Spatiotemporal dynamics of a multi-delayed prey-predator system with variable carrying capacity

Area of Science:

• Ecology
• Mathematical Biology
• Dynamical Systems

Background:

• Prey-predator models are fundamental to understanding ecological interactions.
• Incorporating delays and variable carrying capacities adds realism to these models.
• Previous studies have explored temporal dynamics, but spatiotemporal complexity with delays remains less understood.

Purpose of the Study:

• To analyze the temporal and spatiotemporal dynamics of a delayed prey-predator system with a variable carrying capacity.
• To investigate the impact of delays and system parameters on population dynamics and pattern formation.
• To explore phenomena such as chaos, bistability, and Turing patterns.

Main Methods:

• Detailed dynamical analysis of the temporal system, including stability analysis and bifurcation theory.
• Investigation of chaotic dynamics using maximal Lyapunov exponent and sensitivity analysis.
• Derivation of conditions for Turing instability and analysis of Turing pattern formation in a 2D spatial domain.
• Numerical simulations to validate analytical findings.

Main Results:

• The temporal system exhibits well-posedness, coexistence equilibria, bistability, and Hopf bifurcation.
• The delayed system demonstrates chaotic behavior, confirmed by Lyapunov exponent and sensitivity analyses.
• Turing instability conditions were derived, leading to various spatiotemporal patterns (hot-spot, coldspot, patchy, labyrinth) in 2D.
• Parameter β and delay parameters significantly influence system dynamics and pattern transitions.

Conclusions:

• Delays and variable carrying capacity are crucial drivers of complex dynamics in prey-predator systems.
• The study provides insights into the emergence of chaos and diverse spatial patterns.
• Findings contribute to a deeper understanding of ecological interactions and population dynamics.