HIERARCHAL INDUCTIVE PROCESS MODELING AND ANALYSIS ...
HIERARCHAL INDUCTIVE PROCESS MODELING AND ANALYSIS ...
HIERARCHAL INDUCTIVE PROCESS MODELING AND ANALYSIS ...
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On a biology standpoint α l is the maximum rate of decline and α u is the maximum<br />
rate of growth. Theses bounds give us little information about the model, as it states<br />
that Phytoplankton concentration is contained between zero and infinity. Continue<br />
our analysis with Detritus:<br />
dD<br />
(<br />
)<br />
) (<br />
)<br />
dt = (1 − a 10 )a 9 P +<br />
((1 − a 10 )a 11 Z 2 + (1 − a 10 )(1 − a 12 )H(t)Z<br />
− D(a 17 + a 20 )<br />
Using (22), (23) and (26) we get,<br />
dD<br />
(<br />
) ) (<br />
)<br />
dt ≤ (1 − a 10 )a 9 P 0 e αut +<br />
((1 − a 10 )a 11 K 2 + (1 − a 10 )(1 − a 12 )a 13 K − D(a 17 + a 20 )<br />
Then solving for the upper bound,<br />
dD u<br />
dt<br />
)<br />
)<br />
+ D u (a 17 + a 20 ) =<br />
((1 − a 10 )a 9 P 0 e αut +<br />
(a 11 K + (1 − a 12 )a 13 (1 − a 10 )K<br />
Using Proposition 1,<br />
D u (t) =<br />
lim<br />
t→+∞ Du (t) = ∞<br />
+<br />
(a 11 K + (1 − a 12 )a 13<br />
)<br />
(1 − a 10 )K<br />
+ (1 − a 10)a 9<br />
P 0 e αut<br />
α u + a 17 + a } {{ 20}<br />
β u1 β u2<br />
(a 17 + a 20 )<br />
} {{ }<br />
(D 0 − β u1 − β u2 P 0<br />
)<br />
e −(a 17+a 20 )t<br />
where β u1 = 220.00119 and β u2 = 0.03054<br />
51