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A Reaction-Diffusion Model with Spatially Inhomogeneous Delays.

Yijun Lou1, Feng-Bin Wang2,3,4

  • 1Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong SAR, China.

Journal of Dynamics and Differential Equations
|June 26, 2023
PubMed
Summary
This summary is machine-generated.

This study models population growth with uneven maturation times in varied environments. It predicts species extinction if the basic reproduction ratio is below one, and survival otherwise.

Keywords:
Population dynamicsReaction-diffusion modelSpatially inhomogeneous delayStage-structured model

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Area of Science:

  • Mathematical Biology
  • Ecology
  • Population Dynamics

Background:

  • Population growth is influenced by environmental heterogeneity.
  • Reaction-diffusion models are crucial for understanding spatial population dynamics.
  • Spatially dependent delays present unique challenges in ecological modeling.

Purpose of the Study:

  • To develop and analyze a reaction-diffusion model incorporating spatially dependent maturation durations.
  • To investigate the impact of spatial heterogeneity and delays on population dynamics.
  • To determine conditions for species extinction and persistence.

Main Methods:

  • Construction of a reaction-diffusion model with spatially variable parameters and delays.
  • Rigorous mathematical analysis including well-posedness and stability.
  • Formulation of the basic reproduction ratio (R0).
  • Utilizing a novel functional phase space for stability analysis.

Main Results:

  • Species extinction is predicted when the basic reproduction ratio (R0) is less than one.
  • Uniqueness and global attractivity of a positive equilibrium are established for increasing birth rates when R0 > 1.
  • Species permanence is demonstrated for unimodal birth functions when R0 > 1.

Conclusions:

  • The model provides insights into population dynamics in heterogeneous environments with spatially varying delays.
  • The findings highlight the critical role of the basic reproduction ratio in determining population persistence.
  • The developed methodology is applicable to a wider range of ecological models with spatial heterogeneity and time delays.