Computer Experiments
in Population Ecology (XGROW)
Self-Study Exercises
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This document is a copy of Chapter 8: Computer Experiments in Population Ecology in Studies in Evolution, Ecology and Behaviour: A Laboratory Manual for BIO150Y (Eighth Edition), Departments of Botany and Zoology, University of Toronto, 1997, by Corey A. Goldman.
Objectives:
About the XGROW Computer Program
Schedule:
Preparation:
As surely as individuals are born and die, populations grow and decrease. At any time in its history the size of a population is the product of births and deaths, and immigration and emigration. The rate at which individuals enter the population (births and immigration) and individuals leave (deaths and emigration) will determine whether the population is growing, shrinking, or is stable, and if it is changing, how fast.
Populations seldom undergo unrestricted growth under constant conditions for long periods. Given the capacity for increase of bacteria, this is just as well! More commonly, as populations grow individuals affect one another's access to vital resources such as food and space. This is an example of intraspecific competition (intra = within). As a consequence, birth rates tend to decrease and death rates increase as populations become more crowded -- so-called density-dependent effects on birth and death rates. At some point, the carrying capacity (K) of the environment is reached by the population and there is no more growth (i.e., births balance deaths). The carrying capacity is the number of individuals which can be supported by an environment.
Populations do not live alone, just as individuals do not. When individuals of the same species, or of two different species, depend on a common important resource then competition occurs. Competition can be defined as "interactions between individuals brought about by a shared requirement for a resource in limited supply leading to a reduction in survivorship, growth, and reproduction of individuals." Interspecific competition (inter = between) is the competition between two or more different species for a resource.
Predation is an important interaction between species that affects population levels of species. Predation can be defined as occurring when individuals of one population eat living individuals of another. There are several models for predation, but all models have two parts: the birth rate of the predator increases as the number of prey increases, and the death rate of the prey increases as the number of predators increases.
XGROW is a program which allows you to simulate
The underlying mathematical models for XGROW are detailed in the appendix and in the online Help.
There are two different versions of XGROW available (there are minor differences between the versions, mostly in the look of the interface): XGROW for Windows is recommended because it will run faster on most computers; XGROW on the Web, a Java applet, requires an Internet connection, and can often be slow.
The XGROW for Windows screen is divided into two parts: the graph window on the left and the program controls (buttons and sliders) on the right. From the function bar above the graph window you can: Print a graph or Export a data file, Exit the program, change a species in the Single Species model, and obtain online Help.
Graphs: The graphs show population size (N or lnN) as a function of time. In the Single Species model you can choose to graph population size as arithmetic or natural logarithmic (ln) values of population size by selecting the appropriate button in the Graph Type box; on the arithmetic graphs, the magnitude to which population size is expressed is given at the top of the y-axis (e.g., 000's refers to thousands). The Competition and Predation models display only natural logarithmic values of population size.
Models: Three models can be selected: Single Species, Competition, and Predation. The program starts each time with the Single Species model, with the starting species selected at random. Select Species from the pull-down menu above the graph and click+drag to a new species.
Controls: The Control panel has three functions: Start, Clear, and Reset. "Start" begins your experiment (simulation). "Clear" erases lines from the graphs. "Reset" returns the parameter sliders to the default values (in XGROW, an experiment is set up for you by default). Five simulations can be displayed on a graph at one time, each with a different coloured line. If a sixth simulation is run then the line generated by the first simulation is erased, if a seventh simulation is run then the line generated by the second simulation is erased, etc.
Time: For each species in the Single Species model the time scale along the x-axis (i.e., duration of the experiment) can be varied from the default values. This enables you to "zoom in" on the graph line.
Noise: To more closely simulate population growth, the Noise function randomly fluctuates birth and death rates. The default setting is "ON"; to turn "OFF" select Options from the function bar above the graph window.
Time: To more closely simulate population growth, the Time Delay function builds each arithmetic graph over time. The default setting is "ON"; to turn "OFF" select Options from the function bar above the graph window. The Time Delay function is "OFF" for logarithmic graphs.
Grid: To assist in determining the x,y-coordinates of a data point directly from a graph line, the Grid function superimposes grid lines on the graph.
In each model, particular characteristics of the population (parameters) can be varied, such as birth rate, death rate and the intensity of competition. In each model, a default experiment is set up initially. To change a parameter, click+drag the appropriate slider until the desired value is obtained. Alternatively, click on the variable box to the right of the slider to activate the box and type in the desired variable. Note that each parameter has maximum and minimum values that cannot be exceeded. The parameters are explained below:
Single Species | Interspecific Competition | Predation
and
): These measure the intensity of competition each
species is experiencing from the other relative to that within each species.
For example, a coefficient of = 1.0 for P. aurelia
means that each P. caudatum in the culture is using the same resources
as an aurelia (i.e., interspecific competition is as strong as
intraspecific competition for aurelia). A coefficient of
= 2.6 for caudatum means that each coexisting
aurelia present uses up resources which would support 2.6
caudatum. (Refer to the appendix for a
more complete description of the competition coefficients.)
Prey:
Predator:
You are to conduct computer experiments (simulations) with each of the three models:
Your understanding of these exercises with be evaluated on Quiz 6 (in Lab 8) and Test 4; you will not be evaluated on the equations in the appendix.
XGROW always starts with a simulation in the Single Species Model already set up for you, and ready to run. It automatically selects one of the five species which you can study (human, rat, fruit fly, Paramecium caudatum, Paramecium aurelia, and T-phage virus). You can of course choose the species you want (from the Menu option bar, select Species and click+drag to select the species you want to study).
The simplest form of population growth results if individuals reproduce and die at a constant rate. As you will learn, these conditions seldom hold for long, if ever, in reality, but they are a good way to introduce basic population parameters: individual birth and death rates (b0 and d0), and the intrinsic rate of growth, r. In the Single Species model, b0 and d0 (not r per se) are given; these values cover the realistic range observed in nature for the different species.
Table 10-1.
Simulating unrestricted population growth (r* is estimated from your
graph).
|
One way to summarize the population growth is the intrinsic rate of growth, r, which is the per capita growth rate.
How similar are the two r values?
How does r depend on b0 and d0?
Intraspecific Competition
In the previous experiments, population growth was unchecked and as you observed, population grew at the theoretical maximum rate for the species. These conditions rarely hold for long for any species. Individuals normally interact with each other, often in a negative way -- intraspecific competition. As population density increases, competition for resources is likely to increase. A consequence of competition, is that individuals can no longer reproduce at the maximum rate or die at the minimum rate. What are the consequences of such density-dependence for the growth of a population?
What is the carrying capacity, K?
What type of population growth is there now (i.e., what is the shape of the population growth curve)?
Table 10-2.
Simulating density-dependent population growth (K* is the observed
value estimated from your graph).
Species |
b0 |
d0 |
kb |
kd |
K |
K* |
| ______________________ | _____ | _____ | _____ | _____ | _____ | _____ |
| ______________________ | _____ | _____ | _____ | _____ | _____ | _____ |
How does K depend on the intensity of intraspecific competition?
What happens if you increase the intensity of intraspecific competition? For instance, suppose that habitat conditions change so that food levels decreased for a population.
In the previous simulations you examined how density-dependent effects (intraspecific competition) result in logistic population growth in a single species. But in nature, a species never lives alone. Thus, in addition to intraspecific competition, a population can expect to experience interspecific competition which will affect its growth.
and
affect individual growth is given in
equations 10 and
11 in the appendix.
Note the new population characteristics: (1) Birth and death rates have been
replaced by r. (2) The density coefficients have been replaced by
K. (3) The intensity of competition from the other species is given
as for P. aurelia and for
P. caudatum.
and )
for both species to 1.0.
Table 10-3.
Population growth of two species with and without interspecific competition.
| P. aurelia | = 0
|
= 1.0
|
=
|
|||
| P. caudatum | = 0
|
= 1.0
|
=
|
> 1.0 (i.e., aurelia experiences relatively more inter-
than intra-specific competition), and the coefficient for P. caudatum
at < 1.0. Record your settings in the table above.
You have just simulated an experiment like that which Gause conducted with real protozoans and which led to his formalization of the Competitive Exclusion Principle.
Probably many interspecific interactions are unequal, but relatively weak. And the environment is more favourable for one species than the other, in which case the carrying capacity, K, for each species differs.
=
= 0.5 for both species. Run the new simulation.
Table 10-4.
Population growth of two species which differ in both the intensity of intra-
and interspecific competition.
P. aurelia |
P. aurelia |
P. caudatum |
P. caudatum |
|
|
Simulation |
K |
|
K |
|
Result |
| ___________ | _________ | _________ | _________ | _________ | _________ |
| ___________ | _________ | _________ | _________ | _________ | _________ |
| ___________ | _________ | _________ | _________ | _________ | _________ |
| ___________ | _________ | _________ | _________ | _________ | _________ |
to 1.0. Run a simulation
and check the results. Now what is the outcome?
= 0.5, and set
= 1.0. What is the outcome this time? Evidently, in
some circumstances, strength of interspecific competition alone is not the
only factor controlling the final outcome.
to find a value beyond which
aurelia is excluded by caudatum.
You have just seen with these simulations that the outcome of a two-species interaction depends on the interplay between interspecific competition and intraspecific competition for environmental resources (i.e., carrying capacities).
In a situation where a predator population is feeding on a prey population, the two most important factors influencing the population sizes of each species are the intrinsic growth rate of the prey population and the predator death rate.
Note some new population characteristics for the predator: (1) Satiation -- predators can get full eating prey. (2) Efficiency -- you can vary the efficiency rate at which prey are converted to predator births. (3) Prey capture rate -- you can vary how many prey a predator can catch per day.
Table 10-5.
Simulating population growth of predator and prey (N = population
size attained).
|
What happens to a stable predator-prey relationship such as you have just simulated if the prey's growth rate is perturbed? Suppose, for example, that the hare's basic birth rate is increased.
Now simulate a different scenario. Assume for example, that there is increased hunting pressure by humans on the predator species. As a consequence, predator death rate goes up.
You have just completed some simulations which demonstrate the Volterra Principle, named after the scientist who first described it, and which says that "if predator and prey are in stable coexistence, change in prey growth rate will affect predator population size more and changes in predator death rate will affect prey population size more."
What happens to the stable predator-prey relationship if the quality of the prey's environment is improved. Imagine, for example, that a group of well-meaning citizens decide to supplement the natural diet of wild rabbits with commercial rabbit chow. In effect, they are increasing the K for the prey. Simulate what could happen to prey and predators.
Does food supplementation have any effect on the long-term average population size for the prey?
These simulations have demonstrated a phenomenon known as the "paradox of enrichment." If the carrying capacity of the prey population is increased, what was previously a stable coexistence with its predator species can become unstable and actually result in extinction of either species. In our hypothetical example, the best intentions of the citizens could have back-fired. You will find more details about this paradox in the appendix.
= 1.4, Species B has an
= 0.7. Which species is experiencing the most interspecific
competition?
Complete this glossary by adding the definitions for these terms:
Birth rate:
Carrying capacity:
Competition coefficients:
Death rate:
Density dependent effects:
Exponential growth:
Interspecific competition:
Intraspecific competition:
Intrinsic growth rate:
Logistic growth:
Predation:
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