We give an overview of all codim 1 bifurcations in generic planar discontinuous piecewise smooth autonomous systems, here called Filippov systems. Bifurcations are defined using the classical approach of topological equivalence. This allows the development of a simple geometric criterion for classifying sliding bifurcations, i.e. bifurcations in which some sliding on the discontinuity boundary is critically involved. The full catalog of local and global bifurcations is given, together with explicit topological normal forms for the local ones. Moreover, for each bifurcation, a defining system is proposed that can be used to numerically compute the corresponding bifurcation curve with standard continuation techniques. A problem of exploitation of a predator–prey community is analyzed with the proposed methods.

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The phytoplankton community of eutrophic shallow lakes is often dominated by filamentous cyanobacteria of the family Oscillatoriaceae. In this paper we follow two independent approaches to show that this situation is likely to be one of two alternative stable states of the algal community. First we analyze patterns of cyanobacterial dominance observed in the field, and show that these patterns imply that the algal community is a hysteretic system with two alternative equilibria. Then, we construct a simple competition model to show that hysteresis should in fact be expected from differences in physiology between cyanobacteria and algae. The basic mechanism is that cyanobacteria are the superior competitors under conditions of low light, but also promote such conditions, as they can cause a higher turbidity per unit of phosphorus than other algae. This mechanism of hysteresis offers an explanation for the resistance of cyanobacteria dominance in shallow lakes to restoration efforts by means of nutrient reduction.

Invasion by mats of free-floating plants is among the most important threats to the functioning and biodiversity of freshwater ecosystems ranging from temperate ponds and ditches to tropical lakes. Dark, anoxic conditions under thick floating-plant cover leave little opportunity for animal or plant life, and they can have large negative impacts on fisheries and navigation in tropical lakes. Here, we demonstrate that floating-plant dominance can be a self-stabilizing ecosystem state, which may explain its notorious persistence in many situations. Our results, based on experiments, field data, and models, represent evidence for alternative domains of attraction in ecosystems. An implication of our findings is that nutrient enrichment reduces the resilience of freshwater systems against a shift to floating-plant dominance. On the other hand, our results also suggest that a single drastic harvest of floating plants can induce a permanent shift to an alternative state dominated by rooted, submerged growth forms.D ense mats of free-floating plant have an adverse effect on freshwater ecosystems because they create anoxic conditions that strongly reduce animal biomass and diversity (1). Invasions by introduced exotic species are partly responsible for the increase of floating plant dominance. The problems caused by Eichhornia crassipes, Pistia stratiotes, and Salvinia molesta are notorious: they hamper fish production and navigation in tropical regions around the world (2-4). However, eutrophication is likely to have boosted the spread of free-floating plants, too. In temperate climate zones, it is known that dense beds of duckweeds (Lemnaceae) and small, floating water ferns (Azollaceae) are a symptom of high-nutrient loading in small water bodies, such as ponds and canals (5, 6). Just as in the case of tropical plant beds, the dark and anoxic conditions under thick duckweed cover leave little opportunity for animal or plant life (1).The dependence of free-floating plants on high nutrient concentrations in the water is an obvious consequence of their growth form. They have no direct access to the sediment pool of nutrients, and they have a large portion of their leaf surface exposed to the atmosphere rather than to the water, thereby reducing the possibility of taking up nutrients other than carbon through their leaves. By contrast, rooted submerged macrophytes may take up a large part of their nutrients from the sediment (7, 8) and also use their shoots effectively for nutrient uptake from the water column (9, 10). Although floating plants are obviously superior competitors for light, submerged plants may affect the growth of free-floating plants through a reduction of available nutrients in the water column. Competition is likely to be especially strong for nitrogen. Although phosphorus availability in the water column can be reduced because of uptake by submerged macrophytes (11), many studies show unaltered or even increased ortho-P levels after increased macrophyte cover (12-15). By contrast, submerged nitrogen co...