Beyond Borders: Exploring Existence Theorems in Free Boundary Population Models

With the introduction of mathematics into biological research, patterns arising from biological problems can be effectively studied. The complex interactions involving biological systems and invading pathogens can be described using mathematical frameworks of partial differential equations (PDEs). More specifically, the spread of new or invasive species can be set up as a population model with a freely moving boundary. Proving the existence of solutions to this model helps us scientists to have a solid understanding of how the population works and predict how it changes in the future. Generally speaking, the mathematical modeling of ecological problems is difficult. In this work, we propose a different approach to the understanding of the spreading of species. We have introduced a special case of the well-known Stefan condition as the equation governing the free boundary. Our study therefore seeks to develop rigorous mathematical frameworks underpinning existence theorems and see the role of free boundaries in shaping the behavior of populations. We have employed energy methods with the goal of building a general framework that will be highly portable for use in other, related problems in the modeling of infectious diseases. Invading fungal pathogens like Candida albicans cause over 150 million mucosal infections and nearly 200,000 deaths per year due to invasive and disseminated disease in susceptible populations. This work represents significant advancements in resolving a long-standing mathematical problem and addressing one of the most important health-related issues nations are grappling with.