evolution - Do organisms have a mechanism to control how fast mutations occur?

Sometimes the environment and other species can change rapidly (climate changes, changes in terrains from erosion, new river paths, etc), and it seems like a regular slow mutation rate would be bad, and faster-mutating species would thrive. On the other hand, this fast mutation rate seems wasteful and inefficient when the environment is changing slowly, thus slower mutating species thrive with less waste in this situation.

Organisms capable of responding to the pace of environmental change would seemingly do well in both situations. Does such ability to modulate the rate of mutation/evolution exist in nature?

Answer

The ability of a population to respond to environmental change through evolution occurs over generations, and so the generation time of an organism has to be less than the time scale of the environmental change for such an increase in mutation rate to be beneficial.

For bacteria at least, the answer to your question is yes. Bacterial stress responses involve upregulation of error prone polymerases and thus lead to an increased mutation rate. Check out this paper (links and emphasis added):

Foster PL. 2005. Stress responses and genetic variation in bacteria. Mutat Res 569(1-2):3-11.

Under stressful conditions mechanisms that increase genetic variation can bestow a selective advantage. Bacteria have several stress responses that provide ways in which mutation rates can be increased. These include the SOS response, the general stress response, the heat-shock response, and the stringent response, all of which impact the regulation of error-prone polymerases. Adaptive mutation appears to be [a] process by which cells can respond to selective pressure specifically by producing mutations. In Escherichia coli strain FC40 adaptive mutation involves the following inducible components: (i) a recombination pathway that generates mutations; (ii) a DNA polymerase that synthesizes error-containing DNA; and (iii) stress responses that regulate cellular processes. In addition, a subpopulation of cells enters into a state of hypermutation, giving rise to about 10% of the single mutants and virtually all of the mutants with multiple mutations. These bacterial responses have implications for the development of cancer and other genetic disorders in higher organisms.

This mechanism may be involved in the evolution of antibiotic resistance, for example (emphasis added):

Cirz RT, Chin JK, Andes DR, de Crecy-Lagard V, Craig WA, Romesberg FE. 2005. Inhibition of Mutation and Combating the Evolution of Antibiotic Resistance. PLoS Biol 3(6):e176.

The emergence of drug-resistant bacteria poses a serious threat to human health. In the case of several antibiotics, including those of the quinolone and rifamycin classes, bacteria rapidly acquire resistance through mutation of chromosomal genes during therapy. In this work, we show that preventing induction of the SOS response by interfering with the activity of the protease LexA renders pathogenic Escherichia coli unable to evolve resistance in vivo to ciprofloxacin or rifampicin, important quinolone and rifamycin antibiotics. We show in vitro that LexA cleavage is induced during RecBC-mediated repair of ciprofloxacin-mediated DNA damage and that this results in the derepression of the SOS-regulated polymerases Pol II, Pol IV and Pol V, which collaborate to induce resistance-conferring mutations. Our findings indicate that the inhibition of mutation could serve as a novel therapeutic strategy to combat the evolution of antibiotic resistance.