Chaos and crises in a model for cooperative hunting: A symbolic dynamics approach

Abstract

In this work we investigate the population dynamics of cooperative hunting extending the McCann and Yodzis model for a three-species food chain system with a predator, a prey, and a resource species. The new model considers that a given fraction sigma of predators cooperates in prey's hunting, while the rest of the population 1-sigma hunts without cooperation. We use the theory of symbolic dynamics to study the topological entropy and the parameter space ordering of the kneading sequences associated with one-dimensional maps that reproduce significant aspects of the dynamics of the species under several degrees of cooperative hunting. Our model also allows us to investigate the so-called deterministic extinction via chaotic crisis and transient chaos in the framework of cooperative hunting. The symbolic sequences allow us to identify a critical boundary in the parameter spaces (K, C-0) and (K, sigma) which separates two scenarios: (i) all-species coexistence and (ii) predator's extinction via chaotic crisis. We show that the crisis value of the carrying capacity K-c decreases at increasing sigma, indicating that predator's populations with high degree of cooperative hunting are more sensitive to the chaotic crises. We also show that the control method of Dhamala and Lai [Phys. Rev. E 59, 1646 (1999)] can sustain the chaotic behavior after the crisis for systems with cooperative hunting. We finally analyze and quantify the inner structure of the target regions obtained with this control method for wider parameter values beyond the crisis, showing a power law dependence of the extinction transients on such critical parameters.

Partial control of chaos: How to avoid undesirable behaviors with small controls in presence of noise

“…The model that we have used is an extension of the McCann-Yodzis model [21] proposed by Duarte et al [22], which describes the dynamics of the population density of a resource species R, a consumer C and a predator P . The resulting model is given by the following set of nonlinear differential equations: where the chaotic transient before the extinction is shown.…”

Solving an Infectious Disease Model considering Its Anatomical Variables with Stochastic Numerical Procedures

“…e high rate of efficiency is seen in hunting lions, wild dogs, and hyenas. e mathematical system of this precise predator behavior was modeled and presented for the first time in [10], in which a simple model was applied to describe a cooperation. So far, there have been a few studies of the predator-prey interaction behavior presented in these references: in [11], the hunting effect is considered, in [12], the cooperation effect is considered, in [13], hunting and cooperation effects are considerd at the same time, and finally, in [14], hunting and cooperation effects are considered jointly with the Alle effect.…”

“…It can also be defined as mother-child, where the predator is not diseased by this virus after a direct interaction with the diseased prey. e rate of transmission of the prey population, i.e., infection rate, is represented by δ. e functional form (aP(τ) + λ)P(τ)S(τ) and (aP(τ) + λ)P(τ)I(τ) are the hunting cooperation values [10]. e death rate of the population of the prey is μ, which represents the natural mortality rate of the predator's population.…”

A simple Gause‐type predator–prey model considering social predation

“…16 For example, wolves following bisons, hyenas pursuing buffaloes, African wild dogs (lyncaons) in search of zebras, and so on. 17 These predators and other animals have all evolved the ability to hunt in groups and this behavior is frequently called as cooperative hunting 13,18 or hunting cooperation. 15 This conduct allows them to successfully capture more prey than if they hunt separately.…”