Logistic+Differential+Equations+(BC)

= INTRODUCTION TO POPULATION GROWTH: =

It could be assumed that population can be defined by the function:
where (Figure 1.1)

The graph to the above represents what population growth would look like if it was defined by the above function. However the population cannot continuously be increasing or the planet would become overpopulated so there are constraints on the population such as available food supply, habitat, and living space are just a few of the examples. This introduces the idea of a carrying capacity, M, or a limit that the population cannot exceed for the environment can not support it.

So another equation has been proposed for the rate at which population is increasing/decreasing. Where M is the carrying capacity and k is the growth constant:
(Figure 1.2)

HERE IS THE STEPS REQUIRED TO FIND THE PARTICULAR SOLUTION:


=** SEPERATE THE VARIABLES AND INTEGRATE BOTH SIDES **=

[[image:http://latex.codecogs.com/gif.latex?%5Cint%20%5Cfrac%7BA%28M-P%29%7D%7BP%28M-P%29%7D+%5Cfrac%7BBP%7D%7BP%28M-P%29%7D=kt+C]] Find a common denominator and set equal to the original numerator 1
= IF P=M THEN B=1/M AND IF P=0 THEN A=1/M = = SOOO... =

PLUG IN THE CONSTANTS AND ANTI DERIVE USING THE RULE

= MULTIPLY BOTH SIDES BY M = = =

[[image:http://latex.codecogs.com/gif.latex?ln%5Cleft%20%7C%20P%20%5Cright%20%7C-ln%5Cleft%20%7C%20M-P%20%5Cright%20%7C=Mkt+C]] APPLY RULES OF LOGS, REJECT NEGATIVE VALUES AND RAISE BOTH SIDES BY e
= = = THE LN AND e CANCEL ON THE LEFT LEAVING YOU WITH = = = = NOW FROM HERE ON YOU JUST DO SOME ALGEBRA = = = = = = = =  = = = =  WHERE A IS BASED OFF THE INITIAL CONDITIONS OF THE FUNCTION =

We can determine the value of A by using the below equation which was found when t=0 (initial conditions)
= SOOOO... =

We can also conclude that:


= EXAMPLES: = = Multiple choice Questions: = === 1. If the DEC relocates 3000 beaver onto Strawberry Island when the island can only support 1000, what is happening to the rate at which the beaver population is changing, based of the logistic differential equation. ===

e. The limit does not exist
= Short response Questions (CALCULATOR AVAILABLE): =

POPULATION OF LAREDO, TX

 * = YEARS AFTER 1950 ||= POPULATION ||
 * = 0 ||= 10,571 ||
 * = 20 ||= 81,437 ||
 * = 30 ||= 138,857 ||
 * = 40 ||= 180,650 ||
 * = 50 ||= 215,794 ||
 * = 53 ||= 218,027 ||

CALCULATOR NOT AVAILABLE
=== 3. In 1985 and 1987, the Michigan Department of NAtural Resources airlifted 20 moose from Algonquin Park, Ontario to Erie County in Buffalo, NY. It was originally hoped that the population P would reach carrying capacity in about 30 years with a growth rate of === .

b. Solve the differential equation with the initial condition P(0) = 20
=ANSWERS:= === 1. According to the logistic differential equation,, the rate will be increasing if P<M but in this case the 3000 beavers on the island currently exceeds the carrying capacity of 1000 so the rate is decreasing. The answer is B. ===

=== 2. As the function approaches infinity the part of the equation will be affected because e will be approaching negative infinity so the limit would equal 0 since e to the negative infinity is. Since that part is 0 and it is being multiplied by A or 42.642 then that product equals 0 so your left with 782934/1 so the limit is 782934 or answer C. ===

b. The number it will reach in the future is also the carrying capacity which is the numerator of the equation which is 232740 people.
=== c. The population will reach 225,000 in the year 2010 (60 years after 1950), you can find this answer by plugging in the equation into Y1 of the calculator and looking at the table to see when the value crosses over from under 225,000 to over it. Another way to get a more percise value would be to in addition to Y1 plot a Y2 of 225,000 and then use 2nd_TRACE_intersect_ENTER_ENTER_ENTER. === === d. The value of Mk = .101 and with the value of M being 232739.9 you can easily find the value of k by division, finding k to be equal to 4.340 *10^-7. Now that all the values are know just plug them into the logistic differential equation: ===











=== By using the equation ,, we can figure out what A from the given P(0) = 20, so ===

= WORKS CITED: =

= =