POPULATION DYNAMICS
A MATHEMATICAL APPLICATION OF BIOLOGOCAL SCIENCES
by M. Gaglione
Mathematical applications and models have a wide range of everyday uses. Of
special importance are differential equations. These equations can be used to solve
problems dealing with problems in the physical, biological, and social sciences. The heat
conduction equation is useful in modeling the physical behavior of materials.
Differential equations can explain the effects of a pollutant discharged into a lake. Also,
Newton’s Law of Cooling can aid forensic scientists and detectives in determining times
of death (Boyce, 46). The focus of this paper, however, is modeling in the field of
population biology. Basic concepts and mathematical models of population dynamics,
including exponential and logistic growth, competition, and predation, will be discussed.
A population is “a group of plants, animals, or other organisms, all of the same
species, that live together and reproduce.” (Gotelli, 2) How can the size of a population
change? There are 4 factors that can affect the population. Births in a population
increase the size while deaths decrease the size. Also, new individuals enter the
population (immigrate) or leave the population (emigrate). A very simple equation
represents the statements from above:
N = B – D + I – E
Where:
N = change in population size
B = number of births
D = number of deaths
I = number of immigrants
E = number of emigrants
If a population is assumed to be “closed” meaning that there is no movement between
populations, N = B – D.
The simplest model is when the growth of a population is considered to be density
independent. Density independence means that the birth and death rates are not affected
by the population size (Hastings, 9). Therefore, the rate of births and deaths is
proportional to the number of individuals in the population. A constant r will now be
introduced which is the instantaneous rate of increase or intrinsic rate of increase
(instantaneous birth rate – instantaneous death rate) (Gotelli, 4).
dN = rN
dt
In words, the above equation says that the growth rate (dN/dt) is proportional to r. The
factor r is important because it determines if the population increases exponentially (r >
0), is constant (r = 0), or decreases to extinction (r < 0) (Gotelli, 5). In order to project
the size of a population, the following derivation is required:
 dN/N =  rdt => ln|N| = rt + c => ln(N) = rt + c => N = ert * ec = c1ert => c1ert => c1 = N0
=> N(t) = N0 ert
N(t) = population size at time t
N0 = initial population size
e = base of natural logarithm
t = time
r = intrinsic rate of increase
It is very important to remember that certain assumptions have been made in order to
derive these equations. From an exponential graph, we can see that as t   N(t)  .
In real life situations, it is not possible for a population to grow to this magnitude. This is
evidenced by the following exponential model which predicts a herd that would exceed
the weight of the earth (Gotelli, 9).
Table 1: Exponential growth of a herd of 50 cattle with r = 0.365 cows/(cow*yr).
Year
0
1
2
3
4
5
10
50
100
150
200
Herd Size
50.0
72.0
103.8
149.5
215.3
310.1
1923.7
4.2 * 109
3.6 * 1017
3.0 * 1025
2.5 * 1033
“A Primer of Ecology.” Nicholas J. Gotelli. Sinauer Associates, Inc. 1995. p.8.
Let us recall the assumptions that need to be made in order for a population to fit this
model. First, the population is “closed.” Second, birth and death rates are constant. This
implies that there are unlimited resources for survival (i.e. food, space). Third, there are
no differences between individuals due to age or body size and are reproductive when
born. Finally, there is continuous growth and r is function of current population size
(Gotelli, 10).
Next, will be a discussion on logistic population growth. It has already been
determined that a population cannot grow forever. So what exactly limits the growth of a
population? Food supply, territoriality, cannibalism, competition, predation, parasites,
and diseases can all affect the growth. Density dependence refers to “processes whose
effect changes as the number of individuals within the population changes.” (Hastings,
82) The following implicates how birth rates decrease due to fewer resources.
B = B0 – a N
N = population size
B = per capita birth rate
B0 = birth rate under ideal conditions
a = decrease in birth rate from density dependence
As N increases the lower the birth rate. If N is close to zero the birth rate is close to B0.
The larger a is, the birth rate is dropping more with each added individual. So, the
exponential model is a special case of the logistic model in which a = 0. On the other
hand, the death rate increases as the population grows.
D = D0 + cN
D0 = death rate when population size is close to zero
C = measures increase in death rate from density dependence
While this may represent most populations of species, there are others who do not follow
these descriptions. Namely, the Allee effect describes populations that reproduce, hunt,
and take care of offspring better when they are in groups. So, the birth rate increases and
the death rate decreases.
dN/dt = [(B0 – aN) – (D0 + cN)]N = [(B0 – D0) – (a + c)N]N
Multiply by [(B0-D0)/(B0-D0)] to simplify:
dN/dt = [(B0-D0)/(B0-D0)] [(B0-D0) - (a + c)N]N = rN [1- ((a+c)/(B0-D0)) N]
Carrying capacity: K = (B0-D0) / (a+c)
Logistic Growth Equation: dN/dt = rN [1 - (N/K)]
“A Primer of Ecology.” Nicholas J. Gotelli. Sinauer Associates, Inc. 1995. p.29-30.
The carrying capacity represents the maximum population size that can be supported in a
certain environment. If the population exceeds the carrying capacity (N > K), the growth
rate becomes less than zero and the population decreases to K. The population will reach
stable equilibrium at N = K. This is because when N is (less than/greater) than K the
(birth rate/death rate) is (higher/lower) than (death rate/birth rate) so the population will
(increase/decrease) to N (Gotelli, 31).
In order to project the size of a population, the following derivation is required:
dN/dt = rN [1 - (N/K)]
Separation of variables and integrate from t = 0 to T : dN/N[1-(N/K)] = rdt
Population size as function of time : Nt = K / 1 + [(K – N0) / N0] e-rt
“Population Biology.” Alan Hastings. Springer-Verlag New York, Inc. 1997. P.84-85.
When population is small, the size increases rapidly. Growth then decreases as it
approaches K. Remember that the time needed to reach equilibrium is proportional to r.
Rapidly growing populations will reach K faster (Ibid, 32).
Up until now exponential and logistic models have involved a single species.
There are many different ways in which two species may interact with one another. One
way is competition. Competitive interactions between two species “negatively influence
each other’s population growth rates and depress each other’s population sizes (Ibid,
112).” Competitive interactions can be further divided into 3 categories: exploitation,
interference, and pre-emptive. Exploitation competition is when 2 species have a shared
resource. One species’ use of a resource deprives the other of its necessary food, water,
shelter, etc. Interference competition is the opposite of exploitation. In exploitation,
species directly suppress the other, while in interference, it is through indirect use of
resources. An example would be when an organism has a well-established breeding or
feeding ground. Any organism intruding on the ground could face consequences of
injury. The last category is pre-emptive. Space is the renewable limiting resource (Ibid,
112).
The following equations demonstrate how the growth rate is dependent on both members
of its own species as well as the competing species.
dN1/dt = r1N1 [ (K1-N1-N2) / K1 ]
dN2/dt = r2N2 [ (K2-N2-N1) / K2 ]
The coefficient  represents the effect species 2 has on 1 while  represents the effect
species 1 has on 2 (Ibid, 113).
Finally, a closer look will be taken on the effects of predation on population
growth rates.
Model of prey growth:
V = prey population
P = predator population
dV/dt = rV - VP
The population of the prey would increase exponentially if not for the effect of the
predator. The decrease in the population is proportional to the product of predator and
victim. The constant  represents the encounter rate of predators with prey and V is the
functional response. Functional response refers to the rate of capture as function of prey
density. In order to understand predator growth, we must assume that the predator is an
extreme specialist. That is, the predator only feeds on this prey and there are no
substitutions.
Model of predator growth:
dP/dt = VP - qP
The predator will die exponentially in absence of prey. VP shows the random
encounters of predator and prey.  represents the conversion efficiency. Conversion
efficiency is the “rate at which the predator converts victim biomass into new predator
offspring (Ibid, 142).” V is the numerical response which is the growth of predators as
function of prey density.
This overview of basic ecological and mathematical principles should enable one
to solve simple problems. Not all real life situations involving population dynamics can
be effectively modeled by these equations. Often, there are too many variables facing
ecological systems. However, simplifying the larger problem by making assumptions as
done throughout this paper, can provide useful insight.
Bibliography
Boyce, William E. and DiPrima, Richard C. Elementary Differential Equations and
Boundary Value Problems: Sixth Edition. John Wiley & Sons, Inc. 1997.
Gotelli, Nicholas J. A Primer of Ecology. Sinauer Associates, Inc. 1995.
Hastings, Alan. Population Biology: Concepts and Models. Springer-Verlag, New
York, Inc. 1997.