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We continue to study (see [6]) a renewal equation \(\phi \left( t \right) = {\mathfrak{F}}\phi_{t}\) proposed in [1, 2] to model trees growth. This time we are considering the case when the per capita reproduction rate \(\beta \left( x \right)\) is a non-monotone (unimodal) function of tree’s height x. Note that the height of some species of trees can impact negatively seed viability, cf. [3, p. 524], in a kind of autogamy depression. Similarly to previous works, it is also assumed that the growth rate g(x) of an individual of height x is a strictly decreasing function. As in [5, 6], we analyse the connection between dynamics of the associated one-dimensional map \(F\left( b \right) = \mathfrak{F}b\) , \(b \in {\mathbb{R}}_{ + }\) , and the delayed (hence infinite-dimensional) model \(\phi \left( t \right) = {\mathfrak{F}}\phi_{t}\) . Our key observation is that this model is of monotone positive feedback type since F is strictly increasing on \({\mathbb{R}}_{ + }\) independently on the monotonicity properties of \(\beta\) .

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On the Global Dynamics of a Forest Model with Monotone Positive Feedback and Memory

  • 章节

  • 发表日期：

    2025年

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  • Franco Herrera,
  • Sergei Trofimchuk

摘要

We continue to study (see [6]) a renewal equation \(\phi \left( t \right) = {\mathfrak{F}}\phi_{t}\) proposed in [1, 2] to model trees growth. This time we are considering the case when the per capita reproduction rate \(\beta \left( x \right)\) is a non-monotone (unimodal) function of tree’s height x. Note that the height of some species of trees can impact negatively seed viability, cf. [3, p. 524], in a kind of autogamy depression. Similarly to previous works, it is also assumed that the growth rate g(x) of an individual of height x is a strictly decreasing function. As in [5, 6], we analyse the connection between dynamics of the associated one-dimensional map \(F\left( b \right) = \mathfrak{F}b\) , \(b \in {\mathbb{R}}_{ + }\) , and the delayed (hence infinite-dimensional) model \(\phi \left( t \right) = {\mathfrak{F}}\phi_{t}\) . Our key observation is that this model is of monotone positive feedback type since F is strictly increasing on \({\mathbb{R}}_{ + }\) independently on the monotonicity properties of \(\beta\) .