Predator-Prey Model

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 F1

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 F2

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 F3
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 F4
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.
10.5. Host-Parasitoid Models
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 F5
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 F6
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 F7
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 F8
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 F9 is the maximum fecundity of parasitoid female. Then F10
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 F11
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 F12

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
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) F13
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.
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
per cage Number of
replications Total number of prey
in all cages Total number of prey
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 F14
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?

Next: Lecture 11. Competition and Cooperation
EXEL predator-Pray model
Main Text
EXEL Host-Vector-Pathogen-System