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13 Modeling Biological Systems
13.2 Growth in a Population and Chaos Theory
Anyone who thinks exponential growth continues in a limited world is either a madman or
an economist.
— Kenneth Ewart Boulding
13.2.1 Modeling Population Numbers
Let’s make a simple model of how the number of individuals (rabbits, foxes, or
people) varies with time. Of course, individuals are discrete! But when counts get
large, the change in the count over a short time is relatively small, so we might
expect that a smooth function will fit the variation of the count over time. Smooth
functions can also be approached by testing the behavior of many independent
samples of populations starting with identical conditions, and then averaging over
the samples.
Increases in the count N come about because of births, and decreases by deaths.
In some populations, the birth rate per capita (births per year/population) is constant,
as may be the death rate per capita. In these cases, changes of N , called N, over a
short time t should be proportional to N and to the time t, giving
N = (βN − γ N))t ,
(13.1)
where β is the per capita birth rate and γ is the per capita death rate. If β > γ , then
the population will grow exponentially.
Realistically, the birth rate is not constant, but depends on factors such as the
amount of food available. The amount of food and resources available, in turn,
eventually drops as the population increases. A simple model of this behavior is
to make β = β o − β 1 N for N less than β o /β 1 . Alternatively, one may think of this
expression for β as the first two terms in a Taylor series of β as a function of N . In
consequence, our population number satisfies an equation of the form
dN
dt
= α(1 − N/κ)N .
(13.2)
The value of κ determines at what level the population reaches saturation. One of
the advantages of this model is that it has an analytic solution:
N =
κ
1 + exp (−α(t − t o ))
.
(13.3)
The number N behaves in time as one might expect (see Fig. 13.1). This model for
population growth was invented by Pierre Verhuist in 1844, who named the solution
13 Modeling Biological Systems
13.2 Growth in a Population and Chaos Theory
Anyone who thinks exponential growth continues in a limited world is either a madman or
an economist.
— Kenneth Ewart Boulding
13.2.1 Modeling Population Numbers
Let’s make a simple model of how the number of individuals (rabbits, foxes, or
people) varies with time. Of course, individuals are discrete! But when counts get
large, the change in the count over a short time is relatively small, so we might
expect that a smooth function will fit the variation of the count over time. Smooth
functions can also be approached by testing the behavior of many independent
samples of populations starting with identical conditions, and then averaging over
the samples.
Increases in the count N come about because of births, and decreases by deaths.
In some populations, the birth rate per capita (births per year/population) is constant,
as may be the death rate per capita. In these cases, changes of N , called N, over a
short time t should be proportional to N and to the time t, giving
N = (βN − γ N))t ,
(13.1)
where β is the per capita birth rate and γ is the per capita death rate. If β > γ , then
the population will grow exponentially.
Realistically, the birth rate is not constant, but depends on factors such as the
amount of food available. The amount of food and resources available, in turn,
eventually drops as the population increases. A simple model of this behavior is
to make β = β o − β 1 N for N less than β o /β 1 . Alternatively, one may think of this
expression for β as the first two terms in a Taylor series of β as a function of N . In
consequence, our population number satisfies an equation of the form
dN
dt
= α(1 − N/κ)N .
(13.2)
The value of κ determines at what level the population reaches saturation. One of
the advantages of this model is that it has an analytic solution:
N =
κ
1 + exp (−α(t − t o ))
.
(13.3)
The number N behaves in time as one might expect (see Fig. 13.1). This model for
population growth was invented by Pierre Verhuist in 1844, who named the solution