- WELCOME TO A LESSON ON LOGISTIC GROWTH
USING RECURSIVE FORMULAS.
IN THIS LESSON WE'LL DERIVE
A RECURSIVE LOGISTIC GROWTH EQUATION
AND ALSO FIND POPULATION LEVELS USING LOGISTIC GROWTH.
IN OUR BASIC EXPONENTIAL GROWTH SCENARIO
WE HAD A RECURSIVE EQUATION
IN THE FORM P SUB N = P SUB N - 1 + R,
THE GROWTH RATE EXPRESSED AS A DECIMAL x P SUB N - 1.
IN A CONFINED ENVIRONMENT, HOWEVER,
THE GROWTH RATE MAY NOT REMAIN CONSTANT.
IN A LAKE, FOR EXAMPLE,
THERE IS SOME MAXIMUM SUSTAINABLE POPULATION OF FISH,
ALSO CALLED A CARRYING CAPACITY,
WHERE THE CARRYING CAPACITY OR MAXIMUM SUSTAINABLE POPULATION
IS THE LARGEST POPULATION THAT AN ENVIRONMENT CAN SUPPORT.
CONSIDER A LAKE WITH A CARRYING CAPACITY OF 10,000 FISH.
THE GROWTH RATE MAY LOOK LIKE THAT SHOWN IN THE GRAPH BELOW.
NOTICE HOW WHEN THE POPULATION IS 10,000
THE GROWTH RATE WOULD BE ZERO
BECAUSE 10,000 IS THE CARRYING CAPACITY.
HOWEVER, IF THE POPULATION IS BELOW 10,000,
NOTICE HOW THE GROWTH RATE WOULD BE POSITIVE,
AND THEREFORE THE POPULATION WOULD BE INCREASING.
IF THE POPULATION HAPPENED TO GO BEYOND 10,000
NOTICE HOW THE GROWTH RATE WOULD BE NEGATIVE,
AND THEREFORE THE POPULATION WOULD BE DECREASING.
SO LOOKING AT THIS RECURSIVE EQUATION HERE
FOR EXPONENTIAL GROWTH,
IF WE CAN FIND THE EQUATION FOR THE GROWTH RATE
WE COULD PERFORM A SUBSTITUTION FOR THIS R HERE
AND HAVE A RECURSIVE FORMULA FOR LOGISTIC GROWTH.
IF WE WANT TO FIND THE EQUATION OF THIS LINE HERE,
NOTICE HOW THE Y INTERCEPT IS 0.05.
AND IF YOU WANT TO FIND THE SLOPE
USING THESE TWO POINTS HERE,
NOTICE HOW WE'D HAVE TO GO DOWN 0.05 UNITS
AND RIGHT 10,000 UNITS.
WHICH MEANS, THE SLOPE WOULD BE -0.05 DIVIDED BY 10,000,
AND THIS IS ALL WE NEED
IN ORDER TO FIND THE EQUATION FOR THE GROWTH RATE.
USING THE SLOPE INTERCEPT FORM OF THE LINE
WHERE THE FORM Y = MX + B,
WE COULD WRITE IT AS Y = B + MX INSTEAD.
SO IN OUR CASE WE'D HAVE R = THE Y INTERCEPT,
WHICH IS THE 0.05 + SLOPE,
WHICH IS -0.05 DIVIDED BY 10,000 x THE INPUT VARIABLE,
WHICH IN THIS CASE WOULD BE P, THE POPULATION.
SO THIS IS THE EQUATION FOR OUR GROWTH RATE.
NOTICE HOW IF WE FACTOR OUT 0.05
WE COULD WRITE THIS AS A PRODUCT.
WE'D HAVE 0.05 x THE QUANTITY 1 - P DIVIDED BY 10,000.
THIS IS WHAT WE'LL SUBSTITUTE FOR R IN OUR RECURSIVE EQUATION.
SO BY PERFORMING THIS SUBSTITUTION, WHICH IS HERE,
AND OUR RECURSIVE EQUATION,
WE NOW HAVE A RECURSIVE FORMULA FOR P SUB N.
P SUB N = P SUB N - 1 + THIS PRODUCT HERE x P SUB N - 1.
THIS IS A RECURSIVE EQUATION THAT MODELS LOGISTIC GROWTH.
SO TO GENERALIZE THIS,
THE POPULATION IS GROWING IN A CONSTRAINED ENVIRONMENT
WITH THE CARRYING CAPACITY K,
AND ABSENT CONSTRAINT WOULD GROW EXPONENTIALLY
WITH THE GROWTH RATE R.
THEN THE POPULATION BEHAVIOR CAN BE DESCRIBED
BY THE LOGISTIC GROWTH MODEL GIVEN HERE,
WHICH WE JUST DERIVED.
AGAIN, K IS THE CARRYING CAPACITY
AND R IS THE GROWTH RATE ABSENT CONSTRAINT.
UNLIKE LINEAR AND EXPONENTIAL GROWTH,
LOGISTIC GROWTH BEHAVED DIFFERENTLY
IF THE POPULATIONS GROW STEADILY THROUGHOUT THE YEAR
OR IF THEY HAVE ONE BREEDING TIME PER YEAR.
THE RECURSIVE FORMULA PROVIDED ABOVE MODELS GENERATIONAL GROWTH
WHERE THERE IS ONE BREEDING TIME PER YEAR,
OR AT LEAST A FINDING NUMBER OF BREEDING TIMES PER YEAR.
SO GOING BACK TO OUR FISH EXAMPLE,
LET'S SAY 2,000 FISH INTRODUCED TO A LAKE
WITH A CARRYING CAPACITY OF 10,000.
ABSENT ANY RESTRICTIONS
THE FISH POPULATION WOULD GROW AT 5% PER YEAR.
FIND THE POPULATION AFTER ONE, TWO, AND THREE YEARS.
SO LOOKING AT THE GIVEN INFORMATION,
THE STARTING POPULATION, P SUB 0 = 2,000,
THE CARRYING CAPACITY IS 10,000, SO K IS 10,000.
AND THE GROWTH RATE, ABSENT RESTRICTIONS, IS 5%,
AND THEREFORE R IS 5%, EXPRESSED AS A DECIMAL WOULD BE 0.05.
AND USING OUR RECURSIVE LOGISTIC GROWTH EQUATION,
WE CAN NOW FIND THE POPULATION AFTER ONE, TWO, AND THREE YEARS.
NOTICE R IS 0.05, K IS 10,000.
AND WE NEED P SUB 0 TO FIND P SUB 1.
P SUB 1 = 2,000 + 0.05 x THE QUANTITY 1 - 2,000
DIVIDED BY 10,000 x 2,000, WHICH WOULD GIVE US 2,080.
NOTICE HOW ONCE WE FOUND P SUB 1 WE CAN NOW FIND P SUB 2.
WE NEED P SUB 1 TO FIND P SUB 2
BECAUSE THIS IS A RECURSIVE EQUATION.
JUST LIKE WE USED 2,000 HERE, HERE, AND HERE TO FIND P SUB 1,
WE NEED 2,080 HERE, HERE, AND HERE TO FIND P SUB 2.
AND THEN WE NEED P SUB 2 HERE, HERE, AND HERE TO FIND P SUB 3.
NOW LET'S LOOK AT THIS GRAPHICALLY.
THIS BLUE GRAPH MODELS THE LOGISTIC GROWTH.
NOTICE THE CARRYING CAPACITY OF 10,000 IS HERE.
NOTICE AS THE POPULATION APPROACHES 10,000
THE GROWTH RATE SLOWS.
IN COMPARISON, THE RED GRAPH WOULD BE THE EXPONENTIAL GROWTH
WITHOUT ANY RESTRICTIONS.
IF WE ZOOM IN ON THIS PIECE OF THE GRAPH HERE
WE CAN SEE FOR A SHORT PERIOD OF TIME
THE LOGISTIC GROWTH AND THE EXPONENTIAL GROWTH
ARE VERY SIMILAR,
BUT AS TIME PASSES THE LOGISTIC GROWTH SLOWS
WHILE THE EXPONENTIAL GROWTH RATE DOES NOT.
I HOPE YOU FOUND THIS INTRODUCTION HELPFUL.
THANK YOU FOR WATCHING.