arXiv subjects

Franco Herrera

Publications and source records attributed to Franco Herrera.

4 recordsLinked to original sources

arXiv2026

Sharp threshold dynamics for a bistable age-structured population model

This paper is devoted to the long-term dynamics of solutions to the Gurtin-MacCamy population model with a bistable birth function. We consider a one-parameter monotone family of initial distributions for the population such that for small values of the parameter, the corresponding population density gets extinct as time passes, whereas for large values of them, the solutions exhibit a different behavior. We are interested in the intermediate set of values for the parameters, which are called threshold parameters. We prove the existence of a sharp transition between these two asymptotic dynamics; that is, there exists exactly one threshold value when the age-dependent birth rate of the population has compact support, utilizing the theory of monotone dynamical systems. The case when the birth rate is non-compactly supported is more intricate to deal with, as has been observed in several works, even if the nonlinear birth function is monostable. Nevertheless, the approach used in the present work turns out to be effective to handle a particular birth rate with noncompact support by translating the dynamics of the age-structured model into an integro-differential system.

math.AP↗

arXiv2025

Slowly oscillating periodic solutions in a nonlinear Volterra equation with non-symmetric feedback

In this work we study a nonlinear Volterra equation with non-symmetric feedback that arises as a particular case of the Gurtin-MacCamy model in population dynamics. We are particularly interested in the existence of slowly oscillating periodic solutions when the trivial stationary state is unstable. Here the absence of symmetry of the nonlinearity prevents the use of many traditional strategies to obtain a priori estimates on the solution. Without a precise knowledge of the period of the solution, we manage to prove the forward invariance of a carefully constructed set of initial data whose properties imply the slowly oscillating character of all continuations. We prove the existence of periodic solutions by constructing a homeomorphism between our set and a convex subset of a different Banach space, thereby showing that it possesses the fixed-point property. Finally, in a singular limit of a parameter, we show that this periodic solution converges to the solution of a wellknown discrete difference equation. We conclude the paper with some numerical simulations to illustrate the existence of the periodic orbit as well as the singular limit behavior.

math.AP↗

arXiv2024

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

We continue to study (see arXiv:2401.08618, https://doi.org/10.48550/arXiv.2401.08618) a renewal equation $ϕ(t)=\frak Fϕ_t$ 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 $β(x)$ 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 $ϕ(t)=\frak Fϕ_t$. 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 $β$.

math.DS↗

arXiv2024

Global dynamics of a size-structured forest model

We study a size-structured model proposed in [1] C. Barril, À. Calsina, O. Diekmann, J. Z. Farkas, On competition through growth reduction, e-print arXiv:2303.02981, to describe the dynamics of trees growth in the forest. Our approach to the associated renewal equation is rather different from the methods in [1] and is based on ideas developed in [2] F. Herrera, S. Trofimchuk, Dynamics of one-dimensional maps and Gurtin-MacCamy's population model. Part I: asymptotically constant solutions, Ukrainian Math. J., (in Memory of O. Sharkovsky), 75 (2023), 1635-1651, https://doi.org/10.3842/umzh.v75i12.7678. Assuming relatively weak restrictions on the reproduction, death and growth rates $β, μ, g$, we establish the permanence properties of the semiflow $\frak F^t$ generated by the renewal equation and prove that it possesses a compact global attractor of points $\mathcal A$. Next we show that the opposite types of monotonicity of $β, g$ assure that $\frak F^t$ is also monotone and that in this case $\mathcal A$ coincides with a unique asymptotically stable equilibrium attracting neighbourhoods of compact sets with non-zero initial data.

math.DS↗