CALCULUS FOR MODELLING 02
Variables and Parameters
Variables describe quantities that change within a model; parameters describe numerical characteristics that control how the model behaves.
The main idea
Consider
The population \(P(t)\) is a model variable. The growth-rate parameter \(r\) is usually treated as fixed during one model run.
1. Independent variable
In a dynamical model the independent variable is commonly time:
\[t.\]Other quantities are described as functions of it.
2. State variables
A state variable records the current condition of the system. Examples include population \(P(t)\), infectious individuals \(I(t)\), temperature \(T(t)\), or concentration \(C(t)\).
3. Parameters
Parameters determine characteristics of the model. Common examples are
\[r,\quad K,\quad \beta,\quad \gamma.\]They may represent growth rates, carrying capacities, transmission coefficients or recovery rates.
4. Fixed does not mean universally constant
A parameter may be fixed within a particular model but differ between populations, experiments or time periods. It may also be deliberately made time-dependent in a more detailed model.
5. Initial conditions
An initial condition specifies the state at a chosen starting time:
The parameter \(P_0\) determines which particular solution trajectory is followed.
6. Parameter effects
Changing a parameter can change the speed, equilibrium, threshold or shape of a solution. Studying this dependence is called sensitivity analysis.
7. Estimating parameters
Parameters are often estimated from observations. Their values therefore may contain uncertainty even when the mathematical model treats them as fixed numbers.
8. Model notation
It is useful to distinguish clearly between
9. Example: logistic model
In
\[\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right),\]\(P(t)\) is the changing state, \(r\) controls intrinsic growth, and \(K\) represents carrying capacity.