Population Dynamics Model Of The Markov Skeleton Method

In the thesis, we make use of the method of Markov skeleton processes in order to study the change of single-species population number under the disaster such as earthquakes. The clou recurs to the method of Markov skeleton processes to study the instantaneous distribution of single-species population number on population dynamics.Markov skeleton processes are a kind of comprehensive stochastic processes, which are firstly put forward by Prof.Hou Zhengting and his colleagues in 1997. The processes contain many exist classical stochastic processes models, such as Markov processes,semi-Markov processes,piecewise deterministic Markov processes,Doob processes,regenerative processes,semi-regenerative processes. They have important value in theory and application.In most formal studies, the change of the population number is usually produced by differential equation or difference equation, namely, is discribed by a continuous smooth curve or a ladder curve (the continuous continuation of discrete time) which is right-continuous with left limits, such as the famous single-species Logistic model with density dependence. But when all comes to all or in a relatively long time, for example, when the population number are changed abruptly under the exceptive circumstances, there are distinct limitations in these models. However, we can put the axe in the helve by means of the method of Markov skeleton processes from the aspect of probability theory.In the theory of Markov skeleton processes,we can consider the time of break as a stopping timeτ_n(n≥1) of the series.Then,in order to seek a Markov skeleton process,we supplement new variables.Thus,the timeτ_n(n≥1) at which the population number changes is the discontinuity point of Markov skeleton processes at the time axis.At last,we study the instantaneous distribution of single-species population at random time t. For example, in the model of this thesis, when studying the change of the population number, we take the occurrence of the disaster into account.In this thesis, firstly, we bring forward the expressions of h and q in the backward equation of Markov skeleton processes via Lemma 3.3.1 and Lemma 3.3.2. Sencondly, we put forward Theorem 3.3.1 and Theorem 3.3.2, studying the instantaneous distribution of single-species population under discrete state and continuous condition respectively, and proving that the instantaneous distribution of single-species population number at time t is the smallest non-negative solution of some non-negative linear equation.