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# MATH-M343/S343 Section 2.5 Notes MATH-S343

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This 6 page Class Notes was uploaded by Kathryn Brinser on Wednesday September 21, 2016. The Class Notes belongs to MATH-S343 at Indiana University taught by Michael Jolly in Fall 2016. Since its upload, it has received 4 views. For similar materials see Honors Differential Equations in Mathematics at Indiana University.

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Date Created: 09/21/16

S343 Section 2.5 Notes- Autonomous Equations and Population Dynamics 9-6-16 ???????? Autonomous equation- ???????? = ???? ???? where ????/independent variable does not appear explicitly o Separable; several uses as models for exponential/logistic growth Exponential Growth o Let ???? = ????(????) be population at time ????; rate of change of ???? proportional to current value of ???? ???????? ???????? = ???????? where ???? = rate of growth/decline ???????? = ???????? ???????? 1 ???????????? = ???? ???????? 1 ∫???? ???????? = ???? ∫???? ln ???? = ???????? + ???? ???? = ???? ???????????? ???? because population always > 0, remove absolute value ???? = ???????? ???????? Given initial condition ???? 0 =0???? : ????0= ???????? ???? 0 ???? = ???? 0 ???????? ???? = ????0???? Under ideal conditions, population increases forever; not realistic Logistic Growth o Takes into account that growth rate depends on current population o Replace constant ???? with function of ????:= ℎ ???? ???? ???????? Choose ℎ ???? such that {ℎ ???? ≈ ???? > 0 when y small looks like exponential ) ℎ ???? < 0 when y large to slow growth ℎ ???? = ???? − ???????? is simplest function that satisfies above conditions, where ???? > 0 When ???? → 0, ℎ ???? → ???? when population close to 0, has constant growth rate When ℎ ???? → 0, ???? → ???? as population stops growing, it approaches a constant ???? ???????? o ???????? = ???? − ???????? ????) Verhulst/logistic equation ???????? ???? = (???? − ????)???? convenient to replace ???? wit???? ???? = ????(1 − )???? = ???? ???? ) ′ 2???? o ???? ???? = ???? − ???????? 2 = ????(1 − ???????? o Equilibrium solutions- values of ???? that make= 0; do not correspond to any change or variation ???????? in value of ???? as ???? → ∞ (ie. horizontal asymptotes in graph of ???? vs. ????) ???????? Found in general fo???????? = ???? ???? by identifying zeros/critical points of ???? ???? ???????? ???? To visualize solutions????????o= ????(1 − )????, graph ???? vs. ???? ???? (below) ( ) 0 ( ) ???? 0 = ????(1 − ) ???? = 0 ???? ???? ???? = ????(1 − )???? ???? = ???????? 1 − 1 ) = 0 0 = ???? −2???? ???? ???? ???? = 2???????? ???? 1 = ???? ???? 1 = 2 ???? ???? ???? = ???? 2 ???? ????????2 ????(2) = ???????? − ???? ???? ???? ????2 = ????( 2 − (???? 22 ???????? ???????? = 2 − 4 ???????? = 4 ???? ???????? ∴ vertex of parabola formed by ???? 2s4(), has zeros at 0,0 and ????,0 Concavity/analysis of graph: ???? ???? (−∞,0 ) (0, ) ( ,????) (????,∞ ) 2 2 ???? − + + − ????′ + + − − ???????? or ???? ′′ − + − + ???? ???? Inflection points given by ???? =2: ???????? ???????? = ???? ????) ????2 ???? ???? = ???? ???? )???????? ????????2 ???????? 2 = ???? ???? )???????? this is why product ???????? is used above (explicicomplicated) ???????? ???????? Concave up (0, ) ∪ ????,∞ ) 2 Concave down −∞,0 ∪ ( ,????)???? 2 ???? Only inflection point at ???? = ; as ???? → ???? and ???? → 0, graph gets flat 2 Phase line- representation of ???? axis in which critical points marked and arrows on line indicate whether ???? increasing or decreasing between values o Solving equation: ???????? ???? ????????= ????????(1 − ) ???? ????????2 = ???????? − ???? nonlinear, but separable ????2 = ????(???? − ) ???? 11 = ???? ????− ???? 2 1 ???? ???? ∫ 1 2???????? = ∫ ???????????? can be solved by partial fractions + 1 ????− ???? ???? 1−???????? Easier to solve as Bernoulli: ???? = ????= ???? −1 ???????? −???? − ???????? = ????2 ???????? ???????? = −???? −2 ???????? ???????? = −???? −2 (???????? − ???? )2 ???? = −???????? −1+ ???? ???? = −???????? + ???? ???? ???????? + ???????? = ???? ???????? ???? Let ???? ???? = ???? ∫???????????? = ???????????? ???????????????????? + ???????????? ???????? = ???? ???????? ???????? ???? ???? (???????? ????????)= ???????? ???????? ???????? ???? ???? (???????? ????) = ???????? ???????? ∫ ???????? ∫ ???? ???????? ???? ???????? 1 ???????? = ???? ( ) ???? ???? 1 ???????? = ???? + ???? 1 −???????? ???? = ???? ???????? −1 1 −???????? ???? = +???????????? 1 ???? = 1 −???????? ????+???????? Given ???? 0 = ???? : 1 0 ????0= 1 ???????????? −???? 0 1 = 1 ????+???? 1= + ???? ????0 ???? 1 1 ???? = ???? ????0 1 ???? ???? = 1+( −1)????−???????? ???? ???? 0 = ????????0 ????????0( +( − )????????????) ???????????????? ????0 = 0 −???????? provided 0 > 0; if0???? < 0, there would be a time ???? where ????0+ 0 −???? ???? denominator is 0 as ???? → ∞ As long as ???? > 0, ???? ???? → ???? as ???? → ∞ 0 ???????? Ex. Sketch the graphs of ???? vs. ???? ???? = = −???? + ???? and ???? vs. ???? and analyze them. ???????? ???????? o = ???? 1 − ????2) ???????? o Find zeros: 0 = ???? 1 − ????2) ???? = 0,±1 o Plot ???? vs. ???? ???? =???? : ???????? o Find critical points of ???? ???? : ′ ???? ???? 2 ???? ???? = ????????2 = −3???? + 1 2 0 = −3???? + 1 3???? = 1 2 1 ???? = 3 1 ???? = ± √3 o Analysis of graph: (−∞,−1 ) (−1, −1 ) (−1 ,0) (0, 1) ( 1 ,1) (1,∞ ) √3 √ 3 √3 √3 ???? ????) + − − + + − ???? ???? ) − − + + − − ???????? ′ − + − + − + ???? CC down CC up CC down CC up CC down CC up o Phase line: o Plot ???? vs. ????: ???? ???? → ±1 as ???? → ∞ if ???? 0lose enough to ???? = ±1 These ???? values are asymptotically stable- for each0???? ???? −∞,0 , solution approaches equilibrium solution ???? = ???? ???? = −1 as ???? → ∞, and for each ???? ???? 0,∞ , solution 1 0 approaches equilibrium solution ???? = ????2???? = 1 as ???? → ∞ ???? = 0 is asymptotically unstable- for each0???? ???? −∞,∞ , solution moves away from equilibrium solution ???? = ????3???? = 0 as ???? → ∞ Only way to ensure solution stays near 0 is to set initial value to be exactly 0 Ex. Analyze the graphs of ???? vs. ???? ???? = ???????? = ???? 1 − ???? 2) = ???? − ???? and ???? vs. ???? and analyze them. ???????? o Plot ???? vs. ???? ???? : o Plot ???? vs. ????: ???? ???? o To find concavity of ???? = ???? ???? , find 2: ???? ???????? ???? ???????? ( ) = (???? ( ???? )) ???????? ???????? ???????? = ???? ????(???? ???? ) ( ) ???? ???? ???????? = ???????? ???????? ′ = ???? ????(???? ???? ) ( ( )) = ???????? ′

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