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  • 1
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The equation $$\frac{{\partial u}}{{\partial t}} + \frac{\partial }{{\partial x}}\left( {H(x)f(u) + \left( {1 - H(x)} \right)g(u)} \right) = 0$$ , whereH is Heaviside's step function, appears for example in continuous sedimentation of solid particles in a liquid, in two-phase flow, in traffic-flow analysis and in ion etching. The discontinuity of the flux function atx=0 causes a discontinuity of a solution, which is not uniquely determined by the initial data. By a viscous profile of this discontinuity we mean a stationary solution ofu t +(F δ) x =εu xx , whereF δ is a smooth approximation of the discontinuous flux, i.e.,H is smoothed. We present some results on the stability of the viscous profiles, which means that a small disturbance tends to zero uniformly ast→∞. This is done by weighted energy methods, where the weight (depending onf andg) plays a crucial role.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The equation $$\frac{{\partial u}}{{\partial t}} + \frac{\partial }{{\partial x}}\left( {H(x)f(u) + \left( {1 - H(x)} \right)g(u)} \right) = 0$$ , whereH is Heaviside's step function, appears for example in continuous sedimentation of solid particles in a liquid, in two-phase flow, in traffic-flow analysis and in ion etching. The discontinuity of the flux function atx=0 causes a discontinuity of a solution, which is not uniquely determined by the initial data. The equation can be written as a triangular 2×2 non-strictly hyperbolic system. This augmentation is non-unique and a natural definition is given by means of viscous profiles. By a viscous profile we mean a stationary solution ofu t +(F δ) x =εu xx , whereF δ is a smooth approximation of the discontinuous flux, i.e.,H is smoothed. In terms of the 2×2 system, the discontinuity atx=0 is either a regular Lax, an under-or overcompressive, a marginal under- or overcompressive or a degenerate shock wave. In some cases, depending onf andg, there is a unique viscous profile (e.g. undercompressive and regular Lax waves) and in some cases there are infinitely many (e.g. overcompressive waves). The main purpose of the paper is to show the equivalence between a previously introduced uniqueness condition for the discontinuity of the solution atx=0 and the viscous profile condition.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1432-1939
    Keywords: Key words  Perca fluviatilis ; Planktivory ; Top predator ; Trophic interactions ; Whole lakes
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology
    Notes: Abstract  To examine size-dependent food web interactions in systems with cannibalism, we compared the abundances of zooplankton and phytoplankton over 2 years in four lakes with cannibalistic perch (Perca fluviatilis) of which two also supported the top predator pike (Esox lucius). The abundance of perch 2 years and older was lower in lakes with pike than in lakes with only perch. In contrast, the abundance of small perch (young-of-the-year and 1-year old) was lower in lakes with only perch suggesting that intense cannibalism reduced these size classes to low levels in lakes lacking pike. Functional response experiments with differently sized perch and zooplankton showed that the attack rate of small perch susceptible to cannibalism was much higher than that of large cannibalising perch. The optimal body size of perch with respect to attack rate was also lower for small zooplankton prey than for large zooplankton. The zooplankton communities in lakes with only perch were dominated by the relatively small species Ceriodaphnia quadrangula and Bosmina spp. and total zooplankton biomass was higher in these lakes than in lakes with both pike and perch. In contrast, the mean size of cladoceran zooplankton was largest in lakes with both pike and perch owing to a dominance of the large zooplankton species Holopedium gibberum in these lakes. We relate these patterns to (1) the low foraging efficiency of large perch on small zooplankton and (2) the low abundance of small zooplanktivorous perch (due to cannibalism) in lakes with only perch. The differences in zooplankton community structure also resulted in different seasonal dynamics of phytoplankton between lakes. Cannibalism introduces a vertical heterogeneity to food webs that causes consumer-resource dynamics that are not predictable from linear food chain models.
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  • 4
    ISSN: 1432-1904
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Chemistry and Pharmacology , Natural Sciences in General
    Type of Medium: Electronic Resource
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