## Simulation 7: Introducing biological pest control: a parasitoid

In simulation 6 we simulated the effect of a predator. The predator helps keeping the pest population at a lower level, but the population size of the predator does not depend on the pest population. Now we will introduce a different kind of natural enemy: a very specialized parasitoid whose life cycle is closely linked to that of the pest. The math becomes a bit more complicated.

For the pest our model uses the following parameters.

N1_{0 }= Population size at the start of the simulation.

N1_{t }= Population size at time t (after t days)

r1 = Reproduction factor

K1 = Carrying capacity of the environment

For the parasitoid the model uses these parameters:

N3_{0 }= Population size at the start of the simulation.

N3_{t }= Population size at time t (after t days)

r3 = Reproduction factor

K3 = Carrying capacity of the environment

p3 = Parasitization factor

Population development of the parasitoid (a specialist) is linked to the population density of the pest. We could simulate it with the formula which we have already seen in formula [8] (see Simulation 3).

[11] N3_{t+1} = N3_{t} + N3_{t} × r3 × (1 – N3_{t} / K3)

But in this formula the parasitoid population grows to reach K3, which is a fixed value. We have to link the development of the parasitoid to the population size of the pest. With low population size of the pest, the reproduction of the parasitoid should be slowed down.

We will adjust the daily growth of the parasitoid (N3_{t} × r3) with factor (N1_{t}/K1 – 1). This is a negative number between -1 and 0, which will make sure that the population growth of the parasitoid will decrease towards 0 (or could even become negative) if the pest population gets very low, but it will let the parasitoid population grow to K3 if the pest population is near K1.

N3_{t+1} = N3_{t} + N3_{t} × r3 × (1 – N3_{t}/K3) + N3_{t} × r3 × ( N1_{t }/K1 – 1)

another way to write this formula is:

[12] N3_{t+1} = N3_{t} + N3_{t} × r3 × (N1_{t }/K1 – N3_{t}/K3)

At low pest population levels the growth of the parasitoid population will be zero or even negative. But with higher pest populations the parasitoid can develop faster until it reaches it’s own capacity (K3).

Population development of the pest is also based on formula [8], but includes the effect of parasitization factor p3. This parasitization effect is of course also density dependent; it depends on both the size of the pest population and the parasitoid population. Therefore the effect of parasitization is not just N_{t} × p3 but we multiply this with (N1_{t}/K1) and with (N3_{t}/K3).

[13] N1_{t+1} = N1_{t} + N1_{t} × r1 × (1 – N1_{t }/ K1) – N1_{t} × p3 × (N1_{t }/K1)

× (N3_{t }/K3)

Now try it with different parameters. You can change the default settings, then run the simulation.

### Simulation 7

### Simulation settings

abc

Just like the predator in simulation 6, the parasitoid also keeps pest populations at a level lower than K1. Without the parasitoid 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 parasitoid it cannot reach this level.

The parasitoid removes a part of its host population, on which it depends for its own reproduction. Try different parasitization factors (p3). You will find that the parasitoid population will stabilize below its capacity K3. Only if p3 is 0 (i.e. the parasitoid doesn’t kill the pest) it would reach its capacity K3.

Remember GIGO? Starting with a low pest population and a high parasitoid population is an impossible situation. The parasitoid population collapses immediately.

Let’s continue and combine simulation 6 and 7.

Go to Simulation 8: Biological pest control with predator and parasitoid.

The scripts for these simulations are written in JavaScript. If you have any questions or comments, or if you discover any errors, please contact me.

- Introduction
- Simulation 1 - The development of an insect population
- Simulation 2 - A reproduction factor based on birth and mortality.
- Simulation 3 - Limiting population growth
- Simulation 4 - Simulate use of pesticides
- Simulation 5 - Pesticide applications at a threshold level
- Simulation 6 - Introducing biological pest control: a predator
- Simulation 7 - Introducing biological pest control: a parasitoid
- Simulation 8 - Biological pest control with predator and parasitoid
- Simulation 9 - Biological pest control and chemical control