# Dynamics of a discrete-time predator-prey model with exponential prey growth and saturated response

## TL;DR

This paper studies how predator-prey systems behave when prey grow exponentially and predators have a saturated response, revealing complex dynamics like chaos.

## Contribution

The study introduces bifurcation analysis in a discrete-time predator-prey model with Ricker growth and Holling response, revealing period-doubling and Neimark-Sacker bifurcations.

## Key findings

- The model exhibits transitions from stability to chaos through bifurcations.
- Phase portraits and Lyapunov exponents confirm complex dynamical behaviors.
- Parameter variations significantly influence predator-prey system stability.

## Abstract

Recent studies have increasingly focused on the stability of predator-prey systems incorporating the Holling functional response and Ricker population model. This work investigates the influence of the Holling effect on a discrete-time predator-prey model, demonstrating through bifurcation theory and the central manifold theorem that the system exhibits period-doubling and Neimark-Sacker bifurcations at equilibrium points. Numerical simulations reveal complex dynamical behaviors, with bifurcation diagrams illustrating transitions from stability to periodic oscillations and chaos. Using phase portraits, Lyapunov exponents, and bifurcation analysis, we show how the Ricker map progresses from order to chaos. Our findings enhance the understanding of predator-prey dynamics and provide insights for ecological population management, highlighting the system’s rich behavior under parameter variations.