TSC 220X Spring 2011 Problem Set #5

Transcription

1 Name: TSC 220X Spring 2011 Problem Se #5 This problem se is due in class on Monday 21 March The problem se should be yped. We do no expec Pulizer Prize winning wriing, bu answers should be complee, ariculae, and well-reasoned. Answers comprising liss may be free form in syle and puncuaion, bu essay quesions should follow principles of English usage, syle, and grammar. Loquacious, verbose, garrulous answers are no favored by hose grading your work. Lae problem ses will be docked 50%. Quesion 1 (20 poins). Read he aricle eniled Caasrophic shifs in ecosysems by Maren Scheffer, Seve Carpener, Jonahan A. Foley, Carl Folke & Brian Walker, which appeared in Naure in Ocober of 2001 (p ). Answer he following quesions. a. Wha happens if he sysem in on he upper curve in Figure 1c and he condiions move he he righ beyond F 2? b. If he sysem is on he lower curve in Figure 2b how far mus condiions move o he lef before heir is a movemen o he upper curve? c. Give some examples of sochasic evens in naure. d. Considering Figure 3, he erm resilience is someimes used o refer o he size of he valley. Wha does his mean? e. Consider he following quoe from he aricle, Reducion of nurien concenraions is ofen insufficien o resore he vegeaed clear sae. Indeed, he resoraion of clear waer happens a subsanially lower nurien levels han hose a which he collapse of he vegeaion occurred. Explain his is simple erms using conceps we have discussed in class. f. Explain why we migh discuss Diaedema anilliarum, herbivorous fish, and macroalgae in connecion wih moose and wolves on Isle Royale. Is here a ipping poin for coral reefs? g. Wha are wo possible sable saes for land is suiable for boh grass and rees? Wha migh keep he landscape in one sae or he oher? How migh human acion push he landscape irreversibly oward one sae or anoher? h. Consider he following wo senences from he aricle, A new generaion of coupled climae-ecosysem models demonsraes ha Sahel vegeaion iself may have a role in he drough dynamics, especially in mainaining long periods of we or dry condiions. The mechanism is one of posiive feedback: vegeaion promoes precipiaion and vice versa, leading o alernaive saes. How migh aciviy upse an equilibrium wih moderae levels of vegeaion as compared o deserificaion? i. Now consider Table 1 in he paper. Using he row relaed o woodlands, exemplify he following saemens. Alhough he specific deails of he reviewed sae shifs differ widely, an overview (Table 1) shows some consisen paerns. 1) Firs, he conras among saes in ecosysems is usually due o a shif in dominance among organisms wih differen life forms. 2) Second, sae shifs are usually riggered by obvious sochasic evens such as pahogen oubreaks, fires or climaic exremes. 3) Third, feedbacks ha sabilize differen saes involve boh biological and physical and chemical mechanisms. j. Two free poins. 1

2 Quesion 2 (14 ps). a. Consider an equaion which explains he growh of a populaion N as a funcion of ime. Specifically suppose ha 2e r 2e 2 if r 1 2 (1) graph his funcion Wha happens o b. If you know calculus you can compue he rae of change of wih respec o ime by aking is derivaive. In general we obain. 2e r 2re r r if we subsiue for 2e r (2) e 2 if we subsiue 1 for r. 2 graph Wha happens o c. If we divide equaion 2 by 2e r we obain 2rer 2e r (3) r 1 2 graph Wha happens o d. Consider an equaion which explains he growh of a populaion N as a funcion of ime. Specifically suppose ha (4) where K is he carrying capaciy of he environmen, i.e., a number which will never exceed. If we le K 6 and r 2 1, we hen obain 2

3 2(6)e e e e 2 1 (5) 6e 2 3+ e 2 1 6e 2 2+e 2 graph 6e 2 2+e 2 Wha happens o e. If you know calculus you can compue he rae of change of wih respec o ime by aking is derivaive. In general we obain (K+ 2(er 1))(2rKe r ) ( ) (2re r ) () 2 2rK2 e r + 4rKe 2r 4rKe r 4rKe 2r () 2 (6) 2rKer (K+ 2e r 2 2e r ) () 2 2rKer (K 2) () 2 If we le K 6 and r 1 2, we hen obain 3

4 2rKer (K 2) () 2 2( 1 2 (6)e 2 (6 2) ( ( )) 6+2 e e 2 ( ) 4+2e 2 2 (6)(4)e 2 ( ( )) 2 2 (7) (6)(4)e 2 ( 4 ( ) 2 6e 2 ( ) 2 graph 6e 2 ( ) 2 Wha happens o f. If we divide equaion 6 by 2Ker K+2(e r 1) we obain 2rKe r (K 2) (K+2(e r 1)) 2 K+2(e r 1) 2rKer (K 2) () 2 K+ 2(er 1) (8) r(k 2) We can wrie equaion 8 in a more inuiive fashion by facoring ou r and hen adding and subracing one from he remaining expression. 4

5 r(k 2) (K 2) r + 1 K+ ] 2(er 1) r 1+ (K ] 2 K 2er + 2) 2e r ] r 1 r 1 ] K (9) Wha happens o as N ges closer o K? Compare wih equaion 3. Wha abou when N is very small relaive o K g. If we le K 6 and r 1 2 in equaion 8, we hen obain r(k 2) () 1 ( 2 (6 2) ( )) 6+2 e e e 2 graph 1 Wha happens o h. How migh human behavior affec he growh rae r in he models in equaions 3 and 9. Quesion 3 (16 poins). Consider he aricle Our Man-Made Energy Crisis by Nansen G. Saleri which appeared in he Wall Sree Journal on March 9, a. Based on his aricle, maerials from he course, and oher maerials, answer he quesion Is he world likely o run ou of energy? Discuss supplies of curren sources of energy, poenial new sources, echnological change, human behavior including indusrializaion and populaion growh, pricing, leaving i in he ground, and on. b. Why migh we wan o reduce he consumpion of fossil fuels even if socks may seem o be pleniful? c. Now answer he quesion, Is he world likely o run ou of some key maerials? d. Discuss he relaionship beween your answer o iem a and your answer o iem c and how his relaes o naural resource and environmenal policy. (10) 5