Mathematically, a sequence is an ordered set of numbers. Suppose that a system has a variable \(x\) that is measured at regular intervals (e.g., a population). We often think of our variable as having an initial value and think of subsequent measurements as counting the number of additional observations. An index variable is a variable that represents the count (or index) through the observations. The value of the variable \(x\) for a given index \(n\) is indicated by writing the index value as a subscript on the variable, such as \(x_n\).

As motivated by our agent-based models, we might think of our population as changing through births and deaths. We might also include immigration and emigration. We can think of death and emigration as sources of loss, with individuals who were counted in the previous population and remain in the population for the next observation as survivors. Individuals that are new to the population, either from birth or from immigration, can be considered as recruits. The total population each year is described by the equation \begin{equation*}\hbox{New Population} = \hbox{Survivors} + \hbox{New Recruits}.\end{equation*} If we classified losses as deaths and emigrants and recruits as births and immigrants, we might consider the more detailed equation \begin{equation*}\hbox{New Population} = \hbox{Old Population} - \hbox{Deaths} - \hbox{Emigrants} + \hbox{Births} + \hbox{Immigrants}.\end{equation*}

A mathematical model for a population represents an abstraction of this pattern. First, we will consider each of the terms as variables in a system. Then we will seek mathematical equations that relate these variables in a way that will allow us to predict how the variables change. Because the real system involves stochasticity as well as more variables than we can reasonably include, our equations will only attempt to model enough of the behavior that we deem adequate and necessary to characterize the behavior we seek to describe.

We begin by defining our system. Let \(P\) represent the population. Let \(B\) represent the number of births, or more precisely, the number of individuals in \(P\) that were born since the last observation and have survived for the current census. Let \(D\) represent the number of deaths of individuals that were counted in the previous observation. Let \(E\) and \(I\) represent the number of emigrants (individuals of the previous census who left) and the number of immigrants (counted individuals that arrived since the previous census), respectively. Each of these variables in the system represent sequences. Our population equation becomes an equation involving sequence values \begin{equation*}P_{n+1} = P_n - D_{n+1} - E_{n+1} + B_{n+1} + I_{n+1}.\end{equation*} We often rewrite this equation as a difference equation that only shows the change in the sequence, \begin{equation*}P_{n+1} - P_n = B_{n+1} + I_{n+1} - D_{n+1} - E_{n+1}.\end{equation*}

Our next step is to create equations relating the various terms. In particular, different models correspond to applying different equations that will allow us to find the birth, death and migration rates in terms of the previous population value.

The naive model for reproduction was that the number of births and deaths are proportional to the population. That is, in terms of our sequence, there are constants \(b\) and \(d\) such that \begin{gather*} B_{n+1} = b \cdot P_{n},\\ D_{n+1} = d \cdot P_{n}. \end{gather*} We call the constants \(b\) and \(d\) the per capita birth rate and the per capita death rate, respectively. The phrase per capita literally means per head, and the per capita rates represent the average per individual contribution to birth and death. Rewriting the equations, we could instead define \(b\) and \(d\) as the following ratios, \begin{gather*} b = \frac{B_{n+1}}{P_{n}},\\ d = \frac{D_{n+1}}{P_{n}}, \end{gather*} which we are modeling as being constant. (More advanced models consider these ratios as changing.) We require that \(b \ge 0\) and \(0 \le d \le 1\).

In the absence of migration, the constant per capita rate model leads to a dynamic model \begin{equation*}P_{n+1} = P_n - d \cdot P_n + b \cdot P_n.\end{equation*} We can rewrite this as \begin{equation*}P_{n+1} = (1-d+b) P_n = (1+r) P_n,\end{equation*} where \(r=b-d\) is called the per capita net growth rate. This equation, called the recursive equation, is identical to the equation for compounded interest where \(r\) represents that percent growth rate. Such a model results in a geometric sequence for the population size, \begin{equation*}P_n = P_0 \cdot (1+r)^{n}.\end{equation*} We call this representation of the sequence an explicit formula because we can compute the population value \(P_n\) knowing only index \(n\) and the initial value. A geometric sequence corresponds to either exponential growth (when \(r \gt 0\)) or exponential decay (when \(r \lt 0\)). In the singular case where \(r=0\), the population remains constant.

Recall that constant per capita growth rates where the birth rate exceeds the death rate imply that a population will grow exponentially and without bound. This is inconsistent with the reality that as populations get larger environmental constraints limit their growth. This implies that for populations experiences these constraints are not well-modeled using constant per capita growth rates. (Such a model might work well during the time that the population is far from its limiting size.)

We previously defined the per capita net growth rate \(r=b-d\) where \(b\) is the per capita birth rate and \(d\) is the per capita death rate. Sometimes it is inconvenient or impossible to determine the birth and death rates separately. If we can at least measure the net change in the population, which will be equal to \(\Delta P_{n+1} = B_{n+1}-D_{n+1}\), then the per capita net growth rate will be a sequence defined by \begin{equation*}r_{n+1} = \frac{B_{n+1}-D_{n+1}}{P_{n}}.\end{equation*}

We turn our attention to the case that the per capita net growth rate can be modeled as a function of the population size \(P_n\). This means that if we were to consider a scatter plot of the points \((P_n, r_{n+1})\) that the graph could be reasonably approximated by a function. We generally expect that the graph will be a decreasing function because as the population gets larger, the growth rate should decrease. For a very large population, we even expect \(r \lt 0\) since the population would be greater than its carrying capacity. However, we will always have \(r \ge -1\), with \(r=-1\) corresponding to complete extinction.

In order to develop our models, we will consider parametric formulas. A parametric formula is a function that involves parameters to describe a family of functions that share a common algebraic structure. A parameter is a symbol that represents a constant that appears in the formula. One might think of a parameter as a variable that helps describe the state of a system but does not depend on the state of the system.

The simplest density dependent model to write down would be when \(r\) is a linear function of \(P\), \begin{equation*}r(P) = r_0 - a P,\end{equation*} Here \(r_0\) is called the intrinsic per capita net growth rate and represents the per capita net growth rate when \(P \approx 0\). The parameter \(a\) is the negative slope giving how much the growth rate decreases per individual. We will learn about other parametric models when we consider some alternative representations for the recursive formula defining our sequence.