= Solution
Take selection for darker coats as the premise of the hypothetical example, rather than an established prediction about polar bears. The course of an <adaptive walk> depends on the available variation, its inheritance, fitness effects and ecological context. Warming alone does not specify a pigmentation mechanism, the strength of selection or whether dark coloration would improve total reproductive success.
The initial response can use <standing genetic variation>. If darker <phenotypes> already segregate and have an additive heritable component, a response can begin without waiting for a new <mutation>. The <breeder's equation> $R=h^2S$ describes a short-term one-trait expectation under its assumptions. Here $S$ is the difference between the mean <phenotype> of successful parents and the population mean, while $h^2$ is narrow-sense <heritability> in the relevant environment. Environmental darkening without inherited differences does not supply the same response.
If useful variants are absent, arrival can limit adaptation. In a diploid population with census size $N$ and per-haploid-genome beneficial <mutation rate> $U_b$, the approximate number of new beneficial copies is $2NU_b$ per generation. Most rare variants are lost initially. In a suitable large-population, weak-selection model, an <allele> whose heterozygote has advantage $s$ has establishment probability of approximately $2s$; this is not a general formula for every dominance relation or bottleneck. The <mutation> supply depends on actual reproductive numbers, while <effective population size> controls much of the drift and inheritance history. Long generation times can make a short walk in generations slow in calendar years.
The <genetic architecture> matters. A change in pigment production, hair structure or the spatial regulation of a developmental pathway can have different pleiotropic consequences. A large-effect <allele> may be hidden when recessive, constrained by costs in another trait, or beneficial only after a change at another locus. Under <sign epistasis>, the sign of a step depends on the starting <genotype>; under diminishing returns, later favourable steps tend to have smaller benefits on an already adapted background. One-step-at-a-time fixation is only an approximation for a sexual, potentially polygenic population.
Recombination can assemble favourable variants inherited from different parents, but can also break favourable combinations. <Gene flow> can introduce useful <alleles> or oppose local adaptation if migrants come from a differently selected population. <Population bottlenecks> can remove rare variants and strengthen drift. Spatial differences in substrate, prey, mate choice or thermal conditions can maintain more than one coat <phenotype> instead of producing uniform dark fixation.
Finally, the fitness landscape can move while the population responds. Selection on survival, reproduction, energetic balance and correlated traits must be considered jointly. If demographic decline is faster than adaptive change, the existence of an accessible darker <phenotype> does not guarantee <evolutionary rescue>. \b[No particular number of substitutions, waiting time or final shade follows from the premise alone.] One would need measurements of inherited variation, population structure, generation time and the fitness of alternative <phenotypes> in the changing environment.
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