Ecological consequences of global bifurcations in some food chain models.

G.A.K. van Voorn, B.W. Kooi, M.P. Boer

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

Food chain models of ordinary differential equations (ode's) are often used in ecology to gain insight in the dynamics of populations of species, and the interactions of these species with each other and their environment. One powerful analysis technique is bifurcation analysis, focusing on the changes in long-term (asymptotic) behaviour under parameter variation. For the detection of local bifurcations there exists standardised software, but until quite recently most software did not include any capabilities for the detection and continuation of global bifurcations. We focus here on the occurrence of global bifurcations in four food chain models, and discuss the implications of their occurrence. In two stoichiometric models (one piecewise continuous, one smooth) there exists a homoclinic bifurcation, that results in the disappearance of a limit cycle attractor. Instead, a stable positive equilibrium becomes the global attractor. The models are also capable of bistability. In two three-dimensional models a Shil'nikov homoclinic bifurcation functions as the organising centre of chaos, while tangencies of homoclinic cycle-to-cycle connections 'cut' the chaotic attractors, which is associated with boundary crises. In one model this leads to extinction of the top predator, while in the other model hysteresis occurs. The types of ecological events occurring because of a global bifurcation will be categorized. Global bifurcations are always catastrophic, leading to the disappearance or merging of attractors. However, there is no 1-on-1 coupling between global bifurcation type and the possible ecological consequences. This only emphasizes the importance of including global bifurcations in the analysis of food chain models. © 2010 Elsevier Inc.
Original languageEnglish
Pages (from-to)120-133
JournalMathematical Biosciences
Volume226
DOIs
Publication statusPublished - 2010

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Food Chain Model
Food Chain
Global Bifurcation
food chain
environmental impact
Software
Homoclinic Bifurcation
Population Dynamics
Ecology
Attractor
Cycle
Local Bifurcations
Model
Piecewise continuous
Bistability
Homoclinic
Chaotic Attractor
Bifurcation Analysis
Global Attractor
Predator

Cite this

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title = "Ecological consequences of global bifurcations in some food chain models.",
abstract = "Food chain models of ordinary differential equations (ode's) are often used in ecology to gain insight in the dynamics of populations of species, and the interactions of these species with each other and their environment. One powerful analysis technique is bifurcation analysis, focusing on the changes in long-term (asymptotic) behaviour under parameter variation. For the detection of local bifurcations there exists standardised software, but until quite recently most software did not include any capabilities for the detection and continuation of global bifurcations. We focus here on the occurrence of global bifurcations in four food chain models, and discuss the implications of their occurrence. In two stoichiometric models (one piecewise continuous, one smooth) there exists a homoclinic bifurcation, that results in the disappearance of a limit cycle attractor. Instead, a stable positive equilibrium becomes the global attractor. The models are also capable of bistability. In two three-dimensional models a Shil'nikov homoclinic bifurcation functions as the organising centre of chaos, while tangencies of homoclinic cycle-to-cycle connections 'cut' the chaotic attractors, which is associated with boundary crises. In one model this leads to extinction of the top predator, while in the other model hysteresis occurs. The types of ecological events occurring because of a global bifurcation will be categorized. Global bifurcations are always catastrophic, leading to the disappearance or merging of attractors. However, there is no 1-on-1 coupling between global bifurcation type and the possible ecological consequences. This only emphasizes the importance of including global bifurcations in the analysis of food chain models. {\circledC} 2010 Elsevier Inc.",
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Ecological consequences of global bifurcations in some food chain models. / van Voorn, G.A.K.; Kooi, B.W.; Boer, M.P.

In: Mathematical Biosciences, Vol. 226, 2010, p. 120-133.

Research output: Contribution to JournalArticleAcademicpeer-review

TY - JOUR

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AU - Boer, M.P.

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