The ultimate goal of studying models of population regulation is to predict, project or represent the number of individuals in a single population or an assemblage of populations of interacting species. This section will discuss the basics of models of population regulation. Models of population dynamics specify the rate of change in population abundances over time. The rate of change in population abundance may depend on many features of the population under consideration and its environment. Features of the population that may be relevant are abundance itself (i.e., how many individuals are in the population at this instant in time), and the types of individuals present (e.g., the number of adults and juveniles). Features of the environment that may be relevant are the abundances of other species and the abiotic conditions (e.g., the temperature, humidity, availability of various mineral resources such as phosphorus or nitrogen).

##### What determines the number of individuals in a population?

Population size increases because of births and immigration, and population size decreases because of deaths and emigration. Thus, if we can describe and model these four fundamental processes, we can describe how population size will change over time. The rates by which these four processes occur govern changes in population size: birth rate, death rate, immigration rate and emigration rate. Birth rate is the number of offspring produced per individual per unit time. Death rate is the number of deaths per individual per unit time. Immigration rate is the number of immigrants into the population per individual already present in the population per unit time. Finally, emigration rate is the number of individuals emigrating from the population per individual present in the population per unit time. Notice that I have defined these four fundamental rates as per capita rates, meaning that they are specified on a per individual basis. The difference between per capita and total rates of change will be critical in what follows. For example, the per capita birth rate is the average number of offspring produced per individual per unit time. This quantity would have units of births/(individual·time). In contrast, the total birth rate in the population is the total number of births per unit time, and has units of births/time. The total and per capita rates are intimately related (per capita rates are obtained by simply dividing the total rate by the number of individuals), but assessing whether certain important processes are occurring can only be assessed by one or the other. So in what follows, develop a good understanding of the differences between total and per capita rates, and pay attention to which is being assessed in various contexts. You may also notice here that time plays a critical role in what will follow because we are considering rates. The time scale over which population abundance is followed is critical to how we formulate models. Mathematically, population models are formulated on two different time scales: discrete and continuous time. Mathematically, these are related to one another, and with some notable exceptions, models formulated on these two time scales provide comparable results. Discrete time models operate like a ticking clock: demographic processes occur every time the clock ticks. The length of time between clicks of the clock are frequently specified as the time between one generation, and so the unit of time is equivalent to one generation in this case. Other lengths of time can be specified for discrete models – for example, the time between consecutive census intervals. Age- and stage-structured life history models are examples of population dynamic models formulated in discrete time. Mathematically, discrete-time models are formulated using difference equations. The other formulation of population dynamics models measure time on a continuous scale, and keep track of population size at every instant of time. Thus, population size may change at every instant in time, and so these models are called continuous time models. Mathematically, continuous-time models are formulated using differential equations.

##### A Basic Model of a Single Population – Discrete Time

To build intuition in what is occurring, first consider the most simple model describing the change in size of a single population. This model is formulated in a difference equation framework. When modeling population dynamics, we need to decide what to include in the model and what to abstract away. Because we want an extremely simple model, we will assume that the environment in which this population resides will remain completely constant through time, and without regard to the size of this population. Clearly, this assumption is ludicrous. However, for now we will make it, so that we can understand the mathematical workings of population dyamics. Define N0 as the number of individuals in the population at time t=0 (how we will designate the start time), and assume that time is measured in units of generations. This model may be very applicable to a synchronously developing univoltine insect that has one generation per year; in this case, the fundamental time unit would be one year, since the insect has one generation per year. In our imaginary population, a fixed number of offspring will survive to reproduce in the next generation for each individual present in this generation: call this number λ. Thus, in the next generation, which will be t=1, the number of individuals in the population will be λ N0. We got this number simply by multiplying the number of individuals in the current generation, but the average number of offspring that each will have (note that the parental generation is assumed to all die – i.e., the generations are non-overlapping). One confusing aspect of this is that λ accounts for both survival and reproduction, and so this is not the average number of offspring of indidividuals that survive to reproductive age, but rather the probability of surviving to reproductive age times the average number of offspring produced by each of the survivors. Mathematically, we can write down this scenario as a model of the change in population size from generation 0 to 1:

$N_{1} = \lambda N_{0}$

Now we have a model! Let’s do something with it. Let’s project the population two generations into the future, which would be t=2.

$N_{2} = \lambda N_{1}$

$N_{2} = \lambda \lambda N_{0} = \lambda^2 N_{0}$

We can apply the same logic and mathematics to this model to project any number of generations into the future. Do this yourself for a few more generations to develop a feel for the pattern that emerges. Once you do this, you should arrive at the following equation:

$N_{t} = \lambda^{t} N_{0}$

for projecting the population t generations into the future.

Now consider some of the conceptual issues of this model. In this case, they are very simple, but understanding them in the context of this simple model will help later, when we move to more complex models. What is the per capita growth rate of this population over one generation? In this model, it is simply λ. Remember that λ is defined as the number of individuals in the next generation per individual in the present generation. It is defined in units that are based per individual, which should be a clue to you. What is the total growth rate of the population over one generation? The total growth rate is λ Ni in generation i. Note that this is simply the per capita growth rate times population size. Consider for this model how population size changes over time. If λ > 1, the population will increase each generation by the same multiple. In this case, the population increases geometrically without bounds. However, if λ < 1, the population will decrease each generation by the same multiple until it eventually goes extinct. (Note that λ cannot be smaller than zero, and can be as large as +∞) For this model, a transformation of scale is useful. If we track population size in units of ln(N), many interesting features become apparent, particularly when we analogize this to the continuous-time equivalent model.

$ln(N_{t}) = ln(\lambda^{t} N_{0})$

$ln(N_{t}) = ln(N_{0}) + t ln(\lambda)$

You should recognize this second equation as a line with intercept ln(N0) and slope ln(λ). This means that plotting ln(Nt) against t will give a straight line. The resulting graphs of growing (left) and declining (right) population size against time are as follows. Formulated in this way, ln(λ) is the per capita growth rate of the population. A key feature of this model is that ln(λ) does not change as population size changes. Neither does λ. This means that per capita population growth rate does not change with the abundance of the population. This means that this is a density independent model of population growth.

A demographic rate is density independent if the per capita demographic rate does not change with population abundance (i.e., density).

To visually clarify what density independence means, graph the per capita population growth rate against N. Because ln(λ) does not change with N, the plot of ln(λ) against N is a flat line with slope of zero. This is the characteristic of a density-independent demographic rate; a plot of the per capita growth rate against population size gives a line with slope zero for all N. The intercept of the line is the per capita rate. Now for a little mathematical sleight of hand that will become important when we compare this model to the continuous time analog. Define ln(λ) = r. This means that λ = er. (Do you remember your seventh grade algebra rules for logarithms? If not, this would be a good time to review them.) Thus for our difference equation model, we can write:

$N_{t} = \lambda^{t} N_{0} = (e^{r})^t N_{0} = e^{rt} N_{0}$

or equivalently
$ln(N_{t}) = ln(N_{0}) + rt$

Using the first formulation, we can decompose r, the parameter of overall population growth rate, into per capita birth (b) and per capita death (d) rates such that r = b – d. Let’s do a little algebra by substituting for r in the above equation and see what we can make of it.

$N_{t} = e^{(b-d)t} N_{0}$

From this simple formula, we can see another recurring theme in the analysis of population models. What does it mean if b – d > 0? It means the per capita birth rate is greater than the per capita death rate. What will happen in this case? The population will increase in size! If b > d, then λ > 1, and so the population will be larger in the next generation than it is in this one. Now consider the opposite case. What does it mean if b – d < 0? It means the per capita death rate is greater than the per capita birth rate. What will happen in this case? The population will decrease in size! If b < d, then λ < 1, and so the population will be smaller in the next generation than it is in this one.

##### A Basic Model of a Single Population – Continuous Time

Now let’s construct a model of population growth using exactly the same assumptions but in continuous time. To build a model in continuous time, we must formulate the model as a differential equation. The corresponding differential equation model is:

${dN \over dt} = rN$

For those of you who took a course in differential equations, you will have learned on the first day that the solution to this equation is

$N_{t} = e^{rt} N_{0}$

or equivalently

$ln(N_{t}) = ln(N_{0}) + rt$

This is our discrete time model! Because it is in continuous time, it describes exponential population increase or decline, depending on whether r>0 or r<0, respectively. In this model, the per capita population growth rate is r.
In continuous time models, the mathematical symbology lends itself to designating the total and per capita population growth rate. The total population growth rate is simply the equation dN/dt, whereas the per capita population growth rate is found by dividing the equation by NdN/Ndt = r

${dN \over dt} = rN$ (total population growth rate)

${dN \over N dt} = r$ (per capita population growth rate)

Note again that N does not appear in the equation for the per capita population growth rate. This means that these are density-independent models of population growth. Graphically this means that a plot of ${dN \over N dt}$ against N will be a flat line with slope zero. Note, that the total population growth rate increases with increasing population size. This is because total population growth rate measures the total net change in population size. The slope of the total growth rate is r and the intercept is zero (population size does not change if there are no individuals in the population, and no immigration). Understanding the relationship between total and per capita population growth rate is critical for assessing the consequences of population regulation. These density-independent models of population growth involved no population regulation. In both models, populations would grow without bounds if r>0, and would decline to N=0 if r<0.