Time-dependent localized patterns in a predator-prey model

Area of Science:

• Mathematical Biology
• Chemical Kinetics
• Nonlinear Dynamics

Background:

• Reaction-diffusion models are crucial for understanding pattern formation in biological and chemical systems.
• The Leslie-Gower model, extended to two components, presents complex dynamics.
• Understanding bifurcations and pattern emergence is key to predicting system behavior.

Purpose of the Study:

• To investigate solution branches and pattern formation in a two-component Leslie-Gower reaction-diffusion model.
• To analyze two distinct regimes of pattern emergence and stability.
• To uncover novel mechanisms driving the formation of localized and oscillatory states.

Main Methods:

• Numerical continuation techniques were employed to compute solution branches.
• Two-parameter continuation was utilized to explore parameter space.
• Analysis focused on bifurcations from homogeneous states to patterned states.

Main Results:

• In the first regime, supercritical oscillations were followed by subcritical Turing bifurcations leading to localized states on snaking branches.
• A novel mechanism was identified where disconnected oscillatory states form continuous snaking branches of time-periodic localized states.
• In the second regime, supercritical Turing patterns coexist with subcritical localized states, explained by strongly subcritical sideband Turing states.

Conclusions:

• The study reveals complex pattern formation in reaction-diffusion systems that deviate from expectations based on primary bifurcations.
• Novel mechanisms for the emergence of stable, time-periodic localized states were discovered.
• The findings highlight the importance of numerical continuation in uncovering complex dynamical behaviors in ecological and chemical models.