The logistic equation below is often used for modeling animal populations. dP kP(M-P) (1) The independent variable is t, and the units of t must be specified implicitly, such as months or years or some other duration. The population P (here, the number of animals) is the dependent variable. As such, the logistic equation is an autonomous differential equation, meaning that the independent variable t does not appear explicitly in the equation. If the birth rate in the animal population being modeled is o and the death rate is 80 and the birth rate slows in proportion to the size of the population (modeled by the additional term ), then the rate of change of the population with time is described by: P(o - BP-80) (2) This kind of modulation of the birth rate is common is environments with limited resources. If the units of 0, B1, and 60 are all (months), then we are implicitly specifying that the units of t are months. Letting k = B1 and M = Boo, then eq. (1) results from eq. (2). B1 Given an animal population believed to be characterized by the following logistic equation coefficients, Bo = 0.12 / month B = 0.0005/month S = 0.08/month and an initial population (at t = 0) of 50 animals (that is, P(0) = 50), 1. Separate the variables of eq. (1) and find the general solution (i.e., one that includes the constant C) of the differential equation. 2. Rename the integration constant Po (instead of C) and rewrite your solution, applying the initial condition (P(O)) specified above. 3. How many months from t = 0 are expected for the animal population to reach 60? Round your answer to the nearest tenth of a month. 4. In two years, what is the animal population expected to be? Round your answer to the nearest integer. (This is an example of modeling populations with a continuous dependent variable when the number of animals P can only be a non-negative integer.)