1. KINETIC MODELS Guy SOULAS UMR Œnologie-Ampélologie
Université Victor Segalen Bordeaux 2
351 cours de la Libération, TALENCE Cedex
France

2. Where does first-order come from ?
Reason 1: the experience
Experience shows that many biotic and abiotic processes in environmental compartments such as soil effectively follow single first order kinetics (exponential decay)

3. Reason 2: pragmatism
·    The equation is simple and has only two parameters
·    It is easy to fit the equation to experimental data
·     DT50 and DT90 values are easy to calculate
·     Parameters are theoretically independent of concentration and time … and appropriate for use as input for pesticide leaching models.

4. Reason 3: scientific justifications
·    abiotic hydrolytic processes often follow first-order reaction kinetics
·    biotic degradation processes may be approximated by first-order reaction : ex. when responsible microbial agents (or enzymes) are in excess compared to the chemical (pseudo first order reaction kinetics).

5. : initial substrate concentration; X initial biomass concentration
A phylogeny for the disappearance models
The « Metabolism » case S
<<K s K
>>K
First -
order
Monod without growth
Zero
<<X m X k kS dt dS = +
Logistic growth
Monod with growth
Logarithmic
>>X ) (
: initial substrate concentration; X :
initial biomass concentration

6. Reason 1: heterogeneity
The Gustafsson and Holden assumption:
The soil can be divided into a large number of independent compartments whith distributed first order rate constants.
If pdf = Gamma distribution …

7. Single first order kinetics (SFO)20 40 60 80
100
Time (days)
Concentration (% of initial)
k = 0.005
k = 0.020
k = 0.050

8. But
First-order reaction kinetics may not be obeyed

9. The bi-phasic Gustafson & Holden model (FOMC)
alpha = 0.2 , beta = 5.00
alpha = 0.2 , beta = 1.00
alpha = 0.2 , beta = 0.05
alpha = 1.0 , beta = 5.00
alpha = 2.0 , beta = 5.00
Time 10 20 30 40 50 60 70 80 90
100
Concentration

10. Reason 2: limited availability.
In the soil, pesticides are distributed between a solid phase and a liquid phase where they are available for degradation. This partition induces a bi-phasic pattern of degradation

11. The bi-phasic Hockey Stick model (HS)10 20 30 40 50 60 70 80 90
100
Time
Concentration
k1 = 0.05 , k2 = 0.01 , tb = 10
k1 = 0.07 , k2 = 0.01 , tb = 10
k1 = 0.09 , k2 = 0.01 , tb = 10
k1 = 0.09 , k2 = 0.01 , tb = 15
k1 = 0.09 , k2 = 0.02 , tb = 15

12. The bi-phasic bi-exponential model (DFOP)
k1 = 0.03 , k2 = , M1 = 75
k1 = 0.06 , k2 = , M1 = 75
k1 = 0.09 , k2 = , M1 = 75
k1 = 0.09 , k2 = , M1 = 75
k1 = 0.09 , k2 = , M1 = 90 10 20 30 40 50 60 70 80 90
100
Time
Concentration

13. Reason 3: microbial behaviour
 Different environmental factors affect the activity of the microbial degraders.
 Respective substrate concentration and cell density may induce very different degradation patterns

14. : initial substrate concentration; X initial biomass concentration
A phylogeny for the disappearance models
(1) Metabolism S
<<K s K
>>K
First -
order
Monod without growth
Zero
<<X m X k kS dt dS = +
Logistic growth
Monod with growth
Logarithmic
>>X ) (
: initial substrate concentration; X :
initial biomass concentration

15. True lag phase: The logistic model
a0 = , r = 0.2
a0 = , r = 0.4
a0 = , r = 0.8
a0 = , r = 0.2
a0 = , r = 0.2
100 20 40 60 80
Time
Concentration

16. The Hockey stick model (HS)
Lag phase:
The Hockey stick model (HS) 20 40 60 80
100 10 30 50
Time
Concentration

17. : initial substrate concentration; X initial biomass concentration
A phylogeny for the disappearance models
(1) Metabolism S
<<K s K
>>K
First -
order
Monod without growth
Zero
<<X m X k kS dt dS = +
Logistic growth
Monod with growth
Logarithmic
>>X ) (
: initial substrate concentration; X :
initial biomass concentration