HIERARCHAL INDUCTIVE PROCESS MODELING AND ANALYSIS ...

4.4 Model B
Shifting our focus to Model B we find different model structures. Indeed, (7) yields
a Lokta-Volterra structure which will make for an interesting analysis.
Following the procedure used for Model A, we start our analysis with the Zooplankton
equation (7), the simplest of all five. To find bounds for Z(t) we first need
to find that of H(t).
{
}
a 13 (P − a 16 − a 15 )
H(t) =max 0,
a 14 + (P − a 16 − a 15 )
a 13 (P − a 16 − a 15 )
a 14 + (P − a 16 − a 15 ) ≤ a 13
Thus we get,
0 ≤ H(t) < a 13 (22)
Knowing (22) we can conclude,
dZ
( )
dt = a 12 H(t)Z − a 11 Z 2 − a 18 Z
≤a 12 a 13 Z − a 11 Z 2 − a 18 Z = Z [a 12 a 13 − a 18 − a 11 Z]
} {{ }
Logistic Equation
Setting, a 12 a 13 − a 18 − a 11 Z = 0 we can find the carrying capacity K.
K = a 12a 13 − a 18
a 12
= 42.3156486
48