Description
Title: Inequality and diversity in the ocean: lessons from a scale-invariant population model Abstract: We take a mathematically idealised view of the marine ecosystem, in which the dominant aspect is the coupling of growth and death: predators must consume prey to grow. Predation is size-based: fish eat smaller fish and plankton, zooplankton eats smaller zooplankton and phytoplankton, phytoplankton consumes a common resource. This leads to a model of coupled partial integro-differential equations. Their analysis gives new insight into the relation between two stunning aspects of the marine ecosystem: 1) the abundance of organisms is described by a power-law over 21 orders of magnitude in size and 2) there is a mysteriously high level of biodiversity, known as the paradox of the plankton. Mathematically, the process of a large fish eating a smaller fish and becoming even larger is similar to the coagulation process where two clusters combine to form a larger cluster, described by Smoluchovski's coagulation equation. However the rate kernel describing the feeding preference of fish is very different from the kernels usually studied in coagulation models. It is non-symmetric but has a scaling property and it is the resulting scale-invariance of the model that we exploit to derive analytic results regarding the stationary state, its stability, and the propagation of perturbations. The Smoluchovski equation becomes even more tractable by approximating it by a convection-diffusion equation: the McKendrick-von Foerster equation with an extra diffusion term| Period | 3 Feb 2016 |
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| Held at | University of Strathclyde, Department of Mathematics, United Kingdom |