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### Mathematical Models of Retrogression of Populations under Grazing Condition in the Stipa Steppe

Chen Qing-cheng, Zhao Song-ling and Yang Feng-xiang

Abstract: In the region of North-western China, there are about eight Stipa constructive species which form various communities of Stipa steppe. In this paper, the writers take Stipa breviflora + Agropyron cristatum + Artemisia frigida association as a typical example and make use of the method of mathematical analysis in an attempt to build up mathematical models which may explain the rules of succession of this type by means of mathematical language. Coverage is employed as an indication in representing the dominance of the population of species. In the moderate grazing stage, accompanying the increase in successive grazing, the dominance of the constructive species Stipa breviflora declines by degree and the curve of the population can be indicated by the equation P = 87e–0.74T. Agropyron cristatum, a subdominant in the original community, appears as a dominant species in the second stage, but after this, the 3ominance of its population drops down gradully and the curve can be expressed by the equation P=12T2.7e–0.3T2 Artemisia frigida, an accompanying species in the incipient community, gains the status as a dominant in the heavy grazing stage, and prior to this, its dominance exhibits a progressively raising curve which can be indicated by the equation P = 5.38e0.4T, yet when T > 3, the curve declines as P= 148.16–0.46e0.17T2. Convolvulus ammannii, an accidental in the first stage or an exotic species shows an slowly increase in its population before the excessive grazing stage and the equation in the section of the raising curve is expressed as P=0.745e0.45T, but as T >4, the population degrades abruptly and exhibits as the equation P = 12T2.7e–0.3T2. The community is now approaching to a secondary bare area. As a whole, the retrogressive succession of the Stipa steppe range shows a total characteristic curve of retrogression in its populations due to over grazing. The curve can be represented by the equation P = 132.9e–0.89. The incipient community entirely loses its general nature and it is said to be a heterogeneity community or disclimax. The critical straight line (curve) can be shown by the equation P = 1.1302T and under this critical situation, the total coverage of populations in the community is about 4.5%. At this point, we give the dominant (Convolvulus ammannii) of the community the name of "critical indicative species".

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