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We consider a three-stage discrete-time population model with density-dependent survivorship and time-dependent reproduction. We provide stability analysis for two types of birth mechanisms: continuous and seasonal. We show that when birth is continuous there exists a unique globally stable interior equilibrium provided that the inherent net reproductive number is greater than unity. If it is less than unity, then extinction is the population's fate. We then analyze the case when birth is a function of period two and show that the unique two-cycle is globally attracting when the inherent net reproductive number is greater than unity, while if it is less than unity the population goes to extinction. The two birth types are then compared. It is shown that for low birth rates the adult average number over a one-year period is always higher when reproduction is continuous. Numerical simulations suggest that this remains true for high birth rates. Thus periodic birth rates of period two are deleterious for the three-stage population model. This is different from the results obtained for a two-stage model discussed by Ackleh and Jang (J. Diff. Equ. Appl., 13, 261-274, 2007), where it was shown that for low birth rates seasonal breeding results in higher adult averages.

A selection model with n traits is considered. It is assumed that the mortality function is density dependent and that individuals with "weak" traits are able to disperse to a safe refuge patch and avoid competition with individuals carrying the strongest trait. It is shown that if any subpopulation with a "weak" trait does not have a safe refuge then it will become extinct. Therefore, for survival of n traits n − 1 safe refuge patches are needed. When n − 1 refuge patches are available global stability of the interior equilibrium is proved provided that the fittest trait is sufficiently better than the other traits. Finally, two special cases with linear and Beverton-Holt density dependent mortality functions are studied in detail.

This paper compares the PPC model to a Markov Chain (MC) stochastic deterioration model. First, inspection data from the Société de Transport de Montréal (STM) is gathered and analyzed. Then Transition Probability Matrices (TPM) are developed, and, using Matlab, MC deterioration curves are developed. Comparison between MC and the PPC deterioration curves is performed for subway station walls and slabs. The comparison has shown that the useful service life can be as low as 2 years for components having many inspection history records, and very high as 30 years for components having very few inspection history records. The PPC model has always a higher useful service life estimate. Also, the MC has a ten times higher deterioration rate (0.2 per year) compared to the PPC model (0.02 per year). It can be concluded that the MC deterioration model requires a high amount of inspection data, and it is mathematically difficult to generate since most practicing managers and engineers have no background in Markov Chain modeling.