Phage and bacteria support mutual diversity in a narrowing staircase of coexistence

The competitive exclusion principle states that phage diversity M should not exceed bacterial diversity N. By analyzing the steady-state solutions of multistrain equations, we find a new constraint: the diversity N of bacteria living on the same resources is constrained to be M or M+1 in terms of the diversity of their phage predators. We quantify how the parameter space of coexistence exponentially decreases with diversity. For diversity to grow, an open or evolving ecosystem needs to climb a narrowing 'diversity staircase' by alternatingly adding new bacteria and phages. The unfolding coevolutionary arms race will typically favor high growth rate, but a phage that infects two bacterial strains differently can occasionally eliminate the fastest growing bacteria. This context-dependent fitness allows abrupt resetting of the 'Red-Queen's race' and constrains the local diversity.