Biodiversity and stability of ecosystems with extinctions

Vladimir Kozlov (March 28, 2019)

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Existence and stability of large food webs, where many species share a few of resources, is one of key problems in ecology. We consider an ecological system, where several species compete for few limited resources. Typical examples are plant or plankton ecosystems. Sunlight, water, nitrogen, phosphorus and iron are all abiotic essential resources for phytoplankton and plant species. Resource competition models link the population dynamics of competing species with the dynamics of the resources. The extinction in the model is described with the help of extinction thresholds (if a species distribution reaches the threshold then we remove the species from the system).

A complete description of the system large time behavior is obtained in the case of sufficiently large turnover rates and zero extinction thresholds. This result holds due to two principal properties of our system. First, the system has a typical fast/slow structure for large turnover rates. Second, the system obeys a monotonicity property: if resources increase then species abundances also increase.

If extinction thresholds are non-vanishing, the ecosystem behavior exhibits new interesting effects. The limit equilibrium state still exists but it depends on the initial ecosystem state. This implies in particular that there can a priori exist several distinct equilibrium states.

We establish explicit upper and lower estimates of biodiversity in terms of the fundamental ecosystem parameters (species mortalities, resource consuming rate etc.). These results use no assumptions on the system dynamics (in particular our theorem on global stability).

This is a joint work with S. Vakulenko, Institute of Mechanical Engineering, St Petersburg, Russia, V. Tkachev, Department of Mathematics, Linköping University, and Uno Wennergren, Department of Physics , Chemistry and Biology, Linköping University.