Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response

Area of Science:

• Mathematical Biology
• Ecology
• Population Dynamics

Background:

• Predator-prey models describe species interactions and spatial influences on population densities.
• Prey extinction risks include low birth rates, small habitats, rapid predator growth, and high predation pressure.
• Conservation strategies like protection zones can mitigate prey decline.

Purpose of the Study:

• To investigate the existence of positive steady states in a predator-prey model with reaction-diffusion.
• To analyze the impact of functional response and non-flux boundary conditions on population persistence.
• To determine threshold values for prey survival based on birth rates, growth rates, and protection zones.

Main Methods:

• Analysis of a reaction-diffusion predator-prey model incorporating the Beddington-DeAngelis functional response.
• Application of non-flux boundary conditions to simulate a closed system.
• Mathematical derivation of threshold conditions for the existence of positive steady states.

Main Results:

• Identified a threshold value for prey refuge ability ensuring population positivity.
• Demonstrated that prey survival is guaranteed if birth rates are sufficiently high or predator growth rates are sufficiently low.
• Established conditions under which a protection zone is necessary for positive steady states, and found additional thresholds governing prey persistence.

Conclusions:

• The study provides critical insights into the mathematical conditions for prey survival in spatially heterogeneous environments.
• Results align with mechanistic derivations of the Beddington-DeAngelis functional response.
• Uniqueness and asymptotic behavior of positive steady states were determined for specific parameter ranges.