Read e-book online An invitation to biomathematics PDF

By Robeva R.S., et al.

ISBN-10: 0120887711

ISBN-13: 9780120887712

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When axn > 1, the values xn and xnþ1 calculated from Eq. (1-26) will be on opposite sides of the level 1, causing oscillatory behavior. We emphasize again that the oscillation behavior here is not possible for the continuous logistic growth model. For the Verhulst model, the oscillations are caused by the lag effect described above and not observed in the logistic model (1-24). As the next section will show, logistic equations are capable of generating oscillations when explicit delay is introduced.

We let Sn denote the sum of the first n terms of such a series, so: Sn ¼ a þ ab þ ab2 þ ab3 þ . . þ ab nÀ1 : (1-37) Then: bSn ¼ ab þ ab2 þ ab3 þ . . 1 b n ¼ 0: Applying Eq. (1-40) with a ¼ Ce–Tr and b ¼ e–Tr, the residual concentration after n doses from Eq. (1-34) can be written as: Rn ¼ Cð½eÀTr Šn þ ½eÀTr ŠnÀ1 þ ½eÀTr ŠnÀ2 þ . . þ eÀTr Þ 1 À ðeÀTr Þn : ¼ CeÀTr ð½eÀTr ŠnÀ1 þ ½eÀTr ŠnÀ2 þ . . 1Þ ¼ CeÀTr 1 À eÀTr (1-41) What happens to the residual amounts as the number of doses increases? It appears from Figure 1-25 that, after several doses, the residual values stabilize around a value slightly higher than 8 mg/ml.

3 0. 2 0 C 10 20 30 40 50 Time FIGURE 1-19. Solutions of Eq. 8 and different values of a. 7. 2 D 0 10 20 30 40 50 Time FIGURE 1-19 Cont’d. Population 32 xn | xn+1 – xn | | 1 – xn | 1 xn+1 0 n Time n+1 FIGURE 1-20. Source of oscillatory behavior. When axn > 1, the values xn and xnþ1 calculated from Eq. (1-26) will be on opposite sides of the level 1, causing oscillatory behavior. We emphasize again that the oscillation behavior here is not possible for the continuous logistic growth model. For the Verhulst model, the oscillations are caused by the lag effect described above and not observed in the logistic model (1-24).

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An invitation to biomathematics by Robeva R.S., et al.


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