Density dependence, boundedness, and attraction: detecting stability in stochastic systems

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Abstract
By analogy with deterministic stability, the stability of stochastic ecological systems can be viewed as a tendency for population densities to avoid dynamic boundaries (i.e. boundedness) or to approach a dynamic attractor (i.e. attraction). At the population level, these two views generate predictions consistent with density dependence. I therefore devised two new statistical tests of attraction, the “random-walk attraction test” and the “randomized attraction test”; I then used them successfully, along with randomization techniques that detect boundedness and two autocorrelation methods, to test for density dependence in published sequences of population densities. The attraction tests identify the apparent attractor, the band of densities toward which density tends to shift between generations. Locating the apparent attractor can generate a prediction of the next direction of density change; for data from a dragonfly assemblage, about 80% of these predictions were correct. From the single-population tests, I also developed two multispecies tests of attraction (the multispecies random-walk and randomized attraction tests) and two multispecies tests of boundedness (the multispecies permutation and randomization tests). These detected attraction and boundedness in the dragonfly assemblage and attraction in a collection of laboratory fruitfly populations. An evaluation of the statistical power of the new density attraction tests indicates a strong dependence on the sequence length n and on the number of populations m: power increases with n and particularly with m. Nevertheless, detecting attraction becomes likely even in populations with strong linear density-dependence only with n>30 or for shorter sequences in multispecies assemblages.