Population growth for a single-species population can be modeled in a variety of ways. Populations will grow exponentially if the per capita reproductive rate is constant and independent of population size. Simple difference equations used to model populations with discrete generations can generate complex behaviors including converging on an equilibrium population size, limit cycles, and chaos. The most common model used to estimate growth in populations with overlapping generations is the Verhulst-Pearl logistic equation where population growth stops at the carrying capacity. The Verhulst-Pearl model has restrictive assumptions, and was modified into the 0-logistic model which relaxes the assumptions of the relationship between the intrinsic rate of increase and population size. Time-lag models allow for the future population size to be dependent on a population size in the past. Adding a time lag to a model can destabilize it and cause fluctuations in population size, the amplitude of the fluctuations depending on the length of the time lag. Age- or size-based matrices group individuals into classes with each class having its own natality and mortality rates, and these models can project the future number of individuals for each class. Stochasticity can be parametrized into any model and the outcome will no longer be deterministic and will better reflect the probabilistic nature of biological populations.

See also: Age-Class Models; Allee Effects; Demography; Fecundity; Individual-Based Models; Matrix Models; Parameters; Population Viability Analysis.

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