On the global dynamics of a forest model with monotone positive feedback and memory

TL;DR

This paper investigates a forest growth model with non-monotone reproduction rates and memory effects, analyzing how the dynamics of a one-dimensional map relate to the complex delayed model, revealing monotone positive feedback behavior.

Contribution

It extends previous models by considering unimodal reproduction functions and explores the connection between one-dimensional maps and infinite-dimensional delayed dynamics.

Findings

The model exhibits monotone positive feedback regardless of reproduction rate monotonicity.

The dynamics of the one-dimensional map provide insights into the behavior of the delayed system.

The analysis links the properties of the map to potential forest population outcomes.

Abstract

We continue to study (see arXiv:2401.08618, https://doi.org/10.48550/arXiv.2401.08618) a renewal equation $\phi(t)=\frak F\phi_t$ proposed in [C. Barril et al., J. Math. Biology, https://doi.org/10.1007/s00285-024-02084-x] to model trees growth. This time we are considering the case when the per capita reproduction rate $\beta(x)$ 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, 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. Here we analyse the connection between dynamics of the associated one-dimensional map $F(b)= {\frak F}b,$ $b \in {\mathbb R}_+$, and the delayed (hence infinite-dimensional) model $\phi(t)=\frak F\phi_t$. 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 $\beta$.