F-IF Logistic Growth Model, Abstract Version

Transcription

1 F-IF Logistic Gowth Model, Abstact Vesion Alignments to Content Standads: F-IFB4 Task An impotant example of a model often used in biology o ecology to model population gowth is called the logistic gowth model The geneal fom of the logistic equation is In this equation epesents time, with (t) = 0 e t + 0 ( e t 1) t t = 0, 0 coesponding to when the population in question is fist measued; and ae all eal numbes with being called the ''caying capacity'' while is a gowth ate and is nomally a positive numbe a Explain why the value 0 epesents the population when it is fist measued b Explain why, as time elapses, the population stabilizes, appoaching the value c Explain how the behavio of changes if the gowth ate is inceased o deceased d Below is the gaph of a paticula logistic function, showing the gowth of a bacteia population Using the gaph, identify 0 and 1

2 e Using the values of 0 and fom the pevious pat, sketch the gaph of the logistic function given by Q 0 e Q(t) = 2t + 0 ( e 2t 1) Q Note that is the same as except that the gowth ate has been doubled IM Commentay This task is fo instuctional puposes only and students should aleady be familia with some specific examples of logistic gowth functions such as that given in ''Logistic gowth model, concete case'' This is an impotant example of a function with many constants: 0 the initial population, the caying capacity, and the gowth ate Each of these has a specific meaning which detemines the shape of the gaph and, in case of 0 and, can be eadily estimated using the gaph 0 The goal of this task is to have students appeciate how the diffeent constants (,, and ) influence the shape of the gaph Only has been changed hee, in pat (e), because it is the most abstact of these numbes If the instucto wishes to change the othe numbes, the function used to geneate this paticula gaph is 2

3 (t) = e t,, Note that this is not given in the fom of the logistic equation given above with 0 and It coesponds, afte algebaic manipulation, to the case whee = 1, =, 0 = Showing this identity is a wothwhile algebaic execise which equies caeful 11 manipulation of factions and exponential functions Edit this solution Solution a The population is fist measued when lugging into the expession fo gives t = 0 t = 0 (0) = 0 e ( e 0 1) = 0 e ( e 0 1) = 0 = 0 b As the name suggests, the ''caying capacity'' is the maximum population that the envionment can sustain so it is, in this case, the value that the population appoaches as gows Expanding the expession fo the denominato of,, the t 0 e t 0 0 e t t 0 does not depend on t So as t gows, the denominato is bette and bette appoximated by 0 e t The numeato is 0 e t Taking the quotient of 0 e t by 0 e t gives, the caying capacity So as t gows, the values of become close and close to exponential tem, gows apidly as gows The est of the denominato,, e t c The ate detemines how quickly the exponential function gows Inceasing will incease the ate of gowth of e t This means that the values of will appoach the caying capacity moe apidly since, as seen in pat (b), it is the gowth of the e t tem that make the population appoach If is deceased, then gows moe slowly and the values of appoach moe slowly e t (0) y d The value is the -intecept of the gaph of This value is about half way 3

4 between and and since the units in population fo the gaph ae million this means that thee ae about million bacteia at the beginning of the expeiment The caying capacity appeas to be close to which epesents million bacteia If the actual fomula fo the function is given, then the -intecept can be calculated exactly e 0 (0) = = e 0 11 Since one unit on the gaph epesents bacteia, thee ae a little unde million bacteia when the population is fist measued The caying capacity is 0, 000, 000 as estimated above: since the e t tem, in the population fomula, becomes less and less significant as t gows the population appoaches units o 1 0, 000, e 2t e When the value is doubled, looking at pat (b), what this means is that the exponential tem in the denominato becomes the dominant tem moe quickly, twice as quickly in fact So we expect the population to gow moe apidly at the beginning befoe appoaching the caying capacity This is shown in the gaph below Note that this gaph is pecise because the pecise values of, 0, and wee used: not knowing this infomation the best one can do is daw a cuve with the same initial population, the same caying capacity, and which gows moe apidly initially y 10, 000, 000 4

5 F-IF Logistic Gowth Model, Abstact Vesion Typeset May 4, 2016 at 20:28:21 Licensed by Illustative unde a Ceative Commons Attibution-NonCommecial-ShaeAlike 40 Intenational License