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Simulating Competition and Logistic Growth - YouTube
Channel: Primer
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we've seen how creatures that replicate
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can have their numbers grow
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exponentially without limit but in the
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real world there are limits so a more
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realistic growth curve would look
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something like this sorry buddy
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[Music]
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as we've built up our model in the last
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few videos we've been running
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simulations where the computer steps
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through time and at each time step it
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decides which creatures live die and
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reproduce according to certain odds and
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we both an equation to help us predict
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what we expect to happen from one
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instant to the next in the simulation
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the expected change in the number of
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creatures is equal to the creatures
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replication chance - its death chance
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all times the current number of
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creatures and we can graph this equation
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to help us visualize our prediction the
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most interesting case is when R is
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greater than D for example with a
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replication chance of 10% and a death
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chance of 5% then this graph becomes a
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straight line with a positive slope the
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more creatures there are the more new
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creatures we expect to appear from one
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time to the next this leads to
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exponential growth we went over this
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pretty quickly but an earlier video in
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the series called how to grow
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exponentially it goes through in more
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detail
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ok so that's what the world looks like
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when growth is completely unchecked but
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what should it look like if we want
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growth to level off at some point to
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figure this out let's work backward we
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want the population curve to look
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something like this it's like an
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exponential curve toward the beginning
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but it levels out at a certain point at
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50 creatures for the population to level
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out here we need the expected change per
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time step two go to zero when there are
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50 creatures so this curve is going to
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need to bend downward how can we change
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the equation to make that happen
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this function equation already gives us
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Delta equals 0 when n is 0 but we want
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Delta to also be 0 when n is 50 one way
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to do this is to make the creatures more
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likely to die when there are lots of
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creatures around there's only so much
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space and food in the environment so
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when it's crowded a creature might
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starve
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to do this we'll leave the bass death
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chance alone but we'll include an extra
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term to adjust to the overall death
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chance based on crowding what should
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this term be well we want the term to be
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small when there aren't many creatures
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and we want it to be big when there are
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a lot of creatures a simple way to
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achieve this is to write it as the
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current number of creatures multiplied
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by a constant one and as small the
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effect of carding will be small and when
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and as large the effect will be large
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let's call this constant the crowding
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coefficient just to give it a short name
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its value specifies how much the death
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chance goes up for each creature when we
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add a new creature so if the value is
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say 0.001 that means adding another
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creature increases the death chance of
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all creatures by a tenth of a percent
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the new creature is eating food and
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taking up space so there's less to go
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around for everyone else and when we
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have a lot of creatures this term really
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adds up and because I looked ahead when
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picking these numbers a crowding
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coefficient of 0.001 does cause Delta to
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be zero when n is 50 this is because the
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death chance when adjusted for crowding
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becomes equal to the replication chance
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for creature
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so each creature is just as likely to
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die as it is to reproduce the
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replication and death chances balance
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each other up and we found equilibrium
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again to give you some of the usual
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terminology this equilibrium number is
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called the carrying capacity because
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it's the largest number of creatures
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that the environment can sustainably
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support and this number over time curve
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is called a logistic growth curve as
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opposed to an exponential growth curve
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now that we've decided how to tweak the
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equation and seen how it affects the
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graph let's double check that this
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actually does predict this s-shaped
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logistic growth curve when n is small
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the Delta curve is pretty similar to
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that upward sloping line from the
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exponential case so we'll expect the
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population to look like it's growing
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about exponentially at first in this
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middle region the Delta curve is near
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its maximum and it's mostly horizontal
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so the overall expected growth rate
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doesn't change much the growth rate is
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still high but it's just not speeding up
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anymore
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and finally in this last region the
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growth rate is actually slowing down
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towards zero so we'll expect the
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population to level off and if n goes
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above the carrying capacity which again
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is an equilibrium number the growth rate
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goes negative pushing n back down all
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right let's run a simulation to see
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whether this prediction works it sort of
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works and remember this is all based on
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chance though so to really see how good
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this prediction is we need to look at
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many simulations at once next let's look
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at what happens if new kinds of
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creatures appear through mutation this
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green creature will come out of 1% of
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Blues replications and it'll be slightly
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less good at replication than the blue
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creatures with a replication chance of
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8% but its replication chance is still
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higher than its death chance and this
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orange creature will also come out of 1%
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of glues replications and this one will
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have a lower death chance all three of
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these creatures will share the same
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resources so their Delta equations would
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have a crowding term that includes the
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total number of all kinds of creatures
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if we start a simulation with a few blue
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creatures how do you expect things to go
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[Music]
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as you might have guessed orange
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eventually takes over it's not enough
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anymore for blue to be good at surviving
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in isolation it now needs to be better
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than its competitors to maintain numbers
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one surprising thing in this simulation
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is that Green is doing better than blue
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after 500 time steps you wouldn't expect
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that since it has the worst stats of all
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the creatures but this is a good example
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of how luck is a big part of you
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involving systems the most likely
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outcome doesn't always happen alright
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that's it for the fundamentals of
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limited growth and competition but
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before we say goodbye in this video
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let's take stock of where we are we've
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seen how replication can lead to
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exponentially growing populations we've
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seen how mistakes and replication can
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lead to new kinds of creatures leading
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to diversity and just now we saw how a
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finite pool of resources puts a cap on
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populations and causes competition
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between different types of creatures
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replication mutation and competition
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make up the core of evolution anywhere
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replicators exist even if there's life
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on other planets everything we've said
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so far would apply we're not done yet
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though so far we've been making all the
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decisions ourselves you could say that
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we've been artificially selecting
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successful creatures in the next video
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we'll let go of the reins and let the
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selectmen happen a bit more naturally
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see you then
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[Music]
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[Music]
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