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Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 9475854, 11 pages
https://doi.org/10.1155/2017/9475854
Research Article

An Analysis of Discrete Stage-Structured Prey and Prey-Predator Population Models

School of Business and Economics, The Arctic University of Norway, Campus Harstad, Harstad, Norway

Correspondence should be addressed to Arild Wikan; on.tiu@nakiw.dlira

Received 23 February 2017; Accepted 27 March 2017; Published 16 April 2017

Academic Editor: Hammad Khalil

Copyright © 2017 Arild Wikan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Discrete stage-structured prey and prey-predator models are considered. Regarding the former, we prove that the models at hand are permanent (i.e., the population will neither go extinct nor exhibit explosive oscillations) and, moreover, that the transfer from stability to nonstationary behaviour always goes through a supercritical Neimark−Sacker bifurcation. The prey model covers species that possess a wide range of different life histories. Predation pressure may both stabilize and destabilize the prey dynamics but the strength of impact is closely related to life history. Indeed, if the prey possesses a precocious semelparous life history and exhibits chaotic oscillations, it is shown that increased predation may stabilize the dynamics and also, in case of large predation pressure, transfer the population to another chaotic regime.