The Modeling Process

Mathematical Modeling · Axiom Academy

Discover the systematic journey from real-world problems to mathematical solutions and back again. What is Mathematical Modeling? Mathematical modeling is the art and science of translating real-world problems into mathematical language. It is not a one-way street, but rather a cycle of refinement and improvement. Step 1: Problem Identification and Formulation Every model begins with a question. The first step is to clearly define what we want to understand or predict. Choose a real-world problem to model: The Question: How will a population of bacteria grow over time in a petri dish with limited nutrients? Key Variables: Population size (P), Time (t), Growth rate (r), Carrying capacity (K) What We Want: A function P(t) that predicts population at any time. Step 2: Making Simplifying Assumptions Real systems are complex. To create a tractable mathematical model, we must make assumptions that simplify reality while preserving essential behavior. Select assumptions for your model: Step 3: Building the Mathematical Model Now we translate our assumptions into mathematical equations. This is where the real-world problem becomes a mathematical problem. Verbal Model: "The rate of change of population is proportional to the current population times the fraction of remaining capacity." Mathematical Model (Logistic Equation): With our mathematical model in hand, we now solve it to get predictions. This might involve calculus, algebra, or numerical methods.

This is the written version of the interactive lesson above. See the full Mathematical Modeling course.