Message Board Assignment (Ch7) 8) Often when modelling a single population, a Logistic Growth model is used. A simple logistic growth model has the following form (with a and b positive): P'(t) = aP bP? Provided the bP? term is small, the aP term will overtake it and the population will grow exponentially; the growth rate is proportional to the population. As the population continues to grow, the bP? term will eventually approach the aP term and the growth rate will drop towards 0. This is what logistic growth is; it's a growth rate which grows exponentially for a while but then approaches some carrying capacity which the population cannot easily surpass. You will often see this growth rate in models where the population has some natural upper bound, but not always. Often (as in the standard Lotka-Volterra Predator- Prey population model) this term is excluded for simplicity as the carrying capacity is at such a high level that it is unlikely the populations will approach it. a) Draw the phase diagram for the system, and use it to find and classify all equilibrium values of the system as stable, unstable, or semistable. b) The Verhulst model is 2 more standard logistic growth model: dN N = (1-%) In this model, r represents the maximum population growth rate, N represents the population size, and K represents the carrying capacity (the number of organisms a population can sustain). This differential equation is a separable differential equation, but in order to solve it you will need to use partial fraction decomposition. Assuming that the initial population is given by ng, solve this differential equation. Write your answer as a function N (t) in terms of r, K, ny and t