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The Dynamics of Arthropod Predator-Prey Systems

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Summary

This is a simplified summary of the book, The Dynamics of Arthropod Predator-Prey Systems, by Michael P. Hassell. The following summary is by Justin Baacke and Brita Schneiders. Work shown below can be attributed to Hassell unless otherwise specified.

Hassell starts out by explaining a bit of the history of predator-prey models, starting with Lotka and Volterra. He explains his basic model, based largely on that of Nicholson and Bailey from 1935. Then the rest of the book delves into different extensions, depending on what assumptions you can make. Some examples are interactions with Hyperparasitoids (parasitoids that prey on other parasites), Biological Controls (using parasitoids to control other pests), Functional Responses (versus Numerical Responses), and lastly, Non-Random Search. All of these can be found in Hassell's original book. In this review, we will focus on Hassell's Chapter 4, the Non-Random Search extension.

Let's start with the general model though. Many models have been made to demonstrate predator-prey interactions. The first notable model is probably the one you think of when you hear “predator-prey models”: Lotka and Volterra. Many modifications have been made and definitions refined since they produced this model back in 1910. A more recent idea has been to use parasitoids in place of actual predators in research, since they make for much simpler models, off of which it is easy to springboard and make new extensions.

A parasitoid is unique from a predator in that it really is a parasite, but it is comparable since a parasitoid kills its host (just as a predator kills its prey). Parasitoids come from multiple different phylums. Hassell specifically looks at Arthropod systems (both parasitoid ("predator") and host ("prey") are in the Arthropoda phylum). Check out the Biological Context section of this wiki for more details.

Biological Context

In the mathematical world, parasitoids serve as a good simple model for predators in predator-prey models. This is because, like predators, they kill their host, and they are easy to research in contained spaces. Parasitoids of arthropods (arachnids, insects, crustaceans, etc), specifically, are especially great models because one parasitoid generally gives rise to one offspring for every host it kills, making the average number of predator progeny produced per prey attacked just one. This makes for helpful simplifications in the upcoming model.

Before we explain the model though, a few more background details are necessary. A parasitoid can be thought of as a cross between a parasite and a predator. A parasite is an organism that either harms its host or lives at the expense of the host. Generally, a parasite does not kill its host since it relies on the host for survival. A predator, on the other hand, survives by preying on other organisms, resulting in the death of its prey. A parasitoid is like a parasite in physiology, but like a predator in its ecological outcome of killing its host.

Most parasitoids lead protelean life cycles, meaning they are parasitic in their immature stages, but free living as adults. A parasitoid life cycle can vary quite a bit, but in general, it is as follows: The adult is free living; the female releases its eggs (either in the environment to be ingested, or directly into the next host); the eggs, once in its host, develop into the larval stage; develop into the juvenile stage; then the juvenile or young adult exits its host, killing the host in the process. Below is an image of a more specific example, a hymenopteran life cycle (ants, bees, wasps):


Figure 1: Hymenopteran Life Cycle


Figure 1: An adult female oviposits her egg inside of the host (usually an arthropod), the egg hatches and gives rise to a larva, using much of its host's nutrients, the larva develops into a pupa, usually taking up most of the host's body cavity. The pupa develops into a young adult. When the adult is ready, it emerges from its host, killing the host in the process. Note that this example has a 1:1 ratio of predator progeny produced per prey attacked.


Now given these background details about parasitoids, one can see why a parasitoid functions well as a simplistic model for a predator in a predator-prey model. As previously mentioned, they are easy to research, as they are small and easy to contain in a laboratory setting, and most importantly, in the case of arthropod hosts, they exhibit a one to one ratio of prey attacked to predator progeny produced. With this information established, we can now move on to the mathematics of this system in our Mathematical Model section of this wiki.

History

The original attempts to model predator prey scenarios came via the Lotka-Volterra Predator-Prey Model proposed in 1910. This is generally the first model that comes to mind when one mentions predator-prey models. However, other researchers found it necessary to refine predator behavior assumptions in order to account for various forms of interaction. This is how Nicholson and Bailey produced their model in 1935. The parameters used here are as follows:


                                      N_{e}  Number of parasitoid encounters with hosts
                                      N_{t}   Number of parasitoids at timestep t
                                      N_{a}  Number of hosts parasitized
                                      P_{t}   Number of parasitoids at timestep t
                                       a    Area of Discovery (Likelihood of encounter)

The basic model for parasitoid host interactions came from the Nicholson-Bailey (1935) approach. This model used a simple difference model (as the delay equation models rarely produced polite analytical results) combined with a number of assumptions with regards to the behavior of the given species. Nicholson's original theoretical basis for his work was insect parasitoids. However, his assumptions generalize easily to many predator populations. The main exception to being able to make this generalization is the assumption that the hosts attacked to predator progeny produced is a 1 to 1 ratio.

Nicholson made several assumptions regarding the nature of interaction between the parasitoid under consideration and the host it parasitized.

  • 1. The number of encounters of a parasitoid with a host is directly proportional to the host and parasitoid density:
                                      N_{e}={\text{aN}}_{t}P_{t}
  • 2. The probability of a given host not being attacked is given by the zero term of the Poisson distribution p_{0}=e^{{{\frac  {-N_{e}}{N_{t}}}}} [1]. This therefore produces the following expression for the number of hosts parasitized in a given timestep:
                                      N_{a}=N_{t}\left(1-e^{{{\frac  {-N_{e}}{N_{t}}}}}\right)          
(please note that the above is considered when parasitism can happen multiple times)
  • 3. Each host attacked will produce one parasitoid progeny in the next timestep (this 1:1 ratio is not universal. It must be taken into consideration for the species for whom it does not hold):
                                      {\frac  {N_{e}}{N_{t}}}={\text{aP}}_{t}

With these three assumptions in mind it becomes fairly easy to construct the model:

                                      N_{{t+1}}=\lambda N_{t}e^{{-aP_{t}}}
                                      P_{{t+1}}=N_{t}\left(1-e^{{-{\text{aP}}_{t}}}\right)

and a fairly simple analysis informs us that this has steady states:

                                      N^{*}={\frac  {\lambda Log(\lambda )}{(\lambda -1)a}}
                                      P^{*}={\frac  {Log(\lambda )}{a}}

The major drawback to the Nicholson model is that the steady states are not stable and that it yeilds diverging oscillations when perturbed. This is naturally somewhat disconserting as most real species do not have a tendency to diverge into extinction. However, it provides a nice theoretical springboard from which modifications can be made.

One solution to this problem is that of considering a nonhomogeneous distribution of prey and therefore a non random search path by the predators. This is an entirely reasonable assumption for two main reasons. First of all, numerous examples of such behavior are seen in nature. For example, think of ladybugs and aphids (the tiny green organisms eaten by ladybugs). A ladybug is much more apt to stay on a leaf with an aggregated population of aphids on it than a leaf with only a few. Secondly, laboratory experiments incorporating this model have produced more stable populations than those attempting to create a homogeneous spacial prey distribution.

Mathematical Model

First of all, it is important to understand qualitatively what is going on in an aggregative model. The assumption is made that there will be certain areas with a higher prey density than others and therefore it is more profitable for predators to create a non-random search pattern in which they are more likely to take advantage of this high density patch. An easily explored example of this is in the fruit fly: its feeding sites are more likely to be clumped together. Therefore, when an examination was done, they found that when the fly had not found food for a significant piriod of time, it would take large steps and very small turns resulting in a larger area covered but in little detail. However, shortly after food had been found step length became shorter and average angle turned increased thus causing the fly to explore the area in the immediate radius of the previously found food in far greater detail. See the Example section of this wiki for more explanation.

We will now go into a model which incorporates ideas of aggregation and therefore provides far greater stability in the model. This can be done by considering "n" different patches in which there may be any number of prey. It is important to note that these patches are simply an area in which prey may or may not congregate (such as a leaf plants or groups of plants), but are not necessarily a predefined geometric spacial region. This is because the definition of a patch must be linked with the modified foraging path of the parasitoid. This kind of model was first proposed by Hassell and May in 1973. Please note the additional variables used in this section:

                                      \alpha _{i}    Prey population in patch i
                                      \beta _{i}    Predator population in patch i 
                                      n     Number of patches
                                      \lambda      Prey rate of increase

We will now look at the Nicholson Bailey model with this patch modification, noting the summation over the entirety of the considered space. This allows for the prey to occupy any of the patches rather than being uniformly distributed.

                                      f\left(N_{t},P_{t}\right)=\sum _{{i=1}}^{n}\alpha _{i}e^{{-a\beta _{i}P_{t}}}

Note that in the case where the predators are evenly distributed and \beta _{i}={\frac  {\beta _{t}}{n}} this equation will simplify into the Nicholson Bailey model considered in the History section of our wiki. Hassell and May's analysis concluded that this model would allow for stability with significantly uneven spacial distributions. The exact condition for stability was that:

                                      \lambda \sum _{{i=1}}^{n}a_{i}\alpha \beta _{i}P_{i}e^{{-a\beta _{i}P^{*}}}<{\frac  {\lambda -1}{\lambda }}

The senario considered in Arthropod Predator-Prey Systems and thus analyzed by Hassell and May considered one patch with a high density fraction, \alpha , of the population and the remaining "n-1" patches containing (1-\alpha )/(1-n) as a density fraction of the prey. This allows for an introduction of an aggregation index \mu which is defined as \beta _{i}=c\alpha _{i}{}^{{\mu }} (note c is a normalizing constant).

Here we consider \mu as a measure of how distributed the predators are. If we consider \mu =0 then we have a random search pattern, and if we have \mu =\infty , we are considering a case in which all predators are on the patch with the highest density.

Hassell and May concluded that there were 4 elements contributing to the stability of a given model:

  1. \mu : Aggregation. It was found that higher values as far as predator aggregation contributed to greater stability in a given model. This can even be used to stabilize models which are otherwise highly unstable.
  2. \lambda : Stability. Stability is highly dependent on prey rate of increase. As in all cases, models become unstable when prey rate of increase rises. This was first and most notably demonstrated by Lotka and Volterra in their studies of why increasing the prey fish populations led to a crash in their long-term population.
  3. \alpha : Prey population. As the proportion of the population concentrated on the high density patch increases, higher values of \lambda allow for stability. However, values of \alpha >.5 do not permit stability at low values of \lambda . If \alpha is small and thus there is little difference between the high and lower density patches, lower values of \mu are possible for stability.(Although, it is important to note that if \alpha is "too" low, then the model collapses back into the basic Nicholson model and instability resumes).
  4. n-1: The number of low prey density patches. With a fixed value of \alpha , less predator aggregation is needed for stability because more low density patches are available. This phenomena, in which stability has a direct relation to the number of spatial sub-units, was investigated by Maynard Smith, Roff, and Hilborn in 1974, 1974, and 1975, respecitively.

Example

As mentioned earlier, a prime example of aggregation in predator prey systems is a “psuedo-predator”: the fruit fly. Below are two pictures taken from Hassell's original book. The first is of the path of a fruit fly. It was recorded every three seconds. The second picture graphs the change in angle and step size before and after it had found food. As one can see, the fruit fly tends to collect its movements around areas where food had been found, and therefore where food was more likely to be, given that its "prey" (the melon droplets) tended to collect in those certain areas. Note the difference in step size and angle between between the two sides of the second picture: before feeding and after feeding. The step size is much larger before feeding, and upon finding food, the fruit fly's step size drastically decreases.

Figure 2: Track of a housefly, Musca domestica, in an area with four clumps of sugar droplets. Solid circles show where feeding occurred. The track is marked at 3 second intervals.[2]


Figure 3: Mean angle turned (degrees) and step length (centimeters) per second by houseflies in the one minute periods before and after feeding. Data pooled from 29 experiments (from Murdie and Hassell, 1973).[2]

Analysis

In the basic Nicholson Bailey model we found that the steady states were unstable. In the modified version of the model, in which they created distinct patches, Hassell and May were able to generate stable solutions by allowing for aggregations of the prey and therefore non-random search paths by the predators. This type of behavior has been seen in numerous systems in nature and allows for more sustainable laboratory populations under experimental setups. This introduced three new parameters \alpha , nand \mu . as defined above. By varying these parameters it was possible for stable solutions to develop. It was found that generally, with higher levels of predator aggregation, a higher level of stability would develop. This combined result explains how the Nicholson Bailey model can produce accurate steady states withought producing the stability of realistic systems.

As far as the book The Dynamics of Arthropod Predator-Prey Systems is concerned, this idea of modification of the basic Nicholson Bailey model is fairly consistent with the rest of the book. Hassell makes a strong attempt to classify situations in which any given modification is necessary. While the non-homogeneous prey distribution is a useful model, he also goes into systems with Mutual Interference, enhancements to the predator rate of increase, Polyphagous Predators, Competing predators and Hyperparasitoids, along with a theoretical basis for biological control. As it stands the work provides a pleasant guide to simple modifications one might make to analyze a given system.

References

  • Hassell, Michael P. The Dynamics of Arthropod Predator-prey Systems. Princeton, NJ: Princeton UP, 1978. Print.
  • Nicholson, A. J., and Bailey, V. A. 1935. The balance of animal populations. Part I. Proc. Zool. Soc. Lond., 1935, 551-598.
  • Roberts, Larry S; Schmidt, Gerald D. Foundations of Parasitology. Edition 8. New York, New York: McGraw-Hill, 2009.

Footnotes

  1. The use of the Poisson distribution comes from the fact that a predator encountering prey is a discrete, random event. Therefore the zero term is the probability that the prey (or host) will remain undiscovered. Thus the probability of a predator-prey encounter is one minus the zero term.
  2. 2.0 2.1 Hassell, Michael P. The Dynamics of Arthropod Predator-prey Systems. Princeton, NJ: Princeton UP, 1978. Print.