Box 2.2 (continued)

differential equation (2 lOa.i for the number nt pccipk with the flu. dwn/dr. wijl depend on both the number ot people v.itii the tin. tut), arid the number ol susiepiible individuals in the population. at). Ditfeientidl equations (.an also be vmitLc-u .is limitiuiis ol htnh the dependent \ariahle win and the independent variable t. Such differential equations anse whene\ei we expect a vari-jhlr to change as a function both of its current value and of time. loi example, in a seasonal environment, the budding rate of a tree should depend on the time nf year as well as on the number ot brunches on a tiee. We ian model this bv treating h as some function of tune, bit). rather than a constant. In addition, ditleientiaJ equations might depend on the past state of a variable as well a* (or instead ol) its ciuieiil stale, loi example, in ihe tiee brandling example, the production of new branches at time t might dcpi-nd on the loial number ot branches t days ago, <>r w(f - t>. as these branches are now laige enough to branch again. Revising equation (2.8j gives dn(fi/df - hut - n. Such equations, knimri as "delay dittiivntial equations, ' arise naturally when describing biological processes involving turn- lags.

All ot the above examples ha\e only one independent variable (Lime). Iliese tall into the category known a« 'ordinary differential i-quatinns" |i.)l >11 Manv biolugical piobk-ms involve more than one independent variable le.g., space as well as time), arid suih differential equations art-known as "partial differential equations" iPDH.

and Chapter 4), discrete-time and continuous-time models can sometimes exhibit similar behavior over time, and it is possible to predict when they should behave similarly. Thus, in many cases, one is free to choose between the two.

The next step is to describe the parameters of the model; these are the various quantities that influence the dynamics of the model, but that remain fixed over time as the variables change. As with variables, each parameter is given its own symbol, which you are free to choose. Commonly used symbols for parameters are italicized roman letters (e.g., a, b, c, d, m, and r) and lower-case greek letters (e.g., a, (3, Table 2.1).

A chief difference between discrete-time and continuous-time models is that parameters representing events per unit time are described as the number of events (or fraction of the population undergoing the event) per time step in discrete-time models but as the instantaneous rate of events per unit time in continuous time. In contrast, parameters that do not represent events per unit time (e.g., the probability that an event is one type or another) retain the same definition in the two types of models. We will discuss the difference in parameter units between discrete- and continuous-time models at greater length in Box 2.6, once we have described how their dynamical equations are derived.

Potential parameters for our discrete-time models include (i) the number of new branches that bud off each old branch per day, b; (ii) the fraction of mice in the yard eaten by the cat per day, d, and the number of mice born per mouse per day, b; (iii) the fraction of healthy people that are exposed to a flu carrier per day, c, and the probability of transmission of the flu between a healthy person and a flu carrier upon exposure, a. The analogous parameters in a continuous-time model would be (i) the rate of budding for each old branch, b; (ii) the rate of consumption of mice, d, and the rate of births per mouse, b; and (iii) the rate of contact between a flu carrier and a susceptible person, c, and the probability of transmission of the flu between a carrier and a healthy person per contact, a. These parameters represent events per unit time and so have slightly different definitions for the discrete-time and continuous-time models except a, which always represents the probability of contracting the flu per contact.

As with variables, one should also keep track of any constraints imposed on each parameter. For example, can a parameter be negative? Does a parameter represent a fraction, proportion, or probability, in which case it must fall between zero and one? These constraints might well depend on the type of model. For example, the parameter d in the cat-mouse model is restricted to lie between zero and one in discrete-time models (because it represents the fraction of mice eaten by the cat), whereas the analogous parameter d in the continuous-time model can have any positive value (because it represents the rate of consumption of mice per unit time). This is another common difference in the parameters between discrete- and continuous-time models (described more fully in Box 2.6).

In addition to the absolute constraints on each parameter, it is worth keeping track of the range of parameter values that are biologically reasonable. For example, it is reasonable to assume that the number of new branches that bud off each old branch per day is small for most trees (b « 1). Similarly, the number of mice born per mouse per day (b) will be much less than one (b « 1). We write b « 1 to imply that b is much smaller than one. How much smaller depends on the context, but typically this statement implies that b is 0.1 or less. Having a list of constraints and reasonable ranges for parameters can help in two important ways. First, reasonable parameter values must be chosen to carry out realistic simulations and to plot relevant graphs. Second, results from a model often depend on the values of the parameters, e.g., whether a parameter is positive or negative, large or small, so that making accurate predictions from a model depends on choosing appropriate parameter values.

Before proceeding to the next step, it is a good idea to construct a table of all the variables and parameters in your model, as well as any constraints on these terms. You can later revisit this table to ensure that it includes the variables and parameters needed to capture the essence of the biological process and to address the question of interest. It is very common that the first version of a model includes too many variables and parameters, causing the model to be unnecessarily complex, or too few variables and parameters, causing a model to behave in unintended ways (e.g., populations grow to infinite size, or nobody ever recovers from the flu). If a model displays unintended behavior, then think about whether the biological system being modeled includes other processes that should also be incorporated into the model (e.g., competition, recovery).

Before writing equations down, it is a very good idea to organize your model conceptually with the aid of a diagram or table. Diagrams and tables make it easier to see whether the necessary variables and parameters are included and make it easier to write down dynamical equations (recursion equations or differential equations). We describe three organizational techniques: a life-cycle diagram, a flow diagram, and a table of events.

A graphical technique, which we call a life-cycle diagram, keeps track of the various events occurring during a single time step, along with their order of occurrence. Such diagrams are useful only for discrete-time models, where there is a discrete time period during which various events can occur. As a simple example, consider the tree branching model. Each time step represents a single day, and only one type of event can happen during any given day: the growth of more branches. As result, the life-cycle diagram is extremely simple (Figure 2.2a).

The tree branching model is so simple that a life-cycle diagram is not really required to organize things. Life-cycle diagrams become indispensable when multiple events occur during a single time step. Consider the model of mice being eaten by a cat. Now there are three events that occur each day: mice give birth, mice move in from neighboring areas, and the cat eats mice. In a discrete-time model, one must choose an order for these events, as well as a point in time when the population is censused (e.g., when we count the number of mice, n(t)). For example, Figure 2.2b illustrates the case where events occur in the following order: a census, followed by predation by the cat, mouse births, mouse migration, and finally the next census. These events cause changes to the number of mice, which we describe as n(t) at the census point, n'it) after predation, «"(f) after births, and n"'(f) after migration. Because migration is assumed to occur last in the daily life cycle, the number of mice at the next census, n(t +1), will equal ri'(t). Alternatively, we might instead assume that births happen first, then migration, and then predation, yielding the life cycle in Figure 2.3. As we shall see, the order of events in a life cycle can affect the results of a model, sometimes substantially.

Finally, consider constructing a life-cycle diagram for the model of flu transmission. The time step is again one day, and as with the tree branching model, there is only a single event that can happen during each day: transmission of the flu. There is an additional wrinkle with this model, however, in that there are now two variables that we are tracking (healthy individuals and people with the flu). As a result, we could construct a life-cycle diagram for each of the variables (Figure 2.2c). But because there is only one event per cycle, these life-cycle diagrams are again not very useful (as was the case with the tree branching model).

A life-cycle diagram illustrates the order of events that occur within each time step (for discrete-time models).

A second method for organizing a model, which is often more useful for models containing multiple variables, is a flow diagram. A flow diagram illustrates the interconnections among the variables and provides a schematic picture of how each variable affects its own dynamics as well as the dynamics of the other variables. In a typical flow diagram, each circle represents one variable within the model. Returning arrows that exit and come back to the same circle represent a

A flow diagram illustrates how each variable affects its own dynamics and those of other variables.

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