By Robeva R.S., et al.

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State p1: The graph of f(P) for P near p1 is shown in Figure 1-10. dP > 0; dt so P is increasing toward p1. On the other hand, if P is slightly larger dP < 0 and P decreases, again moving toward than p1, then f(P) < 0, so dt p1. In either case, if P is slightly different than p1, then P moves toward p1. We refer to a point such as p1 as a stable equilibrium point. Suppose P is slightly smaller than p1. Then f(P) > 0, which means State p2: The graph of f(P) for P near p2 is shown in Figure 1-11.

Pn so that xn is the fraction of the maximum K population the environment can sustain. With this notation, the Verhulst model takes the equivalent form: To demonstrate this, let xn ¼ xnþ1 À xn ¼ að1 À xn Þxn ; (1-26) and the carrying capacity of the model in Eq. (1-26) is equal to 1. Equation (1-26) represents the nondimensional form of the Verhulst model from Eq. (1-25). This is due to the fact that Pn and K are measured in the same Pn is nondimensional. This representation has units, so the quantity xn ¼ K 29 30 An Invitation to Biomathematics Chapter One several mathematical advantages.

P(0) < p1; p1 < P(0) < p2; p2 < P(0) < p3; P(0) > p3). dP ¼ f ðPÞ and the graph of f(P) is shown in Figure 1-15. dt Describe what happens if P is close to the equilibrium point p1. (c) Suppose So far, we have only considered questions related to population growth. The techniques described, however, are quite general and can be used to answer a variety of questions related to quantities that change with time, as the following examples illustrate. f(P ) = dP dt 0 f(P ) = 0 P p2 p1 FIGURE 1-13. The graph of f(P) ¼ dP/dt versus P for the logistic Eq.