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1 2 2 1
< or ,ΰ, < ?a, and < or 2a1 < ,a,. (41)
a1 a2 a2 a1
Now the condition that X and Y are positive yields ΰ1ΰ2 > a1a2. Hence the critical point is asymptotically stable. For this case we can also show that the other critical points (0, 0), (1/a1, 0), and (0, 2/a2) are unstable. Thus for any positive initial values of x and y the two populations approach the equilibrium state of coexistence given by Eqs. (36).
Equations (2) provide the biological interpretation of the result that coexistence occurs or not depending on whether ΰ1ΰ2 a1 a2 is positive or negative. The as are a measure of the inhibitory effect the growth of each population has on itself, while the as are a measure of the inhibiting effect the growth of each population has on the other species. Thus, when ΰ1ΰ2 > ΰ1ΰ2, interaction (competition) is weak and the species can coexist; when ΰ1ΰ2 < a1a2, interaction (competition) is strong and the species cannot coexistone must die out.
9.4 Competing Species
Each of Problems 1 through 6 can be interpreted as describing the interaction of two species with populations x and y. In each of these problems carry out the following steps.
(a) Draw a direction field and describe how solutions seem to behave.
(b) Find the critical points.
(c) For each critical point find the corresponding linear system. Find the eigenvalues and eigenvectors of the linear system; classify each critical point as to type, and determine whether it is asymptotically stable, stable, or unstable.
(d) Sketch the trajectories in the neighborhood of each critical point.
(e) Compute and plot enough trajectories of the given system to show clearly the behavior of the solutions.
(f) Determine the limiting behavior of x and y as t and interpret the results in terms of
the populations of the two species.
1. dx/dt = x (1.5 x 0.5y) > 2. dx/dt = x (1.5 x 0.5y)
dy/dt = y(2 y 0.75x) dy/dt = y(2 0.5 y 1.5x)
3. dx/dt = x (1.5 0.5x y) > 4. dx/dt = x (1.5 0.5x y)
dy/dt = y(2 y 1.125x) dy/dt = y(0.75 y 0.125x)
5. dx/dt = x (1 x y) > 6. dx/dt = x (1 x + 0.5 y)
dy/dt = y(1.5 y x) dy/dt = y(2.5 1.5y + 0.25x)
7. Show that
(a1 X + a2Y)2 4(a1a2 a1 a2) XY = (a1 X a2Y)2 + 4a1a2 XY.
Hence conclude that the eigenvalues given by Eq. (39) can never be complex.
8. Two species of fish that compete with each other for food, but do not prey on each other,
are bluegill and redear. Suppose that a pond is stocked with bluegill and redear and let x
and y be the populations of bluegill and redear, respectively, at time t. Suppose further that
the competition is modeled by the equations
dx/dt = x (e1 a1 x a1 y), dy/dt = Σ(ε2 a2 σ a2x).
(a) If e2/a2 > e1/a1 and e2/a2 > e1/a1, show that the only equilibrium populations in the pond are no fish, no redear, or no bluegill. What will happen?
(b) If ^x/ax > f-2/a2 and e1/a1 > e2/a2, show that the only equilibrium populations in the pond are no fish, no redear, or no bluegill. What will happen?
9. Consider the competition between bluegill and redear mentioned in Problem 8. Suppose that e2/a2 > e1/a1 and e1/a1 > e2/a2,so, as shown in the text, there is a stable equilibrium point at which both species can coexist. It is convenient to rewrite the equations of Problem 8 in terms of the carrying capacities of the pond for bluegill (B = e1 /a1) in the absence of redear and for redear (R = e2/a2) in the absence of bluegill.
(a) Show that the equations of Problem 8 take the form
d = e,x >iy), I = Φ1 - iy_ |x
where y1 = a1/a1 and y2 = a2/a2. Determine the coexistence equilibrium point (X, Y) in terms of B, R, y1, and y2.
(b) Now suppose that a fisherman fishes only for bluegill with the effect that B is reduced. What effect does this have on the equilibrium populations? Is it possible, by fishing, to reduce the population of bluegill to such a level that they will die out?
10. Consider the system (2) in the text, and assume that a1 a2 a1a2 = 0.
Chapter 9. Nonlinear Differential Equations and Stability
(a) Find all the critical points of the system. Observe that the result depends on whether
or not a1e2 a2e1 is zero.
(b) If a1e2 a2e1 > J, classify each critical point and determine whether it is asymptotically stable, stable, or unstable. Note that Problem 5 is of this type. Then do the same if
ff1e2 a2e1 < J.