Branching Processes With Common Birth And Death Time In Random Environments

Branching processes form one of the classical fields of applied probability theory.In the Galton-Watson branching processes,each individual is supppsed to have a fixed life length of one unit of time.While the life length of the species in nature is random,which restricts the application of Galton-Watson processes model.In order to fetch up the restricion,Bellman and Harris introduced age-dependent branching processes,where individuals live for a random length of time.However,in both Galton-Watson processes and the age-dependent branching processes,the assumptions that different individuals reproduce independently of one another and according to the same probability distribution seems doubtful because various environmental factors like temperature,food supply,competition etc.may cause a variation of individul behaviour over generations.So the environmental factors need to be considered for the accessible for applications.Allowing environmental changes over generations,we generalize the above two classical branching models and introduce a new process {Z(t)}_t≥0:branching process with common life legth in random environments.In such models,given the environment, individuals still reproduce independently,but the offspring distribution and the random life length of individuals depends on generation.In addition,individuals in the same generaion have common life length.At the same time,we define the processes {Z_N(0,t)}_t≥0,which is related to {Z(t)}_t≥0.In the case of Z_N(0,t),we firstly get the property of the conditional probability generating function of Z_N(0,t)given the environment(?),then we got the expression of E_?(Z_N(0,t)),E(Z_N(0,t))respectively,and show that E(Z_N(0,t))behaves like e^α₁t(t→∞),whereα₁ is the first Malthus parameter defined as the root of E integral from n=0 to∞(e^-α₁xf′₀(1)dG₀(x))=1.Considering the related renewal equations in random environment, we gave a characterization of E_?(Z_N(0.t)))as the unique solution of a renewal eqution in random environments,and we show that it behaves like e^α₂tas t→∞,whereα₂ is the second Malthus parameter defined as the root of E log integral from n=0 to∞(e^-α₂xf′₀(1)dG₀(x))=0.With respect to the processes {Z(t)}_t≥0,We obtained the corresponding results by the similar method lastly.