1. Dynamic plant uptake modeling Stefan Trapp

2. Steady-state considerations: simple & small data need However: often emission pattern is non-steady, e.g.:  non-steady plant growth (logistic)  pesticide spraying  application of manure and sewage sludge on agricultural fields In these scenarios: steady-state solutions are not valid, and dynamic simulation is required. Steady-state vs. dynamic models

3. Three different types of input, namely pulse input ( pesticide spraying) constant input (deposition from air) irregular input (this we cannot solve --> use numerical integration) Dynamic input patterns

4. Dynamic models Designed for - repeated input - dynamic growth - pesticides - manure or sewage appl. - 1 year or 10 year - easy to handle (excel)

5. ● Repeated pulse input from soil or air or constant emission ● Logistic growth of plants (here: summer wheat) ● n variable periods (30 in practice) Dynamic models

6. Dynamic Model Differential equation system In words soil: change of mass = + Input - degradation - uptake into plants roots: change of mass = + uptake from soil - loss to stem - degradation stem: change of mass = + uptake from roots - loss to leaves (fruits) - deg leaves: change of mass = +uptake from stem ± exchange air - degradation fruits: change of mass = +uptake from stem ± exchage air - degradation

7. Differential equation system soil roots (stem) leaves fruits

8. Cascade of compartments Mass balance: "The change of mass in tank 2 is what flows out of tank 1 minus what flows out of tank 2"

9. Differential equation system 1 soil 2 roots 3 stem 4a leaves 4b fruits The system written in a schematic way: Each DE always relates to the DE before, but not to any other DE

10. transfer rate constants k ij (d -1 ) loss rate constant k i (d -1 ) constant external input b (mg kg -1 d -1 ). Structure of the multi-cascade crop model

11. Dynamic Model Same processes, same differential equations, but formulated as matrix 1 is soilk 1 loss rate k 12 transfer rate 2 is rootsk 2 loss rate k 23 transfer rate 3 is stemk 3 loss rate k 34 transfer rate 4 is leaves or fruitsk 4 loss rate b is the input vector

12. Steady-state solution Set dC/dt (left hand) to zero. Then Cascade etc. Conc. = Input / loss

13. etc. … Analytical solution for pulse input, i.e. C(0) ≠ 0

14. Cascade with constant input Analytical solution for all t

15. That's what you always wanted to know about math, wasn't it? Questions?

16. Principle of superposition Concentrations are additive We can thus calculate several subsequent periods with different values, and the output from one period is the input to the next. This allows to simulate non-constant conditions. Our "cascade model" has by default 24 periods to 5 days (= 120 days, i.e. one vegetation period), but this is variable.

17. Principle of superposition Figure: Concentrations are additive

18. Most annual crops show a logistic growth curve initial growth is exponential towards ripening, growth slows down and finally stops Change of plant mass M [kg]: Plant growth kFirst-order rate constant (for exponential growth) [1/d] M max Maximum plant mass[kg] Plant mass as a function of time M0M0 Initial plant mass[kg]

19. Growth and transpiration of plants are related by the water use efficiency (kg plant / L water) or the transpiration coefficient T C (L water / kg plant). Typical values range between 200 and 1000 L/kg dry weight Default value for T C is 100 L/kg fresh weight. Plant growth and transpiration QTranspiration[L/d] TCTC Transpiration coefficient[L/kg dw] In our model, transpiration takes place only when plants are growing

20. Data obtained from agricultural handbooks (summer wheat) Annual seed plant - Initial mass 10 -4 kg (0.1 g for seeds) - Growth rate constant k = 0.1 d -1 (doubling time ≈ 1 week) - Final mass 1 kg data related to 1 m 2 Transpiration coefficient T C = 50 L/kg fw(water content green plans ≈ 90%) Plant growth and transpiration

21. Standard scenario: summer wheat Maximum transpiration Q max is at ½ M max (inflection point) with at time Plant growth and transpiration

22. Annual seed plant Plant growth and transpiration Growth is exponential for t < 70 d Absolute growth & transpiration peak at t = 92 d Growth almost stops for t > 135 d = phase in which fruit or corn ripe leaves decay and plants dry out Biomass M and transpiration Q of summer wheat Dynamic model: Default scenario

23. Reading:

24. Example simulation for a repeated pesticide application Repeated application of insecticide by drip irrigation to soil

25. Comparison to measured data Model result before calibration

26. Comparison to measured data After fit of two parameters (temperature, soil depth)

27. More reading:

28. Lucky You - you survived this part. Questions?