**Jawdat Alebraheem**

Mathematics Department, College of Science Al Zufli, Majmaah University, Majmaah 11952, Saudi Arabia; j.alebraheem@mu.edu.sa; Tel.: +966-53-5479321

Received: 3 September 2018; Accepted: 10 October 2018; Published: 22 October 2018

**Abstract:** The paradox of the enrichment phenomenon, considered one of the main counterintuitive observations in ecology, likely destabilizes predator–prey dynamics by increasing the nutrition of the prey. We use two systems to study the occurrence of the paradox of enrichment: The prey–predator system and the one prey, two predators system, with Holling type I and type II functional and numerical responses. We introduce a new approach that involves the connection between the occurrence of the enrichment paradox and persistence and extinction dynamics. We apply two main analytical techniques to study the persistence and extinction dynamics of two and three trophics, respectively. The linearity and nonlinearity of functional and numerical responses plays important roles in the occurrence of the paradox of enrichment. We derive the persistence and extinction conditions through the carrying capacity parameter, and perform some numerical simulations to demonstrate the effects of the paradox of enrichment when increasing carrying capacity.

**Keywords:** paradox of enrichment; prey–predator system; persistence of predators; extinction of predators
