1. Modeling Logistic Growth and Extinction Sheldon P
Modeling Logistic Growth and Extinction Sheldon P. Gordon

2. The Logistic Model

3. Which Logistic Model? Continuous: P’ = aP - bP2 b << a,
L = a/b = Maximum Sustainable Population
Discrete:
DPn = aPn - bPn2

4. Comparing the Models
Using a = 0.20, b = , and P0 = 1

5. Comparing the Models
Using a = 0.20, b = , and P0 = 20

6. Difference in the Models
Using a = 0.20, b = , and P0 = 20

7. Different Regions of the Plane

8. Biological Principle
Not only is there a Maximum Sustainable Population level L, there is also typically a Minimum Sustainable Population level K.
Whenever a population falls below this level, it tends to die out and become extinct.
How do we model this?

9. Extending the Logistic Model

10. Extending the Logistic Model
The logistic model is:
DPn = aPn - bPn2
= b Pn (a/b – Pn )
= b Pn (L – Pn )
This suggests introducing an extra factor corresponding to the extra equilibrium level at P = K:
DPn = a Pn (L – Pn ) (K – Pn ) or
DPn = -a Pn (L – Pn ) (K – Pn )
This is known as the Logistic Model with Allee Effect.

11. Logistic Model with Allee Effect
DPn = -a Pn (L – Pn ) (K – Pn )

12. A Further Extension The logistic model is: DPn = b Pn (L – Pn )
The Logistic Model with Allee Effect is:
DPn = -a Pn (L – Pn ) (K – Pn )
To account for the appropriate signs, we use a quartic polynomial model:
DPn = -a Pn2 (L – Pn ) (K – Pn )

13. Logistic Model with Extinction
DPn = -a Pn2 (L – Pn ) (K – Pn )

14. Locating the Inflection Points
The inflection points for the quartic model:
DPn = -a Pn2 (L – Pn ) (K – Pn )
occur when DPn is maximal or minimal, which is at those points where the derivative is 0.
This leads to:
-α P [4P (K + L)P + 2KL] = 0.
Concavity changes about P = 0 axis.
Other solutions from quadratic formula: .

15. Some Solution Curves Using a = 10-10, K = 200, L = 2000,
with P0 = 500 and P0 = 1200

16. Some Solution Curves Using a = 10-10, K = 200, L = 2000, P0 = 1600 .
Note: Inflection point at height of about 1518.

17. Some Solution Curves
Now P0 = 180, P0 = 75, and P0 = -50.

18. Estimating the Parameters
For the logistic model:
DPn = b Pn (L – Pn )
Perform quadratic regression on DPn vs. Pn
For the Logistic Model with Allee Effect:
DPn = -a Pn (L – Pn ) (K – Pn )
Perform cubic regression on DPn vs. Pn
For the quartic model:
DPn = -a Pn2 (L – Pn ) (K – Pn )
Perform quartic regression on DPn vs. Pn