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The logistic equation can be modified to include a

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size. The logistic equation can be modified to include a time lag: N ( t -τ) = population size at time t – τ in the past. At intermediate levels, (0.368 < r τ < 1.57), damped oscillations result. r τ is large ( r τ > 1.57), the population fluctuates indefinitely about the carrying capacity. This pattern is called a stable limit cycle . Small populations are at greatest risk. When σ = 0.2, 0.3% of populations went extinct in 70 years. When σ = 0.4, 17% of populations went extinct. When σ = 0.8, 53% of the populations went extinct. Genetic drift—chance events influence which alleles are passed on to the next generation. Allele frequencies can change at random from one generation to the next, and genetic variation is reduced. Alleles may become fixed (frequency 100%) or lost from the population. Genetic drift impacts small populations:1. Reduced genetic variation reduces ability of population to respond to environmental change.2. Harmful alleles can occur at high frequencies. 3. High frequency of inbreeding (mating between related individuals). Inbreeding tends to increase the frequency of homozygotes. Individuals can have two copies of a harmful allele. Reproductive success is reduced. Demographic stochasticity —chance events affect survival and reproduction of individuals. Example: In a population of ten individuals, a storm wipes out six. Allee effects : Growth rate decreases as population density decreases. Environmental stochasticity: Unpredictable changes in the environment that can cause extinction of small populations. Natural catastrophes (floods, fires, etc.) can eliminate or greatly reduce even large populations, and play a role in extinctions. Metapopulation : A set of spatially isolated populations linked by dispersal of individuals or gametes. Characterized by repeated extinctions and colonizations of the small individual populations, but the metapopulation persists. Metapopulations are characterized by repeated extinctions and colonizations of the small individual populations, but the metapopulation persists. p = Proportion of habitat patches occupied at time t c = Patch colonization rate e = Patch extinction rate e p p c p d t d p 1 For a metapopulation to persist for a long time, the ratio e / c must be <1. Some patches will be occupied as long as colonization rate is greater than extinction rate; otherwise, all populations in the metapopulation will become extinct. Habitat fragmentation : Large tracts of habitat are converted to isolated patches, resulting in a metapopulation structure. As patches get smaller and more isolated, colonization decreases and extinction rate increases. If e / c becomes >1, the metapopulation will go extinct. K t N r N d t d N ) ( 1
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