Modeling Logistic Growth and Extinction

Sheldon P. Gordongordonsp@farmingdale.edu

The Logistic Model

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Which Logistic Model?• Continuous:

P’ = aP - bP2

b << a, L = a/b = Maximum Sustainable Population

• Discrete:Pn = aPn - bPn

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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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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

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= 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

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K L K KL LP

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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