## Simulation 7 Introducing biological pest control: a parasitoid

In simulations 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.

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)

b1 = Birth factor

m1 = Mortality factor

r1 = b1-m1 = Reproduction
factor

K1 = Carrying capacity of the environment

For the parasitoids 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)

b3 = Birth factor

m3 = Mortality factor

r5 = b2-m2 = 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}
+ N_{t} × r3 × (1 - N3_{t} / K3)

But in this formula the parasitoid population grows to reach K3,
which is a fixed value. To link the development of the parasitoid to
the population size of the pest we have to multiply K3 with another
factor which includes N1_{t}. We will multiply K3 with N1_{t
}/K1, which is a factor that increases towards 1 as the pest population
growth to its maximum capacity level (K1).

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

At low pest population levels the growth of the parasitoid population will be almost zero. 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)
- N_{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

This simulation shows maximum 300 days.

The simulation ends when the population size is above 10,000,000
or below 1.

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.

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