HIERARCHAL INDUCTIVE PROCESS MODELING AND ANALYSIS ...

(
∴ P 0 + a )
13
α − δ Z 0 e −αlt − a 13
α − δ Z 0e −δt ≤ P (t) ≤ P 0 e αut (20)
P (t) > 0 ∈ (0, +∞)
From a biological standpoint α l is the maximum rate of decline and α u is the maximum
rate of growth. Theses bounds give us little information about the model, as
they simply state that Phytoplankton concentration is contained between zero and
infinity.
Next we look at Detritus:
dD
(
) (
{ }} { )
dt = (1 − a 10 )a 11 P + a 11 + (1 − a 12 ) a 13 (1 − e (−a 14P ) ) (1 − a 10 )Z − D(a 15 + a 18 )
(
)
)
≤ (1 − a 10 )a 11 P +
(a 11 + (1 − a 12 )a 13 (1 − a 10 )Z − D(a 15 + a 18 )
(
)
)
≤ (1 − a 10 )a 11 P 0 e
} {{ αut +
(a
} 11 + (1 − a 12 )a 13 (1 − a 10 ) Z 0 e
} {{ −δt
}
(20)
(16)
− D(a 15 + a 18 )
≥a 13
Using (16) and (20)and simplifying a bit we get a more manageable upper bound.
Solving for upper bound D u (t)
dD u
dt
+ (a 15 + a 18 )D u = (1 − a 10 )a 11 P 0 e αut +
(a 11 + (1 − a 12 )a 13
)
(1 − a 10 )Z 0 e −δt
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