## Simulation 2 A reproduction factor based on birth and mortality

The size of an insect population fluctuates from day to day. Every day some new insects are born and every day some insects die. We have already seen that the population size for one generation depends on the size of the previous generation, and that it is a multiple. In our model the population size of a certain day will depend on the previous day.

Let's consider a birth factor (b) and a mortality factor (m) which represent the birth and mortality per unit of time. For unit of time we could use anything, but we use "day" because we are simulating an insect population (in a model for bacteria we would use "minutes" and in a population model for people we would probably use "years").

If we include b and m in a "difference equation" (an equation that allows us to find values at discrete intervals) it could look like this:

[3] N_{t+1} = N_{t}
+ (b × N_{t}) - (m × N_{t})

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

N_{1} = Population size at time 1 (i.e. after 1 day)

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

b = Birth factor

m = Mortality factor

Note that both b and m must be greater than or equal to zero. The equation can now be rewritten as:

[4] N_{t+1} = N_{t}
+ N_{t} × (b - m)

or

[5] N_{t+1} = N_{t}
+ N_{t} × r

in which r represents a "reproduction factor" (b-m) based on birth and mortality per unit of time.

Let's simulate an insect population based on formulas [4] or [5]. Try different values for birth and mortality. You can change the default settings, then run the simulation.

### Simulation 2

This simulation shows maximum 300 days.

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

Trying different parameters you will see that whenever r>0, we get exponential population growth and if r<0 the population will eventually die out. We have a stable population size when r =0 (or birth equals mortality). At this point we should ask ourselves what are realistic values for b and m.

### How big is the birth factor?

Let's compare simulations 1 and 2 and see if we can find a realistic value for the birth factor in an insect population.

With a mortality factor set to 0, the second simulation is very similar to the first one. A difference is that it shows the population growth at a daily basis and not in steps of one generation at the time. Another difference is that it shows the entire population, while the first simulation shows only the females.

Let's look in simulation 1 at an insect with a life cycle of 20 days, with females laying 200 eggs (of which 50% = 100 eggs develop into new females). Starting with 1 female the population after 3 generations has grown to 1,000,000 females and as there will also be 1,000,000 males the total population size after 3 generations (i.e. 60 days) will be 2,000,000.

Population size after 1 generation = 2 × 100^{1}
= 200

Population size after 2 generations = 2 × 100^{2} = 20,000

Population size after 3 generations = 2 × 100^{3 }= 2,000,000

In simulation 2 this corresponds to a daily birth factor of approximately 0.25893 . Starting with 2 individuals (a male and a female) the population after 60 days (i.e. 3 generations) is just over 2,000,000

Population size after 20 days = 2 × (1+0.25893)^{20}
= 200

Population size after 40 days = 2 × (1+0.25893)^{40} = 20,003

Population size after 60 days = 2 × (1+0.25893)^{60 }= 2,000,437

Here are some more examples of daily birth factors for insect populations with different egg production and different length of life cycles.

Life cycle (days) | Eggs per female | Birth factor (b) | Remark |

100 | 100 | 0.0399 | Slow population growth (long life cycle) |

40 | 50 | 0.08380 | Slow population growth (few eggs) |

25 | 100 | 0.16939 | |

30 | 400 | 0.19317 | |

20 | 200 | 0.25893 | |

10 | 500 | 0.73698 | Fast population growth (many eggs and short life cycle) |

7 | 1000 | 1.42978 | Fast population growth (many eggs and short life cycle) |

The colored values show birth factors that are often found in real insect populations. In our simulation model for insect population growth we should therefore work with (b) roughly in the range 0.1 to 0.5

Keep in mind that the birth factor is not a fixed value for an insect species because the number of eggs per female is never fixed. It will vary a from individual and it also depends on environmental factors (a healthy females will lay more eggs than a weak or malnourished female). The duration of the life cycle is also not fixed and depends on external factors such as food supply and temperature.

### How big is the mortality factor?

The mortality factor can have many different values because mortality depends on many external factors (availability of food, predators, parasitoids, pathogens, sunshine, rain, etc.). What we see in the second simulation is that if mortality factor is equal to the birth factor the population stays at a constant level. If mortality is lower than birth, the population will grow exponentially, and if mortality is higher than birth the population will decline and eventually get extinct. We don't want the population to disappear, so in our simulation model it makes sense to work with values of (m) that are smaller than (b).

If we want to make our simulation more realistic we will have to find ways to avoid exponential growth (in real life this would not be possible because of food shortage) and we have to avoid a complete collapse of the population. Let's make our model a bit more complicated.

Go to Simulation 3: Limiting population growth.

The scripts on this page are written in PHP. If you have any questions or comments please contact me.