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.

The growth of zircon crystals in cooling magmas is modelled by the one-phase Stefan problem, with the growth rate that depends on the magma cooling rate. Some rare elements (like e.g. U, Th, Hf) are incorporated in the crystal at trace concentration. These elements have different temperature-dependent diffusion and partition coefficients. As a consequence their final spatial repartition in the crystal depends on the temperature evolution of the magma during the cooling.

The present work proposes to reconstruct the temperature evolution from the measurements of trace elements concentration in natural zircon. This inverse problem is solved by minimizing the misfit between calculated and measured trace element concentrations. The tangent model to the one-phase Stefan problem provides the sensitivity matrix, and the quadratic cost-function is minimized using Gauss–Newton method. The identifiability and the error on the retrieved parameters are studied in the framework of BLUE (Best Linear Unbiased Linear Estimator). The algorithm is tested on two synthetic datasets, and on real data obtained in a zircon crystal from early Fish Canyon tuff eruption. Reconstructed temperature ranges and cooling duration are in good agreement with available petrological interpretation.