The Unification of Evolutionary Dynamics through the Bayesian Decay Factor in a Game on a Graph.

We unify evolutionary dynamics on graphs in strategic uncertainty through a decaying Bayesian update. Our analysis focuses on the Price theorem of selection, which governs replicator(-mutator) dynamics, based on a stratified interaction mechanism and a composite strategy update rule. Our findings suggest that the replication of a certain mutation in a strategy, leading to a shift from competition to cooperation in a well-mixed population, is equivalent to the replication of a strategy in a Bayesian-structured population without any mutation. Likewise, the replication of a strategy in a Bayesian-structured population with a certain mutation, resulting in a move from competition to cooperation, is equivalent to the replication of a strategy in a well-mixed population without any mutation. This equivalence holds when the transition rate from competition to cooperation is equal to the relative strength of selection acting on either competition or cooperation in relation to the selection differential between cooperators and competitors. Our research allows us to identify situations where cooperation is more likely, irrespective of the specific payoff levels. This approach provides new perspectives into the intended purpose of Price's equation, which was initially not designed for this type of analysis.

Stochastic control of ecological networks.

The paper models the maintenance of ecological networks in forest environments, built from bioreserves, patches and corridors, when these grids are subject to random processes such as extreme natural events. It also outlines a management plan to support the optimized results. After presenting the random graph-theoretic framework, we apply the stochastic optimal control to the graph dynamics. Our results show that the preservation of the network architecture cannot be achieved, under stochastic control, over the entire duration. It can only be accomplished, at the cost of sacrificing the links between the patches, by increasing the usage of the control devices. This would have a negative effect on the species migration by causing congestion among the channels left at their disposal. The optimal scenario, in which the shadow price is at its lowest and all connections are well-preserved, occurs at half of the course, be it the only optimal stopping moment found on the stochastic optimal trajectories. In such a scenario, the optimal forestry management policy has to integrate agility, integrated response, and quicker response time.

Comparing forest governance models against invasive biological threats.

In order to take account of the negative effects of invasive species and pathogens on networked forest areas, we study the dynamics of stochastic closed-loop input-output systems faced with the risk of external random perturbations. The extension of previous works on robustness is carried out by introducing a negative feedback mechanism, such that the output from an element contained in the system behaves as a negative input toward elements to which it is connected. Through the study of an overall network divided into compartments barely connected to one another, we first consider the pathway pertaining to monofunctional zoning. By looking at a single aggregated structure, we then move our focus to the pathway proper to multifunctionality. Our results show that, at significant time scales, the monofunctional-zoning mode of forest governance, generally applied in Australasia, performs robustly against invasive biological threats at all levels of outbreak probability. The multifunctional mode of forest governance, further practiced in Western Europe, is mainly sturdy when the probability of invasion verges into certainty. Should this not be the case, robustness is ensured would disturbers and perturbations be uncorrelated. Accordingly, the monofunctional pathway can afford adopting control strategies for outbreak avoidance, which is only acceptable in case the expected invasion can be halted. For the sake of maintaining low likelihood of invasion, the multifunctional pathway is compelled to applying preventive strategies.

Conditional rehabilitation of cooperation under strategic uncertainty.

In order to address the topic of the tragedy of the commons, more specifically that of tropical forest ecosystems explored as common-pool resources endowed with public-good features and exposed to deforestation risks, we consider game-theoretic population dynamics contingent on various differential equations. We propose an evolutionary model handed down to the Price theorem of selection. In a set of model-players evolving in strategic uncertainty and subject to certain mutation toward cooperation, the Price equation evens out unstructured and structured population replicator dynamics. According to the model outputs, avoiding the tragedy of the commons can be achieved on condition that half of the population temporarily exhibits a cooperative behavior. Furthermore, cooperative model-players ought to be rewarded at a level equivalent to the joint selection of cooperators and competitors issued from the unifying Price identity.

Network connectivity value.

In order to unveil the value of network connectivity, we formalize the construction of ecological networks in forest environments as an optimal control dynamic graph-theoretic problem. The network is based on a set of bioreserves and patches linked by ecological corridors. The node dynamics, built upon the consensus protocol, form a time evolutive Mahalanobis distance weighted by the opportunity costs of timber production. We consider a case of complete graph, where the ecological network is fully connected, and a case of incomplete graph, where the ecological network is partially connected. The results show that the network equilibrium depends on the size of the reception zone, while the network connectivity depends on the environmental compatibility between the ecological areas. Through shadow prices, we find that securing connectivity in partially connected networks is more expensive than in fully connected networks, but should be undertaken when the opportunity costs are significant.