- WELCOME TO A LESSON ON OVERSHOOT AND COLLAPSE
INVOLVING LOGISTIC GROWTH.
WE OBSERVED IN THE PREVIOUS LESSON
THAT LOGISTIC GROWTH MODELS HAVE A CARRYING CAPACITY.
WE SAW THAT LOGISTIC GROWTH OFTEN BEHAVES LIKE THE GRAPH
SHOWN HERE BELOW.
NOTICE HOW IT APPEARS
THE CARRYING CAPACITY WOULD BE 1,500.
THE GROWTH RATE ADJUSTS
AS THE POPULATION APPROACHES THE CARRYING CAPACITY.
HOWEVER, IF THE GROWTH RATE IS VERY HIGH,
IT CAN MOVE THE POPULATION BEYOND THE CARRYING CAPACITY.
THIS IS CALLED OVERSHOOT.
LET'S TAKE A LOOK AT SEVERAL EXAMPLES.
FOR THE POPULATION BELOW THE CARRYING CAPACITY IS 1,500,
THE INITIAL POPULATION WAS 75, AND ABSENT RESTRICTIONS,
THE GROWTH RATE WOULD BE 60% PER YEAR.
SO USING OUR RECURSIVE LOGISTIC GROWTH EQUATION HERE,
FROM OUR PREVIOUS LESSON WE SHOULD BE ABLE TO PERFORM
A SUBSTITUTION FOR R AND K,
AND THEN FIND THE POPULATION LEVELS GIVEN IN THE TABLE.
YOU MAY WANT TO PAUSE THE VIDEO
JUST TO VERIFY YOU CAN FIND THESE POPULATION LEVELS.
ANALYZING THE TABLE,
NOTICE HOW THE POPULATION GRADUALLY INCREASES.
THEN LOOKING AT THE GRAPH HERE THAT HAS ADDITIONAL VALUES,
WE CAN SEE THE POPULATION INCREASES GRADUALLY
AND THEN SLOWS AS IT APPROACHES THE CARRYING CAPACITY OF 1,500.
AGAIN, HERE'S THE CARRYING CAPACITY,
AND THIS GRAPH BEHAVES GENERALLY LIKE EXPECTED
WHEN WE HAVE LOGISTIC GROWTH.
BUT, AGAIN, AS JUST MENTIONED, THIS IS NOT ALWAYS THE CASE.
SO LET'S TAKE A LOOK AT A SECOND EXAMPLE.
FOR THE POPULATION BELOW, AGAIN, THE CARRYING CAPACITY IS 1,500,
BUT IN THIS CASE THE INITIAL POPULATION WAS 1,200
AND ABSENT ANY RESTRICTIONS THE GROWTH RATE WOULD BE 160%.
SO OF COURSE THIS WILL AFFECT
THE RECURSIVE LOGISTIC GROWTH EQUATION GIVEN HERE,
WHICH WE SEE HERE BELOW WITH THE GROWTH RATE
AND CARRYING CAPACITY SUBSTITUTED INTO THE EQUATION.
LET'S BEGIN BY LOOKING AT THE TABLE OF VALUES.
NOTICE HOW P SUB 1, P SUB 3, AND P SUB 5
ALL HAVE POPULATION LEVELS ABOVE THE CARRYING CAPACITY OF 1,500.
THIS IS OVERSHOOT.
BUT NOTICE HOW THE YEAR AFTER THE OVERSHOOT
THE POPULATION GOES BACK DOWN BELOW THE CARRYING CAPACITY.
BUT IT DOES APPEAR OVER TIME THE POPULATION IS GOING TO SETTLE
NEAR THE CARRYING CAPACITY OF 1,500.
LOOKING AT THE GRAPH, AGAIN, HERE IS THE CARRYING CAPACITY,
THE POINTS ABOVE THIS HORIZONTAL LINE REPRESENT OVERSHOOT,
AND OUR GRAPH WOULD LOOK LIKE THIS.
NOT THE TRADITIONAL GRAPH WE WOULD EXPECT
WHEN WORKING WITH LOGISTIC GROWTH.
NOW, LET'S TAKE A LOOK AT ONE MORE EXAMPLE.
AGAIN, THE CARRYING CAPACITY IS 1,500,
THE INITIAL POPULATION WAS 1,100,
BUT ABSENT ANY RESTRICTIONS THE GROWTH RATE WOULD BE 220%.
SO, AGAIN, USING OUR RECURSIVE EQUATION HERE BELOW,
WE SHOULD BE ABLE TO GENERATE THE POPULATION LEVELS
GIVEN IN THE TABLE.
ANALYZING THE TABLE,
NOTICE HOW, AGAIN, P SUB 1, P SUB 3, AND P SUB 5
ALL HAVE POPULATION LEVELS ABOVE THE CARRYING CAPACITY OF 1,500,
ANOTHER EXAMPLE OF OVERSHOOT.
BUT ANALYZING THIS MORE CLOSELY, NOTICE HOW IT DOES APPEAR
THAT THE POPULATION LEVELS ARE BOUNCING BACK AND FORTH
BETWEEN ONE POPULATION LEVEL ABOVE THE CARRYING CAPACITY
AND ONE POPULATION LEVEL BELOW THE CARRYING CAPACITY.
IF WE PLOT THESE POINTS ON THE COORDINATE PLANE,
IT BECOMES EVEN MORE OBVIOUS.
HERE'S THE CARRYING CAPACITY, OUR GRAPH WOULD LOOK LIKE THIS,
AND THIS IS CALLED A TWO CYCLE.
AGAIN, THIS IS BECAUSE THE POPULATIONS
ARE BOUNCING BACK AND FORTH
BETWEEN THE SAME HIGHER LEVEL POPULATION
AND SAME LOWER LEVEL POPULATION.
WELL, IT MIGHT BE TEMPTING TO TREAT THIS
ONLY AS A STRANGE SIDE AFFECT OF MATHEMATICS.
THIS IS ACTUALLY BEEN OBSERVED IN NATURE.
RESEARCHERS FROM THE UNIVERSITY OF CALIFORNIA
OBSERVED A STABLE TWO CYCLE IN A LIZARD POPULATION IN CALIFORNIA.
TAKING THIS EVEN FURTHER,
WE CAN GET MORE AND MORE EXTREME BEHAVIORS
AS THE GROWTH RATE INCREASES HIGHER.
IT IS POSSIBLE TO GET STABLE FOUR CYCLE, AND EIGHT CYCLES,
AND HIGHER.
HERE'S AN EXAMPLE OF A FOUR CYCLE,
AND HERE'S AN EXAMPLE WHERE THERE SEEMS TO BE NO PATTERN
BETWEEN THE HIGHER POPULATIONS AND LOWER POPULATIONS.
IF A POPULATION OVERSHOOTS THE CARRYING CAPACITY BY TOO MUCH,
THIS CAN DEGRADE THE CARRYING CAPACITY
AND LEAD TO A SEVERE DECREASE OF THE POPULATION.
THIS IS CALLED A COLLAPSE.
SO LOOKING AT THE GRAPH HERE BELOW, HERE'S THE OVERSHOOT.
AND BECAUSE THE OVERSHOOT IS SO HIGH
ABOVE THE CARRYING CAPACITY,
NOTICE HOW THIS DEGRADES THE CARRYING CAPACITY
AND RESULTS IN A COLLAPSE,
WHICH IS A DRAMATIC DECREASE IN THE POPULATION,
WHICH WE SEE HERE.
THAT'S IT FOR THIS LESSON.
I HOPE YOU FOUND THIS HELPFUL.