Population Models I: Growth
Note: Your lab report for this activity will consist of numeric and
graphical data gleaned from our population model. As you work through the exercise,
questions and activities are in red. You may work in groups of two! You have two weeks to
complete this assignment. You can see the results of a few runs by pressing the "ask
spider man" button:
Introduction: Mathematical representations of biological systems have long been used by ecologists to model various processes. The models are useful to access how much is understood about a real system. If the model is a good representation of what goes on in the "real world", then the author is fairly confident that they understand the phenomena. If, however, the model doesn't follow the "real world", then it's back to the computer. Some models are so good they have predictive value. Models that predict locust outbreaks in Africa are routinely used to predict when control measures are needed in a region. At home, farmers routinely use models to predict when herbicides, fungicides, and insecticides should be applied to their fields. On a grander scale, climate models are used to shape laws and international treaties. One of the first models written was one to mimic population growth. An example showing the modeling process can be seen in figure 0.
Modeling is the process of simplifying a system we wish to better understand. Systems that can be modeled range from the very simple to the extraordinarily complex. The reason we are using dynamic computer models in this text is to gain a working knowledge of, and an insight into the processes of global change. Because of the complexity of these processes, it is often not enough to read about the workings of these phenomena. We must think about the dynamics of a system, extract critical functioning parts, and attempt to build a model that captures its "essence" by making assumptions to account for external variables.
Modeling should be approached in a logical and ordered manner. Though following guidelines may seem tedious and unnecessary at times, it greatly increases the chances that your model will (a) work and (b) be a relatively accurate representation of the actual behavior of a real-life system. At its essence, modeling is a 5 part process:
Though not always possible, try to test your model against real world data. Although we will not be strictly following every step of this guideline in the development of the models in this text, it is important that you keep them in mind as you develop them now and especially in the future.
The first model we'll explore is the geometric model. According to these equations, a population grows uninhibited over time. The important variables in this simulation are the starting population size (Fig 1 ) and the rate of increase. Several options are available at this level. You can change the starting population size or the amount of reproduction (between 0 and 1.0;here we've set it at .05). You can also change the number of generations over which the simulation runs (the default is 100). The equation for this simulation is: dn/dt = r*N where dn/dT is the change in the population size (N) over time (t). The variable "r" is the rate at which the population increases per generation (between 0.0 and 1.0). This scenario will result in an exponential increase in the population size.
If the starting population size is set to 10 and the rate of increase is set to .05, the following scenario is predicted (Fig 2; Scroll the upper frame until Figure 2 is visible). After 100 generations the population has increased from 10 to approximately 1300 individuals. If we set the number of generations to 500, then the population size will increase to 400,000,000,000 over 500 generations. There are no "stops" to this system. Populations will increase geometrically no mater what the starting variables. For example, let's start with a population of 10,00 (instead of 10). After 100 generations the results are essentially the the same (400,000,000,000,000 individuals; Figure 3; Scroll down for the above pane); by the next generation, our previous inputs would have caught up.
Scroll down the upper window
so that the next figure is visible. If you do not see the application in your
web browser, you will have to download and install the program on your computer.
Download and install the growth program
here
and then run the program as usual. You can adjust the size of
this window and move the program so that both are visible on your system). Click the
growth model selector radio button so that Geometric Growth is selected. Try changing the
values for r, the Starting Population, and Max Gen to see the effect on geometric growth.
You probably found that the only important variable in this simulation is the intrinsic
rate of increase (r). Record and summarize your
observations.
The geometric model is not very satisfying since it doesn't seem to mimic the way real populations behave. Real populations don't expand indefinitely (except, perhaps humans; Figure 4; Read Limits of Human Growth). Instead, they tend to increase for a short time, then level off. Figure 5 depicts the growth of bighorn sheep populations in the Rockies from the early 1800's to about 1940. Note the rapid expansion of the population following it's initial introduction, followed by a leveling-off of the population at about 1.75 million sheep. This is a typical population response seen in most natural populations. It's as if the population has filled to environment. This "filling" of the environment with a particular species is the "carrying capacity" for that species. For birds, the limiting factor in the environment may be the availability of next sites. Other species may be limited by the availability of food or water. Whatever the cause, the environment is capable of supporting a limited number of a particular species. That number is the carrying capacity, or it represents the environmental resistance to further population growth. The carrying capacity of a system may change over time. Droughts, pestilence, and other climatic changes may temporally increase or depress the carrying capacity of the environment. These factors may be responsible for the "bumps" in the above data (more on that later).
TO be realistic, our model needs
to reflect the carrying capacity of an ecological system. Ecologists have settled on the
variable "K" to represent the carrying capacity of the environment for a given
population of organisms. For our simple model, it doesn't matter if the limiting factor is
nest sites, or the availability of food (although different limiting factors could be
easily accommodated. Figure 6 shows the simulation when the Logistic Growth option is selected (Your results may vary; don't worry about
that now. This is a work in progress.). This equation is essentially the same as our first
try, with the exception of the (K-N)/K addition (K is the carrying capacity, N is the
current population density). The (K-N)/K term in the equation "puts the breaks"
on population growth as the population reaches carrying capacity This result is a sigmoid
curve (Figure 7). This model is much more realistic when compared to natural system (Fig
5). Play with the carrying capacity (K), rate of
increase (r) and starting population size (N) to gain an appreciation of how these
variables affect population growth. Record and summarize your observations.
Still, these models are not completely satisfying (they don't have the jiggles seen in Figure 5), so some more tweaking is needed. The current model assumes that a population responds instantaneously to changes in population size. This assumption is unreasonable. If the population has reached it's carrying capacity (K), and 50% or 60% of the females are pregnant, they'll still give birth, causing the population to overshoot the carrying capacity. In addition, some stresses don't assert themselves immediately. For example, as population density increases, it would take some time for individuals to become stressed. Hormones kick in and eventually the individuals are not as capable of reproduction. An example of a change in reproductive rate with increasing population density is shown in Figure 8. Under low density conditions the crustacean Daphnia (water flea) produces 4 offspring per day while under high density conditions no offspring are produced. Note also that survivorship is adversely affected by increasing population density.
This reproductive time lag is represented by variable c in the model; Fig 7). Our population doesn't respond instantaneously, but is lagged by one generation (Figure 9). Under the conditions for this model a lag of 1 doesn't change the simulation appreciably, but if we adjust the lag to 2, the results are very different (Figure 10). A time lag of 2 results in the model population going extinct by generation 87! Note also the wild fluctuations in the population size (around 2000 individuals above the carrying capacity). This destabilizing effect has led to the local extinction of this species during our 100-generation time span.
If we decrease the reproductive
rate, the population becomes less "aggressive" in it's response and doesn't go
extinct within 100 generations (Figure 11; the reproductive rate has been changed from 0.2
to 0.1). In fact, if you set the Max Gen variable to 200, you'll find that the population
will survive until generation 183. Increase the
reproductive rate in increments of .1 to determine it's effect on survivorship. Record and
summarize your observations.
Not only does the reproductive
rate affect the stability of the population, but the carrying capacity comes into play
(Figure 12). The top row in Figure 12 shows the growth of a population with a high
reproductive rate (0.5) under a changing carrying capacity (K=400 and K=50). Note a
reduced carrying capacity causes the population to go extinct sooner. In the bottom
row we have a model of a population with a low reproductive rate. In contrast to the first
scenario, this population persists with a lower carrying capacity. Try several other combinations of very high reproductive
rates coupled with a changing environmental resistance (K) and compare these to low
reproductive rates under the same conditions. Record and summarize your observations.
With a reproductive lag we find that populations with a low intrinsic rate of increase (r) survive longer than those with a high reproductive rate. In addition, the stability of those populations with low reproductive rates is greater (less oscillations in their growth curves). Finally, populations with high reproductive rates also are prone to outbreaks (population increases far in excess of the carrying capacity).
In the real world conditions affecting the reproductive rates of organisms (such as overcrowding, drought, disease, etc) are not likely to last over an extended period of time. In the short-term, populations with low reproductive rates can wait out the environmental problems while those with higher reproductive rates are more likely to go extinct locally. We also find that conditions that adversely affect the carrying capacity can also make a species more prone to local extinction (especially those with a high r). Activities such as deforestation, water polluting, and overuse of pesticides can all decrease the carrying capacity (increase the environmental resistance) and make populations more susceptible to local extinction.
Recognizing the differences between species with high rates of reproduction and those with low intrinsic rates of increase, ecologists often describe a species as being "r-selected" or "K-selected" (Table 1). Those species whose populations levels are controlled by their reproductive rates are defined as r-selected while those whose reproduction is controlled more by environmental resistance are termed K-selected.
Species that are r-selected tend to perceive the environment as variable and unpredictable (Table 1; Click this too). In other words, they see the environment in a coarse-grained manner (Figure 13). In the figure each of the shapes represents a change in the environment while the red line represents the path of the organism through the changing environment in both space and time. In a fine-grained environment the movement of the organism through the environment is perceived by many small changes. In a coarse-grained environment the animal moves through fewer environments, but the conditions vary greatly from one another. Insects are an example of an r-selected species while birds and mammals are often K-selected. It's important to realize that r- vs. K- is comparative. If, for example we compare mice to humans, mice would be r-selected while we would be K-selected. More on this can be found here. Compare and contrast r- vs. K-selected attributes as outlined in Table 1. What r- and K-selected factors are mimicked by our model to this point?
Lags can also be applied to
other portions of the equation with varying success. We can, for example, apply a lag to
the environmental resistance term ((K-N)/K) as shown in figure 14. Click the image to
watch the animation in Figure 14, and note the change in variable w (it ranges from 0 to
10). As with the previous lag, determine the
effect of low vs. high reproductive rates and the effect of changing K on the population.
Record and summarize your observations. How do these results compare with the previous
model? What ecological/behavioral scenario could be used as a real-world example of an
environmental resistance lag?
We're getting close. Real populations show fluctuations in density in response to randomly changing environmental challenges (as well as intrinsic rhythms). In some cases the fluctuations are clearly due to changes in the environment (Figure 15), while at other times the variations are not as clear (figure 16). We can mimic these random effects by randomly adding or subtracting individuals from the current population size:
The magnitude of the random effect is set by the Stochastic variable. View the gif animation in the above frame by clicking on the image (Figure 17). We start with the reproductive rate at r=.2, the carrying capacity at K=400 and a random stochastic effect of 10. Each simulation was then run 10 times. Under this scenario 80% of the simulations were successful (the population did not go extinct over 100 generations.
The stochastic effect was then doubled to 20. We got the same success rate as before (80%). Your simulations will vary from this since this is based on random effects. The results of this model are always different (that's why we run the simulation several times). This is very different from our previous models (all of them are deterministic; there's only one answer as long as you don't change the variables).
For the third run the stochastic effect was again doubled to 40. This time our success rate plummeted to 20%! To test the effect of changing the reproductive rate, it was then set to r=0.6; from our previous analysis we would consider this to be a fairly high intrinsic rate of increase. After another 10 simulations we find that the success rate is now nearly as good as before (70%).
Now it's time for you to test the simulation. Begin with a small stochastic variable (say, 10). Note that the Fast check box under the Show Me! button has been checked and the number has been changed to 50. This allows you to press the button only once, then the program will do 50 simulations using the variables as you selected them. At the same time, it keeps statistics on the runs using the Extinction Counter at the bottom of the application. Press the Show Me! button now. The program then runs 50 simulations. When finished, it shows the results in the Extinction counter.
Now check the results of decreasing the rate of increase to .1 (change r, click "show me". You can now directly compare the results of the two runs since the notepad keeps a record of all the important variables, including the number of successful runs. Only 25 runs are allowed. Then the program recycles. You can restart the program by hitting the refresh button on your browser (or shutting down the program and restarting if your using the stand-alone version).
Discussion: The Real World
Two aspects of the logistic curve make it attractive: (1) it's
mathematical simplicity and (2) it's apparent reality. The equation may be looked at as a
description of how populations increase when in a favorable environment (this is a
flexible way). Or it may be viewed as a strict theory of population growth (this is the
way it was originally proposed). There are two ways to test the theory: A colony of
organisms can be reared in a constant space with a constant supply of food. Their pattern
of growth can then be checked against the logistic curve. A second way of exploring the
equations would be to text and examine the assumptions the logistic curve separately.
Assumptions of the Logistic Equation
Laboratory Tests of the Logistic Theory.
Paramecium aurelia and Paramecium caudatum were used in one of the first laboratory tests on the validity of the logistic theory (Figure 18). The researchers started with 20 Paramecium in a tube with 5cc of salt solution, buffered to a pH of 8. A constant quantity of bacteria were added each day for food (the bacteria will not multiply in the salt solution). The cultures were incubated at 26 C and washed with salt solution each day to remove wastes. The experiment thus tested growth in a constant environment and space. The data for each of the Paramecium species fit the logistic growth curve well (Figure 18). The asymptotic density (K) was 448/cc for P. aurelia and 178/cc for P. caudatum. (P. caudatum is larger then P. aurelia).
Field Data on Population Cycles.
Several problems arise when researchers attempt to follow population growth under field conditions. For most natural populations, growth does not occur continuously and long-lived organisms may show population growth only rarely. In most cases, the populations don't completely fill the environment. There are some ways around these problems. Some researchers have turned to study animals introduced to island habitats (deer and wolves, for example). Others look at species with annual growth, but determine the growth only during a single year. Diatoms, for example, show a spring maximum and grow in an approximately sigmoid manner. Field mouse populations vary wildly from year to year, but the logistic equation can be used to model their growth in a single year. For many organisms, however, logistic models with time lags, or stochastic models are the only ones that can describe the population's behavior.
Crowding Effects
Crowding has been shown to directly affect fecundity for a variety of
species. Figure 19 depicts the number of offspring per female Daphnia under
different population densities. Figure 20 depicts several measures of female success for
the tit mouse under different population densities. Finally, Table 2 shows how house mice
react to increasing density.