Carrying capacity is the maximum number of individuals an environment can sustain indefinitely given the available resources, space, and waste removal. As a population grows toward that limit, resources become scarcer and competition intensifies, causing the per‑capita growth rate to decline. Because births minus deaths slow down with increasing density, the population’s growth approaches zero and the size tends to stabilize near the carrying capacity. If the population overshoots, resource depletion can trigger higher mortality or reduced reproduction, bringing numbers back down toward the limit. Remember that carrying capacity can shift with changes in resources, climate, or technology, so long‑term population size generally centers around this upper bound, even though it may fluctuate around it.

Genetic drift describes random fluctuations in allele frequencies from one generation to the next, driven by chance sampling effects in small populations. When only a small number of individuals contribute genes to the next generation, the alleles that get passed on may not represent the entire parental gene pool. These random shifts can cause certain alleles to become more common or disappear entirely over time, sometimes leading to fixation of an allele independent of its fitness. This contrasts with natural selection, which changes frequencies based on differential survival and reproduction, and with gene flow, which moves alleles between populations. Mutation introduces new alleles but does not describe the random sampling changes in existing allele frequencies typical of drift.

In a closed population, the number of individuals changes only due to births and deaths. There is no movement of individuals across the population boundary, so immigration and emigration are zero. Therefore, the net change in population size is births minus deaths. If births exceed deaths, the population grows; if deaths exceed births, it declines; if they are equal, the population remains the same. The other formulas involve migration or double-counting births and deaths, which aren’t relevant here because there’s no migration and we’re not adding those two numbers together.

Gene flow is the movement of alleles between populations, brought by migrating individuals or dispersal of gametes. When new alleles enter a population, genetic variation within that population increases because there are more alleles for natural selection to act on. At the same time, gene flow tends to make populations more genetically similar to each other, reducing differences in allele frequencies between populations. This combination—increasing variation within populations and decreasing differences among populations—can influence adaptation by supplying new genetic material for selection and spreading advantageous alleles across populations. It can also limit local adaptation if migrants carry alleles that aren’t well suited to a particular environment. The other statements misstate the effects: gene flow does not typically reduce variation within a population; it can increase it. It does not only affect population size; it directly affects genetic variation. And it does not generally increase differences between populations; it tends to lessen those differences.

Genetic variation in populations comes from mutations—random changes in DNA. When a mutation occurs and is passed on to offspring (a heritable, germline change), it creates a new allele in the population's gene pool. This adds new variants that natural selection, genetic drift, and other evolutionary forces can act upon, increasing the overall variation. While many mutations are neutral or harmful and may not spread, the arrival of new heritable DNA changes generally boosts the diversity available for evolution. If a DNA change stays only in somatic cells, it won’t affect population variation.