6.1.3. Population Growth Models. The simplest model for the population growth of an organism is N 0 = rN where N ( t ) is the population at time t and r > 0 is the growth rate. This model predicts exponential population growth N ( t ) = N 0 e rt , where N 0 = N (0). We studied this model in § 1.5 . Among other things, this model assumes that the organisms have unlimited food supply. This assumption implies that the per capita growth N 0 /N = r is constant. A more realistic model assumes that the per capita growth decreases linearly with N , starting with a positive value, r , and going down to zero for a critical population N = K > 0. So when we consider the per capita growth N 0 /N as a function of N , it must be given by the formula N 0 /N = - ( r/K ) N + r . This equation, when thought as a differential equation for N is called the logistic equation model for population growth. Definition 6.1.3. The logistic equation describes the organisms population function N in time as the solution of the autonomous differential equation N 0 = rN 1 - N K , where the initial growth rate constant r and the carrying capacity constant K are positive. We now use the graphical method to carry out a stability analysis of the logistic popu- lation growth model. Later on we find the explicit solution of the differential equation. We can then compare the two approaches to study the solutions of the model. Example 6.1.6 : Sketch a qualitative graph of solutions for different initial data conditions y (0) = y 0 to the logistic equation below, where r and K are given positive constants, y 0 = ry 1 - y K .
230 G. NAGY – ODE april 18, 2014 Solution: The logistic differential equation for population growth can be written y 0 = f ( y ), where function f is the polynomial f ( y ) = ry 1 - y K . The first step in the graphical ap- proach is to graph the function f . The result is in Fig. 39 . The second step is to identify all crit- ical points of the equation. The crit- ical points are the zeros of the func- tion f . In this case, f ( y ) = 0 implies y 0 = 0 , y 1 = K. The third step is to find out whether the critical points are stable or un- stable. Where function f is posi- tive, a solution will be increasing, and where function f is negative a solution will be decreasing. These regions are bounded by the critical points. Now, in an interval where f > 0 write a right arrow, and in the intervals where f < 0 write a left arrow, as shown in Fig. 40 . The fourth step is to find the re- gions where the curvature of a solu- tion is concave up or concave down. That information is given by y 00 . But the differential equation relates y 00 to f ( y ) and f 0 ( y ). We have shown in Example 6.1.4 that the chain rule and the differential equation imply, y 00 = f 0 ( y ) f ( y ) So the regions where f ( y ) f 0 ( y ) > 0 a solution is concave up (CU), and the regions where f ( y ) f 0 ( y ) < 0 a solution is concave down (CD). The result is in Fig. 41 .
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