Evolutionary Behaviour, Trade-Offs and Cyclic and Chaotic Population Dynamics

Andy Hoyle, Roger G. Bowers, Andy White

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


Many studies of the evolution of life-history traits assume that the underlying population dynamical attractor is stable point equilibrium. However, evolutionary outcomes can change significantly in different circumstances. We present an analysis based on adaptive dynamics of a discrete-time demographic model involving a trade-off whose shape is also an important determinant of evolutionary behaviour. We derive an explicit expression for the fitness in the cyclic region and consequently present an adaptive dynamic analysis which is algebraic. We do this fully in the region of 2-cycles and (using a symbolic package) almost fully for 4-cycles. Simulations illustrate and verify our results. With equilibrium population dynamics, trade-offs with accelerating costs produce a continuously stable strategy (CSS) whereas trade-offs with decelerating costs produce a non-ES repellor. The transition to 2-cycles produces a discontinuous change: the appearance of an intermediate region in which branching points occur. The size of this region decreases as we move through the region of 2-cycles. There is a further discontinuous fall in the size of the branching region during the transition to 4-cycles. We extend our results numerically and with simulations to higher-period cycles and chaos. Simulations show that chaotic population dynamics can evolve from equilibrium and vice-versa. © 2010 Society for Mathematical Biology.

Original languageEnglish
Pages (from-to)1154-1169
Number of pages16
JournalBulletin of Mathematical Biology
Issue number5
Publication statusPublished - May 2011


  • Adaptive dynamics
  • Chaos
  • Evolutionary branching
  • Logistic equation
  • Trade-offs


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