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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 4-1-0.5 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....
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- Spring '09
- Derivative, Trigraph, Order theory, Monotonic function, Convex function