By Robeva R.S., et al.

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002(400 – P)P À52 versus P is shown in Figure 1-18 (dashed line). The equilibrium states are approximately 82 and 318 cattle. (b) Classify the new equilibrium states as stable or unstable. f(P) = dP dt 52 0 P 82 318 400 FIGURE 1-18. Models showing the logistic curves with and without harvesting. 002(400 – P)P – 52 (dashed line) versus P. 002(400 – P)P by 52 units. 27 28 An Invitation to Biomathematics Chapter One (c) What happens to the population described by Eq. (1-23) if: (i) P(t0) ¼ 50, (ii) P(t0) ¼ 150, and (iii) P(t0) ¼ 400?

EXERCISE 1-11 Show that the equilibrium states for Verhulst model [Eq. (1-25)] are P ¼ 0 and P ¼ K. In Section VI, we proved that for any value of a > 0 and any nonzero initial population size P(0), the logistic model (1-24) exhibits convergence for t ! 1 to its equilibrium state P ¼ K. For P(0) < K, the population size P(t) is continuously increasing to K when t ! 1 while if P(0) > K, the population size P(t) is continuously decreasing to K when t ! 1 (Figure 1-8). The Verhulst model offers cases of considerably more complex long-term behavior—the system could converge to an equilibrium state through oscillations, exhibit lack of convergence because of periodic oscillatory behavior, or be driven to chaos.

The value C is the dosage, and the 2 in CeÀ2r comes from the fact that it has been 2 hours from the last administered dose. EXERCISE 1-18 We give a dosage of C mg/ml at 2-hour intervals. The elimination constant is r hoursÀ1. There are six doses given. Give an expression for the concentration 30 minutes after the last dose. In general, the drug’s concentration follows the pattern shown in Figure 1-25. If the drug is administered in dosages C at intervals of length T, then at the end of the n-th period the concentration is: Rn ¼ C½ðeÀTr Þn þ ðeÀTr ÞnÀ1 þ ðeÀTr ÞnÀ2 þ .