Population Dynamics
This presentation is about Population Dynamics with Birth and Death. Agent
based modeling of complex adaptive systems is all about populations in one
form or another because we are looking at agents. We are examining
populations of agents. There are numbers, actions and the resulting behavior
that the population of agents we are modeling exhibits.
[Slide 1] What is a population? In general, a population is a group of living things,
individual organisms of the same species living in the same place. Say ants living
in a specific ant colony, or people living in a specific city, state or country. In
agent based modeling it is a group of agents. These agents can represent
anything, those same ants, bees or people. In this video we are talking about
modeling Population Dynamics of living things.
Population Dynamics is a branch of Life Science that talks about the
characteristics of a given population, in terms of its size, age, reproduction and
death. Population Dynamics not only studies the characteristics of a population,
but the changes of that population.
[Slide 2] Population Dynamics or Population Change is only affected by four
things. Two cause the population to increase; birth and immigration.
Immigration is when creatures, or the agents, move into an area. And two things
cause populations to decrease, death or emigration, when creatures or agents
move out of an area.
If you’re looking at the population of fish in a lake, the things that affect the
population are the things that affect the birth and death of fish. Such as
infection, pollution, predators and how the fish move into and out of that lake.
Say through streams, rivers or perhaps by stocking of that lake.
In this video we’re going to focus on birth and death. [Slide 3] Things that give
you positive feedback or things that enhance whatever you’re looking at. It’s sort
of like good things give you more good things. In the schematic below, if you
start with some rabbits and they have all the food they need and nothing eats
them and nothing kills them, then you’ll just get bunnies.
Eventually the bunnies will lead to more rabbits as they age and that will lead to
even more bunnies. The number gets bigger and bigger. Not only does the
number get bigger and bigger, but it gets bigger and bigger faster and faster.
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Why? Let’s think about. If you start with a rabbit and it has two babies, then
each of these two babies eventually become rabbits and they go on to have say
two babies apiece. Then you end up with four rabbits. If those four rabbits have
two babies apiece, then the next time around you get eight rabbits.
Each time you go through the cycle you get more and more rabbits. This type of
growth is called exponential growth and is shown in the simple graph to the
right. You can see the rate at which the population of rabbits is growing
increases faster and faster. Now, in the real world something always interferes
with this process. Otherwise the world would be covered with rabbits.
That something is called negative feedback. [Slide 4] Negative feedback occurs
when things slow down or decrease whatever you’re looking at. In the case of
the rabbit population, negative feedback would be anything that caused rabbits
to die or slow down the birthrate of rabbits. This can occur anytime during the
whole rabbit population cycle, not just where the arrows are.
Some things that might cause negative feedback in this rabbit population loop
would be predators, or perhaps you have too many rabbits and they’re
competing for food. Or too few rabbits and the rabbits can’t find mates. Perhaps
the rabbits have some sort of health problems. They have diseases or illnesses or
perhaps there’s just not enough food to go around.
All these things cause rabbits to die or slow down the birthrate. Therefore, they
would be negative feedback. [Slide 5] How does negative feedback affect that
exponential growth curve we saw before? Well, it limits the growth of
population. Sometimes an upper limit in population occurs, as shown on the left.
This is called a logistic graph. More often the population will fluctuate as shown
on the right, which is the cyclical graph.
[Slide 6] How do you model Population Dynamics? If you were creating a very
detailed model of Population Dynamics, you would include all the things that
affect the birthrate. For instance, the health of the rabbits, the health of the
population, how much food is available to the population and if there are
enough mates for the population. And you would include all the things that
affect the death of the population, which also include health, food, population
and of course, predators.
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Now in a simpler model you would incorporate all those effects into a single
birthrate and a single death rate. That’s what we’re going to accomplish in the
modeling that we do.
[Slide 7] How would you model birth? There are many ways to model birth. You
might model birth as the interaction of agents of the same breed, such as males
and females in a population. You might say, anytime a male and a female in the
population interact with one another there’s a certain probability that
interaction will result in a birth. Or, an agent environment type of interaction.
Say for instance a location. Maybe the species can only give birth in a den. Or,
say if they have the right type of food.
Or, you could model birth based on the changes in the specific agent. For
instance maybe it has to reach a certain age or have a certain amount of energy.
That’s how we’re going to start modeling birth. We’re going to say when each
individual agent’s energy is above the birth threshold energy, then there’s a
certain chance that birth will occur. Once birth occurs energy is lost in that
process. Here’s how it might look in the model. You would ask the turtles to do a
bunch of things I’m sure. Then you would say if energy is greater than the birth
threshold energy, set the energy to be the energy minus the amount of energy
you lose during birth. Hatch however many babies you want to and move them
all forward one. Otherwise, they’ll be sitting right on top of their mother.
[Slide 8] Similarly, there are many ways to model death. Again, agent
interactions could result in death. For instance, a predator prey interaction, or
perhaps your agent encounters an infected agent and dies because it’s infected
by some disease. Also, agent environment interactions could result in death.
Perhaps some location on your net logo plane is deadly. Maybe there’s a cliff, or
maybe there are toxins in the environment; walks over that toxin, it dies.
Or again, maybe changes in the agent itself. For instance if the agent becomes
too old maybe it dies, or it gets a disease. Or, maybe it doesn’t have enough
energy to live. That’s how we usually start modeling death. If the agent’s energy
is less than the death threshold energy, then the agent would die.
Here’s how it might look in the model. You’d ask the turtles to do lots and lots of
things I’m sure, then if the energy of the turtle is less than the death threshold
energy, then the turtle will die. Let’s go take a look at a model. [Slide 9] Here we
are at the wolf/sheep predation model. Let’s go look at the code. You can see
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that we have two breeds, sheep and wolves. Each turtle, that means all the
sheep and all the wolves, have an energy.
When we create the sheep we set the energy to be some value. That’s really
important. Because if say for instance you didn’t set the initial value of the
variable energy to be something other than zero, and your death threshold was
zero, then right away all your agents would die. Let’s go look at the birth and
death of the sheep and wolves.
Here we have the birth of sheep. Now we’re saying if the value of some random
number is less than the sheep reproduction rate, then we’re going to set the
energy to be half of the energy it was. So every time the sheep reproduces it
loses half of its energy. Then we’re going to hatch one sheep and move it
forward. We’re going to do something similar for the wolves. Here we see that if
the energy is less than zero, then whatever the turtle is it dies.
Lets’ go take a look, there we are. We see that the net logo plane is scattered
with sheep and wolves. We’ve turned the grass on in the model. That means that
the grass is also another species. Let’s take a look. Now we can see when we
look at the graph that after awhile the sheep and wolf population are shown by
the blue and red lines. Just keep oscillating back and forth. We get a cyclical
population of agents.
[Slide 10] In summary, a population is group of living things. Population
Dynamics is the study of the characteristics and changes in a population. Positive
feedback results in an increase in the population, and negative feedback results
in a decrease. Some types of population growth include exponential, logistic and
cyclical. You can model birth and death using agent/agent interactions,
agent/environment interactions, or changes in the agent itself.
Thank you.
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