Concern for the quality of public lands has received much attention in recent years. The condition of rangelands administered by the Bureau of Land Management (BLM), and other Federal agencies, has been one of the issues in this concern. The BLM manages public lands that are, for the most part, arid and covered with low scrub and grasses. Private ranchers pay the BLM a fee to be allowed to graze cattle on these lands. The cattle, in eating the vegetation on the land, actually change the balance of species in a region. This exercise deals with one aspect of the problem of the effect of cattle on rangelands. In particular, there is competition in the wild between natural perennial grasses and exotic annual grasses, principally cheatgrass. The perennial grasses are preferable for the cattle, both in terms of nutritive value, and apparently in taste. Thus the cattle tend to eat perennial grasses first, and resort to eating cheatgrass only after most of the perennials are gone. If the area colonized by the perennials becomes too small relative to that colonized by the annuals, it is possible for the perennials to be completely eliminated through competition.
A model has been introduced to allow these levels to be discussed . A part of this model is the topic of this example. Let Ω denote an area in which perennial grasses and annual grasses are in competition. Let g denote the fraction of that area which is colonized by perennial grasses, and let w (for weeds!) denote the fraction that is colonized by annuals. We consider that grasses and weeds may share a patch of ground, so that both w and g may vary from zero to one. The equations describing their dynamics are
The parameters and represent intrinsic growth rates of the grasses and weeds, respectively. The cattle stocking rate is introduced through E. E decreases linearly with the number of cattle on the plot.
We want to find threshold values of g and w dividing the case in which the grasses are able to compete successfully with the weeds, from the case in which the perennial grasses are so sparse that they can no longer compete, and die off in the area Ω. We are interested in the phase plane only in the unit square. In this part of the plane, there are several equilibria: for the following we consider only parameter sets for which there are five. We assume that g is on the horizontal axis, and w is on the vertical axis. Following is a summary of the three equilibria on the axes.
In addition, there are two more equilibria in the interior of
the unit square (see Figure 1).
Problem 1. Find the locations of the two other equilibria when the parameters are given by E = .3, =.27, and =.4. One of these points is a stable node, henceforth denoted Y, while the other is a saddle, denoted X. Ascertain which is which.
Now if the initial ratio of grasses and weeds corresponds to a
point in the phase plane near the interior stable
equilibrium Y, then the
trajectories through that point will approach the stable
equilibrium as time passes. On the other hand, if the point lies
too far from the interior stable node, then the trajectory
through it may approach some other stable node, such as the one
at (0,1). The latter is an undesirable turn of events, since that
equilibrium represents an all-weeds outcome. Apparently there is
some place where the behavior of solutions
Problem 2. Using the previous set of parameters, choose a set of initial conditions that lie on the line w=.2. Use a computer program to plot the orbits that pass through these initial conditions in the phase plane. What happens to the orbits as the initial conditions move to the left? Try to find a curve that divides the set of initial conditions that lie on trajectories that approach the desirable attracting equilibrium, Y, from the those that lie on trajectories approaching the undesirable one at (0,1). How did you get the program to draw that curve?
The curve that separates the regions of different behavior in the phase plane is called a separatrix. The separatrix is composed of the union of the unique orbit that actually joins the unstable node at the origin with the saddle point X, the point X itself, and the unique other orbit that is attracted to the saddle point X. These curves, taken together, comprise the stable manifold of the equilibrium at X. In order to find this stable manifold, we must approximate these orbits with some accuracy. To do this, we focus on the orbit from the origin to X.
What is known about this orbit? We know its endpoints: they
are the origin and X, the coordinates of which depend on
the choice of parameters. These allow us to make a first
approximation to the separatrix.
It turns out that we know more. If we describe the separatrix as a curve in the g-w plane as w = W(g), then we know the value of W(g) and its derivative at the equilibrium point X. In particular, let X = (γ, ω), and let M be the matrix in the linearization of the equation about the point X, so that the linearized equation is
|(g - γ)'||(g - γ)|
|(w - ω)'||(w - ω)|
Since X is a saddle point, M has two distinct
eigenvalues: one positive, the other negative. Let the
eigenvector associated with the negative one be denoted (u,v).
Assume that u ≠ 0. Since the separatrix is
parallel to this eigenvector at the point X, it must
have slope v/u there. We thus make our second
approximation to the separatrix have the form
W(g) = ag + bg2,
where a and b are chosen so as to
make W(γ) = ω, and
W′(γ) = v/u.
Problem 4. Using the same parameters as in Problems 1-3, write two equations that allow you to solve for the coefficients a and b in the approximation for the separatrix. Plot this curve on the same phase portrait from Problem 2, and again compare it with a numerically computed approximation to the separatrix.
The approximation to the separatrix is a quadratic polynomial.
One could, if necessary, obtain coefficients for an approximating
polynomial of higher degree for the separatrix. Such a procedure
is complicated, and is of limited effectiveness.
Problem 5. Leaving all other parameters the same, change the value of to .40, and change to .27. Do Problem 4 using this new set of parameters. How does the approximation to the separatrix compare with the numerically computed separatrix now?
As the last problem shows, there is a problem with our approximation at the origin. The behavior of the separatrix varies there according to the values for and . Whenever then there is a dominant direction for orbits to leave a neighborhood of the origin. In particular, when , then orbits tend to leave the origin tangent to the g axis, while when , then they tend to leave tangent to the w axis. It turns out that the separatrix may be expanded in a series of the form
where . Our final approximation
for the separatrix truncates this series after the first two
terms. We may still use the value and slope of W(g)
at the equilibrium
( γ, ω )
to solve for the coefficients a0
Problem 6. Solve for the
two unknowns a0
to arrive at an approximation for the stable
manifold given the first two terms of the series above. Draw this
curve on the phase portraits from both Problem 4 and
Problem 5. How well does it agree with the curves that you
computed earlier to approximate the
separatrix? How do you expect the error in the approximation
K. D. Cooper and R. Huffaker, The long term bioeconomic impacts of grazing on plant succession in a rangeland ecosystem, Ecological Modelling, 97 no. 1-2 (1997).
for mathematical applications. There are
illustrating how we might use interactive web objects to
help students learn Calculus.