This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 2 Population Dynamics (contd) Logistic Equation with Threshold. Consider the initial value problem dy dt = r y 1 y T , y (0) = y ; in terms of positive constants r and T . The solution is y ( t ) = y T y + ( T y ) e rt . As t the exponential e rt , so we expect y ( t ) 0. However is this really the case? We can determine this by considering the slope field of this differential equation. Consider the function f ( y ) = r y 1 y T . This has two critical points, namely y L = 0 and y L = T . When 0 < y < T we see that f ( y ) < 0 so that y = y ( t ) is a decreasing function. When T < y we see that f ( y ) > 0 so that y = y ( t ) is an increasing function. Figure 1 contains a plot of the slope field. We conclude that lim t y ( t ) = 0 whenever y < T. We call the parameter T the threshold for the differential equation. Whenever the initial position y is less than this threshold the solution y ( t ) 0. Figure 1. Slope Field for y = r y (1 y/T )0.5 0.5 1 1.5 2 2.5 3 3.5 410.5 0.5 1 1.5 2 We conclude the lecture to relating everything weve done so far with population dynamics. Say that we have a population of size P = P ( t ) at time t . We keep track of two properties: There is an environmental carrying capacity K . That is, if P > K then the population will decrease in size because the population has exceeded its resources....
View
Full
Document
This note was uploaded on 05/24/2011 for the course MA 36600 taught by Professor Ma during the Spring '09 term at Purdue UniversityWest Lafayette.
 Spring '09
 MA

Click to edit the document details