Consider the following model,
dx
dt = σ(y −x),
dy
dt = rx −xz −y,
dz
dt = xy −bz,
where x, y, and z are scaled variables and can be negative. The parameters, σ,r,
and b, are positive.
(a) What are nullclines of x, y, and z?
(b) What are equilibria of x, y, and z?
(c) Show the Jacobian matrix of the model.
(d) Show the characteristic polynomial.
(e) Calculate the Jacobian matrix of the model at an equilibrium at the origin.
(f) Show the parameter condition in which the equilibrium at the origin is un-
stable.

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