Modeling Logistic Growth and Extinction Sheldon P. Gordon [email protected].
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Transcript of Modeling Logistic Growth and Extinction Sheldon P. Gordon [email protected].
Modeling Logistic Growth and Extinction
Sheldon P. [email protected]
The Logistic Model
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n
P n
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Which Logistic Model?• Continuous:
P’ = aP - bP2
b << a, L = a/b = Maximum Sustainable Population
• Discrete:Pn = aPn - bPn
2
b << a, L = a/b = Maximum Sustainable Population
Comparing the Models
Using a = 0.20, b = 0.0020, and P0 = 1
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t, n
P
Comparing the Models
Using a = 0.20, b = 0.0020, and P0 = 20
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Difference in the Models
Using a = 0.20, b = 0.0020, and P0 = 20
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Different Regions of the Plane
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?
Extending the Logistic Model
Extending the Logistic Model
The logistic model is:Pn = aPn - bPn
2
= b Pn (a/b – Pn )
= b Pn (L – Pn )
This suggests introducing an extra factor corresponding to the extra equilibrium level at P = K:
Pn = Pn (L – Pn ) (K – Pn )
orPn = - Pn (L – Pn ) (K – Pn )
This is known as the Logistic Model with Allee Effect.
Logistic Model with Allee Effect
Pn = - Pn (L – Pn ) (K – Pn )
A Further Extension
The logistic model is:Pn = b Pn (L – Pn )
The Logistic Model with Allee Effect is: Pn = - Pn (L – Pn ) (K – Pn )
To account for the appropriate signs, we use a quartic polynomial model:
Pn = - Pn2 (L – Pn ) (K – Pn )
Logistic Model with Extinction
Pn = - Pn2 (L – Pn ) (K – Pn )
Locating the Inflection PointsThe inflection points for the quartic model:
Pn = - Pn2 (L – Pn ) (K – Pn )
occur when Pn is maximal or minimal, which is at those points where the derivative is 0.
This leads to:-α P [4P2 - 3(K + L)P + 2KL] = 0.
1. Concavity changes about P = 0 axis.2. Other solutions from quadratic formula:
2 23( ) 9 14 9
8
K L K KL LP
.
Some Solution Curves
Using = 10-10, K = 200, L = 2000, with P0 = 500 and P0 = 1200
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Some Solution Curves
Using = 10-10, K = 200, L = 2000, P0 = 1600 .Note: Inflection point at height of about 1518.
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Some Solution Curves
Now P0 = 180, P0 = 75, and P0 = -50.
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Estimating the ParametersFor the logistic model:
Pn = b Pn (L – Pn )Perform quadratic regression on Pn vs. Pn
For the Logistic Model with Allee Effect: Pn = - Pn (L – Pn ) (K – Pn )
Perform cubic regression on Pn vs. Pn
For the quartic model:Pn = - Pn
2 (L – Pn ) (K – Pn )Perform quartic regression on Pn vs. Pn