Simulation 6 Introducing biological pest control: a predator
In simulations 4 and 5 we have seen how to keep a pest population under a certain level by repeated pesticide applications. We will now see that pest populations can also be controlled by introducing natural enemies of the pest, for example spiders or egg parasitoids.
When bringing natural enemies of the pest into the model we are in fact introducing a new mortality factor. But we cannot simulate this by increasing the value of m in the model, because m is regulated by the factor (K-N)/K which will always stabilize at level K.
Roughly we can consider two types of natural enemies that help control a pest.
The first type is a "generalist", for example a spider or a bird, which feeds on a variety of prey species. Our pest population is only a part of its diet. When the pest population is low, the birds or spiders will feed on other insects. When the pest population increases, the birds or spiders will start eating more of it. But important is that the size of the natural enemy population does not depend on the size of our pest population. Many predators are generalists.
The second type is a "specialist", for example an egg parasitoid or a larval parasitoid. Not all, but many types of parasitoids are very specialized and their behavior and life cycle is very much linked to one or a few related species of host insects. In this case the size of the natural enemy population is directly related to the population of the host. If the pest population is low, the population of specialist natural enemies is also low. When the pest population increases, the population of natural enemies will "follow".
Let's first model a "generalist" natural enemy (a predator).
For the pest our model uses the following parameters.
N10 = Population size
at the start of the simulation.
N1t = Population size at time t (after t days)
b1 = Birth factor
m1 = Mortality factor
r1 = b1-m1 = Reproduction factor
K1 = Carrying capacity of the environment
For the predator the model uses these parameters:
N20 = Population size
at the start of the simulation.
N2t = Population size at time t (after t days)
b2 = Birth factor
m2 = Mortality factor
r2 = b2-m2 = Reproduction factor
K2 = Carrying capacity of the environment
p2 = Predation factor
Note that compared to formula  I have now changed the name of the parameters by adding a number: 1 for pest and 2 for the predator.
Population development of the predator (a generalist) is independent from the pest. We simulate it with the formula which we have already seen in formula  (see Simulation 3).
 N2t+1 = N2t + N2t × r2 × (1 - N2t / K2)
Population development of the pest is also based on this formula, but now we have to include the effect of predation. For this we include predation factor p2. This predation effect is of course density dependent; it dependent on the density of the pest (at low pest populations the predation rate will be lower, because predators feed on alternative preys) and it depends on the population density of the predator. Therefore the effect of predation is not just N1t × p2 but we multiply this with (N1t / K1) and (N2t/ K2).
 N1t+1 = N1t + N1t × r1 × (1 - N1t / K1) - N1t × p2 × (N1t /K1) × (N2t /K2)
At low pest population levels the effect of the predator will be almost zero, but it keeps increasing as the population size increases towards K1.
Now try it with different parameters. You can change the default settings, then run the simulation.
This simulation shows maximum 300 days.
The simulation ends when the population size is above 10,000,000 or below 1.
The effect of the predator is that the pest population is kept at a level lower than K1. Without the predator the pest population will grow to its capacity K1; it can not grow higher because the environment does not have enough capacity (shortage of space, food). With the predator it cannot reach this level.
Now let's introduce a "specialist" natural enemy into the model.
Go to Simulation 7: Introducing biological pest control: a parasitoid.
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