1. Jan 21, 2010

### missavvy

1. The problem statement, all variables and given/known data
A piece of land is seperated in equal parts. There is one species on this land.

P ∈ [0, 1] = fraction of parts suitable for the species
x(t) = fraction of parts currently occupied by the species

The individuals from an occupied part colonize an empty part at rate c, and occupied parts
become extinct at a rate e.

a) Find the steady states(equilibrium solutions) of this equation.
Set c/e = 1/2 and draw the steady states as a function of P ∈ [0, 1].

b) Find the stability condition for each of the steady states.
2. Relevant equations

x'(t) = cx(h − x) − ex.

3. The attempt at a solution

a)

Ok, so I have determined the steady-states:

cx(h-x) = ex
h - x = e/c
so I found that x = h - e/c and x = 0 are the equilibrium solutions.
Now I don't know how exactly to draw the steady states.. If c/e = 1/2, then how would I go about plugging that in for the steady states?

Does this mean to draw a direction field, with 0 and h - e/c as straight lines since they are the steady-states? Or does it mean to draw x(t) as a function ?

b)

x'(t) > 0 = unstable, and x'(t) < 0 = stable
So cx(h - x) - ex > 0, cx(h - x) > ex, h - x > ex/cx, h - e/c > x
and then cx(h - x) - ex < 0, h - e/c < x

then the condition is x > h - e/c for stability, and unstable when x < h - e/c