A predator-2 prey fast-slow dynamical system for rapid predator evolution

Sofia Helena Piltz, Frits Veerman, Philip K. Maini, Mason A. Porter

Research output: Contribution to journalJournal articleResearchpeer-review

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Abstract

We consider adaptive change of diet of a predator population that switches its feeding between two prey populations. We develop a novel 1 fast-3 slow dynamical system to describe the dynamics of the three populations amidst continuous but rapid evolution of the predator's diet choice. The two extremes at which the predator's diet is composed solely of one prey correspond to two branches of the three-branch critical manifold of the fast slow system. By calculating the points at which there is a fast transition between these two feeding choices (i.e., branches of the critical manifold), we prove that the system has a two-parameter family of periodic orbits for sufficiently large separation of the time scales between the evolutionary and ecological dynamics. Using numerical simulations, we show that these periodic orbits exist, and that their phase difference and oscillation patterns persist, when ecological and evolutionary interactions occur on comparable time scales. Our model also exhibits periodic orbits that agree qualitatively with oscillation patterns observed in experimental studies of the coupling between rapid evolution and ecological interactions.
Original languageEnglish
JournalS I A M Journal on Applied Dynamical Systems
Volume16
Issue number1
Pages (from-to)54-90
Number of pages37
ISSN1536-0040
DOIs
Publication statusPublished - 2017

Keywords

  • Lotka-Volterra interaction
  • Fast-slow dynamical systems
  • Geometric singular pertubation theory
  • Planktonic protozoa-algae dynamics

Cite this

Piltz, Sofia Helena ; Veerman, Frits ; Maini, Philip K. ; Porter, Mason A. / A predator-2 prey fast-slow dynamical system for rapid predator evolution. In: S I A M Journal on Applied Dynamical Systems. 2017 ; Vol. 16, No. 1. pp. 54-90.
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abstract = "We consider adaptive change of diet of a predator population that switches its feeding between two prey populations. We develop a novel 1 fast-3 slow dynamical system to describe the dynamics of the three populations amidst continuous but rapid evolution of the predator's diet choice. The two extremes at which the predator's diet is composed solely of one prey correspond to two branches of the three-branch critical manifold of the fast slow system. By calculating the points at which there is a fast transition between these two feeding choices (i.e., branches of the critical manifold), we prove that the system has a two-parameter family of periodic orbits for sufficiently large separation of the time scales between the evolutionary and ecological dynamics. Using numerical simulations, we show that these periodic orbits exist, and that their phase difference and oscillation patterns persist, when ecological and evolutionary interactions occur on comparable time scales. Our model also exhibits periodic orbits that agree qualitatively with oscillation patterns observed in experimental studies of the coupling between rapid evolution and ecological interactions.",
keywords = "Lotka-Volterra interaction, Fast-slow dynamical systems, Geometric singular pertubation theory, Planktonic protozoa-algae dynamics",
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A predator-2 prey fast-slow dynamical system for rapid predator evolution. / Piltz, Sofia Helena; Veerman, Frits; Maini, Philip K.; Porter, Mason A.

In: S I A M Journal on Applied Dynamical Systems, Vol. 16, No. 1, 2017, p. 54-90.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - A predator-2 prey fast-slow dynamical system for rapid predator evolution

AU - Piltz, Sofia Helena

AU - Veerman, Frits

AU - Maini, Philip K.

AU - Porter, Mason A.

PY - 2017

Y1 - 2017

N2 - We consider adaptive change of diet of a predator population that switches its feeding between two prey populations. We develop a novel 1 fast-3 slow dynamical system to describe the dynamics of the three populations amidst continuous but rapid evolution of the predator's diet choice. The two extremes at which the predator's diet is composed solely of one prey correspond to two branches of the three-branch critical manifold of the fast slow system. By calculating the points at which there is a fast transition between these two feeding choices (i.e., branches of the critical manifold), we prove that the system has a two-parameter family of periodic orbits for sufficiently large separation of the time scales between the evolutionary and ecological dynamics. Using numerical simulations, we show that these periodic orbits exist, and that their phase difference and oscillation patterns persist, when ecological and evolutionary interactions occur on comparable time scales. Our model also exhibits periodic orbits that agree qualitatively with oscillation patterns observed in experimental studies of the coupling between rapid evolution and ecological interactions.

AB - We consider adaptive change of diet of a predator population that switches its feeding between two prey populations. We develop a novel 1 fast-3 slow dynamical system to describe the dynamics of the three populations amidst continuous but rapid evolution of the predator's diet choice. The two extremes at which the predator's diet is composed solely of one prey correspond to two branches of the three-branch critical manifold of the fast slow system. By calculating the points at which there is a fast transition between these two feeding choices (i.e., branches of the critical manifold), we prove that the system has a two-parameter family of periodic orbits for sufficiently large separation of the time scales between the evolutionary and ecological dynamics. Using numerical simulations, we show that these periodic orbits exist, and that their phase difference and oscillation patterns persist, when ecological and evolutionary interactions occur on comparable time scales. Our model also exhibits periodic orbits that agree qualitatively with oscillation patterns observed in experimental studies of the coupling between rapid evolution and ecological interactions.

KW - Lotka-Volterra interaction

KW - Fast-slow dynamical systems

KW - Geometric singular pertubation theory

KW - Planktonic protozoa-algae dynamics

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