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Block-pulse integrodifference equations.

Nora M Gilbertson1, Mark Kot2

  • 1Department of Applied Mathematics, University of Washington, Seattle, WA, USA. nmg421@uw.edu.

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|September 13, 2023
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Summary
This summary is machine-generated.

A new hybrid method approximates population dynamics in finite habitats using block-pulse integrodifference equations (IDEs). This approach offers analytical and numerical advantages for understanding population distributions and growth patterns.

Keywords:
Allee effectsBlock-pulse seriesIntegrodifference equationsPopulation dynamicsSpatial ecology

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Area of Science:

  • Ecology
  • Mathematical Biology
  • Population Dynamics

Background:

  • Integrodifference equations (IDEs) model population dynamics across space.
  • Calculating equilibrium distributions for IDEs with increasing growth in finite habitats is challenging.
  • Existing models may lack analytical tractability or efficient numerical solutions.

Purpose of the Study:

  • To develop a hybrid method for approximating equilibrium population distributions of IDEs.
  • To introduce and analyze the block-pulse IDE model.
  • To compare the approximation accuracy and advantages of the block-pulse IDEs against standard growth models.

Main Methods:

  • Approximating the IDE growth function with a piecewise-constant function to create block-pulse IDEs.
  • Deriving analytic expressions for iterates and equilibria of block-pulse IDEs.
  • Analyzing the dynamics and bifurcation structures of one-, two-, and three-step block-pulse IDEs.
  • Using block-pulse IDEs with a numerical root finder to approximate compensatory and depensatory growth models.

Main Results:

  • Block-pulse IDEs provide analytic expressions for population distributions.
  • One-, two-, and three-step block-pulse IDEs exhibit rich dynamics and numerous fold bifurcations.
  • The method accurately approximates equilibrium distributions for compensatory and depensatory growth.
  • The block-pulse IDE approach offers numerical and analytical advantages over original IDEs.

Conclusions:

  • The block-pulse IDE method is an effective tool for approximating population distributions in finite habitats.
  • This hybrid approach simplifies complex population dynamics while retaining essential features.
  • The method facilitates deeper understanding of population regulation mechanisms, including Allee effects.