# Team:Valencia UPV/Modeling/diffusion

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- | $$\frac{\partial c }{\partial t} = D \left(\frac{\partial^2 c }{\partial^2 x} + \frac{\partial^2 c }{\partial^2 y}\right) – \left(v_{x} \cdot \frac{\partial c } | + | $$\frac{\partial c }{\partial t} = D \left(\frac{\partial^2 c }{\partial^2 x} + \frac{\partial^2 c }{\partial^2 y}\right) – \left(v_{x} \cdot \frac{\partial c }{\partial x} + v_{y} \cdot \frac{\partial c }{{\partial y} \right) = D \left( c_{xx} + c_{yy}\right) - \left(v_{x} \cdot c_{x} + v_{y} \cdot c_{y}\right) $$ |

For determining a numeric solution for this partial differential equation are | For determining a numeric solution for this partial differential equation are |

## Revision as of 13:03, 14 October 2014

### Modeling > Pheromone Diffusion

and

Sexually communication among moths is accomplished chemically by the release of an "odor" into the air. This "odor" is the sexual pheromones.

Pheromones are molecules that can undergo a diffusion process in which the random movement of gas molecules transport the chemical away from its source (Sol I. Rubinow, Mathematical Problems in the Biological Sciences, Lecture 9). However, diffusion processes are complex, and modelling them analytically and with accuracy is difficult, even more when the geometry is not simple.

For this reason, we decided to consider a simplified model in which pheromone chemicals obey to the heat diffusion equation. Then, it is solved by the Euler numeric approximation in order to obtain the spatial and temporal distribution of pheromone concentration. See more about heat equation and mathematical expressions for Euler method.

Moths seem to respond to gradients of pheromone concentration attracted towards the source, although there are other factors that lead moths sexually to pheromone sources such as optomotor anemotaxis (J. N. Perry and C. Wall , A Mathematical Model for the Flight of Pea Moth to Pheromone Traps Through a Crop). However, increasing the pheromone concentration to unnaturally high levels may disrupt male orientation (W. L. Roelofs and R. T. Carde, Responses of Lepidoptera to synthetic sex pheromone chemicals and their analogues). See more about the modeling of moth flight paths.

Using a modeling environment called Netlogo, we simulate the approximate moths behavior during the pheromone dispersion process. So, this will help us to predict moth response when they are also in presence of our synthetic plants.

Since pheromones are chemicals released into the air, we have to consider both the motion of the fluid and the one of the particles suspended in the fluid.

The motion of fluids can be described by the Navier–Stokes equations. But the frequent nonlinearity of these equations makes most problems difficult or impossible to solve, since it may exists turbulences in the air flow.

Now attending to the particles suspended in the fluid, an option for pheromone dispersion modeling consists in the assumption of pheromones diffusive-like behavior. That is: pheromones are molecules that can undergo a diffusion process in which the random movement of gas molecules transport the chemical away from its source [1].

There are two ways to introduce the notion of diffusion: either a phenomenological approach starting with Fick's laws of diffusion and their mathematical consequences, or a physical and atomistic one, by considering the random walk of the diffusing particles.

In our case, we decided to hold our diffusion process by the Fick's laws. So it is postulated that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient. However, diffusion processes are complex, and modelling them analytically and with accuracy is difficult, even more when the geometry is not simple (final distribution of our plants in the crop field). For this reason, we decided to consider a simplified model in which pheromone chemicals obey to the heat diffusion equation.

### Approximation

The **diffusion equation** is a partial differential equation which describes density dynamics
in a material undergoing diffusion. It is also used to describe processes exhibiting
diffusive-like behavior, like in our case.

The equation is usually written as:

$$\frac{\partial \phi (r,t) }{\partial t} = \nabla • [D(\phi,r) \nabla \phi(r,t)]$$

where $\phi(r, t)$ is the density of the diffusing material at location r and time t and $D(\phi, r)$ is the collective diffusion coefficient for density $\phi$ at location $r$; and $\nabla$ represents the vector differential operator.

If the diffusion coefficient does not depend on the density then the equation is linear and $D$ is constant. Thus, the equation reduces to the following linear differential equation: $$\frac{\partial \phi (r,t) }{\partial t} = D \nabla^2 \phi(r,t)$$

also called the **heat equation**.

Making use of this equation we can write pheromones chemicals diffusion equation with no wind effect consideration as:

$$\frac{\partial c }{\partial t} = D \nabla^2 C = D \Delta c$$

where c is the pheromone concentration, $\Delta$ is the Laplacian operator, and $D$ is
the pheromone diffusion constant in air.

If we consider the wind, we face a diffusion system with drift and an advection term is added to the equation above.

$$\frac{\partial c }{\partial t} = D \nabla^2 c - \nabla \cdot (\vec{v} c )$$

where $\vec{v}$ is the average *velocity* that the quantity is moving. Thus, $\vec{v}$
would be the velocity of the air flow.

For simplicity, we are not going to consider the third dimension. In $2D$ the equation would be:

$$\frac{\partial c }{\partial t} = D \left(\frac{\partial^2 c }{\partial^2 x} + \frac{\partial^2 c }{\partial^2 y}\right) – \left(v_{x} \cdot \frac{\partial c }{\partial x} + v_{y} \cdot \frac{\partial c }{{\partial y} \right) = D \left( c_{xx} + c_{yy}\right) - \left(v_{x} \cdot c_{x} + v_{y} \cdot c_{y}\right) $$

For determining a numeric solution for this partial differential equation are used the so-called finite difference methods. The technic consists in approximating differential ratios as $h$ is closer to zero, so they are useful to approximate differential equations.

With finite difference methods, partial differential equations are replaced by its approximations in finite differences, resulting in an algebraic equations system. The algebraic equations system is solved in each node $(x_i,y_j,t_k)$. These discrete values describe the temporal and spatial distribution of the unknown function.

Although implicit methods are unconditionally stable so time steps could be larger and make the calculus process faster, the tool we have used to solve our heat equation is the Euler explicit method.

Euler explicit method is the simplest option to approximate spatial derivatives, in which all values are assumed at the beginning of Time.

The equation gives the new value of the pheromone level in terms of initial values in that node and its immediate neighbors. Since all these values are known the process is called explicit.

$$c(t_{k+1}) = c(t_k) + dt \cdot c'(t_k),$$

Now applying this method for the first case (with no wind consideration) we followed the next steps.

1. Split time $t$ into $n$ slices of equal length $\dt$: $$ \left\{ \begin{array}{c} t_0 &=& 0 \\ t_k &=& k \cdot dt \\ t_n &=& t \end{array} \right. $$

2. Considering the backward difference for the Euler explicit method implies that the expression that refers to the current pheromone level each time step is:

$$c (x, y, t) \approx c (x, y, t - dt ) + dt \cdot c'(x, y, t)$$

3. And now considering the spatial dimension, it is applied central differences to the Laplace operator $$\Delta$$, and the backward differences to the vector differential operator $$\nabla$$ ( in 2D and assuming equal steps in x and y directions)

$$c (x, y, t) \approx c (x, y, t - dt ) + dt \left( D \cdot \nabla^2 c (x, y, t) - \nabla \vec{v} c (x, y, t) \right)$$ $$ D \cdot \nabla^2 c (x, y, t) = D \left( c_{xx} + c_{yy}\right) = D \frac{c_{i,j-1} + c_{i,j+1} + c_{i-1,j } + c_{i+1,j} – 4 c_{I,j}}{s} $$ $$ \nabla \vec{v} c (x, y, t) = v_{x} \cdot c_{x} + v_{y} \cdot c_{y} = v_{x} \frac{c_{i,j} – c_{i-1,j}}{h} + v_{y} \frac{\frac{c_{i,j} – c_{i,j-1}}{h} $$

With respect to the boundary conditions, they are null since we are considering an opened-space. Attending to the implementation and simulation of this method, CURSIVA!!dt must be small enough to avoid instability.

**NetLogo** is an agent-based programming language and integrated modeling environment. **NetLogo** is free and open source **software**, under a GPL license.

**NetLogo** is an agent-based programming language and integrated modeling environment. **NetLogo** is free and open source **software**, under a GPL license.

**NetLogo** is an agent-based programming language and integrated modeling environment. **NetLogo** is free and open source **software**, under a GPL license.