# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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1.51 N y y y y ? y y y

t \ N ?*-y v" y y ? y y y y

t \ \ -*-y y y ? y y y y y

t N \ y y y

t t \ v y ? y y

1 - \ y ? ? ? ? ? ? y y

t t t \ V y ? ? ? ? ? y y

t t t \ ? ? ? y y

t 1 / t t N y y y *- y y y y y

t t / / / / Ŧ- *- y y y y

0.5 / / / / y y y y

t t / / 1 y y y y y

t / / X' ^ \ y y y y *- *- y y

t / y y" * X y y

1 1 1 1

0 0.25 0.50 0 .75 1 1 25 x

FIGURE 9.4.3 Critical points and direction field for the system (21).

x = 0, y = 0. Neglecting the nonlinear terms in Eqs. (21), we obtain the linear system

I (;)=(0 0,)Q- (->

which is valid near the origin. The eigenvalues and eigenvectors of the system (22) are

1)

,0,

ri = 1, ^ = (J) ; r2 = 0.5, gŪ = (0) , (23)

so the general solution is

/) = ci(0)e' + Cz(i] e°5^ (24)

Therefore the origin is an unstable node of the linear system (22) and also of the nonlinear system (21). All trajectories leave the origin tangent to the y-axis except for one trajectory that lies along the x-axis.

x = 1, y = 0. The corresponding linear system is

d fu\ /1 A /u

dt \ v) V 0 -0.25) W (25)

9.4 Competing Species

497

Its eigenvalues and eigenvectors are

r, = -1, t(1) =

and its general solution is

r2 = 0.25,

g(2) =

4

3

+ c2

-3e025t.

(26)

(27)

The point (1, 0) is an asymptotically stable node of the linear system (25) and of the nonlinear system (21). If the initial values of x and y are sufficiently close to (1, 0), then the interaction process will lead ultimately to that state, that is, to the survival of species x and the extinction of species y. There is one pair of trajectories that approaches the critical point along the x-axis. All other trajectories approach (1, 0) tangent to the line with slope -3/4 that is determined by the eigenvector ?(2).

x = 0, y = 2. The analysis in this case is similar to that for the point (1, 0). The appropriate linear system is

d ( u

dt \ v

1

0

1.5 0.5

The eigenvalues and eigenvectors of this system are

r1 = ~1, t(1) =

and its general solution is

r2 = -O.5,

g(2) =

c

3 ) et + C2

5) e05t.

(28)

(29)

(30)

Thus the critical point (0, 2) is an asymptotically stable node of both the linear system (28) and the nonlinear system (21). All trajectories approach the critical point along the y-axis except for one trajectory that approaches along the line with slope 3.

x = 0.5, y = 0.5. The corresponding linear system is d dt

The eigenvalues and eigenvectors are

0.5 0.5 \ tu

0.375 0.125 v

r1 = 5 +Z57 = 0.1594,

16

5 V57

r2 =---------- = 0.7844,

2 16

t(1) =

t(2) =

(3 V57)/8

(3 + v^57)/8

so the general solution is

c

1

-1.3187

e0 1594t + c2

0.51687

1

1.3187

1

0.5687

e 0.7844t

(31)

(32)

(33)

Since the eigenvalues are of opposite sign, the critical point (0.5, 0.5) is a saddle point, and therefore is unstable, as we had surmised earlier. One pair of trajectories approaches

1

0

e

v

498

Chapter 9. Nonlinear Differential Equations and Stability

the critical point as t ^ to; the others depart from it. As they approach the critical point, the entering trajectories are tangent to the line with slope (V57 3)/8 = 0.5687 determined from the eigenvector ?(2).

A phase portrait for the system (21) is shown in Figure 9.4.4. Near each of the critical points the trajectories of the nonlinear system behave as predicted by the corresponding linear approximation. Of particular interest is the pair of trajectories that enter the saddle point. These trajectories form a separatrix that divides the first quadrant into two basins of attraction. Trajectories starting above the separatrix ultimately approach the node at (0, 2), while trajectories starting below the separatrix approach the node at Bad math construct!(1,0)Bad math construct!. If the initial state lies precisely on the separatrix, then the solution (x, y) will approach the saddle point as t ^ to. However, the slightest perturbation as one follows this trajectory will dislodge the point (x, y) from the separatrix and cause it to approach one of the nodes instead. Thus, in practice, one species will survive the competition and the other will not.

FIGURE 9.4.4 A phase portrait of the system (21).

Examples 1 and 2 show that in some cases the competition between two species leads to an equilibrium state of coexistence, while in other cases the competition results in the eventual extinction of one of the species. To understand more clearly how and why this happens, and to learn how to predict which situation will occur, it is useful to look

9.4 Competing Species

499

again at the general system (2). There are four cases to be considered, depending on the relative orientation of the lines

1 a1 x a1 y = 0 and 2 a2 y a2 x = 0,

(34)

as shown in Figure 9.4.5. These lines are called the x and y nullclines, respectively, because x is zero on the first and y/ is zero on the second. Let (X, Y) denote any critical point in any one of the four cases. As in Examples 1 and 2 the system (2) is almost linear in the neighborhood of this point because the right side of each differential equation is a quadratic polynomial. To study the system (2) in the neighborhood of this critical point we can look at the corresponding linear system obtained from Eq. (13) of Section 9.3,

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