Math 142 Homework #6
1. 37.4
Answer:
2. The infected population should grow in proportion to the uninfected population.
2. 38.2
Answer:
The growth is governed by
dN
dt =R(N)N
Equilibrium occurs when the time derivative is zero. This can happen when N= 0 and
when R= 0. Since Ris zero at two non-zero Nvalues, there are three equilibrium positions.
At N=๎N1, the growth rate is negative, and the population returns to zero. So N= 0 is
stable.
At the left most root of R(N), the growth rate is positive for values slightly larger than Neq.
This is unstable.
Finally, at the other root of R(N), the growth rate is negative to the right, and positive to
the left. This is stable.
3. Phase plane of linear systems
Sketch the phase plane solutions of the following systems
a.
dx
dt =x+y
dy
dt =x
Answer:
The phase plane equation is
dy
dx =x
x+y
This gives a zero isocline at x= 0 and an infinity isocline at y=โx. When x= 0,
dx
dt =ywhich means that xis increasing when yis positive, and vice versa. When y=โx,
dy
dt =โywhich means yis decreasing when yis positive.
