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490

It is clear that : T(0,0,0) = (0,0). The Prechet derivative

(X,a,u

)

: X

-> Zx

with respect to u

, is given as

(A,

a,u

)v = I -Av + —- / b(x)g'(a + u

)vda - \Qf

(x, a + u

) v ,

V 1-^1 J OD

b(x)g'{a + u

It follows that : T

(0,0,0)v = (-Av,dv/dn), and thus that : T

(0,0,0)v : X

-> Z

is bijective. Therefore the implicit function theorem shows that :

T(\,a,u

) = (0,0)<*u

= u

{\,a), (8)

near

(A,

a, u

) = (0,0). To sum up, problem (1) near the origin

(A,

u) — (0,0) is equivalent

to the equation :

#(A,a) := $(A,a,u

(A,a))

= A / / (x,a + «2

(\,a))dx

+ / b(x)g (a + u

(A,

a)) da = 0,

near the origin (A,u) = (0,0). The reduction is thus complete.

Under the assumption that ^(0) = </(0) = g"{0) = 0, all the partial derivatives

cP<&/d\i~

vanish at the origin up to 2nd order, which means that the so-called Morse

lemma is not applicable. To overcome the difficulty, we impose the additional restriction

that g is analytic at t = 0 in order to verify the assertions of Theorems 2.4 and 2.5, first.

The analyticity of g implies that of u

, via (8) with the aid of [10, Corollary

4.23],

and

consequently so is

<E>(A,

a). Some direct computations give that

$(A, a) = ^(3AA

- 3BAa + Co? + Vi(>,«)),

near

(A,

a) = (0,0). Here A, B and C are positive constants given by

dadX

(0,0) =• 2 / m{x)-y

J D

)dx,

= -^

'°) =

C = 0(0,0) =

g"'(0)J b(x)da

and ipi is a higher order term of the form

oo A;

ipx{\,

a) = 5Z H

kjXa

, c

€ R.

k=3 j=0

491

The requirement for Theorems 2.4 and 2.5 is

3A\

- 3BXa + Ca

V>i(A,

a) = 0,

near

(A,

a) = (0,0). Here we find that

condition (6)

<=>

- Y1AC < 0,

condition (7) <S=> 9B

- Y1AC > 0.

This implies that conditions (6) and (7) determine the sign of the discriminant for

the quadratic function 3A\

—

3B\a + Ca

. Prom this observation, the conclusions in

Theorems 2.4 and 2.5 follow under the analyticity of g.

The extension to the case g without the analyticity assumption is due to a comparison

argument using the super- and sub-solution method as follows. Senn proved in [8] that if

mdx = 0, then the principal eigenvalue Hi{\) of (3) is negative for any

> 0. Denote

by ipi(X) the principal positive eigenfunction of (3). We can easily see under (2) that for

every A > 0 a function

E<pi(\)

is a sub-solution of (1) for any e > 0 small enough.

For any g satisfying (5) and (6) without the analyticity assumption, there exists

<5i

> 0

close to g'"(0) such that

t<™«-/

K.)*>

Let gi (t) = Sit

/6. Then there exists ri > 0 such that

b(x)

(t)<b(x)g(t), te[0,n}. (9)

One sees that gi fulfills (5) and (6) and is analytic at t = 0, so that there is no positive

solution u of (1) with g = g

for any A > 0 and

||M||C2+«(C)

both small enough in the

previous argument. By the well-known bootstrap argument, the same situation holds for

A > 0 and u

C(D) both small enough.

Now we assume to the contrary that the origin (A,u) = (0,0) is a bifurcation point

to the right for the original problem (1) and, without loss of generality, there may exist

positive solutions u^ of (1) such that

||WA||C(D)

—» 0 as A J. 0, so that it follows that

A|IC(D)

< TI for 0 <

< A* with some

> 0. This implies that ux is a super-solution

of (1) with g = gi for any 0 < A < A*, by virtue of (9). We recall the existence of such

sub-solutions of (1) with g = g\ which are sufficiently small. Then the super- and sub-

solution method allows us to obtain that for any A > 0 small the existence of a positive

solution v\ of (1) with g = g\ such that 0 < v\ < u\ on D. This proves that

||«A||C(5)

J. 0, which is a contradiction. The proof of Theorem 2.4 is complete. •

Now we end the proof of Theorem 2.5 in an analogous way. Indeed, it suffices to note

that for any g satisfying (5) and (7) without analyticity, we can take a constant

> 0

such that

r 3n

IDI

"'(0)<5

and / b{x)da< * ' .

492

Put gi{t) =

and the rest of the proof is similar to the proof of Theorem 2.4.

Theorems 2.6 and 2.7 can be also proved in the same way, using super- and sub-

solutions. Finally we refer to [9, Lemma 4.3] for the existence of the minimal positive

solution in Theorems 2.5 and 2.7. •

4 Interpretations

This section is devoted to some ecological interpretation of our main results. Due to [9,

Lemma 4.3], the minimal positive solution of (1) obtained in Theorems 2.5 and 2.7 is left-

side asymptotically stable for the initial-boundary value problem (4) with u

^ 0. Here

we refer to Pao [7, Chapter 5] for the definition of the left-side asymptotical stability.

The critical value v

given by Theorems 2.4 and 2.5 is of ecological interest in the

following sense. We recall ([9, Theorem 2]) that the trivial solution u = 0 of (1) is

unstable for all A > 0 under the condition J_ mdx = 0. Hereby Theorem 2.4 suggests

that, in the case

JdL

b(x)da > u

JdD

the positive minimal stable steady state for the population obeying (4) may be persistent

as the diffusion rate 1/A increases without limitation, even if the initial data u

is small

enough. Meanwhile, Theorem 2.5 suggests that it must be extinct asymptotically as the

diffusion rate goes to infinity, provided

JdD

b(x)da < u

To be specific, for a constant c > 0, we put m{x) = cfh(x) where m 6 C

(D),

rhdx = 0 and Hwll^m = 1. Define a constant

i>§

through the right-hand side of (6)

with m = rh and we can verify that if

then Theorem 2.4 applies, and meanwhile that Theorem 2.5 applies if

0<c<

with B

= J

bda. This assertion suggests that if c = ||m||

^ increases gradually, then

the environment becomes favorable for the species which diffuses at a large enough rate.

References

[1] H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions. In :

New Developments in differential equations (W. Eckhaus ed.), Math. Studies, Vol.

21,

North-Holland, Amsterdam, 1976, pp.

43-63.

493

[2] K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigen-

value problem with indefinite weight function. J. Math. Anal. AppL, 75 (1980),

112-120.

[3] R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights:

population models in disrupted environments. Proc. Roy. Soc. Edinburgh, 112A

(1989),

293-318.

[4] R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights:

population models in disrupted environments II. SIAM J. Math. Anal., 22 (1991),

1043-1064.

[5] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues. J. Funct.

Anal., 8 (1971), 321-340.

[6] P. Hess, Periodic-parabolic boundary value problems and positivity. Pitman Research

Notes in Math. Series, vol. 247, Longman Scientific & Technical, Harlow, 1991.

[7] C. V. Pao, Nonlinear

parabolic

and elliptic equations. Plenum, New York, 1992.

[8] S. Senn, On a nonlinear elliptic eigenvalue problem with Neumann boundary condi-

tions,

with an application to population. Comm. Partial

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Eq., 8 (1983), 1199—

1228.

[9] K. Umezu, Behavior and stability of positive solutions of nonlinear elliptic boundary

value problems arising in population dynamics. Nonlinear Analysis, TMA, to appear.

[10] E. Zeidler, Nonlinear functional analysis and its applications I: Fixed-point theorems.

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