## Population Dynamics

Constrained by certain basic assumptions, a mathematical model can describe salient dynamics of a population.[1] For example, a population may be considered a group of animals of the same species (i.e., potentially interbreeding) in a specific geographic location. A population can interact with others through migration (i.e., immigration and emigration), but in general, members of a population interbreed with each other more frequently than with members of other populations.[2] One of the basic assumptions in most population models is that the members of the population are subject to the same environmental conditions.

In human demographics, epidemiology, and wildlife management applications, models are frequently used to describe and predict the growth of populations. These models consider basic population parameters such as numbers of births, deaths, immigration, and emigration. An example of a simple model describing changes in population size (N) over a time interval (t) is:

where B is the number of births, D is the number of deaths, I is the number of immigrants, and E is the number of emigrants.[1,2] The basic assumptions of the model are simple: numbers increase due to births and immigration, and numbers decrease due to deaths and emigration. Data that could be entered into the model include estimates of births (or recruitment into the adult cohort), death (including harvest or culling), and numbers of animals moving into and out of the population.

Other mathematical models of population growth that are commonly used include those for exponential and logistic growth. Exponential models describe population growth of the type described by Thomas Malthus in his famous description of the geometric increase in population size. Exponential population models rely on estimates of the rate of population growth, which is derived from estimates of individual fecundity. In these models, as the population increases the rate of increase per individual (i.e., fecundity) remains constant, but the increasing number of individuals results in an increasing rate of population growth. In simple terms, exponential growth may be described by the model:

N{t + k) = N {t)Rk where N(t) is the population size at time t, N(t + k) is the population size at time t+k (say after k years or generations), and R is the rate of population growth.[1] Note that R can be positive or negative depending on whether a population is increasing or decreasing in size. The exponential model allows prediction of unrestrained population growth. Because populations are actually limited by resource availability, predictions from these models are usually accurate for only a limited amount of time, until the rate of growth decreases.

The logistic population growth model incorporates information about resource availability and allows for change in the rate of population growth. The standard population logistic equation is:

where dN is the change in population number, dt is the change in time, rmax is the maximum rate of increase, N is the beginning population size, and K is the maximum sustainable population size or carrying capacity.[3] This model is popular in wildlife management applications, but has limitations because it has the unlikely assumption that K remains the same regardless of population size. In many situations, resource availability is dynamic and also affects K.