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Register a new userAn impulsive modelling framework of fire occurrence in a size structured model of tree-grass interactions for savanna ecosystems
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Fires and rainfall are major mechanisms that regulate woody and grassy biomasses in savanna ecosystems. Conditions of long-lasting coexistence of trees and grasses have been mainly studied using continuous-time modelling of tree-grass competition. In these frameworks, fire is a time-continuous forcing while the relationship between woody plant size and fire-sensitivity is not systematically considered. In this paper, we propose a new mathematical framework to model tree-grass interaction that takes into account both the discrete nature of fire occurrence and size-dependent fire sensitivity (via two classes of woody plants). We carry out a qualitative analysis that highlights ecological thresholds and bifurcations parameters that shape the dynamics of the savanna-like systems within the main ecological zones. Moreover, through a qualitative analysis, we show that the impulsive modelling of fire occurrences leads to more diverse behaviors and a more realistic array of solutions than the analogous time-continuous fire models. Numerical simulations are provided to illustrate the theoretical results and to support a discussion about the bifurcation parameters and future developments.
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Many systems in life sciences have been modeled by reaction-diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such that an appropriate formalism is necessary, using, for instance, impulsive reaction-diffusion equations. While several works tackled the issue of traveling waves for monotone reaction-diffusion equations and the computation of spreading speeds, very little has been done in the case of monotone impulsive reaction-diffusion equations. Based on vector-valued recursion equations theory, we aim to present in this paper results that address two main issues of monotone impulsive reaction-diffusion equations. First, they deal with the existence of traveling waves for monotone systems of impulsive reaction-diffusion equations. Second, they allow the computation of spreading speeds for monotone systems of impulsive reaction-diffusion equations. We apply our methodology to a planar system of impulsive reaction-diffusion equations that models tree-grass interactions in fire-prone savannas. Numerical simulations, including numerical approximations of spreading speeds, are finally provided in order to illustrate our theoretical results and support the discussion.
Savannas are dynamical systems where grasses and trees can either dominate or coexist. Fires are known to be central in the functioning of the savanna biome though their characteristics are expected to vary along the rainfall gradients as observed in Sub-Saharan Africa. In this paper, we model the tree-grass dynamics using impulsive differential equations that consider fires as discrete events. This framework allows us to carry out a comprehensive qualitative mathematical analysis that revealed more diverse possible outcomes than the analogous continuous model. We investigated local and global properties of the equilibria and show that various states exist for the physiognomy of vegetation. Though several abrupt shifts between vegetation states appeared determined by fire periodicity, we showed that direct shading of grasses by trees is also an influential process embodied in the model by a competition parameter leading to bifurcations. Relying on a suitable nonstandard finite difference scheme, we carried out numerical simulations in reference to three main climatic zones as observable in Central Africa.
We present and analyze a model aiming at recovering as dynamical outcomes of tree-grass interactions the wide range of vegetation physiognomies observable in the savanna biome along rainfall gradients at regional/continental scales. The model is based on two ordinary differential equations (ODE), for woody and grass biomass. It is parameterized from literature and retains mathematical tractability, since we restricted it to the main processes, notably tree-grass asym-metric interactions (either facilitative or competitive) and the grass-fire feedback. We used a fully qualitative analysis to derive all possible long term dynamics and express them in a bifur-cation diagram in relation to mean annual rainfall and fire frequency. We delineated domains of monostability (forest, grassland, savanna), of bistability (e.g. forest-grassland or forest-savanna) and even tristability. Notably, we highlighted regions in which two savanna equilibria may be jointly stable (possibly in addition to forest or grassland). We verified that common knowledge about decreasing woody biomass with increasing fire frequency is recovered for all levels of rainfall, contrary to previous attempts using analogous ODE frameworks. Thus, this framework appears able to render more realistic and diversified outcomes than often thought of. Our model can help figure out the ongoing dynamics of savanna vegetation in large territories for which local data are sparse or absent. To explore the bifurcation diagram with different combinations of the model parameters, we have developed a user-friendly R-Shiny application freely available at : https://gitlab.com/cirad-apps/tree-grass.
We consider a size-structured aggregation and growth model of phytoplankton community proposed by Ackleh and Fitzpatrick [2]. The model accounts for basic biological phenomena in phytoplankton community such as growth, gravitational sedimentation, predation by zooplankton, fecundity, and aggregation. Our primary goal in this paper is to investigate the long-term behavior of the proposed aggregation and growth model. Particularly, using the well-known principle of linearized stability and semigroup compactness arguments, we provide sufficient conditions for local exponential asymptotic stability of zero solution as well as sufficient conditions for instability. We express these conditions in the form of an easy to compute characteristic function, which depends on the functional relationship between growth, sedimentation and fecundity. Our results can be used to predict long-term phytoplankton dynamic
How large ecosystems can create and maintain the remarkable biodiversity we see in nature is probably one of the biggest open question in science, attracting attention from different fields, from Theoretical Ecology to Mathematics and Physics. In this context, modeling the stable coexistence of different species competing for limited resources is a particularly demanding task. From the mathematical point of view, coexistence in competitive dynamics can be achieved when dominance among species forms intransitive loops. However, these relationships usually lead to species densities neutrally cycling without converging to a stable equilibrium and, although in recent years several mechanisms have been proposed, models able to explain the robust persistence of competitive ecosystems are lacking. Here we show that stable coexistence in large communities can be achieved when the locality of interactions is taken into account. We consider a simplified ecosystem where individuals of each species lay on a spatial network and interactions are possible only between nodes at a certain distance. Varying such distance allows to interpolate between local and global competition. Our results demonstrate that when two conditions are met: individuals are embedded in space and can only interact with other individuals within a short distance, species coexist reaching a stable equilibrium. On the contrary, when one of these ingredients is missing large fluctuations and neutral cycles emerge.
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