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MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 2 Population Dynamics (cont’d) Logistic Equation with Threshold. Consider the initial value problem dy dt = ry ° 1 y T ± ,y (0) = y 0 ; in terms of positive constants r and T . The solution is y ( t )= y 0 T y 0 +( T y 0 ) e rt . As t →∞ the exponential e rt →∞ ,soweexpect y ( t ) 0. However is this really the case? We can determine this by considering the slope Feld of this di±erential equation. Consider the function f ( y )= ry ° 1 y T ± . This has two critical points, namely y L = 0 and y L = T .W h e n0 <y 0 <T we see that f ( y ) < 0 so that y = y ( t ) is a decreasing function. When T<y 0 we see that f ( y ) > 0 so that y = y ( t ) is an increasing function. ²igure 1 contains a plot of the slope Feld. We conclude that lim t →∞ y ( t )=0 whenever y 0
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Unformatted text preview: T the threshold for the dierential equation. Whenever the initial position y is less than this threshold the solution y ( t ) 0. Figure 1. Slope ield 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 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 &gt; K then the population will decrease in size because the population has exceeded its resources. 1...
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