The Bisexual Branching Process

On the basis of some elementary theories about Galton-Watson branching process, the bisexual Galton-Watson branching process and the branching process with random environment are introduced in the paper. According to the bisexual branching process, the bisexual Galton-Watson branching process {zn} with the mating funtion L(x,y) = x, (x,y>0) is studied. Supposing (s) is the reproducing probability generating function of any individual in {Zn}, Fn(s) is the probability generating function ofZn, Q = limp(zn = 0|Z0 > 0) is the extinction probability of{zn} , and a denotes the female birthrate, the concreate conclusions are:1) For all s [0,1), limFn(s) = Q is proved.2) Testifing Q = 1 if and only if if and only if .3) Let q is the extinction probability of the asexal branching process with the probabilitygenerating function , it is held when 4) Get the moments of 5) Proving 6) It is proved that if EZ21< , and m>1, let , then {Wn} is a martingale, so itis convergent.Moreover, added the random environment expressed by the sequence of random variable {Sn} to the process above, {zn} becomes the bisexual branching process with random environment, which has some similar qualities like the common branching process with random environment.