Evolutionary stability is sensitive on the conflict between reproduction and survival: proofs

Evolutionary game theory is a powerful mathematical framework to study how phenotypes evolve by natural selection. Both birth and death are key in classic models in evolutionary games. The conflict between the two is fundamental in life history theory. The conflict between birth and death has been shown to change the evolutionary outcome for continuous traits. However, it is not clear how the conflict reshapes the evolutionary outcome for discrete strategies. An allocation model is proposed, in which part of the payoff is mapped to reproduction and the rest is mapped to illness. For non-evolving allocation, it is proved that the allocation does not change the fixation probability if and only if the illness is an inverse exponential function and the product of reproduction function and illness function is a constant. The necessary and sufficient condition implies that the allocation dramatically alters the evolutionary stability for a wide class of evolutionary processes. This is also verified by alternative construction proofs and numerical examples. Furthermore, the illness and reproduction function also ensures that every allocation is a neutral stable regime, if the allocation evolves to maximize the invasion probability. A deviation can lead to a non-trivial evolutionary branching. These results explicitly show that the reproduction and illness functions are restrictive to ensure the invariance of evolutionary outcome. Thus it implies that the demographic and life history need to be considered together to understand patterns of evolutionary dynamics.