The advantages of stochastic models illustrations

Differences between deterministic and stochastic ecological models have been emphasised many times, explicity or implicitly. The main qualitative deviation in the behaviour of ecodynamic models can be interpreted in terms of survival and extinction. 

Macroscopic ecological level equations have emerged from microscopic genetic equations, and there is a feedback from the macroscopic structure to modify the microscopic dynamics. The appearance and eventual stabilisation of a new mutant might have a crucial role in the evolutionary process. 

Allen Ebeling (1983) have argued for the necessity of a stochastic description of simple ecosystems connected to evolution. They considered interactions among a prey species. A , and a predator species, Y, where / and k denote phenotypes. Elementary steps at the microscopic level modifying the number of interacting species  

To described the temporal evolution of the probability distribution of a new mutant population starting from a pre-existing situation a linearised birth and death equation can be derived  

Evaluating this expression for a time t = njA where n is the number of reproduced generations during /  

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