Dr. Suzanne Lenhart focused on discrete time model.
She started with the population growth of Florida sandhill cranes. She stared with a exponential growth in discrete steps.
She emphasized that order of events is crucial in discrete models.
She modifed the model to x(n+1) = 0.97 (x(n) + 5)
She explained stable models and equilibrium.
She added logistical growth:
x(n+1) = x(n) + R x(n) (1-x(n)/K)
where R is intrinsic growth rate. Here a student explained the carrying capacity K.
She then found the equilibrium population X_bar.
She then asked students what are the unrealistic assumptions for logistical growth and discuss way to improve them. She suggested R and K can be time dependent.
She gave another growth model (Allee model?)
x(n+1) = X(n) + R x(n) (1-x(n)/K)(x(n)-a), 0<a<K.
where $a$ is the critical mass.
Lenhart gave a example of 'augmentation'.
x_k+1 = x_k + r x_k (1- x_k / K1) threatened population
y_k+1 = y_k + s y_k (1 - y_k /K2) captive breed population
She asked students to work in groups to modify the model to "move a percentage of pop $y$ to $x$ 'instantly' at each time step.
After about 5 minutes of discussion, Lenhart worked out solutions on the board. She used 'diagrams' to illustrate two ways to approach the problem: growth + then augment, augment + then growth.
After 50 minutes, Lenhart started another example, vehicular stopping distance.
This case asks students to estimate the distance in feet that car can travel in 2 second with speed at miles per hour. One mile = 5280 feet and one hour = 3600 seconds.
The final form for stop_distance consist of a reactive_distance (c1 * v) and a brake distance (c2 * v^2). The brake distance is reasoned out by using force, moment, and physic principles.
Lenhart concluded the lecture with introducing gender to a population model.
She then asked students what are the unrealistic assumptions for logistical growth and discuss way to improve them. She suggested R and K can be time dependent.
She gave another growth model (Allee model?)
x(n+1) = X(n) + R x(n) (1-x(n)/K)(x(n)-a), 0<a<K.
where $a$ is the critical mass.
Lenhart gave a example of 'augmentation'.
x_k+1 = x_k + r x_k (1- x_k / K1) threatened population
y_k+1 = y_k + s y_k (1 - y_k /K2) captive breed population
She asked students to work in groups to modify the model to "move a percentage of pop $y$ to $x$ 'instantly' at each time step.
After about 5 minutes of discussion, Lenhart worked out solutions on the board. She used 'diagrams' to illustrate two ways to approach the problem: growth + then augment, augment + then growth.
After 50 minutes, Lenhart started another example, vehicular stopping distance.
This case asks students to estimate the distance in feet that car can travel in 2 second with speed at miles per hour. One mile = 5280 feet and one hour = 3600 seconds.
The final form for stop_distance consist of a reactive_distance (c1 * v) and a brake distance (c2 * v^2). The brake distance is reasoned out by using force, moment, and physic principles.
Lenhart concluded the lecture with introducing gender to a population model.
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