Predation

One of the first predator-prey relationships to be closely studied as to their population dynamics was the snowshoe hare - Canadian lynx association. One of the most striking characteristics of this dynamic is the oscillations in both populations (Figure 1). Since then, the predator-prey oscillations have been noted in numerous situations. For this simulation we'll make the prey and predator populations totally dependent on one another for their population dynamics. This'll also keep the equations nice and simple.

• Prey in absence of predation: dN/dt = r₁N
      Where: N is the density of the prey, and r₁ is the innate capacity for increase for the prey.
• Predator: dP/dt = -r₂P
      Where P is the density of the predator population, and r₂ is the instantaneous death rate in the absence of prey.

Putting the predator and prey together:

• Prey: dN/dt = ( r₁ -  k₁P)N
      Where  k₁ is a constant measuring the ability of the prey to escape predation.
• Predator: dN/dt = (- r₂ +  k₂N)P
      Where  k₂ is the skill of the predator in capturing prey.

These equations have a periodic solution that will cause the populations to oscillate (Figure 2). Note that the predator oscillations lag behind the prey oscillations (green lines on Figure 2). An alternate way of showing the population dynamics is shown in Figure 3. The center ovoid depicts the population changes over time for both the predator and prey. This is a stable oscillation with a low magnitude. The larger ovoids show a system with greater magnitude oscillations.

There are four possible oscillation scenarios (the predators are always shown in RED, the prey in BLUE for all the diagrams in this exercise.

• Either the predator or prey could go extinct. If the prey go extinct, the predator will soon follow (depending on the value of r₂; Figure 4).
• A stable oscillation (Figure 5; The prey continues to grow under the blue line for the left graph. The predator grows in the gray area. The arrows show the population trajectory).
• A convergent oscillation (Figure 6).
• A divergent oscillation (Figure 7). Divergent oscillations may eventually lead to extinction of one or both species (The predator in this case).

The parts of the simulation control panel are shown in figure 8. The predator and prey parameters are conveniently grouped as are the simulation control buttons. The output from the simulator is depicted as the familiar density vs. time graph (left) as well as a trajectory graph (right). The trajectory graph is read in a counter clockwise direction. Figure 9 is an animation that shows how the trajectory and density vs. time graphs relater to each other.

Figure 10 shows the outcome of a simulation with the following starting values:

These starting values result in a damped oscillation of both the predator and prey. Note that the trajectory diagram clearly shows the damping. For figure 11, the predator reproductive rate was decreased from 0.2 to 0.1, resulting in a quicker damping. Note also that the predator density can be higher than the prey density under some conditions. Increasing the predator intrinsic rate of increase to 0.4 increases the oscillation (and almost puts the predator out of the picture). Note also that the prey density is higher than the predator. Figure 13 shows the starting conditions that can lead to extinction of the predator.

Predator-Prey Activity

Scroll down to the simulation screen (After Figure 15) or download, install, and run the stand-alone version here.

1. What is the relationship between the predator and prey reproductive rates and an outcome where the number of predators outnumbers the prey density? Under what environmental conditions would you see this situation?
2. Determine the effect of changing the escape constant and/or capture success. What happens when the reproductive rates are low vs. high? Provide a realistic field scenario where you might see the outcomes you observed.
3. Determine the conditions that will lead to a stable situation such as shown in Figure 14.
4. How do environmental uncertainties affect the outcome of your simulations? What type of organisms are more or less sensitive to the environmental perturbations? How does this relate to the real world?
5. Attempt to increase the amplitude of the oscillations shown in figure 15 and discuss your findings (note: Max Gens at 50).
6. Can you produce a divergent oscillation? What are the conditions?
7. Why does this model suck so much? What are the limitations when it attempts to model the real world. What might be done to "fix" it?

NOTE: There are more questions below!

Laboratory and Field Studies of the Predator-Prey Relationship

One of the first studies of the predator-prey relationship was performed on laboratory populations of the protists Paramecium caudatum (prey) and Didinium nasutum (predator). Innitial experiments (Figure 16) always had the same outcome: The Paramecium initially grew well, then the Didinium overfed the prey, wiped them out, and the predators eventually starved to death. Nothing the researchers tried could affect the outcome (only the timing). They tried larger containers, fewer Didinium, adding the Didinium at a different time, etc.

The researchers did manage to change the outcome by growing the Paramecium in oat medium, which forms a scum at the bottom on the vial. Initially, both the predator and prey grow, but eventually Didinium have trouble capturing Paramecium and they eventully die off (Figure 17). The explanation lies in the way that Paramecium swim. Because of their shape, Paramecium tend to take a circular path as shown in Figure 18. When presented with a barrier (such as oat scum), normal Paramecium will back up, eventually turning away from the barrier (Figure 19). Backing up occurs when the stimulation of bumping into a barrier causes Calcium channels to open in the integument. The outflow of calcium causes a depolarization of the cell membrain which causes the cilia to reverse. When the calcium concentrations are re-established, Paramecium will again continue foreward. Because their shape causes them to turn, they will eventually avoid the barrier. Unfortunately, this behavior pattern keeps them in the upper portion of the vial where Didinium hang out. A mutation called pawn destroys the calcium channels so the Paramecium can only move forward (Figure 20). This allows afflicted individuals to move into the scum, and escape the predators. Thus, the predators select against the normal genotype and eventually eat them while the pawn mutation is selected for. Note that the Paramecium never had to "decide" what to do!

They did eventually manage to get a damped oscillation (Figure 21), but had to add one Paramecium and one Didinium every third day. Without this immigration, it wasn't possible to mimic the predictions of the predator prey model.

8. From what you've learned from the model, would you expect this experiment to work using two protists? Explain. What type of organisms should the researchers have used?

Some natural populations do show oscillations. Data for the lynx and snowshoe hare show periodic oscillations (Figure 22). The lynx population density lags behind oscillations of the snowshoe hare. The hare oscillations have been recently attributed to cyclic infestations of parasites.

Natural Regulation of Population Size

The abundance of a species varies from place-to-place since some habitats are inherently good, while others are bad. On top of this, the physical environment is always fluctuating, sometimes in a predictable way (but often not). While seasons and climate patterns are clearly predictable, short term weather patterns often are not. Most models of population growth assume that the parameters of the models (r, K, etc.) are constant, yet these parameters describe components of the environment that do vary over time. Carrying capacity, for example, describes the resources that are available in the environment, and resource availability obviously changes over time.

The possibility of population extinction is increased because of environmental fluctuation. According to the logistic equation, a population will continue with a density of K in a constant environment as long as r is positive. In a changing environment with a moving K having a positive r is no longer a guarantee that the population will not go extinct.

If the population does not go extinct, but the environment fluctuates, the the population will sometimes be above (or below) K and there will a descrepancy between the available resources and the population size.Some organisms, through their very nature, are better at resource tracking; i.e. they are able to track fluctuating resources and predict the future state of the environment.

The spatical pattern of a population may also be affected by fluctuating resources (figure 23).. Most populations occupy an area much greater than the size of an individual organism. In addition, populations are not usually evenly distributed over their range so that members are concentrated in particular areas. The surprising fact is that areas of high population density do not necessarily correspond to high quality habitat. A patch is an area where population densities are higher than normal. These patches may be places where historically the habitat was favorable (so that the animals chase around the favorable habitats and are often a step behind (Figure 23).

Several other agents can work to control population sizes and keep them permanently below the carrying capacity:

• Facultative Agents control populationsize by exherting more influence when the population density increases and less when the density is low. Parasitism and disease would be examples of facultative agents.
• Catistophic Agents are independent of population size. Storms, drought, and temperature extremes are examples of catistrophic agents.
• Non-facultative Agents take a constant number no matter how large the population gets. Many long-lived predators, for example show little change in population density from year to year. No matter how large the prey population gets, they'll continue to take the same number of prey.

Generally, there are four schools of thought as to how populations are controlled:

• Biotic School: Biotic agents are most important in controlling populations. The emphasis is on stability.
• Climate School: Population changes are driven by changes in the weather patterns (Figure 24). The emphasis is on fluctuations.
• Comprehensive School: It depends on the situation. For favorable environments density independent factors are less important while for unfavorablke environments, density dependent factors are important (Figure 25).
• Self-regulatory school. The three previous schools emphasize extrinsic factors (food, weather, other animals), while the self-regulatory school emphasizes intrinsic factors (behavior, endocrinology, stress, etc.).

9. For each of the above schools of thought, describe the type of organism that would fit the school and explain why. (Body lice are of this type because .....)