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 ) = Fb t constructed and analysed in the work [Barril et al ., e-print arXiv:2303.02981 , https://doi.org/10.48550/arXiv.2303 . 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 arXiv:2303.02981 , https://doi.org/10.48550/arXiv.2303.02981 ; 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 arXiv:2401.08618 , https://doi.org/10.48550/arXiv.2401.08618 ]. 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)].