# Global analysis of a predator-prey model with variable predator search rate

## TL;DR

This paper analyzes a modified predator-prey model where predator search rate depends on prey density, revealing conditions for stability and oscillations, and showing how predator adaptation can prevent the paradox of enrichment.

## Contribution

It introduces a variable predator search rate into the Rosenzweig-MacArthur model and provides a comprehensive global analysis of its dynamics and stability properties.

## Key findings

- Coexistence steady state can be globally stable or unstable.
- The paradox of enrichment does not always occur in the modified model.
- Predator adaptation can stabilize the system even when enrichment occurs.

## Abstract

We consider a modified Rosenzweig-MacArthur predator-prey model, based on the premise that the search rate of predators is dependent on the prey density, rather than constant. A complete analysis of the global behavior of the model is presented, and shows that the model exhibits a dichotomy similar to the classical Rosenzweig-MacArthur model: either the coexistence steady state is globally stable; or it is unstable, and then a unique, globally stable limit cycle exists. We discuss the similarities, but also important differences between our model and the Rosenweig-MacArthur model. The main differences are that: 1. The paradox of enrichment which always occurs in the Rosenzweig-MacArthur model, does not always occur here, and 2. Even when the paradox of enrichment occurs, predators can adapt by lowering their search rate, and effectively stabilize the system.