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 (cont’d) 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 we’ve 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
 Spring '09
 MA
 Derivative, Trigraph, Order theory, Monotonic function, Convex function

Click to edit the document details