Hopf and torus bifurcations, torus destruction and chaos in population biology

N. Stollenwerk, PF Sommer, B.W. Kooi, L. Mateus, P. Ghaffari, M Aguiar

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

One of the simplest population biological models displaying a Hopf bifurcation is the Rosenzweig–MacArthur model with Holling type II response function as essential ingredient. In seasonally forced versions the fixed point on one side of the Hopf bifurcation becomes a limit cycle and the Hopf limit cycle on the other hand becomes a torus, hence the Hopf bifurcation becomes a torus bifurcation, and via torus destruction by further increasing relevant parameters can follow deterministic chaos. We investigate this route to chaos also in view of stochastic versions, since in real world systems only such stochastic processes would be observed. However, the Holling type II response function is not directly related to a transition from one to another population class which would allow a stochastic version straight away. Instead, a time scale separation argument leads from a more complex model to the simple 2 dimensional Rosenzweig– MacArthur model, via additional classes of food handling and predators searching for prey. This extended model allows a stochastic generalization with the stochastic version of aHopf bifurcation, and ultimately also with additional seasonality allowing a torus bifurcation under stochasticity. Our study shows that the torus destruction into chaos with positive Lyapunov exponents can occur in parameter regions where also the time scale separation and hence stochastic versions of the model are possible. The chaotic motion is observed inside Arnol’d tongues of rational ratio of the forcing frequency and the eigenfrequency of the unforced Hopf limit cycle. Such torus bifurcations and torus destruction into chaos are also observed in other population biological systems, and were for example found in extended multi-strain epidemiological models on dengue fever. To understand such dynamical scenarios better also under noise the present low dimensional system can serve as a good study case.
Original languageEnglish
Pages (from-to)91-99
JournalEcological Complexity
Volume30
Issue numberJune
DOIs
Publication statusPublished - 2017

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chaotic dynamics
bifurcation
Biological Sciences
stochasticity
food handling
dengue fever
stochastic processes
dengue
timescale
tongue
ingredients
seasonality
case studies
predators
predator
food

Keywords

  • Multi-strain dengue models, Torus bifurcation, Stochastic systems, Stoichiometric formulation, Deterministic chaos Lyapunov exponents

Cite this

Stollenwerk, N. ; Sommer, PF ; Kooi, B.W. ; Mateus, L. ; Ghaffari, P. ; Aguiar, M. / Hopf and torus bifurcations, torus destruction and chaos in population biology. In: Ecological Complexity. 2017 ; Vol. 30, No. June. pp. 91-99.
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Stollenwerk, N, Sommer, PF, Kooi, BW, Mateus, L, Ghaffari, P & Aguiar, M 2017, 'Hopf and torus bifurcations, torus destruction and chaos in population biology' Ecological Complexity, vol. 30, no. June, pp. 91-99. https://doi.org/10.1016/j.ecocom.2016.12.009

Hopf and torus bifurcations, torus destruction and chaos in population biology. / Stollenwerk, N.; Sommer, PF; Kooi, B.W.; Mateus, L.; Ghaffari, P.; Aguiar, M.

In: Ecological Complexity, Vol. 30, No. June, 2017, p. 91-99.

Research output: Contribution to JournalArticleAcademicpeer-review

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T1 - Hopf and torus bifurcations, torus destruction and chaos in population biology

AU - Stollenwerk, N.

AU - Sommer, PF

AU - Kooi, B.W.

AU - Mateus, L.

AU - Ghaffari, P.

AU - Aguiar, M

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N2 - One of the simplest population biological models displaying a Hopf bifurcation is the Rosenzweig–MacArthur model with Holling type II response function as essential ingredient. In seasonally forced versions the fixed point on one side of the Hopf bifurcation becomes a limit cycle and the Hopf limit cycle on the other hand becomes a torus, hence the Hopf bifurcation becomes a torus bifurcation, and via torus destruction by further increasing relevant parameters can follow deterministic chaos. We investigate this route to chaos also in view of stochastic versions, since in real world systems only such stochastic processes would be observed. However, the Holling type II response function is not directly related to a transition from one to another population class which would allow a stochastic version straight away. Instead, a time scale separation argument leads from a more complex model to the simple 2 dimensional Rosenzweig– MacArthur model, via additional classes of food handling and predators searching for prey. This extended model allows a stochastic generalization with the stochastic version of aHopf bifurcation, and ultimately also with additional seasonality allowing a torus bifurcation under stochasticity. Our study shows that the torus destruction into chaos with positive Lyapunov exponents can occur in parameter regions where also the time scale separation and hence stochastic versions of the model are possible. The chaotic motion is observed inside Arnol’d tongues of rational ratio of the forcing frequency and the eigenfrequency of the unforced Hopf limit cycle. Such torus bifurcations and torus destruction into chaos are also observed in other population biological systems, and were for example found in extended multi-strain epidemiological models on dengue fever. To understand such dynamical scenarios better also under noise the present low dimensional system can serve as a good study case.

AB - One of the simplest population biological models displaying a Hopf bifurcation is the Rosenzweig–MacArthur model with Holling type II response function as essential ingredient. In seasonally forced versions the fixed point on one side of the Hopf bifurcation becomes a limit cycle and the Hopf limit cycle on the other hand becomes a torus, hence the Hopf bifurcation becomes a torus bifurcation, and via torus destruction by further increasing relevant parameters can follow deterministic chaos. We investigate this route to chaos also in view of stochastic versions, since in real world systems only such stochastic processes would be observed. However, the Holling type II response function is not directly related to a transition from one to another population class which would allow a stochastic version straight away. Instead, a time scale separation argument leads from a more complex model to the simple 2 dimensional Rosenzweig– MacArthur model, via additional classes of food handling and predators searching for prey. This extended model allows a stochastic generalization with the stochastic version of aHopf bifurcation, and ultimately also with additional seasonality allowing a torus bifurcation under stochasticity. Our study shows that the torus destruction into chaos with positive Lyapunov exponents can occur in parameter regions where also the time scale separation and hence stochastic versions of the model are possible. The chaotic motion is observed inside Arnol’d tongues of rational ratio of the forcing frequency and the eigenfrequency of the unforced Hopf limit cycle. Such torus bifurcations and torus destruction into chaos are also observed in other population biological systems, and were for example found in extended multi-strain epidemiological models on dengue fever. To understand such dynamical scenarios better also under noise the present low dimensional system can serve as a good study case.

KW - Multi-strain dengue models, Torus bifurcation, Stochastic systems, Stoichiometric formulation, Deterministic chaos Lyapunov exponents

U2 - 10.1016/j.ecocom.2016.12.009

DO - 10.1016/j.ecocom.2016.12.009

M3 - Article

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EP - 99

JO - Ecological Complexity

JF - Ecological Complexity

SN - 1476-945X

IS - June

ER -