Информационная система "Конференции"

Abstracts

Stochastic simulation and Monte-Carlo methods

We conside the stochastic model of individuals's community with their transformation and interaction. The model has the following assumptions. Every individual at time $tgeq0$ belongs to one of the classes %$A_1, ldots, A_n$, $ninmathbf{N}$. $A_1, ldots, A_n$, $nin {bf N}$. At time $t=0$ every class $A_i$, $i=1,ldots,n$, contains $x_i^circgeq0$ primary individuals. Every primary individual $x$ from class $A_i$ at time $t = ell_x$ suffers transformation, where $ell_x>0$ is random variable with distribution $L_i^circ$. If $t>0$ is the time of the appearing of individual $x$ in class $A_i$, then $t+ell_x$ is the time of its transformation, where $ell_x>0$ is random variable with distribution $L_i$. As a result of the transformation the collections $eta^x_{i1},ldots,eta^x_{in}$ and $xi^x_{i1},ldots,xi^x_{in}$ of individuals respectively disappear and appear in classes $A_1, ldots, A_n$. We suppose that ${ell_x}_x$ are mutually independent, $L_i^circ$ and $L_i$, $i=1,ldots,n$, are constant distributions, ${(eta^x_{i1},ldots,eta^x_{in})}_{x,,i}$ and ${(xi^x_{i1},ldots,xi^x_{in})}_{x,,i}$ are mutually independent random vectors with constant distributions. Individuals of the classes $A_1, ldots, A_n$ can take part in the interactions of $m$ different types $I_1,ldots,I_m$. We suppose that for $t in (t_1;t_2)$ individuals's transformations from all classes $A_1, ldots, A_n$ don't take place. Then in the time $(t;t+Delta)subset(t_1;t_2)$ the individuals interact by type $I_k$, $k=1,ldots,m$, with probability $lambda_k(x(t))Delta+o(Delta)$, interaction of several such types take place only with probability $o(Delta)$, and probability of absence of interaction equals $1 - sum_klambda_k(x(t))Delta+o(Delta)$, $Delta to 0$. The symbol $x(t) = (x_1(t),ldots,x_n(t))$ means the numbers of individuals belonging to classes $A_1, ldots, A_n$ at time $tgeq 0$, $lambda_k = lambda_k(x(t))$ are nonnegative functions of $n$-dimentional vectors with nonnegative integer coordinates. As a result of the realization $I_k$ at time $t$ the collections $gamma^t_{k1},ldots,gamma^t_{kn}$ and $beta^t_{k1},ldots,beta^t_{kn}$ of individuals respectively disappear and appear in classes $A_1, ldots, A_n$. %The symbols ${(gamma^t_{k1},ldots,gamma^t_{kn})}_{t,,k}$ and ${(beta^t_{k1},ldots,beta^t_{kn})}_{t,,k}$ are mutually independent random $n$-dimentional vectors with nonnegative integer coordinates having constant distributions. Above-mentioned assumptions allows us to formalise the model in terms of Markov random process with special state's space and to construct the formulas describing the rules of state's change. The simulation algorithm was worked out and tested. The algorithm and present version of computer program allows to operate with about million individuals population. The mean value of the algorithm's run time depends on the number of individuals according to the logarithmic law. The model was used to simulate individuals's community with seasonal reproduction and competition. For this model conditions of going out of population's size to constant level were got.

Note. Abstracts are published in author's edition

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