274
Thus, y 1-+ g(y) + fLQ( t, x )cx( x, y) is strictly increasing on [fLO, (0) for fL E [fL*, (0), x E M
and t E R+. Lemma 3 implies that infu ~ fLo, if u is a I-periodic solution of (7) with
fL E [fL*, (0), thus we get for two such I-periodic solutions u,v that
0= 111M {c(x)8du - v](t,x) - div (k(.) grad [u - v](t, ·))(x) + fLQ(t,X)·
[cx(x, u(t, x)) - cx(x,v(t,x))] + g(u(t,x)) - g(v(t,x))}[u(t,x) - v(t,x)]dxdt,
hence u=v.
Using the observations from Lemma 2 and the last assumption in (H'3) one can also
establish that II(fL,·) has at most one fixed point for small fL ~ o. One basically follows
the reasoning in the proof of the previous corollary.
9. Stability and Convergence.
We now employ the implicit function theorem at (fL,19) E \ll, in case that Ao(fL,19) is
nonzero, and Amann's result (cf. prop. 20.7. in [Amann (1976)]), if AO(fL,19) = 0,
in order to derive that \ll is a Jordan curve P 1-+ (fL(P), 19(p)) parameterized over R+.
One actually considers (fL,19) 1-+ 19 - II(fL, 19) and notes that the spectral radius r(fL,19)
of (D2II)(fL, 19) is < 1 (= 1, > 1), iff AO(fL,19) is > 0 (= 0, < 0). It is easy to see
that AO(fL,19) > 0 for fL > 0 small and for large /1. The classification theorem for onedimensional oriented submanifolds yields then the above parameterization, and AO(/1, 19)
crosses through 0 exactly in those points (/1(p), 19(p)), where /1(p) is a strict extremum
of /1. Thus, forward bending curve segments correspond to Ao(fL,19) > 0, backward
bending ones to AO(/1,19) < O. Switching from Ao(fL,19) to r(fL,19) by employing the
relations mentioned above and applying the principal of linearized stability to II(fL,·)
as well as a straightforward adaption of Theorem (23.2) in [Hess (1991)] we obtain
that u(·,·j fL, 19) is asymptotically stable (unstable) provided that (fL,19) is located on a
forward bending (backward bending) curve segment.
Of course, one would also like to know, whether an arbitrary solution curve of (7)
approaches a periodic orbit, or at least, whether this is generically the case. In contrast
to what we discussed for models without seasonal cycle in Section 4, this seems to be
unknown here.
Thus, y 1-+ g(y) + fLQ( t, x )cx( x, y) is strictly increasing on [fLO, (0) for fL E [fL*, (0), x E M
and t E R+. Lemma 3 implies that infu ~ fLo, if u is a I-periodic solution of (7) with
fL E [fL*, (0), thus we get for two such I-periodic solutions u,v that
0= 111M {c(x)8du - v](t,x) - div (k(.) grad [u - v](t, ·))(x) + fLQ(t,X)·
[cx(x, u(t, x)) - cx(x,v(t,x))] + g(u(t,x)) - g(v(t,x))}[u(t,x) - v(t,x)]dxdt,
hence u=v.
Using the observations from Lemma 2 and the last assumption in (H'3) one can also
establish that II(fL,·) has at most one fixed point for small fL ~ o. One basically follows
the reasoning in the proof of the previous corollary.
9. Stability and Convergence.
We now employ the implicit function theorem at (fL,19) E \ll, in case that Ao(fL,19) is
nonzero, and Amann's result (cf. prop. 20.7. in [Amann (1976)]), if AO(fL,19) = 0,
in order to derive that \ll is a Jordan curve P 1-+ (fL(P), 19(p)) parameterized over R+.
One actually considers (fL,19) 1-+ 19 - II(fL, 19) and notes that the spectral radius r(fL,19)
of (D2II)(fL, 19) is < 1 (= 1, > 1), iff AO(fL,19) is > 0 (= 0, < 0). It is easy to see
that AO(fL,19) > 0 for fL > 0 small and for large /1. The classification theorem for onedimensional oriented submanifolds yields then the above parameterization, and AO(/1, 19)
crosses through 0 exactly in those points (/1(p), 19(p)), where /1(p) is a strict extremum
of /1. Thus, forward bending curve segments correspond to Ao(fL,19) > 0, backward
bending ones to AO(/1,19) < O. Switching from Ao(fL,19) to r(fL,19) by employing the
relations mentioned above and applying the principal of linearized stability to II(fL,·)
as well as a straightforward adaption of Theorem (23.2) in [Hess (1991)] we obtain
that u(·,·j fL, 19) is asymptotically stable (unstable) provided that (fL,19) is located on a
forward bending (backward bending) curve segment.
Of course, one would also like to know, whether an arbitrary solution curve of (7)
approaches a periodic orbit, or at least, whether this is generically the case. In contrast
to what we discussed for models without seasonal cycle in Section 4, this seems to be
unknown here.