Abstract
We consider competing functional groups of tree species and develop a model of network response dynamics in order to measure the impacts of perturbations on the population distribution and diversity. The analysis of the equilibrium states relies on the connection between mean field game dynamics and replicator dynamics. We simulate our theoretical results from the data inventoried in French Guiana. Our results show that different types of disturbances modify the competitive interactions by affecting the evolutions of group densities. At the high regimes of disturbance, the canopy shade-intolerant species supplant the canopy shade-tolerant species. Tropical forest managers can thus take advantage of the competitive interactions between the functional groups to stimulate the abundance of marketable timber species. We also validate the hypothesis of maximum diversity at the intermediate disturbance levels.
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Notes
Despite the fact that intra-species interactions can be substantial, Doncaster (2009) showed that the inter-specific impacts are to be considered as of equal magnitude to that of the intra-specific impacts. Besides, the self-loops in the graph implicitly integrate the intra-group competition effects.
When \(\lim (C_{i,j})=1\), we implicitly account for the facilitative or mutual effects.
In detail, we have \(N_{i} \cdot \text {cov}\big (\sum _{i=1}^n N_{i} x_{i}, \sum _{j=1}^n N_{j} x_{j}\big ) = N_{i} \cdot \sum _{i=1}^n N_{i} \cdot \sum _{j=1}^n N_{j} \cdot \text {cov} \left( x_{i}, x_{j} \right) = \sum _{i=1}^n N_{i} \cdot \sum _{j=1}^n \left( 1 - N_{i} \right) \cdot N_{i} \cdot \text {cov} \left( x_{i}, x_{j} \right) = N_{i} \cdot \left( 1- N_{i} \right) \cdot \sum _{i=1}^n \sum _{j=1}^n N_{i} \cdot a_{j,i} = N_{i} \left( 1-N_{i}\right) C_{i,j}\).
According to this branch of game theory, the inter-type interactions must be sufficiently regular for the global phenomena to emerge.
The rest points of the replicator equation are those for which all payoff values are equal (Sigmund 2011). The population game with payoffs \(a_{j,i}\) is a full potential game if there is a continuous differential mapping \(M: \mathbb {R} \rightarrow \mathbb {R}\), that we can use as the inner product of the payoff vector and the direction of motion, such that \(\textit{d} M(N_{i})/ \textit{d}N_{i} = \textit{d}{N}_{i}/\textit{d}t\). The mean field game dynamics is positively correlated if \(\dot{N}_{i} \ne 0 \Rightarrow \langle a_{j,i}\left( N_{i}\right) , \dot{N}_{i} \rangle > 0\). On the contrary, the Euler stationarity would imply \(\dot{N}_{i} = 0\) (Franc et al. 2000).
Basal area is the area occupied by the cross-section of tree trunks and stems at their base.
The figures in brackets are negative values.
For example, \(C_{1,j}= N_{1} a_{1,1} + N_{2} a_{2,1} + N_{3} a_{3,1} + N_{4} a_{4,1} + N_{5} a_{5,1}\).
These trees may simply respond to logging because they are more demanding of light (Poorter et al. 2005).
In game theory, the decisions of agents are said to be strategic complements if they mutually increase their respective payoffs.
The same conclusion was empirically brought forward by Canham et al. (2006) within local neighborhoods, at the species level.
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Acknowledgments
This work was financially supported by the French National Research Agency through the Laboratory of Excellence ARBRE, a part of the Investments for the Future Program (ANR 11 – LABX-0002-01). It was also supported by the French National Forestry Office through the Forests for Tomorrow Chair. We would like to thank the CIRAD Research Center for allowing us to access to the Paracou database, and wish to acknowledge the support of Eric Marcon (Agro ParisTech) and Bruno Ferry (Agro ParisTech). We are also grateful to the editor and the two anonymous referees for their thorough comments and suggestions, which significantly contributed to improving the quality of the paper.
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Dragicevic, A.Z. Functional diversity from network response dynamics. J Bioecon 18, 1–15 (2016). https://doi.org/10.1007/s10818-015-9206-3
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DOI: https://doi.org/10.1007/s10818-015-9206-3