HIERARCHAL INDUCTIVE PROCESS MODELING AND ANALYSIS ...

Thus, an upper bound for Z(t) will be,
lim Z(t) ≤ K = 42.3156486
t→+∞
This is significant, since the Zooplankton concentration will have a maximum of K
and is closer to the type of behavior an ecologist would expect to see in Zooplankton
concentrations. The lower bound of this entity will be zero, since (22) and we know
from biology that Zooplankton concentration cannot be negative. Hence,
∴ 0 ≤ Z(t) ≤ 42.3156486 (23)
However if P (0) ≤ a 16 + a 15 then Z(t) would go to zero as t goes to infinity
because H(t) = 0 hence changing the structure of the equation and driving the
population to extinction.
In order to proceed to the analysis of P(t) we must first find the bounds for M(t).
{
F
M(t) =min
(F + a 5 ) , (1 − e−a 5N ), (e −E P UR (t)
a 2 )(1 − e −E P UR (t)(1+a 3 e(E P UR (t)e1.089−2.12log 10 (a 1 ) ) )
a 1 ))
}
Since M(t) is a minimum function its bounds are,
0 ≤ M(t) ≤ 1. (24)
We now are able to find bounds for P(t),
[ ]
dP [ ]
dt = (1 − E ice (t))a 0 e (0.06933∗E T H 2 O(t))
M(t)P (1 − a 6 )
( )
− (a 9 P ) − H(t)Z − (a 19 P )
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