Diffusive size-structured population model with time-varying diffusion rate

Manoj Kumar; Syed Abbas

doi:10.3934/dcdsb.2022128

Article Contents

2023, Volume 28, Issue 2: 1414-1435. Doi: 10.3934/dcdsb.2022128

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Diffusive size-structured population model with time-varying diffusion rate

Manoj Kumar and
Syed Abbas^,

School of Basic Sciences, Indian Institute of Technology Mandi, Kamand (H.P.) - 175005, India

^* Corresponding author: Syed Abbas
^* Corresponding author: Syed Abbas

Received: December 2021

Revised: May 2022

Early access: July 2022

Published: February 2023

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Abstract

This work is devoted to the study of a size-structured population model with a time-varying diffusion rate. Due to the seasonal variation, it is natural to consider the time-varying diffusion rate. Moreover, introducing the time-varying diffusion rate makes the model more challenging and requires the results of evolution operators for the analysis. Under some assumptions on the time-dependent diffusion coefficient, the existence and uniqueness of mild solution is shown. We apply the method of characteristics and evolution operators to derive our results. Positivity and boundedness of mild solution is also shown. Some examples are provided to illustrate the theoretical findings. We also solve our model numerically to study the size and spatial dynamics of population density.

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Mathematics Subject Classification: Primary: 47J35, 97M10, 65M25.

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Figure 1. Figure shows the behaviour of diffusion coefficient for different values of $ \alpha $

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Figure 2. Diffusion coefficient for different values of $ b $

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Figure 3. Figure depicts size-space profile of population density at fixed time $ t = 0 $

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Figure 4. Size-space profile of population density at fixed time $ t = 0.4 $

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Figure 5. Dynamics of population at fixed time $ t = 0.8 $

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Figure 6. Size-space profile of population density at fixed time $ t = 1 $

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Figure 7. Size-space evolution of population at fixed time $ t = 0 $

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Figure 8. Evolution of population at fixed time $ t = 0.4 $

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Figure 9. Dynamics of population at fixed time $ t = 0.8 $

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Figure 10. Size-space evolution of population at fixed time $ t = 1 $

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Figure 11. Evolution of population at fixed spatial position $ x = 1 $

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Figure 12. Dynamics of population at fixed spatial position $ x = 2 $

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