References
- Grobman DM . Homeomorphism of systems of differential equations. Dokl Akad Nauk SSSR. 1959;128(5):880–881.
- Hartman P . A lemma in the theory of structural stability of differential equations. Proc Am Math Soc. 1960;11(4):610–620.
- Lyapunov AM . The general problem of the stability of motion. Int J Control. 1992;55(3):531–534.
- Goh BS . Global stability in many-species systems. Am Naturalist. 1977;135–143.
- Hofbauer J , Sigmund K . Adaptive dynamics and evolutionary stability. Appl Math Lett. 1990;3(4):75–79.
- Waage P , Guldberg CM . Studies concerning affinity. J Chem Educ. 1986;63(12):1044.
- McCallum H , Barlow N , Hone J . How should pathogen transmission be modelled? Trends Ecol Evol. 2001;16(6):295–300.
- Geritz SAH , Kisdi E , Meszena G , et al . Evolutionarily singular strategies and the adaptative growth and branching of the evolutionary tree. Evol Ecol. 1998;12:35–57.
- Metz JAJ , Staňková K , Johansson J . The canonical equation of adaptive dynamics for life histories: from fitness-returns to selection gradients and Pontryagin’s maximum principle. J Math Biol. 2016;72(4):1125–1152.
- Lotka AJ . Elements of physical biology. Baltimore: Wiliams and Wilkins; 1925.
- Volterra V . Variations and fluctuations of the number of individuals in animal species living together. Journal du Conseil Permanent Internation pour l’Exploration de la Mer. 1928;3(1):3–51.
- Hernández-Bermejo B , Fairén V . Lotka-Volterra representation of general nonlinear systems. Math Biosci. 1997;140(1):1–32.
- Barabás G , Michalska-Smith MJ , Allesina S . The effect of intra- and interspecific competition on coexistence in multispecies communities. Am Naturalist. 2016;188(1):E1–E12.
- Pepper JW , Rosenfeld S . The emerging medical ecology of the human gut microbiome. Trends Ecol Evol. 2012;27(7):381–384.
- Coyte KZ , Schluter J , Foster KR . The ecology of the microbiome: networks, competition, and stability. Science. 2015;350(6261):663–666.
- Begon M , Townsend CR , Harper JL . From individuals to ecosystems. Hoboken: Blackwell Publishers Hoboken; 2005.
- Hellard E , Fouchet D , Vavre F , et al . Parasite-parasite interactions in the wild: how to detect them? Trends Parasitology. 2015;31(12):640–652.
- Bashey F . Within-host competitive interactions as a mechanism for the maintenance of parasite diversity. Philos Trans R Soc B: Biol Sci. 2015;370(1675).
- Mideo N . Parasite adaptations to within-host competition. Trends Parasitology. 2009;25(6):261–268.
- Neilands J . Siderophores: structure and function of microbial iron transport compounds. J Biol Chem. 1995;270(45):26723–26726.
- Wandersman C , Delepelaire P . Bacterial iron sources: from siderophores to hemophores. Ann Rev Microbiol. 2004;58:611–647.
- Ghoul M , West SA , McCorkell FA , et al . Pyoverdin cheats fail to invade bacterial populations in stationary phase. J Evol Biol. 2016;29(9):1728–1736.
- Riley MA , Chavan MA . Bacteriocins. Berlin Heidelberg: Springer-Verlag; 2007.
- Skaar EP . The battle for iron between bacterial pathogens and their vertebrate hosts. PLOS Pathogens. 2010;6(8):e1000949.
- Rath CM , Dorrestein PC . The bacterial chemical repertoire mediates metabolic exchange within gut microbiomes. Curr Opin Microbiol. 2012;15(2):147–154.
- Dobson A , Cotter PD , Ross RP , et al . Bacteriocin production: a probiotic trait? Appl Environ Microbiol. 2012;78(1):1–6.
- Maynard Smith J . Evolution and the theory of games. Cambridge: Cambridge University Press; 1982.
- Alizon S , Lion S . Within-host parasite cooperation and the evolution of virulence. Proc R Soc B: Biol Sci. 2011;278(1725):3738–3747.
- Simonyi E , Kaszás M . Method for the dynamic analysis of nonlinear systems. Chem Eng. 1968;12(4):314–324.
- Zeeman ML . Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems. Dyn Stab Syst. 1993;8(3):189–216.
- Takeuchi Y . Global dynamical properties of Lotka-Volterra systems. Singapore: World Scientific; 1996.
- Berman A , Plemmons RJ . Nonnegative matrices in the mathematical sciences. New York (NY): Academic Press; 1979.
- Jacques M . The growth of bacterial cultures. Ann Rev Microbiol. 1949;3(1):371–394.
- Ruhe ZC , Low DA , Hayes CS . Bacterial contact-dependent growth inhibition. Trends Microbiol. 2013;21(5):230–237.