We study a stochastic individual-based model of interacting plant and pollinator species through a bipartite graph: each species is a node of the graph, an edge representing interactions between a pair of species. The dynamics of the system depends on the between- and within-species interactions: pollination by insects increases plant reproduction rate but has a cost which can increase plant death rate, depending on the densities of pollinators. Pollinators reproduction is increased by the resources harvested on plants. Each species is characterized by a trait corresponding to its degree of generalism. This trait determines the structure of the interaction graph and the quantities of resources exchanged between species. Our model includes in particular nested or modular networks. Deterministic approximations of the stochastic measure-valued process by systems of ordinary differential equations or integro-differential equations are established and studied, when the population is large or when the graph is dense and can be replaced with a graphon. The long-time behaviors of these limits are studied and central limit theorems are established to quantify the difference between the discrete stochastic individual-based model and the deterministic approximations. Finally, studying the continuous limits of the interaction network and the resulting PDEs, we show that nested plant-pollinator communities are expected to collapse towards a coexistence between a single pair of species of plants and pollinators.
Keywords: Ecological mutualistic community, birth and death process, interacting particles, limit theorem, kinetic limit, graphon, integro-differential equation, stationary solution
Sylvain Billiard 1 ; Hélène Leman 2 ; Thomas Rey 3 ; Viet Chi Tran 4
CC-BY 4.0
@article{MSIA_2025__14_1_23_0,
author = {Sylvain Billiard and H\'el\`ene Leman and Thomas Rey and Viet Chi Tran},
title = {Continuous limits of large plant-pollinator random networks and some applications},
journal = {MathematicS In Action},
pages = {23--54},
year = {2025},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {14},
number = {1},
doi = {10.5802/msia.42},
language = {en},
url = {https://msia.centre-mersenne.org/articles/10.5802/msia.42/}
}
TY - JOUR AU - Sylvain Billiard AU - Hélène Leman AU - Thomas Rey AU - Viet Chi Tran TI - Continuous limits of large plant-pollinator random networks and some applications JO - MathematicS In Action PY - 2025 SP - 23 EP - 54 VL - 14 IS - 1 PB - Société de Mathématiques Appliquées et Industrielles UR - https://msia.centre-mersenne.org/articles/10.5802/msia.42/ DO - 10.5802/msia.42 LA - en ID - MSIA_2025__14_1_23_0 ER -
%0 Journal Article %A Sylvain Billiard %A Hélène Leman %A Thomas Rey %A Viet Chi Tran %T Continuous limits of large plant-pollinator random networks and some applications %J MathematicS In Action %D 2025 %P 23-54 %V 14 %N 1 %I Société de Mathématiques Appliquées et Industrielles %U https://msia.centre-mersenne.org/articles/10.5802/msia.42/ %R 10.5802/msia.42 %G en %F MSIA_2025__14_1_23_0
Sylvain Billiard; Hélène Leman; Thomas Rey; Viet Chi Tran. Continuous limits of large plant-pollinator random networks and some applications. MathematicS In Action, Tome 14 (2025) no. 1, pp. 23-54. doi: 10.5802/msia.42
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