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Global dynamics of a size-structured forest model

Franco Herrera, Sergei Trofimchuk

Abstract

We study a size-structured model proposed in [1] C. Barril, À. Calsina, O. Diekmann, J. Z. Farkas, On competition through growth reduction, e-print arXiv:2303.02981, to describe the dynamics of trees growth in the forest. Our approach to the associated renewal equation is rather different from the methods in [1] and is based on ideas developed in [2] F. Herrera, S. Trofimchuk, Dynamics of one-dimensional maps and Gurtin-MacCamy's population model. Part I: asymptotically constant solutions, Ukrainian Math. J., (in Memory of O. Sharkovsky), 75 (2023), 1635-1651, https://doi.org/10.3842/umzh.v75i12.7678. Assuming relatively weak restrictions on the reproduction, death and growth rates $β, μ, g$, we establish the permanence properties of the semiflow $\frak F^t$ generated by the renewal equation and prove that it possesses a compact global attractor of points $\mathcal A$. Next we show that the opposite types of monotonicity of $β, g$ assure that $\frak F^t$ is also monotone and that in this case $\mathcal A$ coincides with a unique asymptotically stable equilibrium attracting neighbourhoods of compact sets with non-zero initial data.

Global dynamics of a size-structured forest model

Abstract

We study a size-structured model proposed in [1] C. Barril, À. Calsina, O. Diekmann, J. Z. Farkas, On competition through growth reduction, e-print arXiv:2303.02981, to describe the dynamics of trees growth in the forest. Our approach to the associated renewal equation is rather different from the methods in [1] and is based on ideas developed in [2] F. Herrera, S. Trofimchuk, Dynamics of one-dimensional maps and Gurtin-MacCamy's population model. Part I: asymptotically constant solutions, Ukrainian Math. J., (in Memory of O. Sharkovsky), 75 (2023), 1635-1651, https://doi.org/10.3842/umzh.v75i12.7678. Assuming relatively weak restrictions on the reproduction, death and growth rates , we establish the permanence properties of the semiflow generated by the renewal equation and prove that it possesses a compact global attractor of points . Next we show that the opposite types of monotonicity of assure that is also monotone and that in this case coincides with a unique asymptotically stable equilibrium attracting neighbourhoods of compact sets with non-zero initial data.
Paper Structure (5 sections, 11 theorems, 66 equations)

This paper contains 5 sections, 11 theorems, 66 equations.

Table of Contents

  1. Introduction and main results
  2. Monotonicity of the operator ${\mathcal{F}}$
  3. Uniform ultimate boundedness and uniform persistence of solutions. Proofs of Theorems \ref{['mainTT']} and \ref{['mainT']}
  4. Conclusion
  5. Appendix

Key Result

Proposition 1

Assume that continuous function $\beta:[x_m,+\infty)\to [0, +\infty)$ has at most polynomial growth and $g: [0,+\infty)\to [0, +\infty)$ is continuous and bounded. Then

Theorems & Definitions (17)

  • Proposition 1
  • Proposition 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • proof
  • Corollary 6
  • proof
  • Corollary 7
  • Theorem 8
  • ...and 7 more