- Email: [email protected]

ABSTRACT A theorem one-dimensional

is presented population

that improves models.

some

Fisher et al. [l] studied the difference x,+1

=

results

on global

stability

equation

f(xt)

(1)

and investigated global stability by Lyapunov functions. Later, Cull [21 gave a very important sufficient and necessary for the global stability of (1): f(f(x))

’ x

for discrete

condition

for all x E [x~,_?).

(2)

Others including Cull [3, 41, Rosencranz [51, and Huang [6, 71have studied this model and have given some sufficient conditions for testing the global stability when the function f(x) behaves in a way that is tested more easily than that indicated in (2). This paper, while related to these earlier results, presents a new sufficient condition for the stability of an equilibrium of (1). We completely analyze the relationship between the shape of f(x) and the stability of the equilibrium of (1) and obtain the following theorem, which includes Cull’s theorem 3 (sufficient condition A) [4] to some extent. Of course, we also consider cases that Cull did not consider, for example, the case that the condition f”(x) < 0 for x in [x,, X) is violated. THEOREM

Consider a population model xt+1= MATHEMATICAL

BIOSCIENCES

OElsevier Science Publishing 655 Avenue of the Americas,

f(x,),

102:121-124

(1)

(1990)

Co., Inc., 1990 New York, NY 10010

121 00255564/90/$03.50

122

NOTE ON GLOBAL

STABILITY

where the function satisfies the following conditions: (a) The function f(x) is differentiable on [O, + m) and f(0) = 0. (b) f(x) > x when x < i, and f(x) < x when x > Z, where X is the unique positiue equilibrium point of (1) with 1f(X)! < 1. (c) There exists x = x* [x* > x,, where x,,, is a maximum point of f(x) in (0, .?)I such that f”(x*) = 0 and f”(x) < 0 when x, < x < x*, f”(x) > 0 when x* < x < f(x,>. (d) f”‘(x) > 0 and f’(x) < 0 for x E [x,, f(x,)). Then we have:

(a) If f’(Z) = - 1, then X is locally and globally stable. (b) If f’(Z) > - 1 and x* 2 2, then X is locally and globally stable; if f’(Z) > - 1 and x* < ??, then X is locally stable (and the global stability of 1 is indeterminate). Proof. According to Cull [4], we know that X is globally g”(x) > 0 for all x E [x,, T), where g(x) = f( f(x))x. We have

g”(x)

=f”(x)f’(f(x>)

If f’(x) = - 1, we will consider (3) x* < z.

stable

if

+ r,

three

(1) If x* = Z, then f”(x) < 0, f’( f(x)) hence g”(x) > 0.

cases: (1) x* = X, (2) x* > 3, < 0, f”( f(x))

> 0 for x E Lx,, X);

(2) When x* > X, then there exists y E[x,,~) such that f(y) = x*. If X, G x G y, then f”(x) < 0, f’(f(x)) < 0, f”(f(x)) > 0; hence g”(x) > 0. If y < x < X, then f”(x) < 0, f’( f(x)) < 0, f”( f(x)) < 0. Since f”‘(x) > 0, f”(x) G f”< f(x)) < 0; hence If”(f(x))l< lf”(x)l. Since f”(x) < 0 [x E [xm, x*)1, ence If’(fbDl> l> If’(x)l> lf’(x)12 and 0 > f’(x) > f’(x) > f’(f(x)); h therefore g”(x) > 0. (3) When x* < Z, let y = f(x*), y > X. If x,,, < x Q x*, then f”(x) < 0, f’< f(x)) < 0, f”( f(x)) > 0. Hence, g”(x) > 0. If x* < x < X, then f”(x) > 0 and x

123

YONGNIAN HUANG

Example 1 f(x)

=(1/8)(42x

where and

+19x3),

f’( 2) = - 7/8,

x=1 f’(x)

-53x2

= (;/8)(42-106x

x, = xi = 0.5724289,

+57x*),

x2 = 1.2872201 such that

f’(xi) = 0, i = 1, 2,

f( xm) A 1.279886, 7(x)=(1/8)(-106+114x), f”‘(x)

x* = 106/114 = 0.9298245,

= 114/8 > 0.

By the lemma of Cull [4], this model is globally stable. In this model, y(Z)

= -7/8>

x*

-1,

Example 2 f(x)=x(x-33/2)[-2-(x-1)-6(x-l)*], X=l,f’(x)=(2x-3/2)[-2-(x-1)-6(x-l)*] +x(x-33/2)[-1-12(x-l)], f’(X) = -l/2, f”‘(x)

f”(x)

= -72x2+120x

x*=O.6292<1,

x**=1.0375,

= -144x

f”‘(x)

+ 120,

-47,

B 0 (x Q x*).

But f(f($>> = 0; hence x is not globally stable. Note that the function of Example 2 can be revised such that have a continuous derivative of the third order in the neighborhood and f(x) > 0, f’(x) < 0 when x > $. These examples are special ones presented by Cull [41; however, the coefficients chosen are from his so that the conclusions obtained here are relevant to the

f(x) will of x = $, cases of different theorem.

I am indebted to Professor T. G. Hallam for his guidance. REFERENCES 1 M. E. Fisher, B. S. Goh, and T. L. Vincent, Some stability conditions for discrete-time single species models, Bull. Math. Bid. 413361-875 (1979). 2 P. Cull, Global stability for population models, Bull. Math. Biof. 43:47-58 (1981). 3 P. Cull, Local and global stability for population models, Biol. Cybem. 54:141-149 (1986).

124 4

NOTE ON GLOBAL STABILITY P. Cull, Stability for discrete one-dimensional 50:67-75

population models, BUN. Math. Bid.

(1988).

G. Rosencranz, On global stability of discrete population models, Math. Biosci. 64:227-231 (1983). 6 Y. N. Huang, A counterexample of P. Cull’s theorem, Kaue Tonbao 31:1002-1003 (1986). 7 Y. N. Huang, A theorem on global stability of discrete population models, Math. Theory &act., 1:42-43 (1987). 5