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Multiple attractors in stage-structured population models with birth pulses.

Sanyi Tang1, Lansun Chen

  • 1Institute of Mathematics, Academy of Mathematics and System Sciences, Academia Sinica, Beijing 100080, People's Republic of China. tsy@math08.math.ac.cn

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This study models population dynamics with a pulsed birth rate, revealing complex behaviors like chaos and multiple attractors. These dynamics arise from density-dependent maturation and seasonal reproduction, impacting population stability.

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Area of Science:

  • Ecology
  • Mathematical Biology
  • Dynamical Systems Theory

Background:

  • Traditional population dynamics models often assume constant birth rates.
  • Many species exhibit seasonal reproduction, a factor frequently oversimplified in ecological models.
  • Understanding population fluctuations is crucial for conservation and resource management.

Purpose of the Study:

  • To formulate and analyze a stage-structured population model incorporating density-dependent maturation and a pulsed birth rate.
  • To investigate the complex dynamics, including equilibria, multiple attractors, and chaos, arising from this model.
  • To elucidate the role of periodic birth pulses in generating diverse population dynamics.

Main Methods:

  • Formulation of a single species, stage-structured model.
  • Analysis of the discrete dynamical system using its Poincaré map.
  • Investigation of equilibria, stability, basins of attraction, supertransients, and chaotic attractors.

Main Results:

  • Demonstrated the existence and stability of nonnegative equilibria.
  • Identified conditions for nonunique dynamics, where multiple attractors coexist.
  • Revealed the occurrence of supertransients and chaotic attractors.
  • Established that periodic birth pulses introduce natural cyclicity, enabling multiple oscillatory solutions.

Conclusions:

  • Pulsed reproduction and density-dependent maturation can lead to complex and unpredictable population dynamics.
  • Minor parameter or initial condition changes can drastically alter population behavior.
  • Seasonal reproduction is a key factor driving rich dynamical behaviors in ecological systems.