Tree-grass interaction dynamics and pulsed fires: mathematical and numerical studies. (English) Zbl 1465.92141
Summary: 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 although 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 we showed that various states exist for the physiognomy of vegetation. Though several shifts between vegetation states appeared to be 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 simulation scheme, we carried out numerical simulations in reference to three main climatic zones as observable in Central Africa.
MSC:
| 92D40 | Ecology |
| 65L12 | Finite difference and finite volume methods for ordinary differential equations |
| 34A37 | Ordinary differential equations with impulses |
| 34D23 | Global stability of solutions to ordinary differential equations |
Keywords:
savanna; tree-grass competition; fires; impulsive differential equations (IDE); bifurcation; nonstandard numerical schemeReferences:
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