Problem1 : From Lotka–Volterra to Complexity: Modeling Multi-Species Interactions

Coordinator: Prof. Paulo Doutor, NOVA School of Science and Technology, Portugal

Summary: The predator–prey model introduced by Lotka [1] and Volterra [2] is a classical example of the application of ordinary differential equations (ODEs) in biology and ecology. As a nonlinear system, its study relies less on explicit solutions and more on the qualitative analysis of its dynamical behavior. Qualitative analysis of ODE systems focuses on equilibrium solutions and bounded trajectories, including periodic orbits, as well as more intricate dynamics arising from bounded but non-periodic behavior. Such phenomena appear across a wide range of models in the natural and physical sciences, from electrical circuits to the spread of infectious diseases.

This workshop centres on models describing interactions among three (or more) species, encompassing not only predator–prey relationships but also competition and cooperation, thereby extending the classical Lotka–Volterra framework. Logistic growth terms will also be incorporated to account for environmental carrying capacity. The analysis will address conditions under which solutions converge to equilibria, whether these equilibria correspond to the extinction of one or more species, and whether the system exhibits sustained oscillatory dynamics, either periodic or irregular.The work will follow the model studied by Peixe and Rodrigues in [3], combining analytical techniques with computational implementation to obtain numerical approximations of solutions and their corresponding geometric representations.

By the end of the Iberian Modeling Week, the working group is expected to develop a publicly accessible website featuring an interactive simulator for this class of models, designed for use by the general public as well as undergraduate and master's students.

References

[1] Lotka, A. J. Analytical note on certain rhythmic relations in organic systems. Proceedings of the National Academy of Sciences, 1920, 6(7): 410–415.

[2] Volterra, V. Fluctuations in the abundance of a species considered mathematically. Nature, 1926, 118(2972): 558–560.

[3] Peixe, T.; Rodrigues, A. Persistent strange attractors in 3D polymatrix replicators. Physica D: Nonlinear Phenomena, 2022, 438: 133346.

Problem 2: Predicting brain tumor subtypes from gene expression data

Coordinator: Dr. Marta Lopes, NOVA School of Science and Technology, Portugal

Brief Description of the Challenge:

Gliomas are a group of brain tumors that are particularly difficult to diagnose and treat due to their complexity and variability between patients. In this challenge, we focus on two types of lower-grade gliomas: oligodendroglioma and astrocytoma. Correctly distinguishing between these two types is critical for improving patient care.

Participants will work with RNA-sequencing data from the Cancer Genome Atlas (TCGA), which captures the activity levels of thousands of genes in each tumor sample. Each patient in the dataset is labeled according to their tumor subtype.

The goal of this challenge is twofold:

(1) to build models that can accurately classify patients into the correct tumor subtype, and

(2) to identify a small set of genes that are most important for discriminating between the tumor subtypes.

Finding a reduced, interpretable number of relevant genes is especially valuable, as it can help researchers better understand the disease and may support the development of novel targeted treatments.

Required background:
Participants need to have basic knowledge of R programming language, data analysis and machine learning.

Problem 3: The Fragility of Learning: Adversarial Attacks on Machine Learning Models

Coordinator: Dr.Rohollah Garmanjani, NOVA School of Science and Technology, Portugal

Summary: Despite their impressive performance, modern machine learning models are known to be surprisingly fragile. A small, carefully designed perturbation added to an input, often imperceptible to a human observer, can cause a model to produce an incorrect and sometimes confidently wrong output. This phenomenon, known as an adversarial attack, has important implications for the deployment of machine learning systems in safety-critical applications such as autonomous vehicles, medical imaging, and fraud detection. In this challenge, participants will explore how adversarial examples can be constructed and why they are effective, by formulating the problem mathematically and analysing the geometry of the model's decision-making process. Participants will be provided with a trained classification model and a set of inputs, and the goal is to develop and compare strategies for crafting perturbations that cause misclassification while remaining as small as possible. The challenge naturally connects to several areas of mathematics and optimization, and can be approached from multiple directions depending on the tools and techniques participants choose to draw upon.

Required background:
A working knowledge of multivariable calculus, linear algebra, and the basics of constrained optimization will provide a sufficient mathematical foundation, while some familiarity with machine learning concepts and a programming language such as Python will be helpful for the computational aspects of the challenge.