Ecological relationships and population dynamics
The model can be expressed by a relatively simple system of two discrete-time difference equations (Eq. 1 and 2), based on the Verhulst-Pearl Logistic equation (chapter 9, Kingsland 1982; Pianka 1974). The change of both populations (\(\Delta H[t]\), \(\Delta P[t]\); see Table 3) depends on an intrinsic growth rate (\(r_{H}, r_{P}\)), the population at a given time (\(H[t]\), \(P[t]\)) and the respective carrying capacity of the environment for each population (\(K_{H}[t]\), \(K_{P}[t]\)), which may vary over time.
\[\begin{equation} \tag{Eq. 1} H[t+1]=H[t]+r_{H}H[t]-r_{H}\frac{H[t]^{2}}{K_{H}[t]} \end{equation}\]
\[\begin{equation} \tag{Eq. 2} P[t+1]=P[t]+r_{P}P[t]-r_{P}\frac{P[t]^{2}}{K_{P}[t]} \end{equation}\]
Human and plant populations engage in a mutualistic relationship, where one species is to some extent sustained by the other (Eq. 3 and 4). The mutualistic relationship is defined in the model as an increment of the carrying capacity of population B caused by population A (\(U_{AB}[t]\)). This increment, expressed as the utility of A to B at a given time is the product of the utility per capita of A to B (\(\bar{U}_{AB}\)) and the population A at a given time (Eq. 5 and 6).
We consider that both populations are sustained also by an independent term, representing the baseline carrying capacity of the environment or the utility gain from other resources, which is time-dependent (\(U_{bH}[t]\), \(U_{bP}[t]\)). While we assume that the growth of the human population has no predefined ceiling, the expansion of the plant population is considered limited as the area over which plants can grow contiguously (\(MaxArea\)), and represented as a compendium of both space and the maximum energy available in a discrete location (Eq. 4).
\[\begin{equation} \tag{Eq. 3} K_{H}[t]=U_{PH}[t]+U_{bH}[t] \end{equation}\]
\[\begin{equation} \tag{Eq. 4} K_{P}[t]=\min(U_{HP}[t]+U_{bP}[t],\, MaxArea) \end{equation}\]
\[\begin{equation} \tag{Eq. 5} U_{HP}[t]=H[t]\cdot \bar{U}_{HP} \end{equation}\]
\[\begin{equation} \tag{Eq. 6} U_{PH}[t]=P[t]\cdot \bar{U}_{PH} \end{equation}\]
Considering that mutualistic relationships involve a positive feedback loop, the population growth at time t improves the conditions for both humans and plants at time \(t + 1\), sustaining their growth even further.
| Domains | Assumptions |
|---|---|
| On interacting populations | A population of humans interacts with a population of plants |
| On population growth | Population growth is a self-catalyzing process, where the population density in the present will contribute to its own increase in the future, depending on an intrinsic growth rate (r) |
| | Population growth is a self-limiting process, where the population density in the present will constraint its own increase in the future, depending on respective carrying capacity of the environment (K) |
| | The logistic growth model is acceptable as an approximation to the dynamics of populations, both human and plant, under constant conditions |
| | The carrying capacity of the environment for a population depends on constant factors and on a time-varying factor (K[t]) |
| On positive ecological relationships | Positive ecological relationships exist, where an individual of one population increases by an amount the carrying capacity of the environment for another population |
| | Coupled positive ecological relationships (i.e., mutualism) exist, where two populations increase the carrying capacities for each other |
| | There is variation in positive ecological relationships, so individuals of one population vary in terms of how much they increase the carrying capacity for the other population |
| On human-plant mutualism | A given plant species yield a positive utility for humans, e.g., as a source of food and raw materials |
| | Humans return a positive utility for this plant species, e.g., by improving soil conditions |
| | The utility given by one population adds value to the carrying capacity for the other, and vice versa |
| | The carrying capacity for humans rely also in other resources, which are independent of the plant species (i.e., the baseline carrying capacity for humans) |
| | The carrying capacity for plants rely also in other conditions, which are independent of humans (i.e., the baseline carrying capacity for plants) |
| | The carrying capacity for plants is eventually constrained by the space available for it to grow contiguously as a population (i.e., maximum area) |