Abstracts

Evolution of Species and Ecosystems: Theoretical Analysis and Computer-Assisted Modeling

In the work presented we consider the one-dimensional model of evolution of populations of the brown frog (Rana temporaria) and the moor frog (Rana arvalis) in a pond. The model is constructed on the basis of long-term observations made by A.S. Severtsov with co-workers1.

Mathematical model consists of two parts. The first one describes frog development in a pond from the stage of fertilized spawn till the stage of mature frog. The second deals with frog vital activity on land. The wintering process is not considered in the model.

Maturation and death of frogs in a pond are described by the system of linear differential equations:
dI0(t,T)/dt= k0(XN0+1(T)+...+XN(T)) - d0I0(t,T),
dIj+1(t,T)/dt= kjIj(t,T)-(kj+1+dj+1)Ij+1(t,T), j=0,..,n-1,
dX1(t,T)/dt= knIn(t,T).

The process of frog ontogenesis in a pond is divided into n elementary stages. Each j-th stage is interpreted as an independent biological age of larvae. At the current moment t, the number of larvae at the j-th stage equals to Ij(t,T). The zero age is identified to the fertilized spawn, whereas the n-th age plainly turns to the stage of a young frog (the variable X1(T)), kj is a constant of the rate of transformation from the j-th age to the (j+1), dj is a constant of mortality rate at the j-th larvae age. In general case, the parameters kj and dj are dependent upon external environmental impacts (water temperature, day period, filling of a pond by water, aeration, etc.).

The living of frogs on land is represented by the discrete part of a model. In all, we discriminate 7 ages of frogs. Population number of frogs depends upon the year Т, which is the macro-variable of the model (Т=1,2,?). We consider that the frogs of all ages live actively in parallel with events occurring in a pond. At the beginning of the year Т, or early in the spring, the number of frogs at each age equals to Xl=Xl(Т), l=1,?,N. When the active living period is over, before wintering, all the frogs, allowing for the dead ones, of all ages, except the last one, turn into the next age. It is postulated that all the frogs of the N-th age are dying. Young generation of frogs grown up in the pond during the current year T is considered to be the frogs of the first age. The resulted number of frogs is viewed as the number of frogs which will be actively living in the spring of the consequent (Т+1)-th year. The ages 1 ,..., N0 are not considered pubertal, because only the frogs of (N0+1)-th age begin spawing.

In this work, our goal was to determine the parameters of the model. It is supposed that some stages discriminated by the model are morphologically indistingishable and can?t be determined experimentally. Analysis of regeneration of parameters is performed for two sets of experimental data. In the first case, it is supposed that the following averaged characteristics of frog ontogenesis in a pond are measured experimentally: average period of the most mass representativeness of each morphologically distinct larvae age, its mean deviation and the level of mortality. In the second case, it is supposed that average duration of each morphologically distinct age is measured directly, as well as dispersion of deviation. Since we do not know a priori the number of stages which constitute each morphologically distinct stage of larvae development, these values are also considered as unknown parameter of the model which should be determined. In the work, we analyze the conditions necessary for determining the model parameters on the basis of experimental data.

Severtsov A.S. and Surova G.S. //Zoologicheskii journal, 1995, vol. 74, p. 80-92 (in Russian)

Note. Abstracts are published in author's edition

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