arXiv subjects

Maicon Sonego

Publications and source records attributed to Maicon Sonego.

3 recordsLinked to original sources

arXiv2026

Boundary-driven patterns in elongated convex domains

We consider the heat equation in a smooth bounded convex domain $Ω\subset \mathbb{R}^2$ with nonlinear Neumann boundary condition $\partial_νu = λ(u - u^3)$. Stable non-constant stationary solutions do not exist when $Ω$ is a ball. We show that this behavior is not a consequence of convexity alone. More precisely, if the inradius of $Ω$ is fixed and its diameter is sufficiently large, then there exists $λ>0$ for which the problem admits such a solution. The result reveals a geometric mechanism for the emergence of stable non-constant stationary solutions in elongated convex domains.

math.AP↗

arXiv2025

Boundary and Interior Control in a Diffusive Lotka-Volterra Model

We investigate the controllability of a generalized diffusive Lotka-Volterra competition model for two species, incorporating boundary controls and an interior multiplicative control. Considering a smooth, bounded N-dimensional domain, we analyze ecologically pertinent scenarios characterized by constraints on both the controls and system states. Our results demonstrate how integrated control strategies can effectively overcome the limitations identified in previous studies. We prove two main results: (1) asymptotic controllability to single-species survival steady states under arbitrary system parameters, ensured by a combination of boundary and interior controls which act jointly to stabilize the system; and (2) finite-time controllability to a specific heterogeneous coexistence steady state via a two-phase strategy - first steering the system near the target with boundary control, then activating an interior multiplicative control in a localized region. The strong synergy between the two control mechanisms is crucial in both cases. We also analyze extinction dynamics and homogeneous coexistence, and support our findings with numerical simulations. The work concludes with perspectives for future research.

math.AP↗

arXiv2024

Control of a Lotka-Volterra System with Weak Competition

The Lotka-Volterra model reflects real ecological interactions where species compete for limited resources, potentially leading to coexistence, dominance of one species, or extinction of another. Comprehending the mechanisms governing these systems can yield critical insights for developing strategies in ecological management and biodiversity conservation. In this work, we investigate the controllability of a Lotka-Volterra system modeling weak competition between two species. Through constrained controls acting on the boundary of the domain, we establish conditions under which the system can be steered towards various target states. More precisely, we show that the system is controllable in finite time towards a state of coexistence whenever it exists, and asymptotically controllable towards single-species states or total extinction, depending on domain size, diffusion, and competition rates. Additionally, we determine scenarios where controllability is not possible and, in these cases, we construct barrier solutions that prevent the system from reaching specific targets. Our results offer critical insights into how competition, diffusion, and spatial domain influence species dynamics under constrained controls. Several numerical experiments complement our analysis, confirming the theoretical findings.

math.AP↗