Lecture 10. Predation and parasitism

Predation and parasitism are examples of antagonistic ecological interactions in which one species takes advantage of another species. Predators (see a picture) use their prey as a source of food only, whereas parasites (see a picture) use their hosts both as a food and as a habitat. Predation and parasitism are stage-specific interactions rather than species-specific. Many species are predators or parasites only on specific stages in their life cycle.

Importance of the study of predation and parasitism:

• In many species predation and parasitism are dominating among ecological processes. Dynamics of these populations cannot be predicted and understood without considering natural enemies.

• Pest species of insects and weeds can be suppressed by introduction of natural enemies or by inundative release of natural enemies (biological control).

• Natural enemies may cause side effects in pesticide applications. The numbers of arthropod natural enemies may be reduced due to pesticide treatment which may result in increasing of pest populations.

10.2 Lotka-Volterra Model

Lotka-Volterra model is the simplest model of predator-prey interactions. The model was developed independently by Lotka (1925) and Volterra (1926):

It has two variables (P, H) and several parameters: H = density of prey P = density of predators r = intrinsic rate of prey population increase a = predation rate coefficient b = reproduction rate of predators per 1 prey eaten m = predator mortality rate

Measuring parameters of the Lotka-Volterra model

The following set of experiments should be done:

1. Keep prey population without predators and estimate their intrinsic rate of increase (r).

2. Put one predator in cages with different densities of prey and estimate prey mortality rate and corresponding k-value in each cage. As we know, k-value equals to the instantaneous mortality rate multiplied by time. Thus, the predation rate (a) equals to the k-value divided by the duration of experiment.

Example: lady-beetle killed 60 aphids out of 100 in 2 days. Then, the k-value = -ln(1-60/100) = 0.92, and a = 0.92/2 = 0.46. Note: if a -values estimated at different prey densities are not close enough to each other, then the Lotka-Volterra model will not work! However, the model can be modified to incorporate the relation of a to prey density. 3. Estimation of parameters b and m:

Keep constant density of prey (e.g., H = 0, 5, 10, 20, 100 prey/cage), and estimate the intrinsic rate of predator population increase (rP) at these densities of prey. Plot the intrinsic rate of predator population increase versus prey density: The linear regression of this line is:

Note: If points do not fit to a straight line (e.g., the intrinsic rate of predator population growth may level off), then the Lotka-Volterra model is not adequate and should be modified. Now , parameters b and m can be taken from this regression equation.

There are two major approaches: analytical and numerical. Analytical methods are complicated and require good mathematical skills. Also, many differential equations have no analytical solution at all. Numerical methods are easy and more universal (however, there are problems with convergence).

The simplest and least accurate is the Euler's method. Consider a stationary differential equation:

First we need initial conditions. We will assume that at time to the function value is x(to).

Now we can estimate x-values at later (or earlier) time using equation:

On this graph we estimate the slope f(x) of the function at point t = to and Actual point at t+Al extrapolate this slope through the entire time interval.

The main source of error in the Euler's method is estimation of derivative at the start of time interval. The direction of actual solution may change drastically during this time interval and numerically predicted point could be far from the actual solution (see the figure).

Euler's method can be improved, if the derivative (slope) is estimated at the center of time interval fit. However, the derivative at the center depends on the function value at the center which is unknown. Thus, first we need to estimate the function value at the middle point using simple Euler's method, and then we can estimate the derivative at the middle point.

k to.satJCxlio))

k is the function value in the center of time interval 1 At. Finally, we can estimate function value at the end of the time interval:

¿S | |

m - ■ | |

yik |

Slope at Hie center of frne rtEJval

Actual and preibcted pants at t+At ahnst corckfe

This is called Runge-Kutta method of second order. The most popular is the Runge-Kutta method of the fourth order. However, for our purposes it is enough to use the second order method.

This method is applied to Lotka-Volterra equations in the following Excel spreadsheet:

First, we estimate prey and predator densities (H' and P', respectively) at the center of time interval:

The second step is to estimate prey and predator densities (H" and P" at the end of time step lit:

f H" = H + At(rH' - aH'P1) \ p" = p + At (bH'P' - rriP')

These two graphs were plotted using the same model parameters. The only difference is in initial density of prey. This model has no asymptotic stability, it does not converge to an attractor (does not "forget" initial conditions).

This figure shows relative changes in prey predator density for both initial conditions. Trajectories are closed lines.

The model of Lotka and Volterra is not very realistic. It does not consider any competition among prey or predators. As a result, prey population may grow infinitely without any resource limits. Predators have no saturation: their consumption rate is unlimited. The rate of prey consumption is proportional to prey density. Thus, it is not surprising that model behavior is unnatural showing no asymptotic stability. However numerous modifications of this model exist which make it more realistic.

Additional information on the Lotka-Volterra model can be found at other WWW sites:

• Interactive Simulation Server (Lund)

• Model Description and Realization on "STELLA"

• Model Realization on "Models"

References:

Lotka, A. J. 1925. Elements of physical biology. Baltimore: Williams & Wilkins Co. Volterra, V. 1926. Variazioni e fluttuazioni del numero d'individui in specie animali conviventi. Mem. R. Accad. Naz. dei Lincei. Ser. VI, vol. 2.

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.

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 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::

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

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:

Hence:

Now we can balance the time budget:

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

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

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:

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 feedback. 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.

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.

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 |

16G |
3 |
35 |
11.6 |
G.G86 |
G.GG31 |

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

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.:

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.

10.4. Predator-Prey Model with Functional and Numerical Responses

Now we are ready to build a full model of predator-prey system that includes both the functional and numerical responses.

We will start with the prey population. Predation rate is simulated using the Holling's "disc equation" of functional response:

The rate of prey consumption by all predators per unit time equals to

The equation of prey population dynamics is:

Here we assumed that without predators, prey population density increases according to logistic model.

Predator dynamics is represented by a logistic model with carrying capacity proportional to the number of prey:

This equation represents the numerical response of predator population to prey density. The model was built using an Excel spreadsheet:

''Excel spreadsheet "predfunc.xls"

You can modify parameters of this model to simulate various patterns of population dynamics. Differential equations are solved numerically, and it may happen that the algorithm (2nd order Runge-Kutta method) will not work for some combination of parameter values. Thus, change parameters with caution. If you suspect that the algorithm does not work properly, reduce the time step (cell A5) until results become independent from the time step.

Simulation results are presented below. This model exhibits more various dynamic regimes than the Lotka-Volterra model.

No oscillations

Damping oscillations converging to a stable equilibrium

Limit cycle

This model can be used to simulate biological control. The goal of biological control is to suppress the density of the pest population using natural enemies. We will assume that the prey in our model is a dangerous pest, and that the predator was introduced to suppress its density. Withour predators the density of prey population is equal to the carrying capacity, K = 500. After a predator with a search rate, a = 0.001, was introduced, the equilibrium population density, N*, declined to the value of 351. Beddington et al. (1978, Nature, 273: 573-579) suggested to measure the degree of pest suppression by the ratio:

For example, if a = 0.001, then q = N*/K = 351/500 = 0.7. Biological control is successful if the value of q is low (at least, <0.5). If we increase the search rate of the predator (i.e., we introduced a more effective predartor species) to a = 0.1, then the pest population is suppressed to the density of 19 (q = 0.038).

It could be expected that more effective predators will cause more suppression of the prey population density. But this is not true, because more effective natural enemies also cause larger oscillations in population density. For example, at a = 0.3, the equailibrium is not stable and populations exhibit periodic cycles (see graph #3 above). Periodically host density reaches the value of 190.

The transition between the stable equilibrium and the limit cycle occurs approximately at a = 0.244. The transition from one type of dynamics to another one is often called "phase transition" (e.g., transitions between liquid and gas phases or between solid and liquid phases). The phase transition in our model will be less ubrupt if we introduce noise. With noise, the system will exhibit oscillations even if the equilibrium is stable. The closer we are to the critical value, a = 0.244, the larger will be these oscillations. These oscillations result from the interaction of predators with prey.

This example illustrates that pest regulation (or control) by natural enemies is an ambibuous notion. First, it does not refer to the type of dynamics (stable equilibrium vs. limit cycle), and second, the excess of "regulation" may cause large oscillations of prey density.

Parasitoids are insect species which larvae develop as parasites on other insect species. Parasitoid larvae usually kill its host (some times the host is paralyzed by ovipositing parasitoid female) whereas adult parasitoids are free-living insects (see images of parasitoids). Most of parasitoid species are either wasps or flies.

Parasitoids and their hosts often have synchronized life-cycles, e.g., both have one generation per year (monovoltinous). Thus, host-parasite models usually use discrete time steps that correspond to generations (years).

Model of Thompson (1922)

The model assumes that female parasitoids lay their eggs randomly on host individuals and do not distinguish between healthy and already parasitized hosts. In this case, the number of parasitoid eggs laid on one host should have a poisson distribution:

where p(i) is the proportion of hosts that get i parasitoid eggs, and M is the mean number of parasitoid eggs per one host.

Survived hosts are those which get 0 parasitoid eggs. The proportion of survived hosts is equal to p(0) = exp(-M).

Variables:

• P = Density of parasitoid females

H = Density of hosts

Parameter

• F = Parasitoid fecundity (no. of eggs laid by 1 female)

PF = Density of eggs laid by all parasitoid females per unit area —

= Average no. of eggs per host individual Then, host survival is

The full model is:

L PurH.fl-sxpt-^Jlq

The first equation describes host survival and reproduction. The numbers of survived hosts are multiplied by Ro which means reproduction.

In the second equation, each parasitized host produce one adult parasitoid in the next generation. P is the density of females only. Thus, the numbers of parasitoids is multiplied by the proportion of females = q.

In the model of Thompson, it is assumed that parasites always lay all their eggs. Thus, realized fecundity equals potential fecundity. This assumption implies unlimited search abilities of parasitoids. In nature, parasites often do not realize their potential fecundity just because they can not find enough hosts. Thus, the model of Thompson may overestimate parasitism rates especially if host density is low.

Model of Nicholson and Bailey (1935)

This model is more realistic than the Thompson's model and is widely used by ecologists. It assumes that parasitoid female is able to examine area a ("area of discovery") during its life time. When a host is found, parasitoid lays only one egg in it. However, the same host can be found again later and then the parasite will lay another egg in it because we assume that parasites do not distinguish between healthy hosts and already parasitized hosts.

Because each encounter with the host results in depositing 1 egg, the realized fecundity equals the product of the area of discovery and host density: F = aH. Substituting this value of F into the Thompson model we get:

In the Nicholson and Bailey model, the potential fecundity of parasites is not limited. Parasites lay an egg at every encounter with the host even if the number of encounters is very large (e.g., if host density is high). Thus, this model may overestimate parasitism rates at high host density.

Model of Rogers (1972)

The model of Rogers applies the model of Holling, which was originally developed for predator-prey systems, to host- parasite systems. It assumes two kinds of limitations in host-parasitoid interactions: limited parasitoid fecundity (as in the model of Thompson) and limited search rate (as in the model of Nicholson and Bailey).

We will use the Holling's disc equation (see section 10.3) to model the functional response of parasitoids. The number of hosts attacked by one parasitoid female is equal to

We can modify this equation by setting T=1 because search rate is considered per life time of parasitoid female. Life time can be coded as 1 because the time step is equal to 1

generation. The ratio ^»is the maximum fecundity of parasitoid female.

Then:

When parasitoid female attacks a host it lays an egg. Thus, realized fecundity F = Ha. Substituting this value of F into the Thompson's model we get:

aP-F

In the model of Rogers, realized fecundity is different from the potential fecundity whereas in previous models this distinction was not present.

All models of host-parasitoid system are unstable: they generate oscillations with increasing amplitude.

This is the dynamics of the model of Nicholson and Bailey.

However, in nature host-parasitoid population never show oscillations with infinitely increasing amplitude. This is not because the models do not capture the mechanisms of host-parasitoid interactions, but because additional ecological processes (e.g., intraspecific competition in hosts or in parasitoids) can partially or completely stabilize the system. It was also shown that spatial heterogeneity and parasitoid dispersal among host patches may also stabilize the population system.

References:

Nicholson, A. J., and V. A. Bailey. 1935. The balance of animal populations. Proceedings of the Zool. Soc. of London. 1: 551-598.

Rogers, D. J. 1972. Random search and insect population models. J. Animal Ecol. 41: 369-383.

Thompson, W. R. 1929. On the relative value of parasites and predators in the biological control of insect pests. Bull. Entomol. Res. 19: 343-350.

10.6. Host-Pathogen Model (Anderson & May)

Host-pathogen models are similar to predator-prey and host-parasite models. Below is the model of Anderson and May (1980, 1981) which describes insect diseases. The host population consist of two portions: susceptibles which are healthy organisms, and infected individuals. The model describes changes in density of susceptibles (S), infected individuals (I) and pathogens (P):

Aqifducton. lAtlid ttjMMKMAiiMkict 41 Ih tuns nit (f] wsiiiîtï* t and do sot twrt ni iifafn tolhsi

/Infectan ni* i* raridad uñng di* aiasMchwi

JP di law. 7«tha nahnfly fwiti nía 14. ardua toinfadion (wrih ratcrgi

law. 7«tha nahnfly fwiti nía 14. ardua toinfadion (wrih ratcrgi

Omnickon afpaltageits fcr/ Imfs (¡tg.. onsamptain) Nibinldacay of pattiqgafti

Infectad wytriiiiswlprfHiqiiiirih lïttA

This model is capable to generate epidemic cycles. It was used to study evolutionary strategies of pathogens.

Models of epidemics in mammalian hosts (including humans) consider immune organisms as a separate category.

Host-pathogen systems may include vectors. For example, malaria is transmitted by mosquitoes. In these systems, hosts become infected only when they have contact with the vector. Thus, the number of pathogens is not that important as the numbers of vectors carrying the patogens. The host-vector-pathogen system can be described as the change in numbers of 4 kinds of individuals: healthy hosts, infected hosts, uninfected vectors, and infected vectors. An example of such a model is given in the following Excel spreadsheet:

•'Excel spreadsheet "vector.xls" References:

Andreson, R. M. and R. M. May. 1980. Infection diseases and population cycles of forest insects. Science 210: 658-661.

Andreson, R. M. and R. M. May. 1981. The population dynamics of microparasites and their vertebrate hosts. Phil. Trans. of the Royal Soc. of London 210: 658-661.

Questions and Assignments to Lecture 10

10. 1. Estimate parameters of type II functional response from 2 experiments in which individual predators were kept in cages of the same size size (1 sq.m.) with different prey density for a period of 2 days.

Number of prey |
Number of |
Total number of prey |
Total number of prey |

per cage |
replications |
in all cages |
killed by predators |

5 |
20 |
100 |
35 |

50 |
5 |
250 |
60 |

10.2. Assume that population dynamics of spruce budworm depends mostly on generalist predators (birds) which density (P) can be considered constant in time and which have a type III functional response. The model of spruce budworm dynamics is:

dl 1 +aM-Th where search rate (a) depends on prey density: a = bN/(N+c)

Parameter values are: r=1.5 per yr; Th = 0.0003yr; P=0.01 birds/sq.m.; b=1500 sq.m./yr; c=30 larvae/sq.m.

Find all equilibrium densities of spruce budworm population. Which equilibria are stable and which are unstable? At what population density birds fail to control spruce budworm population?

10.3. Tachinid fly Parasetigena silvestris is a parasitoid of gypsy moth. Both the host and the parasite have one generation per year. P.silvestris attacks large larvae and emerges from pre-pupae or pupae. The following results were obtained after three years of study:

Year |
Density of host large larvae per sq.m. |
Percentage of parasitism |

1988 |
0.45 |
12 |

1989 |
2.15 |
8 |

1990 |
4.10 |
? |

In 1990, parasitism was not recorded. Estimate expected percentage of parasitism in 1990 according to: 1) the model of Thompson, and 2) the model of Nicholson & Bailey. Estimate parameters of both models. Additional information: P.silvestris has a sex ration of 1:1 and pre-adult mortality of 30%.

10.4. Why the distribution of parasitoid eggs laid on hosts may be different from the Poisson distribution?

10.5. In what host-parasite models (Thompson, Nicholson & Bailey, or Rogers):

• Fecundity is limited

• Search rate is limited

• Host mortality does not depend on host density

• Host mortality does not depend on parasite density?

10.6. In what type of functional response (I, II, or III):

• Predator saturation is considered

• Predator search rate increases with prey density

• The proportion of killed prey is constant

• The proportion of killed prey first increases and then declines with increasing prey density

• The proportion of killed prey always declines with prey density?

10.7. Host density is 5 individuals per square meter; parasitoid density is 1 female per square meter; one parasitoid female can parasitize maximum 100 hosts, and its search rate (area of discovery) is 1 sq.m. per life. What is the proportion of parasitism predicted by each of three models: Thompson, Nicholson & Bailey, and Rogers?

Lecture 11. Competition and Cooperation

Intraspecific competition results in a reduction of population growth rate as population density increases. We already studied several models that consider intraspecific competition: logistic model and Ricker's model. In these models population growth rate steadily declines with increasing population density.

However, in nature competition effect may be completely absent until population density reaches some threshold at which resources become limited.

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