An ant colony simulation made by Matteo Romagnoli for the course Performance Analysis and Simulation of the University of Camerino.
Description and dynamics of the colony
The ant colony, known as a self-organised system, can adapt to the environment through negative and positive feedback; however, it is not clear how ants coordinate. Some studies show that the ant colony is not a hierarchical system (the queen does not command), but each ant has its own task, and moreover the communication between ants is limited to an exchange of deeds (by means of antennas) through which an ant specifies its role.
The ants are really important for the ecosystem because they:
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aerate the soil, allowing water and oxygen to reach plant roots;
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take seeds down into their tunnel to eat the nutritious elaiosomes that are part of the seed; These seeds often sprout and grow new plants (seed dispersal);
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eat a wide variety of organic material and provide food for many different organisms.
So, we literally can't live without them.
The aim of the project is to define and simulate a population model of an ant colony (a new colony or an already defined colony), paying attention to the effects of the variation of the temperature and of the food present in the environment, and see how the ant colony responds to these changes. Indeed, studies show that these variations have effects on the speed of workers, on the development of immature ants and on the reproduction of the queen; the consequences of the effects of the changes can be summarised as follows:
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There is no effect of temperature and food on the ant's energy and dynamics of the colony population.
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Temperature elevations (or diminutions) will increase the risk of ant colonies dying out because they need to consume more energy to sustain themselves.
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Changes of food availability will affect energy and population dynamics of ant colonies.
A lot of factors are involved in the dynamics of an Ant Colony, and since we are going to define a model and simulate it, we need to consider only the major aspect of the ant colony scenario.
We will focus on the work of the ants and the influence of temperature and food availability, without taking into account this factors:
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The temperature inside the nest and outside is the same and does not change during the time;
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The mass of the ant and the pheromones are abstracted away by using the rates;
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There is no distinction between water and food; they are both considered
$\text{Food}$ ; -
It does not exist the season cycle as well as the day-night cycle;
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The queen will lay a constant number of eggs each time;
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The simulation considers only the part of the year when the ant colony is active.
Moreover, we consider an ant colony of Black Garden Ant or Common Balck Ant (species Lasius Niger), that is monogynous, meaning that the colonies contain a single queen, and can reach in size up to 2000 - 7000 ants on average during the years. The ant colony will be active from May to October and will hibernate the other part of the year. In this type of ant colony there are different kinds of ants, each with a particular role: the queen, the males, the workers, the soldiers. Since the males are alive only 1 or 2 days after the mating, and then they die, we can not take them into account. Also the soldiers are not represented because we are studying the temperature/food influence of a single ant colony; this means that there are no conflicts between colonies.
To perform the simulation and provide results, we define a population model representing the rules and behaviours of the system. Since this is a model, and to maintain it simpler, different properties of the real word are abstracted away, like the place, the time (day/night cycle), the change of season... as mentioned before.
We first focus on the agents that interact in the environment and the dynamics inside the colony, and then we will see the influence of temperature and food, and finally we will take a look at the relative population model considering a Population Continuous Time Markov Chain (PCTMC).
Given a colony of Black Garden Ants it is possible to identify and consider four main different ants (or roles):
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The queen: there is only one queen per colony (future queens are not taken in consideration), and her purpose is to lay eggs; the growth of the colony depends mainly on the fertility of the queen and on the efficiency of the worker ants.
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The nurses: they are a particular kind of [worker]{.ul} ants who take care of feeding the larvae and the queen, as well as cleaning the nest; these ants always work inside the nest.
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The foragers: they are the second type of [worker]{.ul} ants that take care of looking for and collecting food outside the nest and making it available to the whole colony; these ants always work outside the nest.
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The larvae: with larvae we mean immature ants (egg, larva, and pupa); the larvae require food to grow and become adult ants; once adult, they will become nurses or foragers based on the request in the colony.
Each ant has a basal metabolic rate which specifies how much energy is consumed; furthermore, doing work (gathering food, or giving to eat, or laying eggs) consumes energy. Of course, an ant dies based on the energies it has: the more tired and hungry the ant is, the less energy it will be, the greater the probability of death.
During its life, a nurse ant can change role and become a forager (or vice versa) according to the needs present in the nest.
In order to define a model, and use it in a simulator, we have to specify the agents and states, the actions they can perform and the rates of the rules.
In particular, the model contains the following agents: Queen,
Nurse, Forager and Larva. These represent the main ants in the
colony and each of them have an energy variable (
We also have to consider other agents like the Nest (or Home) that act as food storage for the colony, the Death Ants (that include the dead foragers, dead nurses and dead queen) and the Death Larvae.
We can summarise all the agents and the interaction previously listed with the following abstract diagram:
The diagram, that is like a graph, contains nodes labelled with the name of the agents (the square brackets inside a node is used to define a parameter) and 2 types of edges:
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The edge with the arrow, that indicate a change of state/agent using a rule; for instance
$a - rule \rightarrow b$ , meaning "the agent a performs the rule and becomes the agent b", like$larva - dies \rightarrow \text{death\ larva}$ , meaning "the agent$\text{larva}$ performs the rule$\text{dies}$ and becomes the agent$\text{death\ larva}$ ". -
The edge with the dot, that indicate the interaction with another agents; for instance
$a - rule - \bullet b$ , meaning "the agent a performs the rule and interacts with the agent b", like$forager - collect\ food \rightarrow \text{home}$ , meaning "the agent$\text{forager}$ performs the rule$\text{collect\ food}$ and interacts with the agent$\text{home}$ ".
As mentioned, 2 factors affect an ant colony, and they are the temperature and the food availability.
Temperature. The ideal temperature for an ant is between 23 and 35 °C (for simplicity we consider the ideal temperature equal to 29 °C and a maximum variation of ± 10 °C from the average, beyond which the temperature has greater influence); at these temperatures ants are more likely to forage for food and keep a colony active (the ant's metabolism is optimal, so it will use less food). The temperature influences the queen reproduction, the nurses and foragers work, the metabolism of all the ants and the death probability of larvae.
The influence of temperature is described by the following mathematical function (that is a Generalised Normal Function):
with parameters
The result is the following curve:
This function ranges between 0 and 1 (0, 1) and it is used in the model to affect the probability of a rule. We can see, also from the picture, that from 23 to 35 the function value is near to 0, but as soon as we move away from it, the function increases exponentially, until the value reaches ~1 at about 5 and 45.
Food. The amount of food present in the environment determines the survival or not of the colony. The consumption of food depends on the role of the ant, the metabolism of the ant and the work of the ant.
Of course, the more food there is in the environment, the less time will be required for harvesting and the more food there will be in the nest. At the end the colony will survive and grow only if there is enough food. As soon as the food in the nest runs out, the ants begin to starve.
The food influences mainly the queen reproduction, the foraging of food and the change of work (nurse ↔ forager); in particular the last one has a trend that can be represented by the following mathematical functions:
Where
This is the relative chart:
This functions are used to manage the change of work between foragers and nurses depending on the food available in the nest:
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In red, the probability that a nurse becomes a forager; it increases as soon as the food storage becomes empty.
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In violet, the probability that a forager becomes a nurse; it increases as soon as the food storage is full.
In this section it is defined the PCTMC of the Ant Colony; the model is
defined as the tuple
Vector Variables
Counting Domain
Initial State
Transition rules
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$queen\ lays\ eggs,\ 1Q\ 1H,\ 1Q\lbrack + 1\rbrack\ 1H\ 15L,\ \left( \lambda_{\text{queenFertility}} - \frac{\text{tI}}{5} \right)\ \cdot \ fS\ \cdot \ e\ \cdot \ cS\ \cdot \ % L$ -
$queen\ consume\ energy,\ 1Q,\ 1Q\lbrack + 1\rbrack,\ \lambda_{\text{queenConsumeEnergy}} + \frac{\text{tI}}{2}$ -
$larva\ consume\ energy,\ 1L,\ 1L\lbrack + 1\rbrack,\ \lambda_{\text{larvaConsumeEnergy}} + \frac{\text{tI}}{2}$ -
$nurse\ consume\ energy,\ 1N,\ 1N\lbrack + 1\rbrack,\ \lambda_{\text{nurseConsumeEnergy}} + \frac{\text{tI}}{2}$ -
$forager\ consume\ energy,\ 1F,\ 1F\lbrack + 1\rbrack,\ \lambda_{\text{foragerConsumeEnergy}} + \frac{\text{tI}}{2}$ -
$nurse\ eats,\ 1H\ 1N,\ 1H\lbrack - 1\rbrack\ 1N\lbrack - 1\rbrack,\ \lambda_{\text{eat}}\ \cdot \ e$ -
$forager\ eats,\ 1H\ 1F,\ 1H\lbrack - 1\rbrack\ 1F\lbrack - 1\rbrack,\ \lambda_{\text{eat}}\ \cdot \ e$ -
$nurse\ feeds\ queen,\ 1N\ 1Q\ 1H,\ 1N\lbrack + 1\rbrack\ 1Q\lbrack - 1\rbrack\ 1H\lbrack - 2\rbrack,\ \lambda_{\text{nurseFeedsQueen}}\ \cdot \ % N$ -
$nurse\ feeds\ larva,\ 1N\ 1L\ 1H,\ 1N\lbrack + 1\rbrack\ 1L\lbrack - 1\rbrack\ 1H\lbrack - 1\rbrack,$
$\left( \lambda_{\text{nurseFeedsLarva}} - \frac{\text{tI}}{2} \right)\ \cdot \ fS\ \cdot \ % N$ -
$nurse\ cleans\ nest,\ 1N\ 1H\ 1DL,\ 1N\ 1H\lbrack + 1\rbrack,\ \left( \lambda_{\text{nurseCleans}} - \frac{\text{tI}}{2} \right)\ \cdot \ % DL$ -
$forager\ collects\ food,\ 1F\ 1H,\ 1F\lbrack + 1\rbrack\ 1H\lbrack + 2\rbrack,\ \left( \lambda_{\text{foragerWorks}} - \frac{\text{tI}}{2} \right)\ \cdot \ fA\ \cdot \ % F$ -
$queen\ dies,\ 1Q,\ 1DA,\ \lambda_{\text{queenDeath}}\ \cdot \ cS$ -
$larva\ dies,\ 1L,\ 1DL,\ \frac{\left( \lambda_{\text{larvaDeath}}\ \cdot \text{\ e} \right)}{3}\ + \ \frac{2\ \cdot \text{\ tI}}{3}$ -
$nurse\ dies,\ 1N,\ 1DA,\ \lambda_{\text{workerDeath}}\ \cdot \ e$ -
$forager\ dies,\ 1F,\ 1DA,\ \lambda_{\text{workerDeath}}\ \cdot \ e$ -
$larva\ becomes\ forager,\ 1L,\ 1F,\ \lambda_{\text{larvaGrows}} \cdot % L\ $ -
$larva\ becomes\ nurse,\ 1L,\ 1N,\ \lambda_{\text{larvaGrows}}\ \cdot % L$ -
$nurse\ becomes\ forager,\ 1N,\ 1F,\ \lambda_{\text{changeOfWork}}\ \cdot (1 - % fS\ + \ sA)\ \cdot \ % N$ -
$forager\ becomes\ nurse,\ 1F,\ 1N,\ \lambda_{\text{changeOfWork}}\ \cdot (% fS\ + \ sA)\ \cdot \ % F$
Where:
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$\text{Food}$ is a constant and represents the capacity of the food storage of the colony. -
$\text{Energy}$ is a constant and represents the hungry levels of an ant. -
$H$ is the Home (or nest) of the colony; the parameter$H\lbrack\rbrack\ = \ fS$ represent the food inside the nest. -
$Q$ is the Queen of the colony; the parameter$Q\lbrack\rbrack\ = \ e$ represent the hungry level of the queen. -
$L$ are the larvae (eggs, pupa, and larva); the parameter$L\lbrack\rbrack\ = \ e$ represent the hungry level of the larva. -
$N$ are the nurses; the parameter$N\lbrack\rbrack\ = \ e$ represent the hungry level of the nurse. -
$F$ are the foragers; the parameter$F\lbrack\rbrack\ = \ e$ represent the hungry level of the forager. -
$\text{DL}$ are the death larvae -
$\text{DA}$ are the death ants (queen, foragers, and nurses) -
$fS\ \in \ {\ x\ \in \ N\ |\ 0\ \leq \ x < 10\ }$ is the food storage level and represents how much food is stored in the nest; this is a [global]{.ul} parameter that depends on$H$ . The higher is$\text{fS}$ , the more food will be in the nest. -
$e\ \in \ {\ x\ \in \ N\ |\ 0\ \leq \ x < 10\ }$ is the energy level of the ant and represents how much an ant is hungry; this is a [local]{.ul} parameter (depends on the ant). The higher is$e$ , the hunger will be the ant. -
$tI\ \in \ {\ x\ \in \ R\ |\ 0\ \leq \ x\ \leq \ 1\ }$ is the temperature influence; this is a [global]{.ul} parameter. The higher is$\text{tI}$ , the more the ants are negatively affected. -
$fA\ \in \ {\ x\ \in \ R\ |\ 0\ \leq \ x\ \leq \ 1\ }$ is the food availability and represents how much food there is in the environment; this is a [global]{.ul} parameter. The higher is$\text{fA}$ , the more food there is in the environment. -
$cS\ \in \ {\ x\ \in \ R\ |\ 0\ \leq \ x\ \leq \ 1\ }$ is the colony size and represents how bigger the colony is with respect to the standard; this is a [global]{.ul} parameter. -
$sA\ \in \ {\ x\ \in \ R\ |\ 0\ \leq \ x\ \leq \ 1\ }$ is the starve ants and represents the percentage of the population that is starving; this is a [global]{.ul} parameter. The higher is$\text{sA}$ , the more ants are starving.
The implementation of the model is made by using the tool [Sibilla]{.ul}.
From the studies and observations we know that an Ant Colony:
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Grows quickly in perfect conditions, in both case of a new colony or an old one;
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Dies if in an hostile environment (like miss of food or pretty low/high temperature), in both case of a new colony or an old one;
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In a environment with middle condition the colony try to survive; in particular at around 12 °C
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a big colony can survive if food is not a problem
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a medium colony only survive for a period of time
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a small colony will die after a while
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We have similar scenarios when food is lacking but the temperature is favourable.
The simulations take place in 1000 time units that correspond to about 6 months (180 days, when the colony is active) with a delta time of 1 unit. In order to make the simulation more general, it is reproduced 10 times (10 replicas for each simulation).
For each simulation we collect the summary statistic of different measures regarding:
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the number and percentage (with respect the population) of foragers;
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the number and percentage (with respect the population) of nurses;
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the number and percentage (with respect the population) of larvae;
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the number of alive ants (only queen, foragers and nurses);
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the number of dead ants (only queen, foragers and nurses);
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the number of dead larvae (only larvae);
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the number of agents (only queen and larvae) that are starving.
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the home's number in a particular state (
$H\lbrack 0\rbrack,\ H\lbrack 1\rbrack\ ...\ H\lbrack 19\rbrack$ )
Each file contains 4 fields, that are the time step (from 0 to 999), the mean of the collected value, the variance and the standard deviation.
During the simulation the temperature and the food availability parameters [do not change]{.ul}.
We take in consideration 12 simulations, 6 simulations with a new colony (consisting of the queen, 20 larvae, 10 foragers and 10 nurses) and 6 simulation with a old colony (consisting of the queen, 20 larvae, 100 foragers and 100 nurses), all of which they start with a nest full of food, and each of which designing a scenario (different temperature and food availability).
Five charts have been defined for each simulation:
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General View representing the trend of the colony during the time (is the colony dying or not?);
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Ants Distribution representing the distribution of the roles in the colony during the time (how much foragers, nurses and larvae are there?);
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Deaths representing the number of death ants and death larvae during the time;
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Food Fluctuation representing the amount of food stored in the nest, and the relative trend, during the time;
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Hunger Colony Distribution representing the amount of ants, foragers and starve agents, and the relative amount of food in the nest (plus the trend), during the time.
A new colony in an ideal environment where
As we can imagine, in ideal conditions, the ant colony will grow
quickly. Since there is a lot of food, the food storage never becomes
empty, moreover remains always half full (as we can see in the food
fluctuation chart), and the number of nurses and foragers are more or
less equally distributed (number of nurses is almost the same as the
number of foragers). In the last graph we can see that the population is
not
starving.
A new colony in an environment where
Also in this case, the ant colony will grow, however it is slower than the first. Since here the problem is the food, the food storage tends to become empty (as we can see in the food fluctuation chart), but at the same time, the number of foragers increases with respect to the nurses. In the last graph we can see that the population is not starving, but sometimes, we can notice some peaks (~5 agents).
A new colony in an environment where
In this scenario, the ant colony will grow really slow. We can observe that the number of death ants overcome the number of alive ants (at the end). Also if the food is not a problem, the temperature requires a constant and large amount of food to eat, and this is visible in the Food Fluctuation graph: the food storage is always empty, because the ants eat all the available food in order to survive.
A new colony in an environment where
In this case the whole ant colony dies after about 200 time units (~36 days). Also if there are more foragers to harvest for food, the request for food is so high that the colony dies. We have to keep in mind that initially the colony has only 20 workers.
A new colony in an environment where
As we can imagine, in these conditions the colony dies after a while (~30 days).
A new colony in an hostile environment where
As we can imagine, in these conditions the colony dies after a while (~30 days).
A old colony in an ideal environment
In this simulation, with the ideal conditions, the ant colony will grow, of course. Since there is a lot of food, the food storage never becomes empty, moreover remains always full (as we can see in the food fluctuation chart), and the number of nurses and foragers are more or less equally distributed (number of nurses is almost the same as the number of foragers). In the last graph we can see that the population is not starving. This scenario is similar to the small colony in the same condition (Simulation 1).
A old colony in an environment where
Respect to Simulation 2, the colony grows faster because of the number of initial workers. As soon as we move till the end (1000 time units), the food storage tends to become empty; for this reason, there is a big gap between foragers and nurses.
A old colony in an environment where
In this scenario, the ant colony will grow slowly. Respect to Simulation 3, in this case, the food storage rarely becomes empty. This means that an old (or bigger) colony is better able to survive than a new (or small) colony.
A old colony in an environment where
In this simulation the ant colony is slowly dying. This is because at the end the queen dies (the larvae are no more generated as we can see in the Ants Distribution graph). However, if the colony is bigger or the time under this condition is shorter, then the colony may survive.
A old colony in an environment where
As we can imagine, in these hostile conditions, the ant colony will die not so fast. The queen dies after 200 time units (~36 days), and then the whole colony dies after 700 time units (~126 days).
A old colony in an hostile environment where
As we can imagine, in these hostile conditions, the ant colony dies
quickly: after 100 time units the queen dies (~18 days), and after that
all the ants in the colony (in around 500 time units, or ~90 days).
We can investigate more on what happens in an [old colony]{.ul} when the temperature varies from 11 °C to 13 °C and food availability is 1.
We can observe that as soon as the temperature overcomes 12 °C the ant colony starts growing; at 12 °C the colony remains stationary and below this temperature, the colony dies.
While in the [new colony]{.ul}, we have this graph:
Essentially, the small colony, in order to survive, requires a temperature 0.5 °C greater than the older one.
By changing the temperature and food availability, the model revealed the impact on the ant colony.
Increasing the temperature can boost the colony speed, because it will speed up the workers, the queen reproduction, the development speed of larvae... However, moving far away from the ideal temperature (increasing or decreasing the temperature) will cause the degradation of the colony, because the metabolism will increase as well as the activities' speed of the ants decrease. Moreover the ant colony in an hostile environment will use more food than another in a moderate environment.
In an environment with the same conditions, a small colony will be disadvantaged over a bigger one. The scarce resources will increase colonies' risk for dying out. Nevertheless, the rich food sources could protect ant colonies from food shortage and starvation.
[https://harvardforest.fas.harvard.edu/ants/life-cycle]{.ul}
[https://onlinelibrary.wiley.com/doi/full/10.1111/gcb.16140]{.ul}
[https://zoologicalstudies.springeropen.com/articles/10.1186/s40555-014-0040-4]{.ul}
[https://www.antwiki.org/wiki/Life_in_an_Ant_Colony]{.ul}
[https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4951116]{.ul}
[https://www.terminix.com/ants/behavior/do-ants-hibernate]{.ul}
[https://www.ted.com/talks/deborah_gordon_the_emergent_genius_of_ant_colonies]{.ul}
[https://dc.etsu.edu/cgi/viewcontent.cgi?article=3757&context=etd]{.ul}
[https://harvardforest.fas.harvard.edu/ants/ecological-importance]{.ul}