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Turing instability and pattern formation in a diffusive predator-prey system with opportunistic predators and weak
Wenjie Li1,2, Wenhao Bai1, Jinde Cao2,3
1Kunming University of Science and Technology, Institute for Differential Equation Theory and Applications, Faculty of Science, Kunming 650600, China.
Physical Review. E
|May 16, 2026
Summary
This study explores Turing instability in predator-prey models. Diffusion can trigger instability, and while noise may destabilize, it can also enhance system stability.
Area of Science:
- Mathematical Biology
- Ecology
- Dynamical Systems
Background:
- Predator-prey models are fundamental in ecology.
- Turing instability explains pattern formation in biological systems.
- Allee effects and diffusion significantly influence population dynamics.
Purpose of the Study:
- To investigate Turing instability in a diffusive predator-prey system.
- To analyze the impact of an opportunistic predator and prey Allee effect.
- To examine the role of diffusion and stochastic noise on system stability.
Main Methods:
- Applied the upper-lower solution method to determine solution existence and bounds.
- Analyzed the conditions for Turing instability in homogeneous steady states.
- Utilized numerical simulations to validate analytical findings.
- Investigated a stochastic reaction-diffusion model with additive white noise.
Main Results:
- Established the existence of positive solutions and derived estimates.
- Demonstrated that diffusion can induce Turing instability.
- Numerical simulations confirmed the analytical predictions.
- Found that white noise can enhance system stability in the stochastic model.
Conclusions:
- Diffusion is a key factor in pattern formation within this predator-prey system.
- The interplay between diffusion, Allee effects, and predator behavior drives complex dynamics.
- Stochastic noise can have a stabilizing effect, contrasting with deterministic predictions.
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