Suppose that we wish to create a mathematical model describing the size of a population of rabbits..

Suppose that we wish to create a mathematical model describing the size of a population of rabbits living in the open fields behind my house. Suppose also that empirical evidence suggests that a small initial population will increase by approximately 10% each year. Let’s assume that we start with Xo rabbits and denote the population in the nth year by Xn. We wish to determine how many rabbits there will be in n years, or equivalently, we wish to determine the value of Xn . Clearly, Therefore, the population of rabbits after n years is determined by applying the function p(x) = l.lx to Xo n

View complete question »Suppose that we wish to create a mathematical model describing the size of a population of rabbits living in the open fields behind my house. Suppose also that empirical evidence suggests that a small initial population will increase by approximately 10% each year. Let’s assume that we start with Xo rabbits and denote the population in the nth year by Xn. We wish to determine how many rabbits there will be in n years, or equivalently, we wish to determine the value of Xn . Clearly, Therefore, the population of rabbits after n years is determined by applying the function p(x) = l.lx to Xo n times. A simple calculation shows that pn(x) = (l.l)n x . Thus, if we start with more than one rabbit, the population grows and keeps growing forever. For example, an initial population of 8 rabbits swells to a population of (l.1)108, which is about 21 rabbits in 10 years. Looking further into the future, we see that according to this model, the same 8 rabbits swell to a population of approximately 54 in 20 years, 939 in 50 years, and 110,245 in 100 years. Given the small size of the field behind my house, this last estimate is clearly too large. While the dynamics of this model are easy to understand—each iterate grows by 10% over the previous iterate–the model’s long-term predictive power is limited since it predicts that the population continues to grow without bound. 0 In general, models that iterate a function of the form p( x) = rx are called exponent ial models. As we demonstrated, exponential models have limited predictive power in population problems since as time passes the predicted population becomes so large that it is no longer realistic. A more sophisticated model for population, which takes into account the limits on growth, uses the logistic function, h(x) = rx(l – x).Models using the logistic function assume there is an absolute limit on the size of the population and designate the actual size of the population as a fraction of the limit. Hence, the size of any population is denoted by a number in the interval [0 , 1] . For example, .25 indicates that the population is 25% of the limit population. If Xo is the population in the first time period, then the population in the next time period is h(xo) = rXo(l – xo). The factor (1 – x) distinguishes models using the logistic function from exponential models. As x approaches 1, this factor approaches o. Thus, as x becomes larger the population grows at a slower rate. If x is large enough, then the population declines. In the following example, we apply this model to the problem studied in Example 1.1.

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