# 随机环境中伴有移民且配对依赖人口数的两性分枝过程（英文）

【摘要】本文首先介绍了两性分枝过程的发展情况,在前人研究的基础上,本文建立了更符合自然界两性生物繁衍规律的模型,配对依赖当前人口数且伴有移民的两性分枝过程.利用上可加函数的性质,得到了平均增长率的极限.在一定的条件,推导出此过程以概率1灭绝的一个充要条件,进而推广了前人的研究成果.

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1 IntroductionDaley[1]introduced the bisexual Galton-Watson branching process as a two-typebranching model{(F*n,M*n),n=1,2,···}defined recursivelyZ*0=N,(F*n+1,M*n+1)=Z*n∑i=1(f*n,i,m*n,i),n=0,1,2,···,Z*n+1=L(F*n+1,M*n+1),n=0,1,2,···, where N is a positive integer and the empty sum is considered to be(0,0).Intuitively,f*n,iand m*n,irepresent the number of females and males produced by the ith matingunit in nth generation{(f*n,i,m*n,i),i=1,2,···,n=0,1,2,···},which is a sequenceof independent and distributed identically(abbreviated i.i.d.),nonnegative,integer-valued random variables.Mating function L:R+×R+→R+is assumed to bemonotonic nondecreasing in each argument,integer-valued on integer arguments andsatisfied L(x,y)≤xy.Consequently,F*nand M*nare the number of females and malesin the nth generation,which form Z*n=L(F*n,M*n)mating units.These mating unitsare reproduced independently through the same offspring probability distribution foreach generation.A branching process is said to be superadditive when its matingfunction L is superadditive,i.e.for any inter n≥2,L(·,·)satisfiesL(n∑i=1(xi,yi))≥n∑i=1L(xi,yi),xi,yi∈R+,i=1,2,···,n.Bisexual Galton-Watson branching processes have received much attention in theliterature[2-4].The extinction problem has been studied by Daley[1],Daley et al[5],Hull[6]and Alsmeyer and R¨osler[7].The main result,proved by Daley et al[5],is basedon the concept of mean growth rate per mating unit,that isr*k:=k-1E(Z*n+1|Z*n=k),k=1,2,···,which was introduced

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