P5, 40pts. Revisiting logistic growth. This problem considers the case where the logistic equation has a slowly varying carrying capacity k(et), where k is a smooth, positive function, but otherwise arbitrary. The equation is 3-(1-844)) y = ry k(et) where y(0) = a and r, a are positive constants. Obtain a first-term approximation of the solution valid for large t. For +10, attempt to determine the second term in the approximation.

What is the carrying capacity for a population modeled by the logistic equation P(t)=250,0001+499e0.45t ? initial population for the model?

Suppose that a particular plot of land can sustain 500 deer and that the population of this particular species of deer can be modeled according to the logistic model as dPdt=0.2(1P500)P. Each year, a proportion of the herd deer is sold to petting zoos. a. Find the function that gives the equilibrium population for various proportions. b. Determine the maximum number of deer that should be sold to petting zoos each year. Hint: Find the maximum sustainable harvest

Plant Growth Researchers have found that the probability P that a plant will grow to radius R can be described by the differential equation dPdR=4DRP2 where D is the density of the plants in an area. Source: Ecology. Given the initial condition P(0)=1, find a formula for P in term of R.