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Population Growth Modeling
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dc V dt dc V dt Population Growth Modeling
description
Population Growth Modeling. Begin with a mass balance on microbial growth. X = population biomass, mg/L V = volume, L Q = flow, L/d k = 1 st order rate coefficient, 1/d t = time, d. Exponential growth model. - PowerPoint PPT Presentation
Transcript of Population Growth Modeling
dcVdt
dc
Vdt
Population Growth Modeling
Begin with a mass balance on microbial growth
in
dXV Q X Q X Reactiondt
dXV V k Xdt
dXk X
dt
X = population biomass, mg/LV = volume, LQ = flow, L/dk = 1st order rate coefficient, 1/dt = time, d
(Mihelcic 1999, Figure 5.4)
Exponential growth model
max
dXX
dt
( )0
ttX X e
when applied to growth rate calculations, the notation for the 1st order rate coefficient (k) is replaced by , termed the specific growth rate coefficient.
Environmental Resistance
(Mihelcic 1999, Figure 5.5)
Logistic growth model
max 1dX X
Xdt K
(Mihelcic 1999, Figure 5.7)
K = carrying capacity, mg/L
Example: carry capacity effects
(Mihelcic 1999, Figure 5.6)
Monod Model
maxs
dX SX
dt K S
(Mihelcic 1999, Figure 5.8)
Consider S,X = f (t)
S = food, mg/LKs = half-saturation constant, mg/L
Low values of Ks indicate an ability to acquire food resources at low concentrations.
Example: resource competition
(Mihelcic 1999, Figure 5.9)
The Yield Coefficient
XY
S
1dS dX
dt Y dt
Y = yield coefficient, mgX/mgS
Consider Y to be the amount of biomass produced per unit food consumed.
The Death (Respiration) Coefficient
d
dXk X
dt
kd = death coefficient , 1/d
It isn’t really death, a singular event, but rather losses of biomass to respiration.
Putting It All Together
max
max
1
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ds
s
dX X Sk X
dt K K S
dS X SX
dt Y K K S
(Mihelcic 1999, Figure 5.10)
These differential equations are solved using numerical methods.
