First-passage times, mobile traps, and Hopf bifurcations.

Area of Science:

• Statistical Physics
• Mathematical Biology
• Dynamical Systems

Background:

• First-passage time problems are crucial in modeling processes like molecular binding and population dynamics.
• Understanding the impact of trap mobility on capture efficiency is key for optimizing search strategies.

Purpose of the Study:

• To investigate if a mobile trap can reduce mean first-passage times (MFPT) compared to a stationary trap on a 1D domain.
• To determine the conditions under which a mobile trap outperforms a stationary one.

Main Methods:

• Analytical computation of MFPT for random walks with mobile traps (randomly moving and oscillating).
• Identification of critical trap speeds separating regimes of improved vs. reduced capture efficiency.
• Establishing a connection between the oscillating trap problem and a moving-sink problem in the Gray-Scott model.

Main Results:

• A stationary trap is more efficient than a very slow-moving trap.
• A sufficiently fast-moving trap significantly reduces capture time compared to a stationary trap.
• The critical speed threshold for mobile trap advantage corresponds to a Hopf bifurcation in the Gray-Scott model.

Conclusions:

• The speed of the trap is critical for optimizing capture times in random walks.
• A surprising link exists between random walk capture problems and bifurcations in reaction-diffusion systems.
• This connection provides a novel method for proving the uniqueness of Hopf bifurcations.