and mortality as infection and removal processes, we apply the model to epidemic spreading and show that a moderate removal rate can counter-intuitively accelerate, rather than delay, the arrival of infection at a new spatial region. Beyond epidemiology, our framework offers broad relevance to systems featuring concurrent particle creation and destruction, such as bacterial growth, tracer diffusion in reactive fluids, or cellular population dynamics in crowded environments.