Coevolutionary dynamics
Undirected variation, which causes part of the population to randomly change to other types, represents the effect of mutation in genetic transmission or of innovation, error, and other mechanisms in cultural transmission. The balance of the subpopulation A of type \(i\) depends on the level of undirected variation (\(v_{A}\)) and on the degree and sign of the difference between the current subpopulation (\(pop_{A_{i}}[t]\)) and the averaged subpopulation (\(1/n_{A}\)), which refers to the completely uniform distribution among types (Eq. 11 and 12).
\[\begin{equation} \tag{Eq. 11} pop_{H}[t]'=pop_{H}[t]+v_{H}\left(\tfrac{1}{n_{H}}-pop_{H}[t]\right) \end{equation}\]
\[\begin{equation} \tag{Eq. 12} pop_{P}[t]'=pop_{P}[t]+v_{P}\left(\tfrac{1}{n_{P}}-pop_{P}[t]\right) \end{equation}\]
By considering inertia, we are assuming that the more frequent a type is, the more likely that it is transmitted. Selection is implemented by assigning a fitness score to each type (\(fitness_{A_{i}}[t]\)), which in turn biases its transmission. Equations 13 and 14 summarizes the combined effect that inertia and selection have on the proportion of population A belonging to type \(i\) (\(pop_{A_{i}}[t]\)) [for a formal similarity of the discrete replicator dynamic and Bayesian inference see Harper2009].
\[\begin{equation} \tag{Eq. 13} pop_{H_{i}}[t+1]=\frac{fitness_{H_{i}}[t]\cdot pop_{H_{i}}[t]}{\sum_{j=1}^{n_{H}}fitness_{H_{j}}[t]\cdot pop_{H_{j}}[t]} \end{equation}\]
\[\begin{equation} \tag{Eq. 14} pop_{P_{i}}[t+1]=\frac{fitness_{P_{i}}[t]\cdot pop_{P_{i}}[t]}{\sum_{j=1}^{n_{P}}fitness_{P_{j}}[t]\cdot pop_{P_{j}}[t]} \end{equation}\]
This mechanism defines how a trait evolves in a single population. However, coevolution can also be represented when the selective pressure on this population is modified by the changing traits of another population. In order to link the two populations, fitness scores of population A are derived from the weight of the contribution or utility of population B (\(U_{BA}[t]\)) in relation to the base carrying capacity of A (\(K_{A}[t]\)) (Eq. 10).
\[\begin{equation} \tag{Eq. 15} fitness_{H_{i}}[t]=\frac{(n_{H}-i)\,U_{bH}[t]+i\,U_{PH}[t]}{U_{bH}[t]+U_{PH}[t]} \end{equation}\]
\[\begin{equation} \tag{Eq. 16} fitness_{P_{i}}[t]=\frac{(n_{P}-i)\,U_{bP}[t]+i\,U_{HP}[t]}{U_{bP}[t]+U_{HP}[t]} \end{equation}\]
As a consequence of this model design, types of both human and plant populations span from a non-mutualistic type (\(i=1\)), which has the best fitness score when there is no positive interaction with the other population (\(U_{BA}[t]\approx 0\)), to a mutualistic type (\(i=n\)), which is the optimum when nearly the whole of the carrying capacity is due to such relationship (\(U_{BA}[t]\approx K_{A}[t]\)).
| Domains | Assumptions |
|---|---|
| On the evolution of traits | A population can be divided into types according to one or more traitsThe distribution of individuals among types can vary in time, due to factors affecting trait transmission |
| On the factors affecting the evolution of traits | Change of the population distribution among types depends on the previous population distribution: the more frequent is a type, the more likely it will be imitated or transmitted to the next generation |
| | Change of the population distribution among types depends on the relative fitness of types: the greater the fitness score associated to a type, the more likely it will be imitated or transmitted to the next generation |
| | Change of the population distribution among types depends on undirected variation |
| On the coevolution of traits related to human-plant mutualism | The utility given by an individual varies within types |
| | The utility given by other resources to a population varies within its types |
| | The fitness of human types is modified by the relative weight of plant utility in the carrying capacity for humansThe fitness of plant types is modified to the relative weight of human utility in the carrying capacity for plants |