Unit 4 Lesson 1 - Population_Growth_Modeling.pptx

This is a powerpoint presentation on Population Growth Model. It describes how a population's size changes over time. The two simplest are exponential growth, which assumes unlimited resources and results in a J-shaped curve, and logistic growth, which accounts for limited resources and produces an S-shaped curve as the population levels off at the carrying capacity

1. Modeling Population Growth Exponential and Logistic Models in Mathematical Modeling

2. Lesson Objectives - Understand the assumptions behind exponential and logistic growth - Derive differential equations from real-world population situations - Solve and interpret solutions to exponential and logistic models - Apply models to predict future population behavior

3. Time (t in hours): 0 1 2 3 4 Population (P): 100 150 225 337 506

4. First Difference ( ΔP ) : 50 75 112 169 Time (t in hours): 0 1 2 3 4 Population (P): 100 150 225 337 506

5. First Difference ( ΔP ) : 50 75 112 169 Time (t in hours): 0 1 2 3 4 Population (P): 100 150 225 337 506 Difference/Original (ΔP / P): 0.5 0.5 0.498 0.501     - the rate of growth of P is proportional to the population P itself.

6. Exponential Growth Symbol : dP /dt = r P Where: - r is the intrinsic grow th rate - P is the population - t is time - The rate of population growth is proportional to the current population size.

7. Exponential Growth Model Differential Equation: dP/dt = kP Solution: P(t) = P₀e^{kt} Where: - P(t) is the population at time t - P₀ is the initial population - k is the growth rate

8. Examples: Exponential Growth Bacterial Culture (Lab Context) A lab starts with 500 bacteria. The culture doubles every 3 hours. a) Write the exponential growth model. b) How many bacteria will be present after 12 hours? c) After how many hours will the population reach 16,000?

9. Limitations of Exponential Growth Exponential growth assumes unlimited resources and space. But in reality: - Food, space, and resources are limited - Overcrowding slows growth A more realistic model includes carrying capacity:

10. Logistic Growth Model

11. Logistic Growth Theory (Observation): Growth rate is proportional not just to the current population but also to how far the population is from its maximum size."

12. Logistic Growth Model Differential Equation: dP /dt = rP (1 - P/K) Where: - r is the intrinsic growth rate - K is the carrying capacity 1-P/K is the “available room” for growth - Population grows rapidly at first, then slows as it approaches K “Growth rate is proportional not just to the current population but also to how far the population is from its maximum size”

13. Logistic Growth Model Differential Equati on : dP /dt = rP (1 - P/K) “Growth rate is proportional not just to the current population but also to how far the population is from its maximum size” General Solution:  

14. Examples: Logistic Growth Urban Population (Real-World Context) A small city has an initial population of 50,000 and a carrying capacity of 200,000. The growth rate is estimated to be 0.1 per year. a) Write the logistic growth model. b) Estimate the population after 5 years. c) After how many years will the population triple its size?

15. Exponential vs Logistic Growth Exponential: - Unlimited growth - No resource constraints Logistic: - Growth slows as resources become limited - Includes carrying capacity

16. Worked Example Example: A population of 100 rabbits doubles every 5 months. 1. Write an exponential model. 2. Predict the population after 1 year. Bonus: Modify with a logistic model assuming a max capacity of 1000.

17. Summary - Exponential model fits early-stage growth - Logistic model adds realism with limiting factors - Both models are tools to understand and predict population trends

18. Seatwork Fish in a Pond (Ecology Context) A fish population in an isolated pond grows exponentially. Initially, there are 100 fish. After 5 months, the population grows to 160. a) Determine the growth rate. b) Predict the population after 10 months. Bacterial Growth with Limited Nutrients (Lab Context) A Petri dish starts with 100 bacteria. The maximum population the dish can support is 10,000. The growth rate is 0.6 per hour. a) Write the logistic model. b) Estimate the population after 4 hours. c) At what population size does the growth rate start to slow significantly?