Practice question

Geometric growth advances by multiplication with a fixed finite rate: N(t+1) = λN(t). Therefore each population census bears the same ratio to the preceding census, and after t intervals Nt = N0λ^t. A fixed ratio is different from a fixed numerical increment. For example, λ = 1.2 adds 20 individuals to a population of 100 but 200 to a population of 1,000 over one interval. Linear growth would add the same absolute number each time. Logistic growth does not maintain a fixed ratio because density dependence makes proportional growth decrease as abundance approaches carrying capacity. Although “constant rate” can be ambiguous, geometric growth specifically keeps the proportional or multiplicative factor constant, not the absolute increase. Values of λ above one produce growth, one gives stability, and below one produce decline. The model is suited to populations with seasonal reproduction, discrete census intervals, or nonoverlapping generations. Environmental variation can make λ differ among years, in which case long-term performance depends on compounded annual multipliers rather than a single fixed value.