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Unformatted text preview: 2.1: Population Models Recall The exponential growth of a population when k > 0 and we assume that the death and birth rates are constant is given by dP dt = kP, P H L = P This has solution given by P H t L = P e kt . DSolve @8 P' @ t D 0.2 P @ t D , P @ D 20 < , P @ t D , t D theDE = 0.2 P; ivp = 88 0, 5 < , 8 0, 30 < , 8 0, 1 << ; Show A VectorPlot @8 1, theDE < , 8 t, 1, 10 < , 8 P, 1, 50 < , FrameLabel fi 8 t, P < , Axes fi True, VectorScale fi 8 Tiny, Tiny, None < , VectorStyle fi Gray, StreamScale fi Full, StreamStyle fi 8 Blue, Thick, "Line" < , StreamPoints fi 8 ivp < , PlotLabel fi "Direction Field", Epilog fi 8 PointSize @ Large D , Point @ ivp D< D , Plot A 20 ª 0.2` t , 8 t, 1, 10 < , PlotStyle fi 8 Red, Thick <E E 99 P @ t D fi 20 ª 0.2 t == 2 2 4 6 8 10 10 20 30 40 50 t P Direction Field 2 Chapter2Section1.nb General Population Dynamics Assume that Β H t L = the number of births per unit population per unit time at time t . Δ H t L = the number of deaths per unit population per unit time at time t . Then, from time t to time t +D t, we have Births » Β H t L P H t L D t Deaths » Δ H t L P H t L D t Hence, we may conclude that the change in population time...
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This note was uploaded on 01/13/2012 for the course MATH 306 taught by Professor Keithemmert during the Spring '11 term at Tarleton.
 Spring '11
 KeithEmmert

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