# Phase transitions in dependence of apex predator decaying ratio in a cyclic dominant system

## TL;DR

This paper investigates how varying the apex predator's decay ratio affects the stability and outcomes of a cyclic dominant system, revealing phase transitions between extinction, coexistence, and dominance states.

## Contribution

It introduces a model with an apex predator into cyclic dominance systems and identifies phase transitions driven by the predator's decay ratio, a novel insight into biodiversity stability.

## Key findings

- Three different population outcomes depending on predator decay ratio
- Identification of discontinuous and continuous phase transitions
- Cyclic dominance can stabilize coexistence over a range of parameters

## Abstract

Cyclic dominant systems, like rock-paper-scissors game, are frequently used to explain biodiversity in nature, where mobility, reproduction and intransitive competition are on stage to provide the coexistence of competitors. A significantly new situation emerges if we introduce an apex predator who can superior all members of the mentioned three-species system. In the latter case the evolution may terminate into three qualitatively different destinations depending on the apex predator decaying ratio $q$. In particular, the whole population goes extinct or all four species survive or only the original three-species system remains alive as we vary the control parameter. These solutions are separated by a discontinuous and a continuous phase transitions at critical $q$ values. Our results highlight that cyclic dominant competition can offer a stable way to survive even in a predator-prey-like system that can be maintained for large interval of critical parameter values.