Special Session 86: Advances in Differential, Difference and Dynamic Equations with Applications in Science and Engineering

Analysis of Initial Conditions of Various Discrete Dynamic Equations and Iterated Function Systems using Visual Heuristics

Chris Ahrendt University of Wisconsin-Eau Claire USA Co-Author(s):

Abstract: Inspired by the fractal images produced by the Electric Sheep project started in 1999 by Scott Draves, we seek to analyze the underlying iterated function systems (IFSs) that are used to produce these fractal images. Since the IFSs involve multiple nonlinear functions selected using a random process, some of the traditional analytic techniques from difference equations run into difficulties. In this work stemming from a recent undergraduate student research collaboration, we will give some background on the Electric Sheep project and how the underlying IFSs are utilized to produce images. The Electric Sheep project focuses on the aesthetics of the resulting images. Here, we seek to develop several visual heuristics (some that are interactive) in order to analyze the mathematical behavior of orbits with various initial conditions for the IFSs that produce these images. We visualize the nullclines using approximations to account for the nonlinear and random aspects of the IFSs. We will develop several heuristics that incorporate ideas involving domain coloring from complex variables in order to construct the visualization for the heuristics. Finally, we apply these techniques to discrete analogs of certain DEs such as the Clairaut equation, the Bernoulli equation, and the Lotka-Volterra system of equations.

Modeling HIV-1 Infection Dynamics Using Mathematical and Statistical Approaches

Elvan Akin Missouri University of Science and Technology USA Co-Author(s):    N.N.Pelen, O. Ozturk, G.R. Olbricht, A.S. Perelson

Abstract: In this study, we examine the dynamics of HIV-1 infection using mathematical models that capture both patient-specific behavior and population-level patterns. The model is formulated as a three-dimensional system describing the interactions between uninfected target cells, infected target cells, and viral load. We first perform mathematical model fits using longitudinal RNA measurements from individual patients, allowing parameters to be estimated directly from clinically observed data. In addition to this mathematical modeling approach, we conduct a statistical analysis across the patient cohort to investigate population-level patterns. Interestingly, the same patients emerge as the best-fitting cases under both approaches, indicating strong agreement between the mathematical modeling results and the statistical population analysis.

Chebyshev Collocation Method for a Three-Dimensional HIV Infection Model

Beyza Cetin Missouri University Science and Technology USA Co-Author(s):    Beyza Cetin, Elvan Akin, Alan Perelson

Abstract: Nonlinear differential equations frequently arise in mathematical models describing viral infection dynamics, where analytical solutions are rarely available. Therefore, reliable numerical methods are required to approximate the behavior of such systems. In this study, we propose a numerical approach based on the Chebyshev collocation method for solving nonlinear systems appearing in HIV infection models. The method approximates the solution components using Chebyshev polynomial expansions and enforces the governing equations at selected collocation points. This formulation transforms the original nonlinear differential system into a system of algebraic equations that can be solved efficiently using standard numerical techniques. Due to the spectral accuracy of Chebyshev polynomials, the proposed approach provides highly accurate approximations with relatively low computational cost. To demonstrate the applicability of the method, we apply the proposed scheme to a three-dimensional HIV infection model describing the interaction between healthy cells, infected cells, and virus particles. Numerical simulations show that the method accurately captures the qualitative behavior of the system dynamics. In addition, the formulation suggests that the proposed collocation framework can be extended to models defined on time scales, providing a potential tool for studying systems involving both continuous and discrete temporal structures.

Inverse Forms of Pachpatte-Type Dynamic Inequalities within Diamond-Alpha Calculus

Billur Kaymakcalan University of Turkish Aeronautical Association Turkey Co-Author(s):    Zeynep Kayar

Abstract: Through the application of concavity, we derive novel reversed diamond alpha Pachpatte type dynamic inequalities. These formulations not only encompass the nabla and delta analogues but also merge both the discrete and continuous frameworks as particular instances. Moreover, in the absence of concavity, they naturally give rise to generalized versions of the diamond alpha Bennett Leindler type dynamic inequalities.

\title{Bifurcation Analysis of a Smith-Type Growth Predator--Prey Model and Its Application}

Md Mutakabbir Khan Missouri University of Science and Technology USA Co-Author(s):    Elvan Akin, Md Jasim Uddin

Abstract: This work develops and analyzes a prey--predator model in which the prey population exhibits Smith-type growth, group-defense behavior, and continuous external replenishment. A forward--Euler discretization produces a two-dimensional map whose positivity, boundedness, and equilibrium stability are established. Bifurcation analysis shows that the coexistence equilibrium may lose stability through Neimark--Sacker or period-doubling bifurcations, leading to quasiperiodic or chaotic dynamics. Numerical simulations, including bifurcation diagrams, phase portraits, and Lyapunov exponents, reveal rich dynamical behaviors. A global sensitivity analysis using partial rank correlation coefficients identifies parameters that most strongly influence long-term species densities. Finally, calibration with COVID-19 and tuberculosis surveillance data demonstrates the potential applicability of the framework under a phenomenological interpretation.

New Theorems for Determining the Global Dynamics of Anti-Competitive Systems

Chris D Lynd Commonwealth University of Pennsylvania USA Co-Author(s):

Abstract: We will present new theorems that can be used to determine the global dynamics of the solutions of anti-competitive systems of difference equations. We will give a brief background on anti-competitive maps and some insights into how to apply the theorems. We will also present some applications of anti-competitive systems of difference equations.

\title{Bifurcation and Chaos Control in a Discrete Predator-Prey Model with Fear Effect and Immigration}

Esrat Nur Louisiana State University USA Co-Author(s):

Abstract: This work investigates the discrete-time dynamics of a predator--prey model incorporating a square root functional response, fear effect, and immigration. Using bifurcation theory and center manifold analysis, the study establishes the occurrence of period-doubling and Neimark--Sacker bifurcations in the positive quadrant. Numerical simulations support the analytical results and illustrate complex behaviors such as high-period orbits, quasi-periodic invariant curves, and chaotic attractors. To control chaotic dynamics, both the Ott--Grebogi--Yorke technique and a state-feedback control method are applied, showing that the system can be guided toward a stable equilibrium. The results provide useful insight into the dynamics and control of nonlinear ecological systems.

Existence Results for Third-Order Nonlinear Dynamic Equations

Ozkan Ozturk Giresun University Turkey Co-Author(s):    Elvan Akin, Neslihan Nesliye Pelen

Abstract: We study a third-order nonlinear dynamic equation on arbitrary time scales, namely, nonempty closed subsets of the real numbers, which unify continuous and discrete analysis within a single framework. Our main focus is on the qualitative properties of nonoscillatory solutions and their quasi-derivatives, especially their asymptotic behavior. By employing improper integral criteria and applying fixed point theorems, we establish the existence of such solutions.

The effect of very fast dispersal on two species competition with drift

Rana Parshad Iowa State University USA Co-Author(s):

Abstract: Classical theory predicts that for two competing populations subject to a constant downstream drift, the faster disperser will competitively exclude the slower disperser. In the current work, we consider a novel model of a much faster dispersing species, modeled via a $p$-Laplacian operator, competing with a slower disperser. We prove global existence of weak solutions to this model for any positive initial condition, in certain parametric regime. Counterintuitively, we show that while the faster disperser always wins - the much faster disperser could actually lose, for certain initial data. Our results have implications for biodiversity, refuge design, and improved biological control, driven by habitat fragmentation and climate change

Modeling Influenza Transmission Across Ecological Scales: Aquatic Birds, Domestic Poultry, and Humans

Naveen K. Vaidya San Diego State University USA Co-Author(s):    Naveen K. Vaidya

Abstract: Climate, especially environmental temperature, plays a critical role in shaping the transmission, evolution, and persistence of highly pathogenic influenza viruses in aquatic birds, thereby driving epidemics in domestic poultry and, eventually, in humans. In this talk, I will present transmission dynamics models of influenza across different scales: aquatic birds, domestic poultry, and humans. Analysis of our data-validated models allows us to identify climate conditions that favor the persistence of highly pathogenic influenza strains among aquatic birds. The models also provide the risk of cross-species transmission and epidemic spread in domestic poultry and humans. Our results indicate that climate functions as a selective filter shaping both the transmission, evolution, and invasion potential of influenza across species.