The name is absent

The application of the system analysis method does not require great mathematical skill and is suitable for all dynamic and
continuous systems. Generally, these models are deterministic and the simulation results are the same if initial values, parameters
and exogenous variables are the same. It is also possible to simulate stochastic behaviors by sampling the rate variable values
from probability distributions or by superimposing a noise to the rate value.

System analysis begins with (a) structure identification, proceeds to (b) identification of information links and rate calculations,
followed by (c) model coding and (d) specifying parameters and initial values so that the simulation can be performed.

A) System structure identification

The first phase pertains to the identifications of the variables that represent the state of the system, at least in relation to
the model purposes. These are named state or level variables. In a relation diagram, the symbol used to represent state variables
is the box (Figure 1). We must also define all the incoming and outcoming fluxes (rate variables) for each state variable. The
model state variables are computed by the state equations. The structure of a state equation is

svar_tjt = svar,,₍_ I + dt × (±rvarι_jt-ι ± rvar₂,_t-ι ÷ ∙ ∙ ∙)

where svar, is a state variable and appears on both sides of the equation, rvar_j- are rate variables added or subtracted to the state
variable, and dt the time step for the rate integration in the state variables. The above equation means that, for each instant t,
the value of the state variable is obtained algebraically by adding the variation rates in the period (t — 1, t) to the previous value
of the state variable at time t — 1. The value of the rate variable in period (t — 1, t) is a function of the state variable values at
time t — 1. If dt is small enough, rvar_jjt-ι is a good approximation of the mean of rvar, in the (t — 1,t) period.

B) Identification of information links and rate calculations

To calculate the state equations, we define the formulations for all the rate variables contained in the state equations. In the
relational diagram, a solid arrow represents the material fluxes and a valve symbol the rate variables (Figure 1). The structure
of the rate equation might be

rvar_t-ι = (1∕TC) × DForce_t-ι

where rvar is the name of a rate variable, TC is the time constant (in time units) and DForce is the part of the equation representing
the quantity to which the rate is proportional. When a rate equation is a function of a state variable, an “information link” is
said to exist. The information links are generally represented by dashed arrows exiting a state-variable box and entering a valve
(the rate variable symbol) (Figure 1).

If an information link controlling a rate derives from the same state variable modified by the rates itself, there is said to be
a “feedback.” If DForce contains a “goal,” the feedback is negative; if a goal does not exist (ex: DForce = svar), the feedback
is positive and the trend of the state variable is auto-catalytic, with an increasing exponential trend.

The concepts that it is possible to find in a rate equation are

1. A goal: A parameter that represents the value toward which the state variable converges.

2. A way to express the “divergence” between the goal and the observed system condition: Depending on the system kind
and the field, this divergence takes different names: error, driving force, potential differential, tension.

3. An apparent condition of the system: This can be the present value of a state variable or (if there are delays in the information
fluxes) the values of that state variable in a previous time. In any case, the apparent conditions of the system (not the real
conditions) determine the rate values. For example, in a prey-predator ecological system, the birth rate of predator is a
function of the prey density. Really, there is a material delay due to the development time: the rate quantity is integrated
to the state variable after a certain time. If the system has control mechanisms (that is has a goal) depending on the state
variable values, it can detect the real “state” of the system with a delay. The state value that enters the rate equation is a
past value of the state variables.

4. An effect of the exogenous variables: Systems can be sensitive to the conditions outside the defined system bound. In
this case the model should also consider some external variables, named “exogenous variables.” They contribute to the
formulation of the rate and auxiliary equations.

5. An indication about the response intensity to the divergence between apparent states and goals: If the dimension of this
parameter is time then it is called a “time coefficient” (TC). Time coefficients, depending on the model, take the names
of “resistance,” “relaxation time,” “average residence time,” “average departure or transit time” or “average life time.” If
the intensity parameter is expressed in quantity per unit time its meanings can be that of “relative growth rate,” “decay

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