An age group model for the study of a population of trees

Laurent Di Menza; Virginie Joanne-Fabre

doi:10.3934/dcdss.2020464

Article Contents

2021, Volume 14, Issue 8: 2823-2835. Doi: 10.3934/dcdss.2020464

This issue Previous Article Numerical simulations of parity–time symmetric nonlinear Schrödinger equations in critical case Next Article The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data

An age group model for the study of a population of trees

Laurent Di Menza^1, , and
Virginie Joanne-Fabre^2,

1.
LMR, UMR 9008, U.F.R. Sciences Exactes et Naturelles, BP 1039, 51687 Reims Cedex 02, France
2.
HABITER, EA 2076, U.F.R. Lettres et Sciences Humaines, BP 30, 51571 Reims Cedex, France

Received: June 2020

Revised: October 2020

Early access: November 2020

Published: August 2021

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Abstract

In this paper, we derive a simple model for the description of an ecological system including several subgroups with distinct ages, in order to analyze the influence of various phenomena on temporal evolution of the considered species. Our aim is to address the question of resilience of the global system, defined as its ability to stabilize itself to equilibrium, when being perturbed by exterior fluctuations. It is shown that a under a critical condition involving growth rate and mortality rate of each subgroup, extinction of all species may occur.

Keywords:

Mathematics Subject Classification: Primary: 34D20; Secondary: 65L05, 92D25.

Citation:

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Figure 1. Time evolution of $R$ in three different cases: (a) $g = 1$; (b) $g = 0.2$ between $t = 3$ and $t = 10$ and $g = 1$ elsewhere, (c) $g = 1$ if $t\le 3$ and $g = 0.2$ elsewhere

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Figure 2. Time evolution of $R$ in two different cases: (a) $m = 1$; (b) $m = 1.2$ between $t = 20$ and $t = 40$ and $m = 1$ elsewhere. The asymptotic state $R_{*, 2}$ is not perturbed by mortality fluctuation

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Figure 3. Time evolution of $R$ in two different cases: (a) $m = 1.2$ for $t\ge 20$; (b) $m = 2$ for $t\ge 20$. In the first case, the asymptotic state $R_{*, 2}$ is perturbed, in the second case, extinction occurs at large times

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Figure 4. Time evolution of $(Y, I, O)$ for the two stable cases $g_Y = 3$, $g_I = 1$, $g_O = 0.5$, $m_Y = 1$, $m_I = 0.8$, $m_O = 0.2$ (left) and $g_Y = 1$, $g_I = 1$, $g_O = 0.6$, $m_Y = 1.5$, $m_I = 1$, $m_O = 0.2$ (right) ($Y$: black, $I$: blue, $O$: red)

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Figure 5. Time evolution of $(Y, I, O)$ when perturbing mortality of intermediate trees : $m_{I, 2} = 1.4$ (left) and $m_{I, 2} = 2$ (right). In the first case, the system drives itself into another stable nontrivial equilibrium; in the second case, the total population extincts ($Y$: black, $I$: blue, $O$: red)

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