Darwin suggested that sexual selection, especially by female choice, may counter natural selection. Theoretical models, such as a Fisherian runaway process, suggest that evolution of preference and preferred phenotypes may drive each other in ever increasing speed.
Because one male may fertilize many females, one could imagine that natural selection against preferred but energetically costly phenotypes may be weak, and the whole process may not slow down fast enough (i.e., be sufficiently self-limiting). If male mortality is high and their number is low, the random fluctuations may easily cause the extinction of population.
Is there any fossil or experimental evidence that this may really happen?
Answer
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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