Once we have a description of a system, and a nice drawing to represent it, we can answer some interesting questions. One of the most common questions is *what is the behavior of the system?* In other words, we usually want to know *what will happen?*.

To answer this, we can use the computer. It is not hard to write a small program to simulate the system. The simulation will be an *approximate* answer, good enough to see the important characteristics of the system’s behavior.

In this document we will consider systems represented by graphs like in the last post “Drawing Systems”, and we will write our computer code using the R language.

## The nodes have values

**Figure 1.** A model of the *cell growth system*. For this example we will use the system represented by the graph of Figure 1.

In this graph each circle represents an item of the system. The label of the circle is the name of the item. In our example we have two circles, named *cells* and *food*. The quantity of each item is represented by a vector with the same name. In this case, the vectors are `cells`

and `food`

.

Each circle has also a value that can change through time, usually the *quantity* or *concentration* of the item in the system. This value has the same name as the circle label. In our example an item as the value `cells`

and the other the value `food`

.

In these kind of systems each item has a second value, corresponding to the *change* of the item’s quantity in each time-step. The name of this value starts with `d_`

, followed by the item label. In our example these values are called `d_cells`

and `d_food`

.

The *process* are represented by boxes and are much simpler. They only have one constant value, which is a rate. The value name is made with the label of the box and the suffix `_rate`

. So the box called *replication* has a constant named `replication_rate`

.

In summary, each circle has two vectors, and each box a fixed number.

## Discrete time

In the computer the time is represented by an integer that increases step by stepIn other words, we use *discrete time*. . The units are arbitrary, so we can think of anything between microseconds and millennia, including years, days, hours, seconds, weeks, etc. We only assume that each time-step has the same duration.

For example in our case we can think that each time-step is one hour. Time 1 corresponds to the initial condition, that is, the beginning of the experiment. Time 2 is one hour later. Time 25 is one day later.

We can use any symbol to represent time. Most people uses letters `i`

, `j`

or `k`

, since these are the letters traditionally used as index of vectors and matrices. In this example we are going to represent time with `i`

.

The items of the system, represented by circles in the graph, have two values that change with time. We represent these variables by vectors, one element for each time-step. So the number of cell at the time `i`

is `cells[i]`

and its change at the same time is `d_cells[i]`

.

## Finding the equations

To simulate the system we need to find the formulas. We start with the boxes, because they are the easy ones.

For each box in the graph we get a single term: the **multiplication** of the rate constant and each of the *quantity* variables of the circles connected by incoming arrows. If there are several incoming arrows from the same circle, then the variable is multiplied several times. The outgoing arrows are not important in this part.

In our example the formula for *replication* box is `replication_rate * cells[i-i] * food[i-1]`

. Here we use the index `i-1`

because we do not know yet the value of `cells[i]`

or `food[i]`

. We will calculate them later. Metaphorically, at the begin of *“today”* we only know *“yesterday”*.

The formula for the circles is made with the **sum** of all the terms of boxes connected with incoming arrows, **minus** all the terms of boxes connected with outgoing arrows. This value is assigned to the *change* variable of the circle.

In our example the variable `d_food[i]`

gets assigned the value `-replication_rate * cells[i-i] * food[i-1]`

, since the *food* circle has only one outgoing arrow. The *cell* circle has two incoming arrows from *replication* and one outgoing arrow to *replication*. Therefore the variable `d_cells[i]`

gets the value

```
replication_rate * cells[i-i] * food[i-1]
+ replication_rate * cells[i-i] * food[i-1]
- replication_rate * cells[i-i] * food[i-1]
```

which, after simplification, is `replication_rate * cells[i-i] * food[i-1]`

resulting on a final result of one positive incoming arrow. In summary, the formulas for the *change* variables are:

```
d_food[i] <- -replication_rate * cells[i-i] * food[i-1]
d_cells[i] <- replication_rate * cells[i-i] * food[i-1]
```

Finally, the *quantity* variables have to be updated. Each *quantity* variable is the cumulative sum of the change variables. In our example we have

## Initial conditions

The last missing piece necessary for the simulation are the initial values of the circles’ variables. The value of `cells[`

*today*`]`

depends on the value of `cells[`

*yesterday*`]`

, and we can only calculate that for time `i`

greater than or equal to 2.

In our case we will use the variables `cells_ini`

and `food_ini`

to store the values corresponding to `cells[1]`

and `food[1]`

. For the *change* variables, we can assume that they are initially zero.

## Complete code

Our simulation will be a function. The inputs are:

`N`

, the number of simulation steps,`replication_rate`

, the rate of change for the given time`cells_init`

, the initial quantity of cells`food_ini`

, the initial quantity of food

and the output will be a *data frame* with the vectors `cells`

and `food`

. Thus, we know that the code in R should be something like this:

```
cell_culture <- function(N, replication_rate, cells_init, food_ini) {
# create empty vectors
# initialize values
for(i in 2:N) {
# update `d_cells` and `d_food`
# update `cells` and `food` as a cumulative sum
}
return(data.frame(cells, food))
}
```

To create an empty vector in R we only need to know the vector sizeStrictly speaking, we do not need the *exact* vector size, but the program is faster if we use the correct number. , which in this case is `N`

. Therefore we ca write

Here we use a *trick* in R that allows us to assign the same value to two variables. The line

is equivalent to

The first component of each vector needs an initial value. Our function receives `cells_ini`

and `food_ini`

as inputs. To make thing simple we assume that the *change* values start at zero, so we write

Putting all together, we get

```
cell_culture <- function(N, replication_rate, cells_init, food_ini) {
# create empty vectors
cells <- d_cells <- rep(NA, N)
food <- d_food <- rep(NA, N)
# initialize values
cells[1] <- cells_init
food[1] <- food_ini
d_cells[1] <- d_food[1] <- 0
for(i in 2:N) {
# update `d_cells` and `d_food`
d_cells[i] <- +replication_rate * cells[i-1] * food[i-1]
d_food[i] <- -replication_rate * cells[i-1] * food[i-1]
# update `cells` and `food` as a cumulative sum
cells[i] <- cells[i-1] + d_cells[i]
food[i] <- food[i-1] + d_food[i]
}
return(data.frame(cells, food))
}
```

# Other examples

This modeling and simulation technique is not limited to a particular field of science. The same rules apply to may other systems, small and big. That is why we can use examples from very different areas and still learn something useful for out own interest. Let’s see other examples

## Water formation

**Figure 2.** A model of the *water formation reaction*. A chemical reaction combines hydrogen and oxygen to produce water. To simplify we assume that the reaction is this: \[2H+O\leftrightarrow H_2O\]

This system has 3 items: Hydrogen, Oxygen and Water, represented with the letters H, O and W. This reaction is reversible, so we represent it with two irreversible reactions at the same time: water formation (*r_1*) and water decomposition (*r_2*). We can represent this system with the graph in Figure 2. The R code to simulate this system is:

```
water_formation <- function(N, r1_rate=0.1, r2_rate=0.1,
H_ini=1, O_ini=1, W_ini=0) {
# first, create empty vectors to fill later
W <- d_W <- rep(NA, N) # Water, quantity and change on each time
H <- d_H <- rep(NA, N) # Hydrogen
O <- d_O <- rep(NA, N) # Oxygen
# fill the initial quantities of water, hydrogen and oxygen
W[1] <- W_ini
H[1] <- H_ini
O[1] <- O_ini
d_W[1] <- d_H[1] <- d_O[1] <- 0 # the initial change is zero
for(i in 2:N) {
d_W[i] <- r1_rate*H[i-1]*H[i-1]*O[i-1] - r2_rate*W[i-1]
d_O[i] <- -r1_rate*H[i-1]*H[i-1]*O[i-1] + r2_rate*W[i-1]
d_H[i] <- -2*r1_rate*H[i-1]*H[i-1]*O[i-1] + 2*r2_rate*W[i-1]
W[i] <- W[i-1] + d_W[i]
O[i] <- O[i-1] + d_O[i]
H[i] <- H[i-1] + d_H[i]
}
return(data.frame(W, H, O))
}
```

## Predator-Prey population dynamics

**Figure 3.** A model of the *Lotka-Volterra system*. This is a system with several applications, known as “*Lotka-Volterra system*”. It was discovered and first analyzed by Alfred J. Lotka, and Vito Volterra, and appears again and again in ecology and population dynamics.

Here we have two items: cats and mice, and three processes: birth (of mice), catch (of mice, also reproduction of cats) and death (of cats). The system is represented by the graph in Figure 3 and simulated by the following R code:

```
cat_and_mouse <- function(N, birth_rate, catch_rate, death_rate) {
mice <- d_mice <- rep(NA, N)
cats <- d_cats <- rep(NA, N)
mice[1] <- 1
cats[1] <- 3
d_mice[1] <- d_cats[1] <- 0
for(i in 2:N){
d_mice[i] <- birth_rate*mice[i-1] - cats[i-1]*mice[i-1]*catch_rate
d_cats[i] <- -death_rate*cats[i-1] + cats[i-1]*mice[i-1]*catch_rate
mice[i] <- mice[i-1] + d_mice[i]
cats[i] <- cats[i-1] + d_cats[i]
}
return(data.frame(mice, cats))
}
```

## Money in the bank

This one is an exercise. Consider the following system.

What will be the money in the bank after 12 month if the interest rate is 1/12? What if the interest is 1/100 and the total time is 100 months?

# References

Koch, Ina. *“Petri Nets - A Mathematical Formalism to Analyze Chemical Reaction Networks.”* Molecular Informatics 29, no. 12 (December 17, 2010): 838–43. doi:10.1002/minf.201000086.

Baez, John C., and Jacob Biamonte. *“Quantum Techniques for Stochastic Mechanics,”* September 17, 2012. http://arxiv.org/abs/1209.3632.