Think of it as exponential decay. If the half life of a completely non-functional gene is only 2 hours, then after a week only one in 10^25 bacteria have it—i.e. none. Trace amounts of functionality (like trace amounts of antibiotics) will slightly extend the half life—but this causes outsized effects at the tails. If the half life of an antibiotic resistance gene when there’s trace antibiotics gets extended to 3 hours, then after a week one in 10^17 bacteria still have it—i.e. still probably none, but maybe a few that can carry the gene from farm to farm or hospital to hospital.
One other mechanism that would lead to the persistence of e.g. antibiotic resistance would be when the mutation that confers the resistance is not costly (e.g. a mutation which changes the shape of a protein targeted by an antibiotic to a different shape that, while equally functional, is not disrupted by the antibiotic). Note that I don’t actually know whether this mechanism is common in practice.
Think of it as exponential decay. If the half life of a completely non-functional gene is only 2 hours, then after a week only one in 10^25 bacteria have it—i.e. none. Trace amounts of functionality (like trace amounts of antibiotics) will slightly extend the half life—but this causes outsized effects at the tails. If the half life of an antibiotic resistance gene when there’s trace antibiotics gets extended to 3 hours, then after a week one in 10^17 bacteria still have it—i.e. still probably none, but maybe a few that can carry the gene from farm to farm or hospital to hospital.
One other mechanism that would lead to the persistence of e.g. antibiotic resistance would be when the mutation that confers the resistance is not costly (e.g. a mutation which changes the shape of a protein targeted by an antibiotic to a different shape that, while equally functional, is not disrupted by the antibiotic). Note that I don’t actually know whether this mechanism is common in practice.