Synthetic microbial consortia are conglomerations of multiple strains of genetically engineered microbes programmed to cooperatively bring about population-level phenotypes. By coordinating their activity, the constituent strains can display emergent behaviors that are difficult to engineer into isogenic populations. To do so, strains are engineered to communicate with one another through intercellular signaling pathways. As a result, the regulatory networks that control gene transcription throughout the population are sensitive to the extracellular concentration of the signaling molecules, and hence the relative densities of constituent strains. Here, we use computational modeling to examine how the behavior of a synthetic microbial consortium results from the interplay between the population dynamics governed by cell growth and the internal transcriptional dynamics governed by cell-to-cell signaling. Specifically, we examine a synthetic microbial consortium in which two strains each produce signals that down-regulate transcription in the other. Within a single strain this regulatory topology is called a "co-repressive toggle switch" and can lead to bistability. We find that in a two-strain synthetic microbial consortium the existence and stability of different states depends on the population-level dynamics of the interacting strains. As the two strains passively compete for space within the colony, their relative fractions can fluctuate and thus alter the strengths of intercellular signals. These fluctuations can drive the consortium to alternative equilibria. Additionally, if the growth rates of the strains depend on their transcriptional states, an additional feedback loop is created that can generate relaxation oscillations. These findings demonstrate that the dynamics of microbial consortia cannot be predicted from their regulatory topologies alone, but also is determined by interactions between the strains.