The stochastic evolutionary game on robust linear biological network under natural selection

Evolution is the physical, genetic or behavioral change in populations of biological networks over time as a result from natural selection. Natural selection works within genetic, developmental and environmental constraints to shape biological networks in ways that make them appear adapted to their environment. Network robustness is the persistence in the stability of phenotypic trait of biological networks under genetic variations and environmental disturbances in the evolutionary process. In this study, a population of evolutionary biological networks are described by a stochastic dynamic system with intrinsic random parameter fluctuations due to genetic variations and external disturbances due to environmental changes in the evolutionary process. The worst case effect of environmental changes on the phenotype stability of evolutionary biological network is considered as evolutionary level of stochastic biological network, and the fitness of biological network is inverse to evolutionary level. Therefore, the maximum fitness of genetic network to the desired phenotypic trait can be formulated as a stochastic minmax evolutionary game problem. Based on the linear stochastic evolutionary game strategy, we find that if the network robustness can provide larger genetic robustness to buffer neutral genetic variations and larger environmental robustness to resist environmental disturbance in the evolutionary process, then the phenotypic traits of biological networks could be more selected by natural selection in evolution. If the accumulated neutral genetic variations are large enough and environmental disturbances are strong enough so that the network robustness is no longer to confer enough genetic robustness and environmental robustness for them simultaneously, the phenotype robustness might be broke down and phenotypic trait may be pushed from one equilibrium point to other equilibrium point to change phenotypic traits through the hidden neutral genetic variations ha- bored in network robustness by the adaptive evolution.