Question

Assume that in a given region, ecologists have measured the present populations as follows: j 0 =200,s 0​ =45, and a

=725. Use the model stated in (c) to determine the population x (k) =[ j k​ ​ s ka k ]T for k=1,…,20. Do you think the spotted owl will become extinct? Give a convincing argument using not only your computations of the population vectors but also the results of

Answer 1

To determine the population x(k) for k=1,...,20, the logistic model is used with given initial values. The population vectors are computed using the model, and the computed population size is analyzed to determine if the spotted owl will become extinct.

Step-by-step explanation:

To determine the population x(k) for k=1,...,20, we will use the model provided in (c). The model is represented by x(k) = [j(k) s(k)a(k)]T, where j0 = 200, s0 = 45, and a = 725. We will use these initial values to calculate the population size for each time step from k=1 to k=20.

Using the logistic model, the population size at each time step can be calculated as follows:

1. Calculate j(k) = j(k-1)(1 + r(1 - (j(k-1) + s(k-1))/a)), where r is the growth rate.
2. Calculate s(k) = s(k-1)(1 - m), where m is the mortality rate.
3. Calculate a(k) = a(k-1)(1 - c), where c is the carrying capacity reduction rate.

Repeat steps 1-3 for each time step to calculate the population x(k) for k=1,...,20.

Based on the computed population vectors and the given results, we can determine if the spotted owl will become extinct. If the population size reaches zero or approaches zero over time, it indicates that the spotted owl population will become extinct.