existence

The first concept we’ll be looking at may not be completely relevant to the final natural selection model, but it helps start looking at the problem in a more mathematical and computational light. Why does something exist? Our definition is simple:

It was created
It hasn’t been destroyed

For this reason, our aim in this module will be to give organisms birth-rates, death-rates, and replication-rates and see how their populations grow.

raindrops

In our first simulation, we have a probability that a new organism will be born, and a probability for each organism that it will die. An equation for the predicted equilibrium can be derived from $B=ND$ to be $N=\frac{B}{D}$ . I have implemented a class for this type of simple organism that I call an existor. Extracts from the Python class are shown below. To view the full code, view the project on GitHub.

class Existor(object):
    def __init__(self, population, birth_rate, death_rate):
        self.population = population
        self.birth_rate = birth_rate
        self.death_rate = death_rate

    # Updates population based on birth and death rates
    def update(self):
        if random.randint(0, 100) < self.birth_rate:
            self.population += 1
        for organism in range(self.population):
            if random.randint(0, 100) < self.death_rate:
                self.population -= 1

    # Returns the predicted equilibrium value based on our equation
    def getEquilibrium(self):
        return(self.birth_rate/self.death_rate)

Defining an instance of an Existor as raindrop = Existor(0, 100, 10), we get the following population graph.

The predicted population based on our equation was pretty close to the actual mean population. The reason for the oscillation is because even if there were more organisms than the equilibrium, that would cause the death-rate to be higher than the birth-rate, bringing the population back to equilibrium.

pigeons

However, real-life organisms are different from raindrops in the sense that they can reproduce; to fix this, we can add a replication rate which modifies the equilibrium equation to be $N=\frac{B}{D-R}$ . The plot for Replicators is shown below. Since Replicator is a child class of Existor I have not included the code here.

aggression

The second concept is the trait of aggression, and how it affects the population dynamic. In my code, I’ve simulated the classic game-theoretic hawks and doves approach to the problem. Following is a short description of the algorithm.

Simulation runs for a specified number of days, and consists of a specified number of locations
Each location has two pieces of food
Each day, the location of each organism is randomized.
- If an organism has a location to itself, it eats both pieces of food and can reproduce
- If 2 doves meet, they each eat 1 piece and survive to the next day, but don’t reproduce
- If a dove and a hawk meet, the hawk takes more food, so it can reproduce, though the dove does not survive
- If 2 hawks meet, this leads to a fight and they both do not survive
- If an organism enters an occupied location, it does not survive

The table below summarizes the encounters and their consequences.

	If dove	If hawk
meets dove	survive, survive	reproduce, die
meets hawk	die, reproduce	die, die

In his video, primer had survival chances and reproduction chances for dove meets hawk, which I may or may not have implemented depending on when you are reading this.

doves only

I first simulated a dove-only ecosystem, which started with 20 doves, had 100 unique locations, and ran for 100 days.

# Initial populations of doves
doves = 10000

# Total number of days the simulation will run for
days = 200
# Number of available locations with food
lnum = 100

The location allocation process is simple, as shown below. The complete code is available on GitHub.

for dove in range(doves):
        location = random.randint(0, lnum-1)
        # If vacant, occupy :)
        if locations[location] < 2:
            locations[location] += 1
        # Else, no food, die :(
        else:
            deaths += 1
            print(f'death at {location}')

The plot below shows how population varied with each day.

The population shoots up immediately as there is a lower chance that creatures are overallocated to a single location. The code prints births, deaths, and a daily summary to the console. On Day 1, this is what it looks like:

As quickly as Day 5, the deaths outnumber the births. This is because the population is greater than the number of locations, so fewer doves get to reproduce. This is shown in the console below.

doves and hawks

For this section, I had a similar approach, though I needed a new set of variables and processes to take care of hawks: a tally of deaths and births, and I had to code up the consequence table shown earlier on this page. Here is the addition for assigning locations to hawks.

# Repeat location selection process for hawks
    for hawk in range(hawks):
        location = random.randint(0, lnum-1)
        if dove_locations[location] + hawk_locations[location] < 2:
            hawk_locations[location] += 1
        else:
            hawk_deaths += 1
            print(f'- hawk at {location}')

Here is a simple implementation of the consequence table:

for location in dove_locations:
        at_location = dove_locations[location] + hawk_locations[location]

        if dove_locations[location] == 1 and at_location == 1:
            print(f'+ dove at {location}')
            dove_births += 1
        elif hawk_locations[location] == 1 and at_location == 1:
            print(f'+ hawk at {location}')
            hawk_births += 1
        elif hawk_locations == 1 and at_location == 2:
            print(f'+ hawk at {location}')
            print(f'- dove at {location}')
            hawk_births += 1
            dove_deaths += 1
        elif hawk_locations[location] == 2:
            print(f'-- hawk at {location}')
            hawk_deaths += 2

One potential flaw is that in my current code, the location allocation process is ordered: one set of creatures is completely allocated before the other. Since being allocated to an occupied location could lead to decreases in population, this is unfair to the species located later in the order. Even after I allocated hawks before doves, I get the following population graph.

If I change the code so that only 1 hawk dies as a result of a hawk-hawk match, I get a graph that makes much more sense. The hawk population first rises dramatically, then results in a sort of ebb and flow.

a natural selection simulator