Transitions in population dynamics: Equilibria to periodic cycles to aperiodic cycles

Brian Dennis, R. A. Desharnais, J. M. Cushing, R. F. Costantino

Research output: Contribution to journalArticlepeer-review

97 Scopus citations


1. We experimentally set adult mortality rates, μ(a), in laboratory cultures of the flour beetle Tribolium at values predicted by a biologically based, nonlinear mathematical model to place the cultures in regions of different asymptotic dynamics. 2. Analyses of time-series residuals indicated that the stochastic stage-structured model described the data quite well. Using the model and maximum-likelihood parameter estimates, stability boundaries and bifurcation diagrams were calculated for two genetic strains. 3. The predicted transitions in dynamics were observed in the experimental cultures. The parameter estimates placed the control and μ(a) = 0.04 treatments in the region of stable equilibria. As adult mortality was increased, there was a transition in the dynamics. At μ(a) = 0.27 and 0.50 the populations were located in the two-cycle region. With μ(a) = 0.73 one genetic strain was close to a two-cycle boundary while the other strain underwent another transition and was in a region of equilibrium. In the μ(a) = 0.96 treatment both strains were close to the boundary at which a bifurcation to aperiodicities occurs; one strain was just outside this boundary, the other just inside the boundary. 4. The rigorous statistical verification of the predicted shifts in dynamical behaviour provides convincing evidence for the relevance of nonlinear mathematics in population biology.

Original languageEnglish (US)
Pages (from-to)704-729
Number of pages26
JournalJournal of Animal Ecology
Issue number5
StatePublished - Sep 1997
Externally publishedYes


  • Nonlinear population dynamics
  • Transitions in dynamic behaviour
  • Tribolium

ASJC Scopus subject areas

  • Ecology, Evolution, Behavior and Systematics
  • Animal Science and Zoology


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