The name is absent

Step 1: Model development

The modeling process consists of the following steps: 1) identification of the relevant system variables for the representation
of the system (state variables); 2) identification of the fluxes (rate variables) which determine the “speed” of the state variable
variations. The “state equations” link the rate variables to the state variables; 3) development of the relations for the rates. Rates
are computed as functions of the state and exogenous variables (not other rates). For a clearer analysis of the system, the rate
equations can be also functions of auxiliary variables.

The third step is the most important for the modeling process. In fact, the evolution of the state variables derives from the
integration of the rate equations, starting from certain “initial conditions.”

Example of model development

Let us consider a classical Volterra model on prey-predator interactions in an ecological system (see Maynard 1974 for
details) and assume that the prey is a herbivorous. Two state variables describe the system conditions: the prey density in the
ecosystem (X, as number of individuals per unit area) and its predator density (Y, as number of individuals per unit area). For
the model development, we make the following assumptions:

1. The birth rate of the prey, without predator and without limiting factors, is a constant fraction of the prey density.

2. The environmental limits determine a maximum value for the prey density that is not possible to exceed. This limit is
generally indicated as “carrying capacity” (C^,_c).

3. The predation rate is a fixed fraction of all the possible encounters between prey and predator (represented by the product
.V × У).

4. The death rate of predator is a constant fraction of the predator density.

5. The birth rate of predator is a fraction of Y density but the predator birth rate tends to zero when the X density approaches
the Y density.

6. In addition, to show the use of exogenous variables, we make a further assumption: if the prey is a herbivorous, the carrying
capacity of the environment (C_c) depends on the annual grass production of the range. C_c, in turn, could be dependent
on the annual ratio (total rainfalls/average temperature) by the equation C_c = 500 + K₁ × (rain∕temp). Rain and temp are
exogenous variables.

The differential equations describing the change with time of X and Y are

where 1∕K_hx, 1∕K_mx, 1∕K_my, and 1∕K_hy are the time coefficients of the rates.

Figure 2 shows the relational diagram of the described system, following the hydraulic analogy proposed by Forrester
(1968). Let the initial values for prey and predator density in the area be 1000 and 100 individuals, respectively. K_hx is the
birth coefficient of the prey population, expressed as fraction of X birth per year, and fixed to 0.05. The birth coefficient for
predator is K_hy = 0.006 and represents the efficiency of the prey-predator conversion. The mortality coefficients for prey and
predator are, respectively, K_mx = 0.001 and K_my = 0.05. The coefficient for grass production (K_i) equals 80.

Considering a time lag, from the predation time to the new predator births, due to the development time (Devt), we can
build a more realistic model by introducing a delay in the birth rate of the equation (2). The births are delayed in respect to the
predation (material delay); the integrating rates are then dependent on the past values of X and Y. Equation (2) becomes

(2) dY∕dt =K_my × Y+K_by ×Y ×(X_t

-Devt Yt-Devt)

variation Y             effect of predation

rate of    death            on predator birth

У       rate           with material delay

where t is the time index and Devt the material delay time due to the development time of predator. We assume that the material
delay for the predator birth is 0.5 year.

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