__Simple population cascade (=SPC) in a honeybee colony__

This is our the first basic application.

#### Queen laid eggs

The idea is that - in an healthy colony - only the queen is laying eggs that become adult (female) bees. The queen lays eggs with a given rate that has a
give variability. Both parameters can be set on the first page called "Parameters".

*The variety in the egg laying rate is modeled by a random generator which produces (due to multiplication of 2
random numbers between -1 and +1) values towards 0 much more frequently than towards -1 or +1. This "trick" solves the problem that EXCEL produces random numbers with a uniform distribution instead
with a normal distribution.*

Please note that this is the only place where we introduced a random generator, just to demonstrate the variability and how it effects other parameters. If you want to see the pure
system dynamics without any noise, just set the egg-laying rates' variability to zero.

#### And worker laid eggs ?

We can neglect worker laid eggs because they are laid rarely and in almost all cases removed very soon after hatching. If any would reach the adult age,
they would be certainly be drones and as long as we neglect the role of drones we can neglect worker laid eggs.

#### What happens with queen laid eggs ?

The eggs hatch to one day old unsealed larvae after three days and are__ shifted__
to the column representing two day old larvae one day later. Of course not all larvae are shifted, a percentage of them disappear (are dead). These percentages ("mortalities") are also parameters that can be set. This game of shifting is continued until the larvae gets capped after 5 days. The capped period lasts for some days, than "young bees" are hatching out of the cells.

#### What kinds of adult bees do we have ?

Per definition we distinct between "young bees" (younger than 15 days after hatching out of the cell) and "old bees". For simple
elaboration of this very basic model, we can assume that young bees will prefer to nurse and old bees would prefer to forage. Therefore we can set different mortality rates for both age groups.

#### And what results do we get from this ?

The most important result we get is that we can face the enormous stability this system represents. Due to the fact that the animals do not reproduce by
themselves and all reproduction is done by one queen, population numbers of each separate age class strongly depend on:

- the egg laying rate in the past
- the time span the specific age class represents
- the mortalities for each class

We also already see a sort of puffer effect the later developmental stages represent. Starting from eggs, where we face a strong fluctuation per day,
these fluctuation disappears class by class with increasing age. At last (class "old bees") we do see almost no daily fluctuations, just long-term tendencies.

Of course, we do not have any feedback mechanisms in this model yet, but we already can explore the basic system dynamics by changing the system
parameters.

#### Let's see some screen shots now:

This is the way we feed some simulation parameters and initialization values into the model. You can
see that we switched of the mortality rates of all stages, what means that all individuals live up to the maximum of their life spans.

As we can see here, this leads to the buildup of the hives population up to a maximum in about 150
days. Eggs and unsealed larvae reach their maximum number quickly. The variation we see are due to the variability of the egg laying rate we set in our simulation parameters. Almost no fluctuation is
visible in the group of old bees.

The proportion between the age-groups are all approaching to a fixed point, due they are only
influenced by the egg laying rate and the number of days a bee belongs to this group. In later elaboration we will see this changing due to feedbacks, mortalities and cannibalism.

Here we can see that the empty space in the hive decreases. Nevertheless the queen keeps an egg laying rate that fluctuates around a fixed value.

__SPC elaborated with limited space feedback__

The idea of this elaboration is that the colony is restricted in space.

#### Concept

You can set this space on the parameters page. You can also set a "minimum percentage
free"-parameter there. This means that when the hives free available cells reach this border, the queen stops it's egg laying. After enough sealed brood has hatched and therefore enough free space
is available again, the queen restarts it's egg laying. But it takes some (3) days until the queen can reach it's full egg laying rate.

#### Results

We can see this situations as breaks in the egg numbers and can follow them through the next developing stages.

When space gets scarce, the egg laying rate drops down to 20% and the queen needs some days to reestablish full production.

This leads to a short-time drop in the number of eggs and - a little bit later- to a small drop in unsealed brood.

But there is no visible effect to the number of adult bees.

__SPC with seasonal changes in egg-laying rate__

The basic idea of this elaboration is that the egg laying rate of the queen increases during spring,
reaches a plateau in summer and decreases again towards autumn (in temperate regions). This leads to severe changes in the populations demographic dynamics.

We see how the size of the brood nest increases towards late spring (day 120) and summer and
decreases later again. In winter (day 260+) we see almost no new brood anymore. Adult bees show the same dynamics. Due to the fixed life span (50 days) for all bees, the colony dies out in winter now.

We will have to fix this problem with further elaboration (see below), where we increase life span of adult bees towards winter.

Now we can see how strong this little change was to proportions of young to old bees and young
bees (potential nurses) to unsealed larvae. Assuming that more nurses per larva mean better condition for each single larva, we see bad conditions for spring reared larva, stable conditions for
summer reared larva and very-good conditions for autumn reared larva.

Here we see how the brood nest fills this (small) hive in summer and empties towards winter again.
But please remember, we do not have any storage of pollen and honey in this model so far.

__SPC elaborated with changes in adults life-span__

As we have seen in the previous example, we will have to adjust the maximum life span of bees in
autumn. We did this by changing the life span through a feedback with the proportion of available nurses (young bees) and consumers (unsealed larvae). So bees have normally a life span of 50 days,
but this time span can be enlarged up to 140 days if there are more nurses per larvae available.

#### Why ?

- Well on the one hand, each nurse will have to perform less nursing activities in her life when there are more nurses per larva around.
- On the other hand, larvae will be better nursed in this situation.

So the idea is, they might start with more reserves into their adult life and age slower if their
metabolic rate is slower due to less jelly production in their nurse phase.

In this elaboration you can set an optimal nurse to larva ratio. At this condition the life span is
expanded to the maximum 140 days. More available nurses per larva do not further extend the future life span.

Lets see what we get out from this sub-model:

We can see how the size of the brood nest increases towards summer and decreases towards
autumn again. But due to the enlarged life span of autumn-raised bees, the colony does not die out throughout winter. If you think this are too much bees for overwintering, remember that we have set
the mortalities to zero so far. You can adjust them in the parameters setting page and -for sure- will find fewer bees then.

Again we see a cyclic dynamic in the proportions of one age-cast to another. Due to the life-span
adjustments, we find a lot of old bees for very few young bees in autumn.

The chart of available empty space didn't change significantly in this elaboration.

__Do you need ideas for further elaboration ?__

- You can try to add further feedback, for example less old bees per young bees could mean less forages and therefore less input.
- You can try to simulate wax manipulants that start to enlarge the colony by building additional cells, once the empty space gets scarce
- Of course you might add the collection of pollen and/or honey
- ..