By Helmut Strasser

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**Example text**

Thus, in this case, xn and xnþ1 will always be on opposite sides of the maximum level 1, causing oscillatory behavior. Table 1-5 presents several values from Eq. (1-26) of xn, for n ! 8, and for different values of a to demonstrate the dependence of the long-term behavior of the process upon the value of a. 5, the system oscillates above and below 1 before settling into the equilibrium state of 1. 10, the system oscillates between two values. As a increases further, the system will oscillate among four values.

A roller coaster model of equilibrium points. Positions 2 and 3 represent stable equilibria while positions 1 and 4 represent unstable equilibria. com. ) 21 Chapter One An Invitation to Biomathematics 22 EXERCISE 1-8 dP P ¼a 1À P¼ (a) For the logistic model (1-12), we had dt K a 0 0 ðK À PÞP ¼ a ðK À PÞP; where a ¼ a/K. So in the example K discussed above, f(P) ¼ a0 (K – P)P. The graph of f(P) is shown in Figure 1-13. Classify the equilibrium states for the logistic model as stable or unstable.

When we graph P versus t (population vs. time), these values divide the graph into two regions— values of P larger than the carrying capacity K and values of P smaller than K (see Figure 1-8). Suppose we begin a new culture with a very dP small quantity of yeast. Because the population is small, P(t) < K, then dt is positive, and P(t) will increase (see the curve labeled P1). This does not give the complete information that the solution of the logistic curve gave, but it gives valuable information for very little effort.