TL;DR:
There is a dearth of actual experimental evidence. However:
there is at least one study that confirmed the process ([STUDY #7] - Myxococcus xanthus; by Fiegna and Velicer, 2003).
Another study experimentally confirmed higher extinction risk as well ([STUDY #8] - Paul F. Doherty's study of dimorphic bird species an [STUDY #9] - Denson K. McLain).
Theoretical studies produce somewhat unsettled results - some models support the evolutionary suicide and some models do not - the major difference seems to be variability of environmental pressures.
Also, if you include human predation based solely on sexually selected trait, examples definitely exist, e.g. Arabian Oryx
First of all, this may be cheating but one example is the extinction because a predator species specifically selects the species because of selected-for feature.
The most obvious case is when the predator species is human. As a random example, Arabian Oryx was nearly hunted to extinction specifically because of their horns.
Please note that this is NOT a simple question - for example, the often-cited in unscientific literature example of Irish Elk that supposedly went extinct due to its antler size may not be a good crystal-clear example. For a very thorough analysis, see: "Sexy to die for? Sexual selection and risk of extinction" by Hanna Kokko and Robert Brooks, Ann. Zool. Fennici 40: 207-219. [STUDY #1]
They specifically find that evolutionary "suicide" is unlikely in deterministic environments, at least if the costs of the feature are borne by the individual organism itself.
Another study resulting in a negative result was "Sexual selection and the risk of extinction in mammals", Edward H. Morrow and Claudia Fricke; The Royal Society Proceedings: Biological Sciences, Published online 4 November 2004, pp 2395-2401 [STUDY #2]
The aim of this study was therefore to examine whether the level of
sexual selection (measured as residual testes mass and sexual size dimorphism) was related to the risk of extinction that mammals are currently experiencing. We found no evidence for a relationship between these factors, although our analyses may have been confounded by the possible dominating effect of contemporary anthropogenic factors.
However, if one takes into consideration changes in the environment, the extinction becomes theoretically possible. From "Runaway Evolution to Self-Extinction Under Asymmetrical Competition" - Hiroyuki Matsuda and Peter A. Abrams; Evolution Vol. 48, No. 6 (Dec., 1994), pp. 1764-1772: [STUDY #3]
We show that purely intraspecific competition can cause evolution of extreme competitive abilities that ultimately result in extinction, without any influence from other species. The only change in the model required for this outcome is the assumption of a nonnormal distribution of resources of different sizes measured on a logarithmic scale. This suggests that taxon cycles, if they exist, may be driven by within- rather than between-species competition. Self-extinction does not occur when the advantage conferred by a large value of the competitive trait (e.g., size) is relatively small, or when the carrying capacity decreases at a comparatively rapid rate with increases in trait value. The evidence regarding these assumptions is discussed. The results suggest a need for more data on resource distributions and size-advantage in order to understand the evolution of competitive traits such as body size.
As far as supporting evidence, some studies are listed in "Can adaptation lead to extinction?" by Daniel J. Rankin and Andre´s Lo´pez-Sepulcre, OICOS 111:3 (2005). [STUDY #4]
They cite 3:
The first example is a study on the Japanese medaka
fish Oryzias latipes (Muir and Howard 1999 - [STUDY #5]). Transgenic males which had been modified to include a salmon growth-hormone gene are larger than their wild-type counterparts, although their offspring have a lower fecundity (Muir and Howard 1999). Females
prefer to mate with larger males, giving the larger
transgenic males a fitness advantage over wild-type
males. However, offspring produced with transgenic
males have a lower fecundity, and hence average female
fecundity will decrease. As long as females preferentially
mate with larger males, the population density will
decline. Models of this system have predicted that, if
the transgenic fish were released into a wild-type
population, the transgene would spread due to its mating
advantage over wild-type males, and the population
would become go extinct (Muir and Howard 1999).
A recent extension of the model has shown that
alternative mating tactics by wild-type males could
reduce the rate of transgene spread, but that this is still
not sufficient to prevent population extinction (Howard
et al. 2004). Although evolutionary suicide was predicted
from extrapolation, rather than observed in nature, this
constitutes the first study making such a prediction from
empirical data.
In cod, Gadus morhua, the commercial fishing of large
individuals has resulted in selection towards earlier
maturation and smaller body sizes (Conover and Munch
2002 [STUDY #6]). Under exploitation, high mortality decreases the
benefits of delayed maturation. As a result of this,
smaller adults, which mature faster, have a higher fitness
relative to their larger, slow maturing counterparts
(Olsen et al. 2004). Despite being more successful
relative to slow maturing individuals, the fast-maturing
adults produce fewer offspring, on average. This adaptation,
driven by the selective pressure imposed by
harvesting, seems to have pre-empted a fishery collapse
off the Atlantic coast of Canada (Olsen et al. 2004). As
the cod evolved to be fast-maturing, population size was
gradually reduced until it became inviable and vulnerable
to stochastic processes.
The only strictly experimental evidence for evolutionary
suicide comes from microbiology. In the social
bacterium Myxococcus xanthus individuals can develop
cooperatively into complex fruiting structures (Fiegna
and Velicer 2003 - [STUDY #7]). Individuals in the fruiting body are
then released as spores to form new colonies. Artificially
selected cheater strains produce a higher number of
spores than wild types. These cheaters were found to
invade wild-type strains, eventually causing extinction of
the entire population (Fiegna and Velicer 2003). The
cheaters invade the wild-type population because they
have a higher relative fitness, but as they spread through
the population, they decrease the overall density, thus
driving themselves and the population in which they
reside, to extinction.
Another experimental study was "Sexual selection affects local extinction and turnover
in bird communities" - Paul F. Doherty, Jr., Gabriele Sorci, et al; 5858–5862 PNAS May 13, 2003 vol. 100 no. 10 [STUDY #8]
Populations under strong sexual selection experience
a number of costs ranging from increased predation and
parasitism to enhanced sensitivity to environmental and demographic
stochasticity. These findings have led to the prediction that
local extinction rates should be higher for speciespopulations
with intense sexual selection. We tested this prediction by analyzing
the dynamics of natural bird communities at a continental
scale over a period of 21 years (1975–1996), using relevant statistical
tools. In agreement with the theoretical prediction, we found
that sexual selection increased risks of local extinction (dichromatic
birds had on average a 23% higher local extinction rate than
monochromatic species). However, despite higher local extinction
probabilities, the number of dichromatic species did not decrease
over the period considered in this study. This pattern was caused
by higher local turnover rates of dichromatic species, resulting in
relatively stable communities for both groups of species. Our
results suggest that these communities function as metacommunities,
with frequent local extinctions followed by colonization.
This result is similar to another bird-centered study: Sexual Selection and the Risk of Extinction of Introduced Birds on Oceanic Islands": Denson K. McLain, Michael P. Moulton and Todd P. Redfearn. OICOS Vol. 74, No. 1 (Oct., 1995), pp. 27-34 [STUDY #9]
We test the hypothesis that response to sexual selection increases the risk of extinction by examining the fate of plumage-monomorphic versus plumage-dimorphic bird species introduced to the tropical islands of Oahu and Tahiti. We assume that plumage dimorphism is a response to sexual selection and we assume that the males of plumage-dimorphic species experience stronger sexual selection pressures than males of monomorphic species. On Oahu, the extinction rate for dimorphic species, 59%, is significantly greater than for monomorphic species, 23%. On Tahiti, only 7% of the introduced dimorphic species have persisted compared to 22% for the introduced monomorphic species.
...
Plumage is significantly associated with increased risk of extinction for passerids but insignificantly associated for fringillids. Thus, the hypothesis that response to sexual selection increases the risk of extinction is supported for passerids and for the data set as a whole. The probability of extinction was correlated with the number of species already introduced. Thus, species that have responded to sexual selection may be poorer interspecific competitors when their communities contain many other species.
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