Logistic Growth with MATLAB
Summary
Purpose: Many populations grow in a manner that is important to the existing population. That makes sense. The larger the population, the larger the growth rate. This type of unconstrained growth can be modeled with the differential equation, the exponential growth model. Not all population growth is unconstrained. There are many situations in which populations increase up to a point and then level off. This leveling off usually occurs because of limited food, space, or other resources. Functions can be used to model growth when you must account for a decreasing growth rate due to overcrowding or when there is an upper bound on the size of the population, which is called the Carrying Capacity.
CollapseLearning Goals
- To model growth with logistic equations, interpreting parameters such as growth rate and carrying capacity.
- To use MATLAB to simulate differential equations, observing population dynamics.
Context for Use
Logistic Model in Population and Resource Management With MATLAB
Topic: Logistic Differential Equation and Real-Life Modeling
Description and Teaching Materials
Class Duration: 45 minutes
Materials: MATLAB, logistic model diagrams, MATLAB ODE solver scripts
Lesson Outline:
1. Introduction to Logistic Growth (5-10 minutes)
- Present the logistic growth model and its applications in ecology, economics, and resource management.
- Highlight White Station High School's success in applying this model in the M3 Mathematical Modeling Competition.
2. Mathematical Foundation of the Logistic Model (10 minutes)
- Review and solve the logistic differential equation $\frac{dP}{dt} = rP(1 - \frac{P}{K})$
with Separation of Variables
- Discuss how parameters r (growth rate) and K (carrying capacity) affect the model.
3. Implementing the Logistic Model in MATLAB (20 minutes)
- Step 1: Solve the differential equation using MATLAB's ODE solver.
r = 0.5; K = 100;
logistic = @(t, P) r*P*(1 - P/K);
[t, P] = ode45(logistic, [0 20], 10);
plot(t, P)
title('Logistic Model of Population Growth')
- Step 2: Analyze the logistic curve, exploring carrying capacity and its implications in modeling.
4. Connecting Optimization to Growth Models (10 minutes)
- Show the optimization process in logistic growth: maximizing growth rate at P = \frac{K}{2}
syms P
logistic_eq = r*P*(1 - P/K);
dP = diff(logistic_eq, P);
solve(dP == 0, P)
- Discuss applications in various fields.
5. Q&A and Summary (5 minutes)
Assessment
Homework: Exercises from Section 9.4 (#1,3,5,7,17,23) on logistic models