Reading some posts and analyzing dynamics of relationships going on between some 'species' on this board I couldn't resist copy-pasting this lecture from University of California, Davis website [http://www.ucdavis.edu/]
I will be away from the board for a while.
Enjoy!
E.
Direct link: http://trc.ucdavis.edu/catoft/EVE101/Lec7pred.htm
Lecture 7: Predator-prey interactions
0. Introduction
I. Introduction
A. Background: premises and terms
B. Patterns in nature: ?Hypotheses
II. Simple models
A. Form of model
B. Functional Responses
C. Behavior of model
III. Realistic refinements of model
A. Prey K
1. Paradox of enrichment
2. Predator regulation of prey and predator efficiency
B. Time lags: predator-prey instrinsic time scales
revised Feb. 7, 2001
Lecture 7 -- Introduction to two-species population interactions
& Predator-Prey interactions
O. Introduction to two-species population interactions
I. Background. We could go into more detail focusing on a single species population, but in fact that focus is entirely arbitrary. No species exists in a vacuum. All animals eat food that is often another living organism, that in turn occurs in its own population. Even plant populations are better understood if nutrients (N, P, K) are treated as chemo-stat populations (bound in the plant and unbound in the soil). And of course, few populations escape having predators or parasites. In addition, most organisms depend on cooperative relationships with other organisms, for example the gut flora that helps many animals digest food, and so on.
In the simple models we already went over, where an organism gets its food and how that translates into the birth rate is all subsumed in the single symbol b. We don't model the other population which is the food that goes into b. Similarly, we don't know what the sources of mortality are: we could put them simply into d, including death from predators, herbivores or disease. In addition, mutualistic relationships with other populations could contribute to b or reduce d.
In other words, we are talking about population interactions. All populations interact with other populations. In most cases, food is really in the form of another population of some kind, and death includes death from interacting with other species. To understand the interaction we are focusing on , we need to understand the population(s) it interacts with as well.
II. Types of population interactions
First, let's take an overview of all population interactions, to put the rest of this unit in context. Population interactions will be the focus for the rest of the quarter. We specifically mean interspecific population interactions--interactions between populations belonging to two different species. We can classify all interactions based on the net effect on the individual members of each species or as a population-level effect:
+, benefit: an individual benefits from an interaction with an individual of another species; it eventually gives birth as a direct consequence of this benefit or it eventually survives longer as a direct consequence of this benefit: population growth rate increases; population size increases
-, harm: an individual is harmed by an interaction with another individual. Again, this harm causes fewer births or more deaths; population growth rate decreases; population size is lower than when the interaction is not occuring.
0, no effect: some interactions involve no effect on one of the two species, and no harm or benefit to the other. We are not interested at this point in two populations that have no effect whatsoever on each other, as we are taking two populations at a time. Later, when we build food webs and communities, we will be very interested in populations that are unaffected by other species in the group.
Species 1 Species 2
"Exploitation" =
Trophic interactions
Predator-Prey
Herbivore-Plant
Parasite-Host
+
individuals of species 1 benefits
growth rate of population increases
population size increases
-
individuals of species 2 are harmed
growth rate of population decreases
population size decreases
Competition -
individuals of species 1 are harmed
growth rate of population decreases
population size decreases
-
individuals of species 2 are harmed
growth rate of population decreases
population size decreases
Mutualism +
individuals of species 1 benefits
growth rate of population increases
population size increases
+
individuals of species 2 benefits
growth rate of population increases
population size increases
Examples:
+,- predation:
classical predators that kill individuals and eat them; carnivores, insectivores etc.
herbivores: eats pieces of plants, this harms the plant in some way, population growth is slowed; leaves, seeds, etc. , less photosynthetic material, birth rate of plant is lowered.
parasites don't kill the host outright, necessarily--they might cause it to be weak and die for other reasons, or might lower the birth rate.
-, - interspecific competition: includes any organisms sharing resources, going hand-in-hand with intraspecific competition--if resources are limiting, there is not enough for individuals of the same (intra-) and different (inter-)species using this resource: Competition may be for food, nutrients, water, space, nesting sites and other features of habitat, and may involve atagonistic interactions (such as territoriality) or not.
+, + mutualism: includes pollination and fruit dispersers, microorganisms that live in the gut and help to digest cellulose; ants that tend aphids. Mutual benefits involve resources (food, nutrients, water shared), protection from predators, shelter.Many food webs and biomes exist as we know them only because of underlying mutualisms (such as mycorrhizae, which are fungus-tree mutualisms, without which the tropical rainforests and many coniferous forests could not exist).
Predator-Prey Interactions
I. Introduction: Reading in Molles: Chapter 11 and parts of Chapter 6
A. Background. We will use the term predation loosely, to mean the general +, - phenomenon. We have predators in the traditional sense like lions eating antelopes or a bird eating worms or a lizard eating crickets, etc. However, other kinds of +, - interactions are qualitatively the same, such as cows grazing on grass or tapeworms in the intestines of wolves or viruses causing small pox outbreaks.
Your text Molles uses the term "exploitation" in the same general sense that I'm using "+,-" to mean simply that individuals in one population benefit somehow by harming individuals in another population. In Chapter 11, Molles starts by impressing you with how complicated systems of exploitation are. While I agree that Nature is wonderfully complex, I will start by reducing this complexity of exploitation to the simplest form we can make it, and we'll then build up complexity from there.
Fundamental properties: (classical predator like a lion)
1. individuals of the prey get killed and eaten; This source of mortality slows population growth of the prey (lower r); and the prey population size is smaller than when no predator is present. If the predator is not present, the prey population is increased by births (they might have their own prey which causes this population to grow) and the prey population is decreased by deaths not having to do with the predator in question.
2. individual predators eat prey and get energy and nutrients. The predator needs these prey to survive (they would starve without them) and to reproduce (energy above maintenance goes into reproduction). This causes population to grow, because individuals are living and giving birth. Population size is much bigger with the prey than without (depends on if it is the only prey; very few predators eat just one prey species....well get to that later).
Terms: This a trophic or trophic-level interaction. Trophic refers to food. When energy in the form of food passes from one population to another, we call this a trophic interaction.
The population that the energy comes from is called the lower trophic level (prey, host, plant). The population that the energy goes to is called the upper trophic level (predator, parasite, herbivore).
When we speak of these "levels", we can visualize the concept of a food chain, with species linked by their trophic interactions, in order of who eats whom. We can conceptualize energy "flowing" from one trophic level to another along this "chain".
Regulation: Can predators keep the prey population below their carrying capacity? The number of prey is lower when in the presence of predator than in the absence of predator. In other words, the predator itself is having the largest impact on the prey population, as opposed to resources or some other factor (e.g., weather). It's not obvious in any given ecological setting whether the predator of interest is actually regulating its prey or if something else is. This is most obvious in cases of parasites and diseases, where prey doesn't necessarily have to die as in the case of a classical predator.
B. Examples of real predator prey interactions.
Here we emphasize the population level. It's easy to watch TV programs of single acts of predation. This is really a behavioral focus (as in optimal foraging) or an evolutionary focus (as in predator-prey coevolution, the Red Queen effect), whereby we look (in the present) at ways in which predators have evolved to capture prey and prey have evolved to escape predators.
Now, for examples we turn to long-term data sets, where we keep track of numbers of predators and numbers of prey in a certain place over many generations worth of time:
1. Laboratory example: (Fig. 11.18, p. 268) The early experiments of the Italian biologist Gause, who watched closed populations of Didinium (rotifer, a predator) eating Paramecium (protist, a prey) in a glass jar. Gause found that in the laboratory predator and prey could not coexist indefintely. Without his intervention, one or both species went extinct in the laboratory cultures. This was not very promising, because ecologists at the time wanted to understand the conditions that allow predator and prey to coexist--and we still do!
Outcome of the interaction: extinction!
In addition to easily documented laboratory experiments, there are numerous records of predators or prey of economic interest:
2. Removal of predators and response of prey: Kaibab plateau deer vs. cougars wolves & coyotes: example of steady-state regulation of deer population which we infer from what happens when the predator is removed. The deer population was regulated by predators; predators were removed; the deer population increased until it was regulated by resources; the deer population apparently exceeded its long-term carrying capacity; humans intervened as the new predator and keep deer well below their carrying capacity.[The Kaibab Plateau is on the north rim of the Grand Canyon, in Arizona]
Outcome of the interaction (before predator "control"): coexistence of predator and prey: regulation of the prey by the predator: roughly steady-state population sizes.
3. Hudson Bay fur company, or other hunting and forestry records show long-term cyclic behavior of populations: A population cycle is defined as having a constant period (time between any two points in the cycle, for example, between two peak population sizes) and amplitude (the difference between the highest or peak and lowest population sizes). In real life cycles, the amplitude varies a lot, but the periods are regular
A. Famous Canada lynx, snowshoe hare: 9 to 10-year period.
B. Muskrat, fox, etc. 9 to 10-year period, but the amplitude is especially variable
C. Red Grouse shot on Scottish moors -- a nematode-grouse interaction with a 9 to 10-year period
D. Pine moths and Larch moths, 9 to 10-year period
E. Lemmings --grass -lemming interaction, 3 to 4 year period (Shelford, famous ecologist studied these)
F. Shrews-insects?; mice-mouse food or mouse predator? 3 to 4 year period
Remarkably, many cycles are on either a 3 to 4 year period or a 9 to 10 year period, even considering vastly different organisms. What could be involved to create such a universal pattern?? Also, most of these examples are of populations at high latitudes. What kind of environment occurs at high latitudes and what effects could this environment have on populations?
Outcome of the interaction: coexistence of predator and prey; regulation of prey by predator: population sizes exhibit true "cycles" with a regular period and roughly regular amplitude.
4. Humans-plagues & parasites. Human populations were apparently regulated by parasites for long stretches of human history. Historical records show irruptive population behavior, not on a regular cycle: plague comes and population size goes down; after the plague, population growth resumes its former exponential rate.
Outcome of the interaction: coexistence of "predator" and "prey" and regulation of prey by predator over time, although local interactions might involve extinction of prey or predator or escape by prey of regulation by predator; population sizes of prey and predator tend to be locally "irruptive"
5. New predators introduced into new prey populations: humans overexploiting fisheries, of various kinds of real fish and of whales: We see downward trend in the prey population (?new level of regulation?), and potentially extinction. Examples of extinction from human predation (as opposed to competition or habitat destruction) include: passenger pigeon, Eskimo curlew, dodo, great auk, some of the Galapagos tortoises, giant ground sloth and mammoths. Humans have caused the near extincition of many fisheries.
Outcome of the interaction: often extinction (lack of coexistence); predator drives prey extinct by over-exploitation.
Summary:
Review examples of population behavior in chapters 9 and 11, and transparencies shown in lecture.
Patterns of real populations resulting from predator-prey interactions that we will want to explain:
1. Extinction
simple laboratory experiments
introductions of new predators
2. Coexistence
constant population sizes of prey and predator
variable population sizes
irruptive
true cycles
regulation
What are the categories of explanation?
Hypotheses for what causes particular population behaviors can be placed in two broad categories:
1. Extrinsic factors: from outside the population; environmental factors that affect the population; immigration and emigration of individuals from outside this local population
2. Intrinsic factors: generated from the very characteristics of the population itself parameters of birth rates, death rates, and other characteristics of the two populations and importantly the interaction between the two populations.
For example, what makes some populations show periodicity (regular) fluctuations? Is the periodicity extrinsic, that is caused by an outside fluctuation like sun spots or El Niño? Or is the periodicity intrinsic, that is caused by the nature of the predator-prey interaction itself? We really need to analyze some models to generate some hypotheses and possible mechanisms for periodicity of population sizes.
II. A simple model
A. Form of the model. Once again, to learn how the two populations will behave when they interact, you need to use a simple model. Only a model will allow us to tally all the effects of prey births, prey deaths and predator births as a result of this interaction.
See your handout (page 25):
1. Verbally, we want to keep track of what is in each box: contributions to births and deaths, of prey and predator respectively. Then we put symbols for each of these ideas into a simple model to see what this model can tell us about predator-prey population interactions. Important points:
1. We want to keep track of the number of prey and predators that result from their interaction with each other;
Prey:
2. The prey can grow as any population would in the absence of the predator. We don't care in this viewpoint about whether the prey's food is a population or not. We're just focusing on one interaction, taking two populations in a vacuum instead of one as we did earlier..
3. As a result of the interaction with the predator, prey die when they are captured and eaten. We could express this as part of little r and the death rate, but we're going to pull it out here for the sake of understanding it and look at how this death rate depends on the number of predators:
The capture rate is of special interest to us, because a lot of the predator's behavior goes in here. How the predator captures prey will determine how many it captures under certain conditions. We call this the: functional response, i.e. all the predator's behaviors that determine how many prey it will capture per unit time. (units: number of prey captured/predator/time).
Also, the prey are captured in proportion to how many predators there are, so we need to plug in the number of predators.
Predator:
4. The predator population grows in proportion to how many prey are captured per unit time. The number of new predators with time is called the numerical response. The numerical response is said to be a function of the functional response. The number of prey per unit time eaten will be converted into offspring. The conversion rate efficiency will be less that 100% because the prey will be used for the mother's maintenance and the offspring's growth. In other words, typically many prey will be needed to produce one predator offspring.
5. The numerical response is a more explicit way of writing b in r = b-d. You multiply times the number of predators there times the percapita birth rate to get the number of new predators.
6. The predators die in the absence of the prey. This can be viewed in one of two ways. This can be viewed as the death rate, similar to the d in r = b-d. Or, it is sometimes viewed as rate at which predators starve to death if there are no prey. Let's just consider it to be the simple death rate, which will usually be independent of the prey.
2. Mathematically We will simply give this ideas symbols and use algebra & calculus to understand the population behavior. See handout p. 25)
Sidenote about symbols: Your text Molles gives this model different symbols (p. 265). I'm using another set of common symbols that are either traditional (e.g., N and r for prey or a for functional response) or pneumonic, which means that they are the first letter of what we call them (e.g., P for predator, c for conversion rate, d for death rate). I find that using bunches of subscripts (e.g., Nh and Np for number of prey or host and predator or parasite) makes it harder for me to remember and keep track of the symbols--you are welcome to learn the book's symbols if you prefer them.
Additional points about the model:
1. The prey will grow exponentially in the absence of the predator. We are only going to do this at first to see what happens. We want to isolate the effect of the predator, to see if the predator can regulate the prey population. So we don't want to put in density-dependent regulation of the prey. Thus this r is the familiar intrinsic growth rate, rmax of the prey.
2. We will use a to mean the capture rate. It's constant, which may be unrealistic; it a minute we'll consider more realistic but more complicated forms of the capture rate. Formally, aN is the functional response. Here prey and predators ***p into each other randomly (NP), just like the optimal forager that we already considered.
3. Prey will therefore die at a rate aNP, which gives you the probability of a prey individual encountering a predator (NP) and the probability that once encountered, the prey will be killed by the predator (a). In a sense, this meets the criterion of regulation, because the total number of prey killed depends on the predator density.
3. The symbol c will be our conversion rate. So caN is the numerical response. This is the per capita rate at which predator offspring are produced by each predator, or the predators birth rate b.
4. The symbol d will be the predator death rate. So the predator's intrinsic growth rate, rpred = caN-d. Thus the prey controls the predators population; we can ask, when does the predator control the prey?
B. Functional response.
Before we look into the behavior of this model, let's consider the functional response. The functional response is one of the most important features of predation, because it is the predator's behavior and our chance to consider the biology.
3 general kinds of functional responses Molles pp. 148-50; Fig. 6.21; Lecture handout p. 26
We could write equations for any of these, but the only one I want you to remember is the simplest case, aN or Type I functional response. [IMPORTANT NOTE: don't confuse these with survivorship curves!!!!!] Functional response curves are plotted in terms of number of prey captured per predator per day versus prey population density.
Type I: is linear. In its purest form, the predator never satiates. The more prey there are, the more it eats. This does not seem to apply to most predators, although it may apply to many some of the time (see your book p. 149, Fig. 6.21 top).
Type II: The number of prey levels off, because the predator satiates (is full and needs to digest), or it takes the prey a lot of handling time with each prey. At some point handling time or digestion time is the limiting factor. Even if there are more prey than that, the predator is eating them at the fastest rate it possibly can, so at that point the prey caught per predator per unit time levels off at a maximum rate. However, the important point is that approaching the asymptote, the slope of the function is maximum at lowest prey density and only decreases (gradually) from there.
Type III. The number of prey taken levels off for exactly the same reason in the type II curve. However, something different occurs at low prey density. At low prey density, the predator may ignore the prey or can't find it. Then at some higher prey density the predator catches it at a higher rate. This is exactly the prey switching we already covered. Prey switching may be caused by a search image. Or, it may be that there are a fixed number of hiding spots, and when prey get more common, they can't all hide, so the predator finds and eats them at a faster rate. In other words, the slope of the functional response is not maximum at lowest prey densities but rather at intermediate prey densities. While this may seem like a fine point to you, the resulting population behaviors are vastly different for Type II and Type III functional responses.
C. Behavior of the model: (lecture handout p. 27)
1. The Volterra model uses a type I functional response (lecture handout p. 26)--the simplest, with the least biology. As before, we start out with the simplest possible model to see what happens. We will add more complexity only if it's necessary to explain what real predator-prey populations do.
2. You want to know what happens at equilibrium. At equilibrium, nothing is changing, so the growth rate of both populations is zero, averaged over time. dN/dt = 0 and dP/dt = 0. Why do you want to solve for this equilibrium? Answer: because you want to know how the interaction comes out. You want to know what happens in the long term, whether the predator and coexist and whether the predator can regulate the prey population. So you see how your model behaves when numbers of predators and prey are at a steady state.
a. The equilibrium number of predators, P*. When dN/dt = 0, then rN - aNP = 0. You solve for the number of predators, P. Because we're defining this as the "equilibrium" number, we use the notation P*. We use the prey population equation to solve for the number of predators because the amount of food (prey population size) determines the number of predators around.
dN/dt = rN - aNP = 0
rN = aNP
r = aP
P* = r/a
b. The equilibrium number of prey, N*. When dP/dt = 0, then caNP - dP = 0. You solve the predator equation for the equilibrium number of prey because we are looking for the conditions under which the predator regulates prey population. In other words, the predator population determines the number of prey around.
dP/dt = 0, caNP - dP = 0
caNP = dP
caN = d
N* = d/ca
Summary: We are asking: can predator and prey coexist AND can the predator regulate the prey population?. So, we solve the model of 2 coupled equations for their joint solution, with both populations at steady state (the average number of predators and prey are neither increasing nor decreasing), with members of both populations around (coexistence; neither population goes extinct), with numbers of prey being determined by the numbers of predators and vice versa. Once we do that, we need to see how the model behaves, to see if it captures the qualities of real predator and prey interactions (that we summarized above).
3. To understand the behavior of the model visually, we can plot out the numbers of prey against the numbers of predators. This is called a phase plane, which is when the state variables (N and P) are plotted against one another. Time is not on the plot, but you can see time when you draw arrows from one point on the plot to another (Fig. 11.16, p. 266; lecture handout p. 27 W99).
On a phase plane, you plot the zero-population growth isoclines. Zero population-growth (ZPG) isoclines on a phase plane are where the growth rate of each population is zero. There is a prey ZPG (dN/dt = 0) isocline and a predator ZPG (dP/dt = 0) isocline. On one side of this line the population size increases (dN/dt > O or dP/dt > 0) and on the other it decreases (dN/dt < 0 or dP/dt < 0).
An "isocline" a line where everything is equal (iso= equal, cline = line). You look at isoclines every night on the news during the weather report. Isoclines on a weather map plot everywhere the temperature is equal to some one value, like 70o, or everywhere the barometric pressure is some high value or some low value. On a topographic map you are looking at elevation isoclines. So don't let the word "isocline" confuse you.
Look at the graph to see what regions the prey is increasing and what regions the prey is decreasing, and similarly for the predator. To make this part less confusing, refer to your lecture handouts (pp. 29-30 W99) on understanding phase plane trajectories. Where the two ZPG isoclines cross, you have an equilibrium point (if there is one). In other words, at this point, both populations are jointly at steady state, neither increasing or decreasing, that is dN/dt = 0 and dP/dt = 0.
In this Volterra model things are more complicated, however. There is not only one possible equilbrium point, but there is the possibility of stable cycles (Fig. 11.16 a, p. 266):
1. There is an equilibrium point at the crossing of the two lines. If the system is here, population sizes of the prey and predator will be constant.
2. However if the population sizes are disturbed from this point, they are pushed into cycles. The prey population size increases and decreases with a regular period and amplitude. The predator population size also increases and decreases with a regular period and amplitude, but the peaks (highest population size) and troughs (lowest population size) of the predator lag a bit behind those of the prey (Fig. 11.16a).
See the picture of nested cycles and plot of population size in lecture on the phase planes (see also the lecture handout; book Fig. 11.16b) . In the Volterra model, the trajectories look like eggs or ellipses. You read it like this: When you have this number of prey, you get this number of predators, and when you get the number of predators changes, you end up with that number of prey it translates into population size plotted against time. In other words, these lines on the phase plane can be viewed as trajectories in time.
This model is so far too simple to be very realistic, but it does produce one important behavior exhibited by more realistic models:
Predator-prey interactions have a natural tendency to oscillate (cycle).
We haven't seen cycling behavior before in the simple models we've gone over in class. Populations may cycle for other reasons, but predator-prey oscillations are common in nature as we saw in the examples. These cycles arise from the very nature of the interaction: the predator population size depends on the size of the prey population and the prey population size likewise depends on the size of the predator population. This interdependency produces a pattern of cycles: The predator population pushes down the prey population; fewer prey makes the predator population go down; when the predator population is down, the prey can increase; when the prey increase, the predators can increase and so on.
The mechanism of the tendency to oscillate in the model is an implicit time lag. The predator is lagging behind the prey, thus the two populations in effect overcompensate or overreact to any change in the other.
Any kind of time lag, whether due to predation or other factor will produce cycles in population size.
Even with the simplest possible model, we now have an hypothesis for population cycles. (see III D. as well): We can hypothesize that time lags of both intrinsic (due to age structure, time to grow to maturity, time to react to the other population) and extrinsic (seasonal environment means that breeding takes place only one part of the year, and so on) origin. We can pursue this hypothesis in more complex models and with data collected from populations in lab and in nature.
More complex models than we will go over in this course predict that the period of the cycle should be approximately 4 x the time lag, T, i.e., 4T. This prediction is interestingly consistent with the observed periods of high latitude cycles, 3-4 years for small mammals suggests a 9-month to 1-year time lag and 9-10 year cycles of large mammals suggest a time lag a little over 2 years. These time lags are consistent with the life cycles of these species and the seasonal environment in which they live. So you don't need something exotic like sunspots to explain these cycles; you need only a seasonal environment and some realistic demographic parameters in your model.
Next we explore some more realistic features to add to the simple model that we just went over.
III. Realistic refinements of the simple model
Four factors that influence the inherent tendency of predator-prey interactions to oscillate are:
A. Prey carrying capacity (K)
B. Predator efficiency
C. Type II or Type III funcational responses
D. Prey vs. predator, relative values of their intrinsic rate of increase
A. Prey carrying capacity. We started out with no limit on the prey population because we wanted to see what the predator alone would do. Obviously, most prey species will have a carrying capacity set by their resources, so what effect does this have on the tendency to oscillate?
Prey carrying capacity counteracts the tendency to oscillate (lecture handout: p. 28).
The reason is very simple and very intuitive: A prey carrying capacity puts a ceiling (i.e., a limit from outside) on the prey populations oscillations, and so this also puts a limit on the predator populations oscillations and the ability of the predator to drive prey oscillations.
Lecture handout p. 28:
Mathematically, this appears as a sloping prey isocline, with an intercept at N=K.
P* = r(1-N/K)/a this makes the line slope down
You can see how the oscillations ***p into a ceiling and this causes them to cycle inward. So, after a disturbance, we see decreasing oscillations to return to a constant population size. We call this damping oscillations. By damping, we mean that they get smaller through time.
Paradox of enrichment
This effect leads to something called the paradox of enrichment. The prey zero-population-growth isocline gets a steeper slope and intercepts the N-axis (K) at lower N, then the prey and predator equilibrium population sizes are closer to the prey's carrying capacity. Near the prey's carrying capacity, the oscillations are very small to begin with and population sizes settle very quickly to constant numbers of prey and predator.
Say you somehow enriched the preys habitat. Then you would move the prey's carrying capacity much higher than the current population sizes. This would allow room for the oscillations to occur. The population sizes could swing widely again, just as if there were no prey K.
The paradox of enrichment was first discovered in aquatic systems, in predator-prey interactions between algae and daphnia and other species which eat the algae. When pollution enriches the aquatic habitats, the algae and daphnia populations would go wild. Some extinctions even occurred, during the extreme amplitudes of the fluctuations.
See the biggest egg on the Volterra model's phase plane.
This pattern is called a paradox. Importantly, the use of this term (which is judgmental!) reflects human values. We think that fertilizer and more food is good and extinction is bad. But this is just the way we as humans would see it. Otherwise it is a simple and straightforward relationship: high input of resources for the prey perturbs the interaction and removes the stabilizing effect of a low ceiling (carrying capacity) on prey numbers.
B. Predator efficiency.
The prey K is imposed from the outside on the interaction. But exactly the same effect as the paradox of enrichment can come from within the predator-prey interaction itself.
The predator may be more or less efficient, as values of a, the simple Type I funcational response, varies . If the predator is very good at finding and catching prey (high a), then the predator population can depress the prey population well below its carrying capacity. See your handout Playing with predator-prey ZPG isoclines (pp. 29-30).
Once again, the further the prey is held below its K, the more room for oscillations. More importantly, we can see the meaning and effects of regulation of the prey population by the predator population: The difference between N* and K is the degree of regulation.
The higher the value of a, the smaller the value of N*=d/ac, the more N* is below K, and the greater the degree of regulation.
C. OPTIONAL An even more realistic model is to incorporate a type II functional response and K, and vary the parameters of the response to see what happens. I won't go into the mathematical details because this is the topic of a more advanced course, but you get stable cycles with a constant period and amplitude. This type of cycle is stable because it is resilient to outside disturbances. No matter what happens, the predator and prey populations return to oscillating with a constant period and amplitude.
D. Prey and predator intrinsic growth rate.
This effect has a different mechanism. If prey and predator intrinsic growth rates are vastly different, the system may get out of synchony, causing cycles.
As the prey intrinsic growth rate gets faster (all else equal), the prey can shoot ahead of the predator. In other words, if the prey intrinsic growth rate is sufficiently faster than the predator populations, then the predator is slower in catching up to changes in the prey population size.
And vice versa also happens. Predator can overrun prey, knock it way down, then it takes the prey a long time to recover.
Either effect exaggerates the time lag. The bigger the time lag, the bigger amplitude and longer period of the cycles.
SUMMARY of predator-prey interactions.
To revisit the real examples, we find that they are still difficult to explain, because they may incorporate any or all these effects. For only a few real examples, do we understand the exact and full cause(s) of the cycles. However, there is no doubt that real populations cycle. Your book, Molles, gives some nice explanations of cycling populations on pages 263-7.
Interestingly, most of the cycles that are really conspicuous are Arctic, boreal and north temperate. In other words, they occur in highly seasonal and extreme physical environments.
Cycles are less conspicuous in the tropics, although they do occur there too. One interesting cycle is the cycle of the disease yellow fever. It is caused by a virus which is vectored by a mosquito. The host is primates, monkeys and of course humans. This cycle occurs on a classical 10-year periodicity. In the tropics, remember that the physical environment is constant and much less extreme than the Arctic, but as we will see, the biotic environment is harsh. Still, no one knows the exact cause of the yellow fever cycles.
Most cycles are either 3-4 or 9-10 or 11 years. These correspond in simple naive models to 1-year and 2-year time lags, respectively. Thus a roughly 1-year time lag causes a 4 year cycle, and a 2-year time lag causes a 10 year cycle in the simplest of models. Trophic-level interactions will often have 1 and 2 year time lags as a natural phenomenon. This matches the Earths seasonality and that most organisms take 1 or 2 years to reach sexual maturity.
Now the question of Arctic carnivores: Are these the driving predators in a predator-prey interaction , that is do they regulate the prey by pushing it well below its carrying capacity? OR are they a third trophic level following a predator-prey cycle of herbivores and plants? This is still a controversial area.
Many of the herbivores have high rmax: lemmings, hares, forest insects. Their intrinsic growth rates are much higher than the plants renewal rates. This difference builds in a time lag, for plants to recover, and herbivores to overshoot. Snowshoe hares cycle on island without lynx.
The answer to cycling populations in nature is that these cycles are caused by a combination of instrinsic and extrinsic factors in which time lags occur and are felt within the population's natural dyamics. Importantly, extrinsic factors alone cannot explain the observed cycles. There are no regular 3-4 year or 9-10 year cycles in any environmental factor that is known. For example, the lynx-hare cycles were thought to be caused by sunspot irruptions, which are periodic, but the sunspot periods are longer than 10 years and so soon get out of phase with the lynx-hare cycles. In fact, the various populations themselves are often out of phase, suggesting that an extrinsic factor cannot be the cause. Rather, the cycling is a result of the instrinsic features of population growth that are affected by the seasonal periodicity of most places on Earth. However, populations can still cycle under completely constant laboratory conditions, suggesting that time lags of any origin are responsible for population cycles.
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