Understanding the rift between update rules in Evolutionary Graph Theory: The intrinsic death rate drives star graphs from amplifying to suppressing natural selection

Evolutionary graph theory is the study of evolutionary dynamics in structured populations. A well-known problem in evolutionary graph theory is that the spread of mutation (measured by fixation probability) is impacted by the graph type chosen and the update rule. For example, the star graph is an amplifier of natural selection under the birth-death with fitness on birth (Bd) update rule but a suppressor of natural selection under the death-birth with fitness on birth (dB) update rule. A continuous-time EGT model has been found to replicate Bd and dB results as special cases. Using this model, we show that changing the natural (intrinsic) death rate can cause a shift from Bd to dB dynamics. Assuming the mutant is advantageous, we show that if the natural death rate is greater than $\frac{1}{\sqrt{N}}$ the star is a suppressor, where $N$ is the number of nodes. As $N \longrightarrow \infty$, the natural death rate required to drive the star to a suppressor tends towards zero, so as the size of the graph increases, the star graph is likely to be suppressing for any non-zero natural death rate.