199

CHAPTER 2

25.

Linearity and Nonlinearity

(a–e)

SECTION 2.6

Systems of DEs: A First Look

199

When the two nullclines intersect as they do in the figure, then there are four equilibrium ⎛a ⎞ ⎛ d⎞ points in the first quadrant: (0, 0), , 0 , 0, , and ( x , y ) , where ( x , y ) is the ⎟ ⎜ ⎟ ⎜ e e e e f ⎠ ⎝b ⎠ ⎝ intersection of the lines bx + cy = a , ex + fy = d . Analyzing the sign of the derivatives in the four regions of the first quadrant, we find

⎛a

⎛ d⎞ ⎞ are stable and the , 0 and 0,

⎜b ⎝

⎜ ⎝

⎟ ⎠

f

⎟ ⎠

other two unstable. Hence, only one of the two populations survives, and which survives depends on the initial conditions. See Figures. For initial conditions in the upper region y survives; for initial conditions in the lower region, x survives. (a)–(b)

y

y

(c)–(d)

d/f

d/f h-nullcline

h-nullcline

a/c

a/c

υ -nullcline

υ -nullcline x

x d/e

d/e

a/b

Nullclines and equilibria

a/b

Typical trajectories when the nullclines intersect and the slope of the h-nullcline is more negative.



Unfair Competition

26.

x′ = ax (1 − bx ) − cxy y′ = dy − exy Setting x′ = y′ = 0 , we find three equilibrium points:

( 0, 0 ) , ⎛⎜ b , 1

e −bd ⎞ ⎞ ⎛d 0 ⎟ , and ⎜ , a . e ce ⎟ ⎠ ⎝ ⎠

The point (0, 0) corresponds to both populations becoming extinct, the point

⎛1 ⎜b ⎝

, 0

⎞ ⎟ ⎠

corresponds

e −bd ⎞ corresponds to either a ⎜e ce ⎟⎠ ⎝ 1 d stable coexistent point or an unstable point. If we take the special case where > , e.g., b e to the second population becoming extinct, and the point

⎛d

,a

Solutions manual for differential equations and linear algebra 2nd edition by farlow ibsn 9780134689

Solutions Manual for Differential Equations and Linear Algebra 2nd Edition by Farlow IBSN 9780134689548 Full clear download( no error format...

Solutions manual for differential equations and linear algebra 2nd edition by farlow ibsn 9780134689

Solutions Manual for Differential Equations and Linear Algebra 2nd Edition by Farlow IBSN 9780134689548 Full clear download( no error format...