10.3. Functional and Numerical Response

   Holling (1959) studied predation of small mammals on pine sawflies, and he found
   that predation rates increased with increasing prey population density. This
   resulted from 2 effects: (1) each predator increased its consumption rate when
   exposed to a higher prey density, and (2) predator density increased with
   increasing prey density. Holling considered these effects as 2 kinds of responses
   of predator population to prey density: (1) the functional response and (2) the
   numerical response.

Modeling Functional Response

   Holling (1959) suggested a model of functional response which remains most
   popular among ecologists. This model is often called "disc equation" because
   Holling used paper discs to simulate the area examined by predators.
   Mathematically, this model is equivalent to the model of [1]enzime kinetics
   developed in 1913 by Lenor Michaelis and Maude Menten.

   This model illustrates the principal of time budget in behavioral ecology. It
   assumes that a predator spends its time on 2 kinds of activities:
    1. Searching for prey
    2. Prey handling which includes: chasing, killing, eating and digesting.

   Consumption rate of a predator is limited in this model because even if prey are
   so abundant that no time is needed for search, a predator still needs to spend
   time on prey handling.

   Total time equals to the sum of time spent on searching and time spent on
   handling::

                                       [eq9.gif]

   Assume that a predator captured Ha prey during time T. Handling time should be
   proportional to the number of prey captured:

                                      [eq9a.gif]

   where Th is time spent on handling of 1 prey.

   Capturing prey is assumed to be a random process. A predator examines area a per
   time unit (only search time is considered here) and captures all prey that were
   found there. Parameter a is often called "area of discovery", however it can be
   called "search rate" as well.

   After spending time Tsearch for searching, a predator examines the area = a
   Tsearch, and captures aHTsearch prey where H is prey density per unit area:

                                      [eq10.gif]

   Hence:

                                      [eq11.gif]

   Now we can balance the time budget:

                                      [eq12.gif]

   The last step is to find the number of attacked prey Ha:

                                      [eq13.gif]

   The graph of functional response that corresponds to this equation is shown
   below:

                                     [gfunc1.gif]

   This function indicates the number of prey killed by 1 predator at various prey
   densities. This is a typical shape of functional response of many predator
   species. At low prey densities, predators spend most of their time on search,
   whereas at high prey densities, predators spend most of their time on prey
   handling.

   Holling (1959) considered 3 major types of functional response:

                                     [gfunc2.gif]

   Type I functional response is found in passive predators like spiders. The number
   of flies caught in the net is proportional to fly density. Prey mortality due to
   predation is constant (right graph on the previous page).

   Type II functional response is most typical and corresponds to the equation
   above. Search rate is constant. Plateau represents predator saturation. Prey
   mortality declines with prey density. Predators of this type cause maximum
   mortality at low prey density. For example, small mammals destroy most of gypsy
   moth pupae in sparse populations of gypsy moth. However in high-density
   defoliating populations, small mammals kill a negligible proportion of pupae.

   Type III functional response occurs in predators which increase their search
   activity with increasing prey density. For example, many predators respond to
   kairomones (chemicals emitted by prey) and increase their activity. Polyphagous
   vertebrate predators (e.g., birds) can switch to the most abundant prey species
   by learning to recognize it visually. Mortality first increases with prey
   increasing density, and then declines.

   If predator density is constant (e.g., birds, small mammals) then they can
   regulate prey density only if they have a type III functional response because
   this is the only type of functional response for which prey mortality can
   increase with increasing prey density. However, regulating effect of predators is
   limited to the interval of prey density where mortality increases. If prey
   density exceeds the upper limit of this interval, then mortality due to predation
   starts declining, and predation will cause a positive feed-back. As a result, the
   number of prey will get out of control. They will grow in numbers until some
   other factors (diseases of food shortage) will stop their reproduction. This
   phenomenon is known as "escape from natural enemies" discovered first by
   Takahashi.

Estimation of Parameters of Functional Response

   Experiments should be done as follows: predators are kept in large-size cages
   individually. Large-size cages are important because search abilities of
   predators should be limited. Different number of prey are released in these
   cages. Each prey density should be replicated to get sufficient accuracy. More
   experiments should be done with low prey density than with high prey density
   because the error of mortality estimates depends on the total number of prey.
   Experiments are usually set for a fixed time interval. At the end of experiments,
   survived prey are counted in each cage.

   This is an example of experimental data:
   No. of
   prey per
   cage
   H No. of
   replications Total
   prey
   killed Average no.
   of prey killed
   Ha           1/Ha       1/(HT)
                5          20     50 2.5  0.400 0.1000
                10         10     40 4.0  0.250 0.0500
                20         7      55 7.9  0.127 0.0250
                40         5      45 9    0.111 0.0125
                80         3      38 12.6 0.079 0.0062
                160        3      35 11.6 0.086 0.0031

   Cage area was 10 sq.m., and duration of experiment was T=2 days. Holling's
   equation can be transformed to a linear form:

                                      [eq14.gif]

   [gfunc4.gif] The linear regression has the following coefficients:

   y = 3.43 x + 0.0612

   Th = 0.0612 T = 0.1224 days = 2.9 hours

   a = 1/3.43 = 0.29 cages = 2.9 sq.m.

   Another possible method of parameter estimation is non-linear regression. It may
   give better results at high prey density than the linear regression method.

   Type III functional response can be simulated using the same Holling's equation
   with search rate (a) dependent on prey density, e.g.:

                                      [eq15.gif]

Numerical Response

   Numerical response means that predators become more abundant as prey density
   increases. However, the term "numerical response" is rather confusing because it
   may result from 2 different mechanisms:

    1. Increased rate of predator reproduction when prey are abundant (numerical
       response per se)
    2. Attraction of predators to prey aggregations ("aggregational response")

   Reproduction rate of predators naturally depends on their predation rate. The
   more prey consumed, the more energy the predator can allocate for reproduction.
   Mortality rate also reduces with increased prey consumption.

   The most simple model of predator's numerical response is based on the assumption
   that reproduction rate of predators is proportional to the number of prey
   consumed. This is like conversion of prey into new predators. For example, as 10
   prey are consumed, a new predator is born.

   Aggregation of predators to prey density is often called "aggregational
   response". This term is better than "numerical response" because it is not
   ambiguous. Aggregational response was shown to be very important for several
   predator-prey systems. Predators selected for biological control of insect pests
   should have a strong aggregational response. Otherwise they would not be able to
   suppress prey populations. Also, aggregational response increases the stabilility
   of the spatially-distributed predator-prey (or host-parasite) system.

References:

   Holling, C.S. 1959. The components of predation as revealed by a study of small
   mammal predation of the European pine sawfly. Canad. Entomol. 91: 293-320.
   Holling, C.S. 1959. Some characteristics of simple types of predation and
   parasitism. Canad. Entomol. 91: 385-398.

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   [5]Alexei Sharov 1/12/96

References

   1. http://dept.physics.upenn.edu/courses/gladney/mathphys/subsection4_1_6.html
   2. http://alexei.nfshost.com/PopEcol/lec10/lotka.html
   3. http://alexei.nfshost.com/PopEcol/lec10/predat.html
   4. http://alexei.nfshost.com/PopEcol/lec10/fullmod.html
   5. http://alexei.nfshost.com/~sharov/alexei.html