Welcome to your training disk!
This disk enables the reader training and practical application of the knowledge that the student will
get from reading the main text in the book. The exercises are in EXCEL and are easily accessible by
clicking the links here. When in EXCEL, the reader gets back to this text by clicking the
corresponding link that appears in the sheet at the bottom right. The EXCEL files are optimized for
the display resolution 1280 by 1024 pixels. Using a display with a lower resolution may sometimes
result in not the whole area of interest being on screen. In such case we suggest the reader to zoom
the EXCEL files to some smaller value, e.g., 75%. To fit all the explanatory frames into one visible
part of the EXCEL sheet, parts of the simulations are sometimes hidden behind the explanatory
frames.
The EXCEL files usually contain the following:

The simulation itself in columns A and B, sometimes C (when the dynamics or more
populations is simulated).

A yellow-highlighted region containing the numbers of individuals at the beginning – at time
zero, and all the parameters of the model. We find it convenient to separate these model
inputs from the main simulation, as it allows the user to easily change them without changing
the formulas in the main simulation.

A figure with the result, which uses the data in columns A and B (sometimes C) as inputs.

An explanation frame to the right of this figure, where it is explained, what the program
simulates, what is the meaning of the resulting figure in the active sheet, if this is not quite
straightforward and what are the exercises the reader is expected to perform here.

Another explanation frame below this figure, where it is explained, how the program was
created – what is the meaning of the formulas in the sheet. This helps the reader to create
his/her own programs in EXCEL.
Of course, there exist special programs for simulations of population systems. However,
programming in EXCEL allows adapting the simulation for any particular situation and – last but not
least – it is not necessary to buy the special programs, which are usually not free. We believe that
after having performed all the exercises presented here, the reader will be able to produce own
EXCEL simulations of population dynamics.
Files contained in this disk
This disk contains the following files, each of which can also be run independently:
Introduction.doc
This file
PG1.EXE
“Game of Life” – a population game using cellular automata
Discrete exponential.xls
Simulation of the discrete exponential model
Discrete logistic.xls
Simulation of the discrete logistic model
Time delay.xls
Simulation of the discrete logistic model with time delay
Continuous models - 1 species.xls
Simulation of several continuous models for one species:
exponential, logistic, logistic with delay, Allee effect,
deterioration of environment, harvesting, biocontrol
Continuous models - 2 species.xls
Simulation of several continuous models for two species:
mutualism, competition, predator-prey
Chapter 1: Game of life
In Chapter 1, we introduce the reader into the “Game of Life”. You can access the program
that simulates this game here. In some systems, the following message can appear: “Some files can
contain viruses or otherwise be harmful to your computer. It is important to be certain that this file
is from a trustworthy source. Would you like to open this file?” This is because you are trying to
open an .exe file. Do not worry and click “OK”, this file is innocent. For getting back to this text,
you have to stop the program by pressing Ctrl-Break. Start with 5 individuals and run simulations in
different starting configurations. Explain the dynamic behavior you observe and the causes of that
behavior. Repeat with different numbers of starters. What is the single most important conclusion
from this exercise?
Chapter 2: The elementary population model
2.1. Exponential model
In Chapter 2, we examine the dynamic behavior of single-species populations and develop an
elementary feedback model of the population system. The elementary population model, that of
exponential growth of the population, is defined in the main text in equation (2.4):
Nt  Nt 1  RNt 1 .
(2.4)
Here Nt denotes the number of individuals in the population at time t, and R represents the
individual, or per capita, rate of increase over the time period t – 1 to t. From this equation we can
calculate the density of the population at the end of a particular time increment from its density at the
beginning of the interval and the individual rate of increase.

Use the corresponding EXCEL file in this disk to perform the simulations required in
Exercise 2.1.

Run simulations for different initial numbers of individuals in the population, N0, and for
different values of the per capita rate of increase, R.

How does the outcome depend on the choice of N0 and R?

Where do you see the value of N0 in the graph?

How does increase/decline in the value of R perform in the resulting growth of the
population? What can you conclude about the growth of the population, if R > 0, R < 0, R =
0?
You can access the EXCEL file that simulates the exponential growth here.
2.2. Logistic model
Later in the chapter, we allow the individual rate of increase, R, to be dependent on the
density of the population, as described in equation (2.5) in the main text:
R  Rm  sN .
(2.5)
This results in the logistic model, described in equations (2.6) and (2.7):
Nt  Nt 1  ( Rm  sNt 1 ) Nt 1 .
(2.6)
Nt  Nt 1  Rm (1  Nt 1 / K ) Nt 1 .
(2.7)
Note that the equations (2.6) and (2.7) are equivalent, as we have set s  Rm / K . Due to this
substitution, both parameters in the equation (2.7) have a clear biological meaning: Rm, is the
maximum individual rate of increase (maximum value of R under the assumption that there is a very
low number of individuals in the population, so that competition for a scarce resource does not yet
occur, and K is the carrying capacity of the environment - the population density where all living
space is fully utilized and there is no more room for additional growth.
Using the simulation model in EXCEL in this disk, solve Exercise 2.2. Then confirm that the
population will show:

asymptotic approach to equilibrium when Rm ≤ 1,

damped stable oscillations when 1 < Rm < 2,

neutrally stable oscillations with amplitude determined by the initial displacement when
Rm = 2,

unstable, often non-periodic behavior when Rm > 2.
You can access the EXCEL file that simulates the logistic growth here.
You will find out that in some cases the logistic growth described by equation (2.7) does not
give biologically meaningful predictions – for some large values of Rm is the predicted population
size, Nt, negative. To get rid of this problem, the following modification of the logistic equation is
often used:
N t  N t 1e Rm (1 Nt 1 / K ) .
(2.7a)
Because the “regulatory term” Rm (1  N t 1 / K ) is now in exponent, the value e Rm (1 Nt 1 / K ) is never
negative and so is the resulting prediction on Nt. The simulation of equation (2.7a) can be found
here, in sheet “Modified logistic”, and the comparison of the numbers of individuals, Nt, predicted
by equations (2.7) and (2.7a) are given in the same file (accessible here) in sheet “Comparison”. We
suggest you to run the simulations of equation (2.7a) for various values of the initial numbers of
individuals, N0, and for various values of the parameters Rm and K and see, how the systems behaves
in response to their changes. You will find out that:

the predictions of the number of individuals are always biologically reasonable, i.e., positive,
independently of the choice of the initial numbers of the individuals and of the parameters;

with increasing Rm, the system predicts first asymptotic approach to equilibrium, then
damped stable oscillations, followed by neutrally stable oscillations. For which value of Rm
does this occur? Try to calculate this critical value first, before you perform the simulations!
When Rm increases beyond this critical value, we can observe unstable oscillations and
finally a non-periodic solution, which is known as “chaos” in biological literature. Those
interested in the mathematics of this phenomenon are referred to the following papers:
R. M. May in Science, vol. 186, p. 645, 1974.
R. M. May in Nature, vol. 261, p. 459, 1976.
R. M. May and G. F. Oster in the American Naturalist, vol. 110, p. 573, 1976.
C. Sparrow in J. Theor. Biol., vol. 83, p. 93, 1980.
J.G.P. Gamarra and R.V. Solé in Ecology Letters, vol. 3, p. 114, 2001. (Here one of the most
popular data sets in ecology, that of lynx returns, is analyzed in order to look for
evidence for a bifurcation process and chaos.)
2.3. Logistic model with delay
Finally the concept of time delay is introduced at the end of chapter 2, in equation (2.9b):
Nt  Nt 1  Rm (1  Nt 2 / K ) Nt 1 ,
(2.9b)
which was then generalized to an arbitrary delay in equation (2.11b):
Nt  Nt 1  Rm (1  Nt T / K ) Nt 1 .
(2.11b)
A simulation program in EXCEL that shows the predictions of the numbers of individuals based on
equation (2.11b) can be found here. Use this program to test the stability criterion for equation
(2.11b):
y / x  RmT  1 ,
(2.10)
where T is the length of the time delay, and stability is a quality of both Rm and T, which claims that
the system is stable, if y/x < 1. We suggest you to run the simulations of equation (2.11b) for various
values of the initial numbers of individuals, N0, N1, …, NT-1, and for various values of the parameters
Rm and K and see, how the system behaves in response to their changes.
2.4. Continuous models
2.4.1. Introduction
Note 2.4 introduced the reader to continuous models. Why to bother with these? We should, because
in nature we can also find two clearly distinct groups of organisms. Butterflies can serve as an
example of the first one. Their adults can usually be found on wings only during a certain period of
the year – say, for instance, in July. These adults mate, reproduce and die. Their offspring undergo
egg, larval and pupal stages – and appear next July as new adults. However, because their parents
have died the previous year, next July we find a completely different set of adults – none of those
from the previous year. We say that such species have non-overlapping generations. In contrast,
aphids have overlapping generations. At any instant, we can find all age groups in their colonies:
newborn larvae, large larvae, and young as well as senescent adults. Aphids – unlike butterflies –
have neither a distinct “reproduction period”, nor a period of “mass death”. We can therefore
approximate this situation by assumption that their populations change continuously. Thus while
discrete models, which are used throughout the main text of this book, are more appropriate for
modeling population dynamics of the former group of organisms, those with non-overlapping
generations, represented by butterflies in our simplified example. Continuous models, on the other
hand, are more appropriate for modeling population dynamics of organisms with overlapping
generations, represented here by aphids. Nothing is absolute here, however. For example, if we take
into consideration only the population numbers of aphids in – say – July, and look for a rule that
predicts their numbers the next July, discrete models should be used. In any particular situation,
therefore, we should carefully consider, which type of model is appropriate in the situation we are
interested in.
As it was shown in Note 2.4, in continuous models we assume that the numbers of
individuals change continuously, not in discrete time instants, as in discrete models. To make it
easier for the reader to distinguish, when we speak about discrete and when we speak about
continuous models, we will denote the number of individuals in the latter ones by x(t) and will
assume that x(t) changes continuously – mathematically, x(t) is a continuous function of time, t. For
our convenience, we usually omit “t” in the bracket (mathematically – the argument of the function
x) if this does not lead to confusion, and write only x instead of x(t). As a consequence, population
growth can be described by a smooth curve. Thus, in continuous models, population growth is
described as the change in population size (dx) that occurs during a very small interval of time (dt),
and the population growth rate is then dx/dt in this case. The value dx/dt is a convenient
mathematical tool, called the first derivative, sometimes also written as x’(t), or only x’, if this
omission of the argument t does not lead to confusion. It represents the rate of change in population
density within an infinitesimally small interval of time. Population growth is then described by a
differential equation, in which the first derivative is always on the left-hand side of the equation and
its right-hand side (called RHS below) describes a rule, how the number of individuals changes in
dependence on the present number of individuals. Thus, for example, the equation of exponential
growth, derived in Note 2.4, is mathematically expressed as
dx
 rx ,
dt
(note that we have replaced N by x and R by r here, to denote we are speaking about a continuous
model; as explained in the main text, r  log e ( R  1) ). This equation can be understood as: “the rate
of change of the number of individuals at time t is proportional to the number of individuals at time
t”, or – using corresponding symbols – dx/dt is proportional to x with the constant of proportionality
equal to r.
2.4.2. Numerical simulation - Euler’s method
The simplest method, how to simulate population growth in continuous models, is known as
the Euler’s method (a simulation illustrating the use of this method can be accessed here). The way
how the continuous models were derived (see Note 2.4) is used to derive the corresponding formula
– the change in population size
dx = x(t+dt) – x(t),
during an infinitesimally small interval of time, dt, is described by the right-hand side (RHS) of the
differential equation:
(x(t+dt) – x(t))/dt = RHS.
Now, if we make the time increment (the step), dt, very small instead of infinitesimally small, one
can expect that the resulting population growth curve will not change dramatically. In other words, if
we calculate the population growth as
(x(t+h) – x(t))/h = RHS,
where the step, h, is small, or – which is the same:
x(t+h) = x(t) + h.RHS,
we obtain a good approximation of the differential equation model of population growth. Starting
from x(0), we can calculate iteratively – by repeated usage of this formula – the values
x(h), x(2h), x(3h) ...
Note that h = 1 leads us back to discrete models! In this disk, we will use the method just
described with the step, h, equal to 0.1, for simulations. You will see on examples that this is a
satisfactory approximation of population growth described by continuous models.
If the right-hand side of the model predicts large changes of the population during a short
interval of time, our simple method may not be accurate enough to make a good prediction of the
population growth. In such cases more sophisticated – and more precise – formulas for simulation of
continuous models, like Runge-Kutta, predictor-corrector, or a smaller step, h, of the iteration should
be used. You will see, however, that this is not necessary in our simple models. For those interested
in more sophisticated simulation methods for differential equation models, we recommend the book
A First Course in Numerical Analysis: Second Edition by A. Ralston and P. Rabinowicz, published
by Dover Publ. Inc., 2003.

How does the accuracy of the forecast obtained by the Euler’s method, compared with
the exact solution, depend on the step of the simulation, h? To find this out, change the
size of the step, h, and watch, how this affects the accuracy of the simulation. The exact
analytical solution of the equation in the grey box is in column B and shown as the blue
line in the figure, while the prediction obtained from the simulation by the Euler’s
method is in column C and shown as the red line in the figure.
2.4.3. Exponential growth
In the file dealing with continuous models, which can be accessed here, we present several
models of continuous growth. The first one is the continuous analogue of exponential population
growth:
dx
 rx .
dt
It assumes that the population growth rate, dx/dt, is proportional to the population size, x, with the
constant of proportionality equal to the population growth rate, r. In other words, in this model, the
per capita growth rate, (dx/dt)/x, is constant. This is the case, when for example the per capita birth
rate, b (number of newborn individuals per one existing individual per unit time) and the per capita
death rate, d (proportion of individuals that die per one unit of time) are constant and independent of
the population numbers and age structure and when there is no migration into and/or out of the
population studied. In this case, similarly to the discrete analogue explained in detail in the main
text, r = b – d. The assumptions of the exponential growth seem to be too restrictive for this to have
any practical use, but the contrary is true for three reasons:
1. In real populations, these values change with age of the individual, as older individuals are more
likely to die during a unit time interval and they are less fecund. However, it has been shown that
the age structure of the population very often tends to stationary values, which means that the
proportion of individuals of a certain age in the population approaches an asymptotic value and
does not change substantially later (see the book A Primer of Ecology by N. J. Gotelli, published
by Sinauer Associates, Inc., Sunderland, Massachusetts, 1998 for a detailed analysis of this
phenomenon). In such situations the mean per capita birth and death rates are constant, when
averaged over the whole population.
2. The birth and death rates also depend on the population numbers, as the intraspecific competition
for food, space etc. between individuals within larger populations is more severe, which usually
results in their larger death rates and lower fecundity. However, if the number of individuals in
the population is far below the carrying capacity of the environment, K (see the main text for the
explanation of this constant), ignorance of intraspecific competition does not have a dramatic
effect on the prediction of population numbers in the future.
3. When the number of individuals grows rapidly, then its numbers of births and deaths may be
much larger than the number of emigrants and/or emigrants per unit time. In such case, ignoring
migration does not affect the predictions of population numbers dramatically.
To summarize: In rapidly growing populations, which are far below their carrying capacity
and which consist of a mixture of individuals of different ages, the model of exponential growth may
give a good prediction of population numbers in the future. This is the case of many insect pests
during their outbreaks, spread of infectious diseases etc.
Exponential growth has a great advantage in its simplicity. If this equation is integrated
following simple rules of calculus (see the book A Primer of Ecology by N. J. Gotelli, published by
Sinauer Associates, Inc., Sunderland, Massachusetts, 1998 for a detailed analysis), the result can be
used to project, or predict, the population size:
x(t )  x(0).e rt
where x(0) is the initial population size, x(t) is the population size at time t, r is the instantaneous
growth rate of the population, and e is the base of the natural logarithm (e ~ 2.717). Knowing the
starting population size and the intrinsic rate of increase we can use this equation to forecast
population size at some later time. Alternatively, we can use the Euler’s simulation procedure for the
same purpose. How the accuracy of this method depends on the step of simulation, h, and the
instantaneous growth rate, r, is shown here. Using the corresponding EXCEL file in this disk,
answer the following questions:

How does the forecast of the population numbers depend on the values of the instantaneous
growth rate, r, and on the initial number of individuals, x(0)?
2.4.4. Logistic growth
When the number of individuals is close to the carrying capacity of the environment, the
intraspecific competition cannot be ignored any more and the logistic model is appropriate. The
continuous version of the discrete logistic equation from the main text is
dx
x

 rx1   ,
dt
 K
The term in the bracket, 1- x/K, represents intraspecific competition. When the population consists of
only few individuals, i.e., when x is small, then this term is close to one and the growth is
approximately exponential: dx/dt ~ rx. The larger the number of individuals, the smaller is this term,
and therefore the smaller is the growth rate of the population.
When the number of individuals is equal to the carrying capacity, K, the number of
individuals does not change in the model, as dx/dt is zero in this case. We say that the carrying
capacity, K, is the equilibrium, or steady state, of the system. In the continuous, differential equation
systems, which we will use in this book and CD, the equilibrium can be easily calculated by setting
the right-hand side of the model equation equal to zero and solving it for x – the population size.
When the number of individuals is larger than the carrying capacity, K, population growth
rate is negative and vice versa. Thus, in the language of the main text, its state variable, x, tends to
return to or towards the steady state following an environmental disturbance, and this steady state is
stable (see the section 1.6. of the main text).
Simulation of this equation using the Euler’s method can be found here. If this equation is
integrated following simple rules of calculus, we obtain
Kx0 e rt
.
x(t ) 
K  x0 (e rt  1)
where x(0) is the initial population size, x(t) is the population size at time t, r is the instantaneous
growth rate of the population, K is the carrying capacity of the environment, and e is the base of the
natural logarithm.

Try to change model parameters in the EXCEL file here. Note that the number of individuals
approaches the carrying capacity, K, as time proceeds, that this approach is quicker, if r is
larger, and that population grows at the maximum speed when x(t) =K/2.
2.4.5. Logistic growth with delay
The feedback term in the bracket in the logistic equation, 1- x/K, represents intraspecific
competition. In some systems, this feedback loop may be delayed. For example, large density of
aphids on a plant may result in deterioration of plant food quality and consequently lower food
(phloem sap) intake of the adults, which results in lowering the rate of development of the embryos
inside the mothers and finally in lower fecundity of adult aphids some time later. The net result is
that the feedback term in the bracket does not depend on the present density of aphids, x(t), but on
their density some time before, x(t - T). The equation is then modified to:
dx(t )
x(t  T ) 

 rx (t )1 
.
dt
K 

In order to avoid confusion, we explicitly write the arguments here – i.e., we write x(t) instead of x,
as before. Simulation of this equation can be found here.
Similarly to the analogous discrete logistic equation with delay, increasing delay and/or
increasing instantaneous population growth rate, r, results first in damped oscillations, limit cycles,
and then oscillations with increasing amplitude, which finally lead to extinction of the population.

Test this claim by manipulating the model parameters in the yellow-highlighted region.
2.4.6. Allee effect
In the models of exponential and logistic growth, the growth rate is the largest when the
population is small. This is not always true. For example, animals hunting in groups need a certain
minimum size of the population (equal to the minimum size of the group), for the population not to
go extinct, as too small group size might not be sufficient to overdue and kill a large prey (wolves
and deer may serve as an example). In other situations, too small population density might lead to
problems in finding the mating partner. Then the population growth rate might become even
negative, when the population size is small. Note that the plot of the population growth rate, x’(t),
against its size, x(t), is a parabola, and the right-hand side of the corresponding equation is a
x
rx 2

quadratic function, rx1    rx 
. Quite logically then (at least for those who know, how the
K
 K
plots of polynomial functions – quadratic, cubic and higher-order polynomials – look like), an
appropriate function to model Allee effect is a cubic function:
dx
x  x


 rx1    1 ,
dt
K  C


where 0 < C < K. As said in section 2.4.3., the equilibrium can be calculated by setting the righthand side of this equation equal to zero and solving it for x – the population size. We obtain three
equilibria: x = 0, x = C, and x = K. Check the EXCEL model of this system here to check that the
first and last of these, 0 and K, are stable, while that in between these, C, is unstable. In other words,
the number of individuals declines, if the population size is smaller C or larger than K, while it
increases when the population size is between C and K. This is exactly, what the Allee effect
predicts.
2.4.7. Deterioration of environment
The feedback term in the bracket in the logistic equation, 1- x/K, is dependent on the
instantaneous number of individuals in the population. In certain systems, however, fecundity may
be reduced and mortality enhanced by the amount of waste material or excrements produced by the
population during its whole existence (human beings might possibly be one example), by its
cumulative effect on the host populations (for example, aphids affect negatively their host plants,
and this effect seems to be proportional to their cumulative density), or simply may be proportional
to the cumulative density of the population, rather than to its instantaneous size. In such situations,
the regulatory, or feedback term should be dependent on the cumulative density of the population,
the number of “individual-days”, or – mathematically – on the integral of the population numbers
t
from time zero to t, i.e.:
 x( )d . The resulting equation is then
0
t


dx
 rx.1  a. x( )d  ,
dt
0


where a is a scaling constant. Simulation of this equation can be found here.

Try to change model parameters in the EXCEL file here. Note that the number of individuals
initially increases, then declines and eventually approaches zero, independently of the
parameter values.

How does the size and timing of the peak depend on the parameters r, a, and on the initial
size of the population?

How does the asymptote (the maximum value, to which the curve is approaching, when time
increases infinitely) of the integral (biologically: “total deterioration”, or “cumulative
density”) in the lower figure depend on the parameters r, a, and on the initial size of the
population?

Which biological conclusions can you draw from the answers to the points above?

Assume some organisms competing for a limiting resource (nitrogen in soil, phloem sap, host
organisms etc.). Assume that each individual takes a units of the limiting resource per unit
time. The total amount of the resource increases exponentially due to extrinsic effects - for
example, the host organisms increase exponentially. Modify the model of deterioration of
environment accordingly and see, whether the population would also go extinct under these
assumptions.
2.4.8. Harvesting
Harvesting of the population (fishing, selective logging of trees in a forest, shooting or
poaching of wild animals for meat, fur or other parts of the body etc.) may dramatically affect its
numbers. There is a whole array of possible models dealing with this situation. Here we will present
only two simplest cases – constant harvesting of a logistically growing population and the
“biocontrol” case, which will be explained in the next section. The reader is invited to modify this
basic model for other situations (exponential growth, logistic growth with delay, non-constant
harvesting etc.).
The model for constant harvesting of a logistically growing population is – as the reader, now
familiar with several other models and their construction probably expects:
dx
x

 rx.1    H ,
dt
K

where H is the number of harvested individuals per unit time. The easiest way, how to analyze the
behavior of this model, is to look at the plot of dx/dt against x. When H is zero, this plot is a curveddown parabola. As H increases, this parabola moves by the value H downwards, as the reader can
check and see in the EXCEL file here. The parabola then has two intercepts with the x-axis, defined
by setting the right-hand side of this equation equal to zero and solving for x. The solutions are:

H 
.
x1, 2  K2 1  1  4

Kr


As we have seen above, these points are equilibria of the system, i.e., the points for which dx/dt = 0,
or – in other words – the population size does not change. The reader can verify in the EXCEL file
that these equilibria approach each other, as H increases. The plot of the number of individuals
against time shows that between these two equilibria, the population numbers increase (as the values
of dx/dt are positive – see also the plot of dx/dt against x), while when the number of individuals is
larger than the larger equilibrium, or smaller than the smaller equilibrium, the numbers of
individuals decrease with time.
Note one important biological consequence. If the population is initially, before the
harvesting begins, at its carrying capacity, and the level of harvesting increases slowly, then the
population attains a smaller and smaller equilibrium (the larger of the two), that one defined by

H 
.
x1  K2 1  1  4

Kr


H
becomes negative, there is no more
Kr
equilibrium of the system, as the square root in the corresponding equation does not have a real
value, the whole parabola in the plot of dx/dt against x is below the x-axis, and therefore the numbers
of individuals decrease continuously – the population goes extinct eventually. Mathematically, this
phenomenon is called a bifurcation, and – more importantly for us – biologically this means that
with increasing harvesting the population numbers will initially not respond too dramatically, but
when a certain level of harvesting is exceeded, the whole population suddenly collapses and goes
extinct – with obvious conservation consequences.
However, when the value of H exceeds Kr/4, then 1  4
2.4.9. Biocontrol
Instead of constant harvesting, one can imagine a situation, when the amount of harvested
individuals is proportional to the present number of individuals – the more individuals are present in
the population, the easier it is to catch them and the more of them can be harvested. Besides human
harvesting, this may be the situation of a biocontrol agent (predator or parasite) released with the aim
to control a pest species. The rate of catch of its victim is likely to be proportional to the density of
the victim – to the likelihood that the predator will find it. In this case, the harvesting equation can
be modified in this way:
dx
x

 rx.1    Hx
dt
K

The equilibria of this system can again be calculated by equating the right-hand side of this equation
to zero (thus assuming that the rate of change of the population, dx/dt is equal to zero) and solving
for x; the solution is then:
x1  0
 H
x2  K 1  
r 

and the corresponding simulation can be found here.
Chapter 4: Interactions between two species
In the main text, the graphical methods developed in Chapter 3 are applied to the analysis of
interactions between two species, including mutualistic, competitive, and predator-prey systems, in
Chapter 4. We present discrete models in the main text, while in this disk we show simulations of
interactions between two species for continuous systems. Note, however, that because of the Euler’s
method used in the simulations, setting the step equal to 1 takes you back to their discrete analogs.

Explore carefully all three sheets in the EXCEL file with simulations of two species
systems (accessible here), and see, how changes of initial values (numbers of individuals
at time zero) and changes of model parameters affect model predictions. Note that in
some cases the model predictions are quite wild, or the numbers of individuals vanish
quickly. This is because the two species models are more complicated, such is also their
behavior and they do not always make a biological sense. One has to employ own
intuition to see, where is the problem with the model in such cases.
4.1. Continuous models
Generally speaking, almost all two species models we shall deal with can be expressed in the
following form:
dx1
 f1 ( x1 )  x 2 12 ( x1 , x 2 )
dt
dx 2
 f 2 ( x 2 )  x1  21 ( x1 , x 2 )
dt
where xi is the number of individuals (abundance) of species i (i = 1, 2), dxi/dt are the corresponding
derivatives (see 2.4.1), fi are functions describing the population growth of species i if it is alone and
does not interact with the other species (see chapter 2 for various models of these), and  ij are
interaction terms, describing the interactions between the two species: they tell you, how the
abundance of one species affects changes in the numbers of individuals of the other species. Each of
these interaction terms is a function of x1 and x2 – formally: ij = ij(x1, x2), which means that the
strength of interaction always depends somehow on the abundances of both species. Note that each
of these terms is multiplied by the number of individuals of the interacting species (x2, describing
population dynamics of species 2, in the first equation, and x1, describing population dynamics of
species 1, in the second equation). Thus, ij describes the interaction strength per one individual of
the interacting species. This is quite convenient for the analysis of such systems, as absence of the
interacting species causes vanishing of the whole interaction term and the dynamics of the species in
question is then governed only by the function fi. E.g., if species 2 is absent, i.e., if x2 = 0, then the
second term at the right-hand side of the first equation vanishes and the dynamics of species 1 can be
described by
dx1
 f1 ( x1 ) .
dt
The three types of models are distinguished by the signs in the equations: in mutualistic
systems, both signs are positive (as the abundance of each species positively influences changes in
abundance of the other one); in competitive systems, both signs are negative (as the abundance of
each species negatively influences changes in abundance of the other one), and in predator-prey
systems one sign is positive and one is negative, as the abundance of predators negatively influences
the number of individuals of prey (predator eats prey), while prey is eaten, provides food for the
predator, and therefore its abundance positively affects changes in predator numbers. Any forms of
the fi functions described in the section 2.4. of this text can be used; only in the predator-prey
systems it is reasonable to restrict the within-species growth to the negative exponential, as – in the
absence of prey – the number of predator individuals necessarily declines due to its starvation, and
the most reasonable model for this is exponential decline.
In the simulations, we assume logistic growth of each species in the absence of the other one
(except of the predator-prey systems, where negative exponential growth is always assumed for the
predator and an alternative with exponential growth of prey is also considered).
In each of these models, the system dynamic trajectories and the equilibrium lines (often
called isoclines) are plotted in the figures, where each axis denotes the number of individuals of one
species. The point with coordinates equal to the initial conditions (numbers of individuals of each
species at the beginning of the time interval considered) are indicated by a blue point – starting point
of the system dynamic trajectory, which then connects all points, representing numbers of
individuals of each species at all consecutive instants following the initial one – see also, e.g., Fig.
4.14. of the main text for an example of the system dynamic trajectory.
As described in the main text (p. 75), the equilibrium line (isocline) of species i connects all
points, in which the numbers of individuals of the two species are such that species i neither grows,
nor declines in numbers. You have seen in the main text, how to find equilibrium lines in discrete
models. In continuous models, this is even easier: the species corresponding to this isocline neither
grows, nor declines in numbers, therefore the derivative of the function describing dynamics of this
species is zero anywhere in this isocline – and such must be also the right-hand side of the
corresponding model equation, as it is equal to this derivative. Thus the equilibrium lines are
obtained as follows: the right-hand side of each of the model equations is set equal to zero, and then
the number of individuals of the species 2 is expressed as a function of the number of individuals of
species 1. An example of this calculation is given below in the section dealing with mutualistic
systems.
4.1.1. Mutualistic (cooperative) systems
In cooperative systems, the abundance of each of the species positively affects the changes in
numbers of individuals of the other species. We assume the simplest possibility: the strength of this
interaction (and therefore the influence of the abundance of one species on the rate of change of
individuals of the other species) is proportional to the numbers of individuals of each of the species
(so-called Lotka-Volterra interactions, named after the people who first introduced them into models
of population dynamics). We assume that the interspecific interactions affect the population carrying
capacity. This results in the following system of equations:

x1
x1'  r1 x1 1 
 K 1   12 x 2





x2

x 2'  r2 x 2 1 
K


x
2
21 1 

Here, x1 and x2 represent the numbers of individuals of species 1 and 2. Each of them is assumed to
be growing logistically in the absence of the other species:

x 
x1'  r1 x1 1  1 
 K1 

x 
x 2'  r2 x 2 1  2 
 K2 
Presence of species 2 causes an increase of the carrying capacity of species 1, which is proportional
to the number of individuals of species 2: 12 x2 (see the term in the denominator of the first model
equation), and vice versa.
If we set x1'  0 , or – in other words – the right-hand side of the first equation equal to zero,


x1
  0 , we obtain an expression, which describes all possible pairs of population
r1 x1 1 
 K1  12 x2 
sizes (numbers of individuals), for which species 1 does not change in numbers (which is the
meaning of x1'  0 ). We can now express x2 as a function of x1: either x1=0 (and then necessarily


x  K1
x1
  0 , or x 2  1
r1 x1 1 
, which results in the same. See the reproduction plane
 12
 K1  12 x2 
approach in the main text for a detailed explanation. A similar calculation may be performed for the
other species. Thus the isoclines of the systems can be expressed as:
x2 
x1  K 1
 12
x 2  K 2   21 x1
The corresponding simulation can be found here.
4.1.2. Competitive systems
In competitive systems, the abundance of each species negatively affects the changes in numbers of
individuals of the other species. We assume that – contrary to the mutualistic systems – this
performs in the numerator of the model equations. I.e., interspecific competition is expressed at the
same place as the intraspecific one – in the numerator of the fraction in the bracket on the right-hand
side (see below). This leads to the model

x   12 x 2
x1'  r1 x1 1  1
K1





x   21 x1 

x 2'  r2 x 2 1  2
K2


and the model isoclines are
x2 
K1  x1
12
x2  K 2   21x1
The corresponding simulation can be found here.
4.1.2. Predator-prey systems
The general form of the predator-prey model is
dx1
 f1 ( x1 )  x 2 g ( x1 , x 2 )
dt
dx 2
 f1 ( x 2 )  x 2 h( x1 , x 2 )
dt
The simulation program, which can be accessed here, allows you to choose between two types of
population growth of the prey (function f), exponential and logistic, four types of functional response
(function g) – Lotka-Volterra and the three Holling-type responses depicted in Fig. 4.19 of the main
text, and two types of numerical response (function h) – constant conversion efficiency of prey to
predator biomass and logistic predator growth. Thus the possibilities are:
Prey growth curve:
1. Exponential
f1(x1) = rx1
2. Logistic
f1(x1) = r x1.(1- x1/K)
We use exponential growth in situations, when intraspecific competition due to large density of
individuals can be ignored.
Predator growth curve:
1. Negative exponential
f2(x2) = -dx2
We always assume that predator will exponentially decline in numbers due to starvation, if there is
no prey available.
Functional response:
0. Lotka-Volterra
g(x1,x2) = ax1
1. Holling type I
g(x1,x2) = ax1, if x1 < S/a
g(x1,x2) = S,
if x1 > S/a
2. Holling type II
g(x1,x2) =
aSx1
ax1  1
3. Holling type III
g(x1,x2) =
aSx12
ax12  1
The Lotka-Volterra type of functional response assumes that there is no restriction on the number of
prey individuals that the predator is able to eat per unit time. This is of course unrealistic, as the
predator needs some time for handling the prey: catching, killing and eating it. This type can be a
reasonable approximation of reality in cases, when prey never becomes abundant and when the
predator spends majority of time by searching for the prey. In such case, the assumption of linear
dependence between the number of prey available and the number of prey eaten (i.e., that doubling
of the number of prey available results in doubling of the number of prey eaten) may be realistic and
for the simplicity of the corresponding equation even preferable, as it leads to simpler and therefore
more easily tractable models.
In the Holling type I functional response it is assumed that the number of prey attacked
increases linearly with prey density and then suddenly stops when the predators are satiated. This
type of response seems to be rather rare in nature, but may be characteristic of some filter feeders,
which spend little or no time pausing after each prey is captured; that is, they do not need to stop
hunting in order to kill and devour their prey. On the other hand, the type II response is typical of
predators that pause after each prey is captured and, therefore, their rate of attack declines as the
density of their prey increases. This type of response seems to be typical of many invertebrate
predators, but it should be noted, however, that most of the data come from laboratory experiments.
Type III functional responses are characteristic of predators that attack their prey at an increasing
rate as prey density rises, but then the rate of attack declines as handling time becomes a factor in
determining how fast prey can be caught. It is generally thought that type III responses are typical of
general predators, particularly vertebrates, which switch their attack to a particular prey species
when it becomes more abundant; that is, they learn to look for, or develop a “searching image” of,
the more abundant species in their repertoire of prey. However, these responses have also been
found in some insect parasitoids, and they may be more common in nature than was previously
supposed.
Numerical response:
1. Constant conversion efficiency
h(x1,x2) = q. g(x1,x2)
2. Logistic predator growth
 x 
h(x1,x2) = Zx 2 1  2 
x1 

The assumption of constant conversion efficiency means that the predator converts a certain,
constant proportion q of the prey biomass eaten into its own biomass. The assumption of logistic
predator growth means that the predator grows logistically and that its carrying capacity is
proportional to the number of prey available (and therefore x1 is in the denominator of the
corresponding equation).
Epilogue
After having been through all the simulation exercises above, we are sure you now are able to
develop your own models of population dynamics.