Logistic Growth: Using Recursive Equations

Logistic Growth: Using Recursive Equations


– 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.  

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2 thoughts on “Logistic Growth: Using Recursive Equations”

  • Mardon Usmanov says:

    Thank you very much, I am very appreciative of this video and I believe that it shall get me through my quiz tomorrow, which I was very clueless on before I found this video! Glad my teacher recommended it for my class

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