Cylindrospermopsis raciborskii, an invasive freshwater cyanobacterium, originated from the tropics but has spread to temperate zones over the last few decades. Its northernmost populations in Europe occur in North German lakes. How such dramatic changes in its biogeography are possible and how its population dynamics in the newly invaded habitats are regulated are still unexplained. We therefore conducted a long-term (1993-2005) study of two German lakes to elucidate the mechanisms behind C. raciborskii population dynamics and to identify the abiotic constraints on its development. Our data revealed that pelagic populations of C. raciborskii thrived for three months during the summer, contributing up to 23% of the total cyanobacteria biovolume. Population sizes varied greatly between years without exhibiting any distinct long-term trends. In the annual lifecycle, C. raciborskii filaments emerged in the pelagic habitat when the temperature rose above 15-17 degrees C. At that time, mean photosynthetically active radiation in the mixed water column (I (mix)) overstepped its maximum. Rates of population net increase were highest at the beginning of the season (0.15-0.28 day(-1)), declined continuously over time, and were significantly positively correlated with I (mix). This indicates that the onset of the pelagic population is temperature-mediated and that I (mix) controls its growth. Since I (mix) peaks before the population onset, the time of germination is of crucial importance for successful development. To test this hypothesis, we designed a model to simulate pelagic population size, starting at different dates in the annual cycle. Moving the population onset forward by 30 days resulted in a doubling of the population size. We therefore conclude that an earlier rise in water temperature associated with climate change has promoted the spread of C. raciborskii to the temperate zone. Earlier warming permits earlier germination, thereby shifting the pelagic populations to a phase with higher I (mix), which advances growth and the population establishment.

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This paper continues the series of publications about applications of partial ordering. The focus of this publication is the derivation of approximate analytical expressions for the averaged rank and the ranking probabilities. To derive such combinatorial formulas a local partial order is suggested as an approximation. The performance of the approximation is rather high; we therefore conclude that three very simple descriptors of the local partial order seem to be sufficient to get a rough impression of the linear order, induced by the averaged ranks and the ranking probabilities of empirical partially ordered sets. Linear order derived from the partial order, ranking probabilities, and other characteristics are considered as parts of a so-called "General Ranking Model" (GRM). Following the local partial order, the averaged rank of an object x can be estimated applying the following simple formula: Rk(av) = (S+1)*(N+1)/(N+1-U). S is the number of successors of the object x, N is the total number of objects (of the quotient set), and U is the number of objects incomparable with x. More complex formulas for the ranking probabilities are given in the text. A list of abbreviations and symbols can be found in Tables 3 and 4.

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