Topics:

• Modeling continuous systems using Ordinary Differential Equations (ODE).
• Application: Volterra-Lotka predator-prey model. Wikipedia info
• Newton's Second Law as a second-order differential equation. Wikipedia info
• Application: Linear Harmonic oscillator. Wikipedia info
• Converting a second-order equation to a pair of first-order equations.
• Lecture notes:   Lecture 1   Lecture 2  

Assignment:

1. Logistic model with Harvesting: Supposed that a fish population grows according to Logistic differential equation but also harvested at a constant rate:
   x' = r x1 (1 - x1/k) - H
   where r is the growth rate, k is the carrying capacity and H is the harvesting rate. In the Xi vs Time view, first set r=1, k=2, and H = 0, and plot about a dozen inital conditions and get the familiar picture.
   Now increase the harvesting rate from 0 to 0.75 with increments of 0.05 and create a gallery of 16 pictures. Note that as the harvesting is increased, some initial conditions will start becoming negative, and eventually all your solutions might become negative. This is extinction cuased by overfishing. Explore the values of H and initial conditions when extinction happens. Note that for a given H, some small initial conditions might go extinct while larger initial conditions might survive. However, for large H, all initial conditions cannot avoid extinction. Please report your findings from a biological viewpoint.

2. Another predator-prey model: Recall that in class we considered the predator-prey system
   x1' = ax1 - bx1 x2
   x2' = -cx2 + dx1 x2
   where we assume that in the absence of the other, each species grows exponentially. We observed that both populations oscillate for almost all initial conditions.
   Now, let us modify this model by assuming that prey (x1) grows according to the logistic model, leaving other assumptions the same as before. Then the model becomes:
   x1' = rx1(1 - x1/k) - bx1 x2
   x2' = -cx2 + dx1 x2
   For simplicity, let us fix r=1, k=1, b=1, c=1, and d=1. In the Phaser portrait view, plot many initial conditions and determine what happens to prey and predator.
   Now, leaving the other parameters fixed, change c from 0 to 2. Describe what happens to the populations.
   Next, vary d from 0 to 2 while keeping the other parameters fixed at 1. Describe your findings.

3. Harmonic oscillator and Euler: Consider the harmonic oscillator x''=(-k/m)x. Convert this 2nd-order system to a pair of first order differential equations. Using k = 2, m=1, the initial position=3 and initial velocity= 0, and step size h = 0.2, compute BY HAND two steps of Euler's algorithm. Check your numbers on Phaser.