Global stability for a forest model with unimodal fertility and monotone growth rates

The main object of our studies is an infinite delay model b(t) = Fbt constructed and analysed in the work [Barril et al., e-print <ns0:ext-link ext-link-type="uri" ns4:href="https://arxiv.org/abs/2303.02981">arXiv:2303.02981</ns0:ext-link>, <ns0:ext-link ext-link-type="uri" ns4:href="https://doi.org/10.48550/arXiv.2303">https://doi.org/10.48550/arXiv.2303</ns0:ext-link>. 02981; Barril et al., J. Math. Biol. 88 (2024) 66] dealing with the growth of trees competing for light (with b(t) being interpreted as the population growth rate at time t). In [Barril et al. e-print <ns0:ext-link ext-link-type="uri" ns4:href="https://arxiv.org/abs/2303.02981">arXiv:2303.02981</ns0:ext-link>, <ns0:ext-link ext-link-type="uri" ns4:href="https://doi.org/10.48550/arXiv.2303.02981">https://doi.org/10.48550/arXiv.2303.02981</ns0:ext-link>; Barril et al. [J. Math. Biol. 88 (2024) 66], the action F is defined in terms of two nonlinear monotone functions: (increasing) per capita reproduction rate β(x) of an individual of height x and (decreasing) growth rate g ∈ C(R+) for the observed species of trees. As a consequence, the functional F is also monotone [Herrera and Trofimchuk, e-print <ns0:ext-link ext-link-type="uri" ns4:href="https://arxiv.org/abs/2401.08618">arXiv:2401.08618</ns0:ext-link>, <ns0:ext-link ext-link-type="uri" ns4:href="https://doi.org/10.48550/arXiv.2401.08618">https://doi.org/10.48550/arXiv.2401.08618</ns0:ext-link>]. However, by admitting that the height of some species of trees can negatively impact the seed viability [Caraballo-Ortiz et al., J. Trop. Ecol. 27 (2011) 521–528], we should also consider hump-shaped fertility functions β. Our key finding is that in spite of this form of β, the functional F can still possess a kind of weak monotonicity property for a specific class of growth rates g. This fact assures the global attractivity of a unique positive steady state, in this way answering one of the open questions in [Herrera and Trofimchuk, 2023 MATRIX Annals, Springer (2025)].