Two species are interactive as a predator and a prey.

Two species are interactive as a predator and a prey. While the predator consumes as muchas it can find when food is scarce, it is not unreasonable that, during periods of abundance,the predator satiates and then feeds at a maximum per capita rate B, independentof the prey. Let x(t) and y(t) be the population sizes, at time t, of prey which feeds onan unlimited food source of its own, and the predator which feeds on the prey. Then if x(t) is suffiently large, y(t) increases and vice versa. We also assume that x is the solefood source for the predator. This verbal description yields the following mathematical model

dx/dt = rx [1 −(x/K)]−Bxy/(A + x ) eq (1)

dy/dt = sy [1 −y/(νx)],

where r, K, s, ν, A, and B are all positive constants.

(a) How many steady states are there for the system?

(b) Using parameter K as the "unit" for population sizes x and y, and use B−1as the"unit" for time, show that the above pair of equations can be simplified into

du/dτ = ku(1 − u) − uv/(a + p1) eq (2)

dv/dτ = σv [1 −v/(νu)]

What are the k, a and σ in terms of the original parameters in Eq. (1)?

(c) Analyzing the type and stability of the non-trivial steady state of Eq. (2).

(d) Is there a possibility of a limit cycle? You should include a plot as well as analytical arguments. You should include a plot as well as analytical arguments.