S-shaped growth

1. S-shaped growth Flowers

2. Logistic Equation dA / dt = r * A * (K-A) / K r net growth rate for non-stressed growth K carrying capacity Ao initial population (or Area) If A<<K then (K-A) / K ~1 And we have exponential growth

3. dA / dt = r * A * (K-A) / K = r*A - r*A*(A/K) If A<<K then (K-A) / K ~1 And we have exponential growth

4. r net growth rate for non-stressed growth K carrying capacity Ao – initial population (or Area)

7. S-shaped growth FlowersVerifys

8. Problems with Logistic Equation • Simple One example of s-shaped growth • The linear decrease in growth rate is often not true in real life. • The logistic equation fails to highlight the mechanisms responsible for limited growth. i.e. is it a decrease in birth rate or an increase in death rate.

9. Sales of Widgets Stella

10. Exponential growth • Exponential decay (goal seeking) to zero • Exponential decay (goal seeking) to a non-zero level • S-shaped growth

11. Exponential growth • Exponential decay (goal seeking) to zero • Exponential decay (goal seeking) to a non-zero level • S-shaped growth

12. Exponential growth Exponential decay (goal seeking) to zero Exponential decay (goal seeking) to a non-zero level S-shaped growth

13. Exponential growth Exponential decay (goal seeking) to zero Exponential decay (goal seeking) to a non-zero level S-shaped growth

14. Growth rate=rate*(1-stock/100) Exponential growth Exponential decay (goal seeking) to zero Exponential decay (goal seeking) to a non-zero level S-shaped growth

15. Growth rate=rate*(1-stock/100) Exponential growth Exponential decay (goal seeking) to zero Exponential decay (goal seeking) to a non-zero level S-shaped growth

16. Exponential growth Exponential decay (goal seeking) to zero Exponential decay (goal seeking) to a non-zero level S-shaped growth

17. Exponential growth Exponential decay (goal seeking) to zero Exponential decay (goal seeking) to a non-zero level S-shaped growth

18. The graphs above used growth rates of 5 or 10% and carrying capacities of 100 or 80. Which of the series have a carrying capacity of 80? Series 1 Series 2 Series 3 Series 4

19. The graphs above used growth rates of 5 or 10% and carrying capacities of 100 or 80. Which of the series have a carrying capacity of 80? Series 1 Series 2 Series 3 Series 4

20. Series 1 • Series 2 • Series 3 • Series 4 The graphs above used growth rates of 5 or 10% and carrying capacities of 100 or 80. Which of the series have a carrying capacity of 80 and an intrinsic growth rate of 5 %?

21. Series 1 • Series 2 • Series 3 • Series 4 The graphs above used growth rates of 5 or 10% and carrying capacities of 100 or 80. Which of the series have a carrying capacity of 80 and an intrinsic growth rate of 5 %?

22. The graphs above used growth rates of 5 or 10% and carrying capacities of 100 or 80. Which of the series have a carrying capacity of 100 and an initial value of 4? Series 1 Series 2 Series 3 Series 4

23. The graphs above used growth rates of 5 or 10% and carrying capacities of 100 or 80. Which of the series have a carrying capacity of 100 and an initial value of 4? Series 1 Series 2 Series 3 Series 4