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Stability and Bifurcation in Discrete Dynamical Systems: Application to Population Dynamics

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Prof. Dr. Rafael Luís

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Department of Mathematics, University of Madeira, Campus Universitário da Penteada, 9020-105 Funchal, Portugal
Interests: difference equations; discrete dynamical system; populations dynamics; stability; bifurcation

Special Issue Information

Dear Colleagues,

This Special Issue focuses on the study of stability and bifurcation in discrete dynamical systems, with a particular emphasis on their applications to population dynamics since stability and bifurcation are key concepts for understanding how biological populations respond to environmental changes, species interactions, and other dynamic factors. Additionally, this Special Issue aims to explore the symmetric and asymmetric properties of autonomous or nonautonomous population models, with or without evolutionary dynamics.

In this Special Issue, original research articles and reviews are welcome. Research areas may include (but are not limited to) the following:

Mathematical modeling of populations (development of discrete models to describe population dynamics and local and global stability analysis of equilibrium points or periodic orbits);
Bifurcation (analysis of bifurcations in population models and their impact on dynamic behavior and classification of bifurcation types and their implications for population stability);
Symmetric and asymmetric properties (study of symmetric and asymmetric dynamics in population models, exploration of how symmetry and asymmetry influence stability and bifurcation, and how these factors affect population dynamics).

I look forward to receiving your contributions.

Prof. Dr. Rafael Luís
Guest Editor

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Research

22 pages, 10564 KB

Open AccessArticle

Bifurcation and Global Dynamics of Continuous and Discrete Competitive Models for Genetic Toggle Switches

by Carmen R. Ferrara and Mustafa R. S. Kulenović

Symmetry 2026, 18(4), 629; https://doi.org/10.3390/sym18040629 - 9 Apr 2026

Viewed by 221

Abstract

We investigate the asymptotic behavior of a proposed ordinary differential equation (ODE) model for Genetic Toggle switches from Gardner et. al. and I. Rajapakse and S. Smale:

\frac{d x}{d t} = \frac{a}{1 + y^{m}} - x

and [...] Read more.

We investigate the asymptotic behavior of a proposed ordinary differential equation (ODE) model for Genetic Toggle switches from Gardner et. al. and I. Rajapakse and S. Smale:

\frac{d x}{d t} = \frac{a}{1 + y^{m}} - x

and

\frac{d y}{d t} = \frac{b}{1 + x^{n}} - y

where

a, b, m, n > 0

and

x (t), y (t) \geq 0

. We also investigate the asymptotic behavior of the Euler discretization of this system:

x_{n + 1} = a_{1} x_{n} + \frac{b_{1}}{1 + y_{n}^{m}} = f (x_{n}, y_{n})

and

y_{n + 1} = a_{2} y_{n} + \frac{b_{2}}{1 + x_{n}^{n}} = g (x_{n}, y_{n}),

where

1 - h = a_{1}

1 - k = a_{2}

a h = b_{1}

and

b k = b_{2}

a_{1}, a_{2} \in (0, 1)

and

h, k > 0

are steps of discretizations. Here, x and y represent protein concentrations at a particular time in both genes and

a, b, m, n > 0

, respectively, above. We will apply the theory of competitive maps to find the basins of attractions of different equilibrium points and period-two solutions of systems of difference equations. Full article

(This article belongs to the Special Issue Stability and Bifurcation in Discrete Dynamical Systems: Application to Population Dynamics)

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31 pages, 2940 KB

Open AccessArticle

Global Dynamics and Bifurcation of an Evolutionary Beverton-Holt Model with the Allee Effect

by Emma D’Aniello, Saber Elaydi, Eddy Kwessi, Rafael Luís and Brian Ryals

Symmetry 2025, 17(11), 1811; https://doi.org/10.3390/sym17111811 - 27 Oct 2025

Cited by 2 | Viewed by 1080

Abstract

We study the global dynamics and bifurcation structure of an evolutionary Beverton–Holt model with Allee effects, a framework that couples ecological constraints with adaptive trait evolution. The model accounts for density dependence, mate limitation, and predator saturation, while traits evolve according to selection gradients that influence reproduction and competition. From an ecological perspective, we show that weak Allee effects create bistability between extinction and survival, while strong Allee effects generate a critical threshold below which populations collapse and above which they persist at carrying capacity. Evolutionary feedback further reshapes these outcomes by shifting thresholds, modifying stability regions, and producing multiple long-term attractors. Biologically, this reveals how demographic pressures such as scarce mates or high predation interact with trait evolution to determine persistence or extinction, and how adaptive responses may rescue populations facing critical density barriers. Our rigorous analysis and simulations demonstrate that eco–evolutionary processes not only alter classical Beverton–Holt outcomes but also provide insight into mechanisms underlying species persistence, extinction risk, and invasion success under Allee constraints. Full article

(This article belongs to the Special Issue Stability and Bifurcation in Discrete Dynamical Systems: Application to Population Dynamics)

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