A predator-prey model with optimal suppression of reproduction in the prey
type: Item
instance of: scholarly article
title: A predator-prey model with optimal suppression of reproduction in the prey
zbMATH Open document ID: 0851.92020
DOI: 10.1016/0025-5564(95)00082-8
author: Mats Gyllenberg
author: Ilkka Hanski
author: Torsten Lindström
published in: Mathematical Biosciences
publication date: 1996-08-20T00:00:00Z
review text: The main objective of this paper is to formulate and analyze a predator-prey model with the possibility of suppressed reproduction in the prey. We commence with a single-species model in which we assume a trade-off between the reproductive effort and competitive ability. If no behavioral adjustment of the rate of reproduction is allowed for, the type with the reproductive rate maximizing the equilibrium population size prevails, regardless of whether the equilibrium point is stable or not. However, when behavioral adjustments are allowed for, the winning type suppresses reproduction, to some extent, whenever the equilibrium is unstable, and this behavior decreases the mean amplitude of population oscillations. We then extend this model to include predation, assuming a trade-off between the reproductive effort and an individual's ability to avoid predation. Optimal adjustment of reproduction by the prey adds to population stability, as measured by the mean amplitude of oscillations. Our paper is organized as follows. In Section 2 we introduce the notions of fixed and adjustable strategy in the single-species case. We find the best fixed strategy and investigate the asymptotic behavior of solutions to a model describing competition between fixed and adjustable strategies. In Section 3 we introduce a new discrete predator-prey model, which can be regarded as a combination of the \textit{W. E. Ricker} model [J. Fish. Res. Board Can. 11, No. 5, 559-623 (1954)], the \textit{A. J. Nicholson} and \textit{V. A. Bailey} model [Zool. Soc. (Lond.) Proc. 3, 551-598 (1935)], and the continuous Lotka-Volterra model. A mechanistic explanation of the model is included in Section 3. In Section 4 we investigate the dynamics of this discrete predator-prey model, and this study culminates in an important stability theorem. In Section 5 we investigate the dynamical consequences of the optimal suppression of reproduction in the prey. The analytical study of the predator-prey model is supp…
Mathematics Subject Classification ID: 92D40
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 897910
zbMATH Keywords: Nicholson-Bailey model
zbMATH Keywords: Ricker model
zbMATH Keywords: predator-prey model
zbMATH Keywords: suppressed reproduction
zbMATH Keywords: single-species model
zbMATH Keywords: reproductive effort
zbMATH Keywords: competitive ability
zbMATH Keywords: equilibrium
zbMATH Keywords: population oscillations
zbMATH Keywords: best fixed strategy
zbMATH Keywords: adjustable strategies
zbMATH Keywords: continuous Lotka-Volterra model
zbMATH Keywords: discrete predator-prey model
zbMATH Keywords: stability theorem
MaRDI profile type: Publication
full work available at URL: 0025-5564(95)00082-8
OpenAlex ID: W2047000841
Wikidata QID: Q52304637
cites work: Behavioral stabilization of host-parasite population dynamics
cites work: Q3895297
cites work: Q4067388
cites work: Dynamics in one dimension
cites work: Continuous versus discrete single species population models with adjustable reproductive strategies
cites work: The subcritical collapse of predator populations in discrete-time predator-prey models
cites work: Discrete Dynamical Systems in Dimensions One and Two
cites work: Q3751432
cites work: Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study
cites work: Nonuniqueness of limit cycles of gause-type predator-prey systems
cites work: Global stability of Gause-type predator-prey systems
cites work: Uniqueness of limit cycles in Gause-type models of predator-prey systems
cites work: Q3993502
cites work: Q3896875
description: scientific article; zbMATH DE number 897910