Question

DIFFERENTIAL EQUATIONS PROBLEM

Consider a population model that is a generalization of the exponential model for population growth (dy/dt = ky). In the new model, the constant growth rate k is replaced by a growth rate r(1 - y/K). Note that the growth rate decreases linearly as the population increases. We then obtain the logistic growth model for population growth given by dy/dt = r(1-y/K)y. Here K is the max sustainable size of the population and is called the carrying capacity and r>0.

i) Find all equilibrium solutions.

ii) Sketch the phase line.

iii) Determine whether each equilibrium is asymptotically stable, semistable, or unstable.

iv) Based on this information, sketch solution curves in the t-y plane.

Answer 1