Modelling Systems : Competing Species

In an unmanaged tract of forest area, hardwood and softwood trees compete for the available land and water.  The more desirable hardwood trees grow more slowly, but are more durable and produce more valuable timber.  Softwood trees compete with the hardwoods by growing rapidly and consuming the available water and soil nutrients.  Hardwoods compete by growing taller than the softwoods can and shading new seedlings.  They are also more resistant to disease.  Can these two types of trees coexist on one tract of forest land indefinitely, or will one type of tree drive the other to extinction?  Model the problem by the system of differential equations:


   dx1/dt = r1*x1 - a1*(x1)^2 - b1*x1*x2 and
   dx2/dt = r2*x2 - a2*(x2)^2 - b2*x1*x2  

where x1 and x2  represent the hardwood and softwood populations, respectively (tons/acres), and r1, r2, a1, a2, b1, and b2 are positive constants with:

r2/a2 < r1/b1   and     r1/a1 >= r2/b2


a) Locate each of the equilibrium points (x1,x2) in the state space x1 >=0,  x2>= 0
b) Sketch the vector field.
c) Classify each equilibrium as stable or unstable.
d) Suppose that we start out with an equal amount of hardwood and softwood trees.  What does this model predict about the future of the two species?