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A structured population model with diffusion in structure space.

Andrea Pugliese1, Fabio Milner2

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Summary
This summary is machine-generated.

This study introduces a stochastic population model using partial differential equations. It proves a bifurcation result based on a net reproduction number, showing distinct population equilibria and observing the Allee effect in simulations.

Keywords:
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Area of Science:

  • Mathematical Biology
  • Population Dynamics
  • Epidemiology

Background:

  • Structured population models are crucial for understanding disease dynamics.
  • Stochastic individual dynamics add complexity to traditional models.
  • Partial differential equations (PDEs) and ordinary differential equations (ODEs) are used to model population structures.

Purpose of the Study:

  • To develop and analyze a structured population model with stochastic individual dynamics.
  • To investigate the existence and stability of equilibrium solutions.
  • To explore the role of a net reproduction number and the Allee effect.

Main Methods:

  • A PDE of advection-diffusion type for infected individuals structured by pathogen density.
  • An ODE for susceptible/recovered individuals with a source term linked to diffusion and advection.
  • Mathematical analysis for global solution existence and bifurcation.
  • Numerical simulations to observe equilibrium stability and population dynamics.

Main Results:

  • Existence of a global-in-time solution is proven.
  • A net reproduction number ([Formula: see text]) is defined, determining the existence of trivial or non-trivial equilibria.
  • Numerical simulations demonstrate stabilization towards positive equilibria when [Formula: see text] and trivial equilibria when [Formula: see text].
  • Simulations also illustrate the Allee effect, which enhances population growth at low densities.

Conclusions:

  • The model provides a framework for studying structured populations with stochasticity.
  • The net reproduction number is a key parameter for predicting population persistence.
  • Numerical findings suggest complex dynamics, including the Allee effect, which warrant further analytical investigation.