Logistic Growth Model Explained (Population Modeling Example) скачать в хорошем качестве

Logistic Growth Model Explained (Population Modeling Example)

In this video, we study the logistic differential equation, a first-order separable differential equation used to model population growth with a carrying capacity. We begin with the logistic growth equation: dP/dt = rP(1 − P/K) and explain the meaning of each parameter: P(t) = population at time t r = intrinsic growth rate K = carrying capacity (the maximum population an environment can sustain indefinitely) P₀ = initial population You’ll learn how to: Recognize the logistic equation as a first-order separable ODE Separate variables and integrate Use partial fractions Apply an initial condition Solve for the explicit population function Interpret the long-term behavior of the solution Understand why the solution approaches the carrying capacity We work through a full deer population example: Initial population: 100 deer Carrying capacity: 1000 deer Growth rate: r = 0.6 per year The final solution shows how the population evolves over time and approaches the carrying capacity as t → ∞. This topic is essential in a first course in differential equations and is widely used in biology, ecology, and applied mathematics. 📘 Free guided notes are available here: https://www.understandthemath.com/ord... ODE Videos    • Ordinary Differential Equations Full Cours...      • How to Classify a Differential Equation (O...      • Slope Fields Explained Visually (Step-by-S...      • How to Analyze Autonomous Differential Equ...      • How to Solve Separable Differential Equati...      • Solving First-Order Linear ODEs with an In...      • How to Solve Exact Differential Equations      • How to Solve Homogeneous Differential Equa...      • How to Solve Bernoulli Differential Equations      • Modeling with First-Order Differential Equ...      • Euler’s Method Explained (Numerical Approx...      • Differential Equations Midterm Review | Fi...   Connect & Support TikTok:   / understandthemath   Instagram:   / understandthemath   Facebook: https://www.facebook.com/profile.php?... Support: https://buymeacoffee.com/understandth... Timestamps 00:00 Introduction 00:28 Logistic Model and Solution 02:31 Deer Population Example More About Understand The Math Understand The Math offers clear, accessible, and engaging mathematics instruction for students and lifelong learners. I'm Cheryl Hile, the founder of Understand The Math, with a Ph.D. in Engineering Science and Applied Mathematics and over 25 years of university teaching experience. My goal is to provide accurate, high-quality math instruction through step-by-step explanations and carefully worked-out examples. Each video includes free guided notes, linked in the description. These notes help students follow along, highlight key concepts, and practice problems, making review easier and more effective. #LogisticEquation #PopulationGrowth #DifferentialEquations #SeparableDifferentialEquations #MathematicalModeling #EngineeringMath #AppliedMathematics #UnderstandTheMath