2010-04-07 - MAT 1332 CALCULUS FOR LIFE SCIENCES JING LI...

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Unformatted text preview: MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI Contents 1. Review: Systems of differential equations I: linear case, phase plane 1 2. Systems of differential equations I: examples, linear stability of zeros 1 2.1. Examples 1 2.2. Linear stability analysis 8 2.3. summary 11 1. Review: Systems of differential equations I: linear case, phase plane • General and particular solutions of linear system of differential equations • Stability of zero • Phase plane 2. Systems of differential equations I: examples, linear stability of zeros 2.1. Examples. Let’s look at more examples of linear systems of differential equations in this section. Linear system: Romeo & Juliet “Every love affair has its ups and downs over time, ... so it can be modeled by differential equations.” When one thinks of love, the classic tale of Romeo and Juliet comes to mind. So let’s took at the example of Romeo & Juliet. We begin with the general model: dx dt = ax + by dy dt = cx + dy or d dt • x y ‚ = • a b c d ‚• x y ‚ = A • x y ‚ with A = • a b c d ‚ where a,b,c,d are constants, and x ( t ) = Romeo’s love/hate for Juliet at time t , (the “temperature” of Romeo’s feelings for Juliet) y ( t ) = Juliet’s love/hate for Romeo at time t , (the “temperature” of Juliet’s feelings for Romeo) A negative temperature could be interpreted as hostility, a zero temperature as indifference, and a positive temperature as attraction. • x and/or y > 0 love exists • x and/or y < 0 love does not exist • x and/or y = 0 indifference Date : 2010-04-07. 1 2 JING LI Example 1. Cautious love dx dt =- x + 1 2 y dy dt = 1 2 x- y with A = •- 1 1 2 1 2- 1 ‚ Solution: MAT 1332: CALCULUS FOR LIFE SCIENCES 3 Example 1 –2 –1 1 2 y –2 –1 1 2 x Figure 1. Example 1: the top line is the solution to the initial condition...
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This note was uploaded on 03/19/2011 for the course MAT 1332 taught by Professor Munteanu during the Spring '07 term at University of Ottawa.

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2010-04-07 - MAT 1332 CALCULUS FOR LIFE SCIENCES JING LI...

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