Discrete-Time Models in Biology

This entire section of the course focuses on how we can use mathematics to mimic 
biological situations in which it is reasonable to view the underlying biological variables 
as changing at discrete time intervals. We have already seen several examples of this. 
When we discussed matrix models for populations structured by age, using the Leslie 
model, we projected the age structure forward one time period by multiplying the vector 
of the population's age structure by the transition matrix to get the population structure 
one time unit later. Another example arose when we analyzed simple population genetics 
models in which we determined the gene frequency in the population at the next 
generation from the gene frequencies in the previous generation.

The types of models we will discuss now are designed for populations with discrete 
generations (e.g. many insects and plants), in which we can think of the next generation 
as being produced dependent upon the size of the current generation. Thus the state 
variable that describes the population will typically be x sub n or the population size (or 
density) at generation n. These models are in some sense easier than the matrix models 
such as the Leslie model since at each time period we are only following a single state 
variable rather than a vector of state variables (the age structure). Even though they are 
simpler, these models still serve as the basis for many very practical applications, 
including many models in fisheries and insect pest management. 

The key assumption in discrete-time models is that there is a fixed time period, the time 
step (often a year, or a generation time), which is an appropriately chosen so that the 
population size at the next time step is a simple function of the population size at a 
previous (sometimes several previous) time steps. There are many situations in which this 
is clearly an approximation, for example human populations breed and reproduce 
continuously, not at fixed time steps. The next section of the course will show how to 
produce models for continuously breeding populations (this is part of calculus), but in 
some cases discrete-time models are an adequate representation of even continuously 
breeding populations to be useful. Thus much of human demography (the study of the 
structure of human populations) utilizes discrete-time matrix models (typically using 5 
years as the time step, so a human population is broken into age classes 0-4 years old, 5-9 
years old, etc.), even though humans breed continuously.

We will start with the simplest possible population model in discrete time, in which we 
view each individual in the population as replacing themselves with b individuals in the 
next generation. What do we expect to happen in this case as time passes? Obviously if b 
> 1 we expect that the population will increase and if b < 1 the population will decrease 
since each individual is not completely replacing themselves. Here you may wonder what 
it means if we let  b = 1.5 since it is not clear that .5 individuals makes sense (what does 
half an individual mean – the right half of a baby if Solomon really did let it be cut in 
two?). At least two interpretations make sense here. One is that the state variable for the 
model is density (individuals per m^2 is one potential measurement of density) so that b 
= 1.5 means that for each unit of density currently present, the population has 1.5 units of 
density at the next time step. Another interpretation is that b= 1.5 represents the average 
number of new individuals produced per individual present the previous generation, 
where this average is taken over the entire population. Thus, b = 1.5 would arise if half 
the individuals produce one offspring and die, and the other half produce two offspring 
and then die. The average reproduction rate is thus 1.5 individuals per time step.

This simple model is described by letting x sub n be the population size at generation n so 
that x sub (n+1) =  b  x sub n and clearly then x sub n = (x sub 0) b ^n where x sub 0 is 
the initial population size. What happens here as we look at many generations? If b > 1, 
then x sub n gets larger and larger, while if b < 1 then x sub n gets as close as you might 
want to zero by choosing n large enough. This model is called Geometric Growth or 
Malthusian Growth (after Thomas Malthus, who first described the implications of 
geometric growth on human populations as eventually outstripping the potential for 
agricultural production of food). You have already seen geometric growth since when we 
plot x sub n versus n on semi-log paper, we get a straight line. Geometric growth is the 
same as exponential growth, except in discrete time.

If we write down the list x sub 0, x sub 1, x sub 2, x sub 3, ... then we get the list of 
population sizes indexed by the generation. Mathematically, this list is called a sequence 
- it is determined by a rule (a function) that assigns some number (here it is population 
size) to each non-negative integer (the generation). As we argued above, if the sequence 
is geometric (formed from the geometric growth model above, then the sequence gets 
closer and closer to zero (but never quite reaching it) as n gets larger if b < 1. We say 
then that the sequenced has a limit, which is zero. If b > 1, then x sub n continues 
growing, never approaching anything, so no limit exists. 

The idea of a "limit" therefore means that a sequence of numbers x sub n approaches a 
particular value as n gets larger and larger. If the sequence represents population size, 
then the population size approaches some "limiting value", an equilibrium value, as time 
goes on. We have seen an example of this in the Leslie model, in which the population 
structure approached the structure of the eigenvector of the dominant eigenvalue. Just 
think of the limit as the long term population size, if it exists. Clearly, in the real world 
there are numerous factors which cause populations to vary in time, so an equilibrium 
size is just an approximation.