Answer Happy

Baleen whales (population Y) eat krill (population X). Both krill and whales have density dependent growth rates. And, their respective carrying capacities are K and Kw dX. The resulting dimensional model equations are: = dx X aX bXY dt K x(1-1) (1-2) dY dt Y 1 Y dX where a, b, c, d, K are all constant. a) Nondimensionalize the model equations using the definitions t X 2 = X* Y y = Y* T = so the dimensionless system is: Ax (1 – x) – xy dx dt dy dt = = By(1 - х x + 0 . Express each of X*,Y*,t*, a and B in terms of the original parameters (a, b, c, d, K). b) Show that if a > 0, the model has a (physical strictly positive) coexistence equilibria. c) Show the coexistence equilibria is stable. There is a lot of algebra in this one but the determinant of the Jacobian greatly simplifies! d) Is the whale extinction state (y=0) stable?