Population diversity

The HPC model contemplates a vector (pop) of length n, containing the population fractions of each type. The number of types are population-specific and are given as two parameters (\(n_{H}\), \(n_{P}\)). These types include all possible variations within a population so that this vector amounts to the unity (\(\sum_{i=1}^{n}{pop_{A_{i}}}=1\)).

To account for multiple types, we replace Eq. 5 and 6 with Eq. 7 and 8, where the utility of population A to B at any given time (\(U_{AB}[t]\)) is calculated by summing up the utility per capita of each type (\(\bar{U}_{A_{i}B}\)) proportionally to the share of population of the respective type (\(pop_{A_{i}}[t]\)), and multiplying the result by the population at a given time. The baseline carrying capacity (\(U_{bA_{i}}[t]\)) is calculated in a similar manner, though using the utility that each type is able to gain from other resources (\(U_{bA_{i}}\)) (Eq. 9 and 10).

\[\begin{equation} \tag{Eq. 7} U_{HP}[t]=H[t]\sum_{i=1}^{n_{H}}{pop_{H_{i}}[t]\cdot \bar{U}_{H_{i}P}} \end{equation}\]

\[\begin{equation} \tag{Eq. 8} U_{PH}[t]=P[t]\sum_{i=1}^{n_{P}}{pop_{P_{i}}[t]\cdot \bar{U}_{P_{i}H}} \end{equation}\]

\[\begin{equation} \tag{Eq. 9} U_{bH}[t]=\sum_{i=1}^{n_{H}}{pop_{H_{i}}[t]\cdot U_{bH_{i}}} \end{equation}\]

\[\begin{equation} \tag{Eq. 10} U_{bP}[t]=\sum_{i=1}^{n_{P}}{pop_{P_{i}}[t]\cdot U_{bP_{i}}} \end{equation}\]

Types relate to population-specific values of utility per capita (\(\bar{U}_{A_{i}B}\)) and baseline carrying capacity (\(U_{bA_{i}}\)). These values are defined by linear interpolation between pairs of parameters representing the values corresponding to types \(1\) and \(n\) (e.g., if \(n_P=10\), \(\bar{U}_{P_{1}H}=1\) and \(\bar{U}_{P_{n}H}=10\), then \(\bar{U}_{P_{5}H}=5\)). The shares of population within types follow a one-tail distribution rather than a normal distribution, which would be more adequate but less straightforward to use in a theoretical model. Under this circumstance, the distribution of population within types will always be biased towards the intermediate types.