5. In the next class, we will be returning to population models, now involving interaction between two populations. In order to understand these more complicated models, it '3 important to have a rm grasp of population models for a single population in isolation. This problem recaps the dierent population models we've looked at so far. (a) When a population lives in an environment with abundant resources and little strife, the rate at which the population grows is modeled as being proportional to the population size. i. Suppose a population of bees behaves in this way. Write a differential equation for B(t), the population of bees at time t. ii. Suppose that there are 900 bees at time t = 0 and 1200 bees at time t = 1. How many bees are there at time t? (b) When a population loses its food source for some reason, we often model the population as dying off at a rate proportional to its size. Write a differential equation modeling this situation; please explain the meaning of any variables you use in your differential equation. What is the general solution of this differential equation? (c) Give a brief explanation of the logistic equation (imagine that you are explaining it to a friend who has forgotten it completely). What phenomenon does the logistic equation model? What does the term carrying capacity mean? What does a logistic equation look like, and how does the carrying capacity gure in the logistic equation