More complex dynamics in a discrete prey-predator model with the Allee effect in prey

Area of Science:

• Ecology
• Mathematical Biology
• Dynamical Systems

Background:

• Prey-predator models are crucial for understanding ecological interactions.
• The Allee effect in prey populations introduces complex dynamics.
• Previous analyses of this model had limitations in local stability and bifurcation findings.

Purpose of the Study:

• To correct errors in the local stability analysis of a discrete prey-predator model with an Allee effect.
• To uncover and analyze complex dynamical properties and bifurcations beyond those previously studied.
• To investigate the impact of the Allee effect on population dynamics and stability.

Main Methods:

• Correction of local stability analysis for fixed points.
• Derivation and analysis of codimension-1 (transcritical, period-doubling) and codimension-2 (1:2 resonance) bifurcations.
• Numerical simulations to illustrate theoretical findings and discover new dynamics.

Main Results:

• Corrected local stability of the unique positive fixed point ($E_*$).
• Identified new bifurcations: transcritical at $E_1$, transcritical and period-doubling at $E_2$, period-doubling and 1:2 resonance at $E_*$.
• Numerical simulations confirmed theoretical results and revealed novel dynamics, including the significance of 1:2 resonance.

Conclusions:

• The discrete prey-predator model with an Allee effect exhibits richer dynamics than previously understood.
• The occurrence of 1:2 resonance bifurcation signifies strong system instability.
• This instability suggests potential for rapid, sudden increases in both predator and prey populations.