Tree-Grass interactions dynamics and Pulse Fires: mathematical and numerical studies

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.Comment: 51 pages, 7 figure

Epidemiological Models and Lyapunov Functions

We give a survey of results on global stability for deterministic compartmental epidemiological models. Using Lyapunov techniques we revisit a classical result, and give a simple proof. By the same methods we also give a new result on differential susceptibility and infectivity models with mass action and an arbitrary number of compartments. These models encompass the so-called differential infectivity and staged progression models. In the two cases we prove that if the basic reproduction ratio $\mathcal{R}_0$ ≤ 1, then the disease free equilibrium is globally asymptotically stable. If $\mathcal{R}_0$ > 1, there exists an unique endemic equilibrium which is asymptotically stable on the positive orthant

Bifurcation and Stability in a Delayed Predator-Prey Model with Mixed Functional Responses

The model analyzed in this paper is based on the model set forth by Aziz Alaoui et al. [Aziz Alaoui & Daher Okiye, 2003; Nindjin et al., 2006] with time delay, which describes the competition between the predator and prey. This model incorporates a modified version of the Leslie-Gower functional response as well as that of Beddington-DeAngelis. In this paper, we consider the model with one delay consisting of a unique nontrivial equilibrium E* and three others which are trivial. Their dynamics are studied in terms of local and global stabilities and of the description of Hopf bifurcation at E*. At the third trivial equilibrium, the existence of the Hopf bifurcation is proven as the delay (taken as a parameter of bifurcation) that crosses some critical values

Bioeconomic Sustainability and Resilience of Savanna

Our paper investigates the bioeconomic sustainability and resilience of savanna social-ecological systems (SES). A stylized dynamics of an exploited grass-tree systems is thus considered accounting both for the competition between trees and grass along with logging and grass harvesting activities. Regarding sustainability, we rely on bioeconomic viability goals including consumption security for grass, profitability of logging and coexistence of tree-grass states. A first analytical result relates to the elicitation of sufficient sustainability conditions through the non-emptyness of the so-called viability kernel. Such sufficient conditions rely on coupled MSY (maximum sustainable yield)-MEY (maximum economic yield) reference states-controls. A larger viable set including these MSY-MEY equilibria is also identified. The resilience of such viability states-controls for savanna SES facing shocks such as fire is then put forward from both recovery through stability analysis and resistance viewpoints. Simulations inspired from savanna systems in Cameroon exemplify the analytical findings

An impulsive modelling framework of fire occurrence in a size-structured model of tree–grass interactions for savanna ecosystems

International audienceFires and mean annual rainfall are major factors that regulate woody and grassy biomasses in savanna ecosystems. Within the savanna biome, conditions of long-lasting coexistence of trees and grasses have been often studied using continuous-time modelling of tree–grass competition. In these studies, 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 interactions that takes into account both the impulsive 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 bifurcation parameters that shape the dynamics of the savanna-like systems within the main ecological zones. Through a qualitative analysis, we show that the impulsive modelling of fire occurrences leads to more diverse behaviors including cases of grassland, savanna and forest tristability and a more realistic array of solutions than the analogous time-continuous fire models. Numerical simulations are carried out with respect to the three main ecological contexts (moist, mesic, semi-arid) to illustrate the theoretical results and to support a discussion about the bifurcation parameters and the advantages of the model

International Workshop on Differential Equations in Mathematical Biology Global analysis of malaria intra-host model

Intra-host models of malaria describe the dynamics of the blood-stage of the parasite and their interaction with host-cells, in particular red blood cells (RBC) and immune effectors. During the past decade there has been considerable work on the mathematica