Team:KU Leuven/Project/Ecological/Modelling

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tree ladybugcartoon

Ultimately our project aims to reduce crop loss because of aphid infestation. Given the time span of the competition, it is impossible to conduct a field experiment for this. Therefore we attempt to predict the effect of our pheromones on the ecology through a series of modeling steps.

The first step is to calculate the concentration of the pheromones released in the environment. When a colony of E. coligy in what form so ever is placed in a field, the produced substances will be transported in the air by diffusion and convection. Diffusion is always present, whereas the source of the convection term is the wind. In order to establish a realistic model, certain parameters are needed. Therefore approximate diffusion coefficients and air velocity were searched. Production of the pheromones by the colony was coupled with the other modeling approach <link to Sander’s text>.

Convection-diffusion equation

Diffusion coefficients

Because we found no measured diffusion coefficients of the pheromones in literature, estimations were made with a calculator based on methods described in (Lyman, Reehl, & Rosenblatt, 1982). Using the average supplied by the calculator, the results are 4.62 x 10-6 m2/s for E-beta-farnesene and 6.33 x 10-6 m2/s for methyl salicylate. The conditions supplied were a pressure of 1 atm and a temperature of 15 °C.


Plot of the wind profile

Figure 1 ǀ Wind profile for a crop height of 2 m and a wind speed of 3.39 m/s at a height of 10 m.

Wind speed

Because of friction and obstacles on the earth’s surface, wind speed varies with altitude. Generally, the velocity decreases with increasing altitude. For the part above the crops we can use the logarithmic wind profile is appropriate (Goudriaan, 1977, p. 96). The formula for this profile is

Wind profile

with u representing the velocity. Here d accounts for an upward shift above a vegetative cover. The relation d=0.63 x zc is suggested, where zc is the height of the crops. The length z0 is called the roughness length and is often supposed to be about one tenth of zc.

For the part inside the canopy, the profile is exponential.

Wind profile inside canopy

As can be noticed, the wind speed decreases exponentially with extinction factor a when approaching ground level and thus going deeper into the canopy. uc is the speed at height zc and can easily be calculated by the formula of the logarithmic wind profile.

Evidently, the wind direction changes in time. Most regions however tend to have a dominant wind (“Prevailing winds,” 2013), at least during a certain time period. This is the reason our model only incorporates convection in one direction, which greatly simplifies calculation. Furthermore we can also find average wind speeds, measured by KMI in Belgium (“Maandelijkse normalen - KMI,” 2013). These are measured at 10 m above the ground, making it possible to calculate u*/k used in the log law. Now, all parameters for convection are known when the crop height is given. The wind profile was entered in the software using a piecewise function and self-defined parameters, making it easy to change wind speed and crop height. An example of the profile is plotted in Figure 1.

Boundary conditions

In order to solve our model, boundary conditions need to be specified.

Inflow and outflow

The faces perpendicular to the wind direction were specified as an inflow (with a concentration of 0 mol/m3) respectively an outflow.

No flux

The face representing the ground will have no flux through it, because we assume that the diffusion coefficient of our pheromones is much larger in soil than in air. Furthermore, all faces but one of the volume representing a container with bacteria are given zero flux as well.

Flux

The upper face of this container is the only place where pheromones are released in the air. Under the assumption that the pheromone production and vaporization is at steady state, we can set the flux at this surface equal to the production rate of the entire colony.

Once we know the production per cell it is straightforward to calculate the output of the whole colony by using an average cell density in the appropriate growth phase. This output represented as an amount per time can then be converted to a flux through the contact surface with the air, which can be entered in the simulation program.

<PART SANDER>


Geometry for the model

Figure 2 ǀ Geometry for the model: the blue dot is the container with E. coligy, a is an influx plane, b an open boundary and c a face with no flux.

Analysis


Slice of the geometry showing the concentration

Figure 3 ǀ Slice of the geometry showing the concentration.

To be written

Software

At first, we used Mathworks Matlab with the Partial Differential Equations Toolbox, but this software was limited to 2D geometries and more stringent boundary conditions. Later on, we noticed that the Department of Engineering of our university provided COMSOL Multiphysics. This program was very suitable for our purposes, as it provides a model for “Transport of Diluted Species”. Our complete model can be downloaded here.


COMSOL Multiphysics

Goudriaan, J. (1977). Crop micrometeorology : a simulation study. Wageningen University, Wageningen.
Lyman, W. J., Reehl, W. F., & Rosenblatt, D. H. (1982). Handbook of chemical property estimation methods: environmental behavior of organic compounds. McGraw-Hill.
Maandelijkse normalen - KMI. (n.d.). Meteo. Retrieved August 22, 2013, from http://www.meteo.be/meteo/view/nl/65239-Home.html
Prevailing winds. (2013, August 21). In Wikipedia, the free encyclopedia. Retrieved from http://en.wikipedia.org/w/index.php?title=Prevailing_winds&oldid=569524163