282
is the generator of a positive analytic semigroup in C(M) and
Q
fO
L:{}>-+-J.t-03a(·,W,w)
P(s){}(s,·)ds
C
-T
is an order positive bounded linear operator from C([-T,O] x M) into C(M). Setting vt(s) = v(t + s) for v E C([-T,oo),C(M)), t E [0,00) and s E [-T, 0], we
can consider the infinitesimal generator A of v' + Bvt(O) = Lv!, i.e. dom(A).{¢ E C 1 ([-T,0],C(M)) : ¢(O) E dom(B), ¢/(O) = B¢(O) + L¢} and A¢ := ¢' for
¢ E dom(A). It turns out -this is the result of Kerscher and Nagel- that the spectral
bound of A and that of B + L 0 I have the same sign, where I denotes the natural
embedding from C(M) into C([-T,O] x M) (functions of position alone are in time
constant functions). Observing that J~T f3( s) ds = 1 yields therefore the statement.
13. S-shapedness in Case of Seasonal Resolution.
Let us now turn to the case, where the seasons are resolved, i.e. our model equation is
C(X)Otu(t,X) - div(k(·) gradu(t,·))(x) =
(22)
= J.t Q(t,x)[I- a(x,u(t,x), [OTP(S)u(t + s,x)ds)]- g(u(t,x)).
The possible global climates will be associated with the stable I-periodic solutions of
(22). It turns out that compared with the situation for the "undelayed" (7) we need to
restrict the asymptotic behavior of the emission function g even more in order to obtain
the a priori bound from below. Also, the assumptions for a and Q have to be adapted.
We provide once more the full list of assumptions for the convenience of the reader.
Modified Basic Hypotheses.
(AO) M is a 2-dimensional connected compact oriented Riemannian manifold without
boundary, T E (0,00);
(AI) c, k E C 2 (M) are positive; f3 E CCXl([ - T, 0]), f3( - T) = 0, f3( s) > ° for s E (-T, 0]
and J~Tf3(s)ds = 1; Q E C 2 (R+ x M), Q(-,x) is I-periodic, Q(t,x) ::: q(t) for
x E M and t ::: 0, where q E C 2 (R+) is nonnegative and I-periodic, and q(O) > 0;
(A2) a E C 2 (M x R+ x R+), infa > 0, sup a < 1,
03a(X,y,Z) < ° for x E M,y E R+ and z > 0;
is the generator of a positive analytic semigroup in C(M) and
Q
fO
L:{}>-+-J.t-03a(·,W,w)
P(s){}(s,·)ds
C
-T
is an order positive bounded linear operator from C([-T,O] x M) into C(M). Setting vt(s) = v(t + s) for v E C([-T,oo),C(M)), t E [0,00) and s E [-T, 0], we
can consider the infinitesimal generator A of v' + Bvt(O) = Lv!, i.e. dom(A).{¢ E C 1 ([-T,0],C(M)) : ¢(O) E dom(B), ¢/(O) = B¢(O) + L¢} and A¢ := ¢' for
¢ E dom(A). It turns out -this is the result of Kerscher and Nagel- that the spectral
bound of A and that of B + L 0 I have the same sign, where I denotes the natural
embedding from C(M) into C([-T,O] x M) (functions of position alone are in time
constant functions). Observing that J~T f3( s) ds = 1 yields therefore the statement.
13. S-shapedness in Case of Seasonal Resolution.
Let us now turn to the case, where the seasons are resolved, i.e. our model equation is
C(X)Otu(t,X) - div(k(·) gradu(t,·))(x) =
(22)
= J.t Q(t,x)[I- a(x,u(t,x), [OTP(S)u(t + s,x)ds)]- g(u(t,x)).
The possible global climates will be associated with the stable I-periodic solutions of
(22). It turns out that compared with the situation for the "undelayed" (7) we need to
restrict the asymptotic behavior of the emission function g even more in order to obtain
the a priori bound from below. Also, the assumptions for a and Q have to be adapted.
We provide once more the full list of assumptions for the convenience of the reader.
Modified Basic Hypotheses.
(AO) M is a 2-dimensional connected compact oriented Riemannian manifold without
boundary, T E (0,00);
(AI) c, k E C 2 (M) are positive; f3 E CCXl([ - T, 0]), f3( - T) = 0, f3( s) > ° for s E (-T, 0]
and J~Tf3(s)ds = 1; Q E C 2 (R+ x M), Q(-,x) is I-periodic, Q(t,x) ::: q(t) for
x E M and t ::: 0, where q E C 2 (R+) is nonnegative and I-periodic, and q(O) > 0;
(A2) a E C 2 (M x R+ x R+), infa > 0, sup a < 1,
03a(X,y,Z) < ° for x E M,y E R+ and z > 0;