Cooperative interactions, their stability and evolution, provide an interesting context in which to study the interface between cellular and population levels of organization. Here we study a public goods model relevant to microorganism populations actively extracting a growth resource from their environment. Cells can display one of two phenotypes - a productive phenotype that extracts the resources at a cost, and a non-productive phenotype that only consumes the same resource. Both proliferate and are free to move by diffusion; growth rate and diffusion coefficient depend only weakly phenotype. We analyze the continuous differential equation model as well as simulate stochastically the full dynamics. We find that the two sub-populations, which cannot coexist in a well-mixed environment, develop spatio-temporal patterns that enable long-term coexistence in the shared environment. These patterns are purely fluctuation-driven, as the corresponding continuous spatial system does not display Turing instability. The average stability of coexistence patterns derives from a dynamic mechanism in which the producing sub-population equilibrates with the environmental resource and holds it close to an extinction transition of the other sub-population, causing it to constantly hover around this transition. Thus the ecological interactions support a mechanism reminiscent of self-organized criticality; power-law distributions and long-range correlations are found. The results are discussed in the context of general pattern formation and critical behavior in ecology as well as in an experimental context.
Bibliographical noteFunding Information:
This research is supported in part by the Israel Binational Science Foundation personal grant (NB) (grant number 1566/11 ).
- Microbial cooperation
- Microorganism population
- Spatio-temporal patterns
- Stochastic processes