Wednesday, June 12, 2013

Notes, discrete time modeling, NimbioS REU course, day 1

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. 





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