INTERSPECIFIC COMPETITION AND COMPETITIVE EXCLUSION

Transcription

1 9 INTERSPECIFIC COMPETITION AND COMPETITIVE EXCLUSION Objecives Program he Loka-Volerra model of inerspecific compeiion in a spreadshee. Undersand he compeiive exclusion principle and how i relaes o he model. Use he model o explore compeiive exclusion and coexisence. Deermine under wha condiions wo compeing species can coexis, in erms of heir compeiion coefficiens, carrying capaciies, and inrinsic raes of increase. Suggesed Preliminary Exercise: Logisic Populaion Models INTRODUCTION Our previous models of populaion dynamics considered only one populaion. As informaive as hose models were, i should be obvious ha real populaions do no exis in isolaion, bu share habias wih populaions of oher species. In many cases, coexising species will inerac by inerspecific compeiion, predaion, parasiism, muualism, or oher ecological ineracions. More realisic models mus ake such ineracions ino accoun. In he 1920s, Vio Volerra and Alfred Loka (1932) independenly developed models of inerspecific compeiion (compeiion beween wo species), and invesigaed he condiions ha would permi compeing species o coexis indefiniely. In his exercise, you will build a discree-ime version of heir coninuous-ime models. An imporan ecological generalizaion, he compeiive exclusion principle, has grown ou of he Loka-Volerra model and from oher sources. This principle saes ha wo species canno coexis unless heir niches are sufficienly differen ha each limis is own populaion growh more han i limis ha of he oher. In oher words, if here is oo much niche overlap, one species will compeiively exclude he oher. In realiy, wheher wo species coexis depends no only on heir compeiive ineracions wih each oher, bu also on heir ineracions wih he abioic environmen and wih oher species no included in his simple model. Neverheless, as wih oher models in his book, he compeiive exclusion principle has proven fruiful in simulaing research and undersanding ecological ineracions in he naural world.

2 126 Exercise 9 Model Developmen To review, he geomeric model of populaion growh, N +1 = N + RN, includes no effec of compeiion. The populaion increases by RN in every ime inerval, wihou any limiaions such as migh be imposed by finie resources. The logisic model of populaion growh includes inraspecific compeiion (compeiion beween individuals of he same species). To keep hings (relaively) simple, we will develop our model of inerspecific compeiion beginning wih his form of he logisic model: N N RN K N + 1 = + K Equaion 1 where K is he carrying capaciy, or larges susainable populaion. The value of K is se by available resources and by each individual s resource demand. This version of he logisic model has inraspecific compeiion buil ino i in he erm (K N )/K. This erm reduces he populaion growh rae in response o he addiion of each new member of he populaion, represening he reducion in per capia birh rae, and increase in per capia deah rae, caused by compeiion for limied resources. You can review Exercise 8, Logisic Populaion Models, for more informaion abou his model. The Loka-Volerra model of inerspecific compeiion builds on he logisic model of a single populaion. I begins wih a separae logisic model of he populaion of each of he wo compeing species. Populaion 1: N N R N K 1 N 1, 1, + 1 = 1, + 1 1, K1 Populaion 2: N N R N K 2 N 2, 2, + 1 = 2, + 2 2, K2 Noe he use of subscrips 1 and 2 o denoe which species populaion is being modeled. Each populaion has is own rae of increase R and carrying capaciy K, and hese may differ beween he wo species. Nex we build inerspecific compeiion ino each of hese equaions. In he model of populaion 1 above, we assume ha each new member of populaion 1 reduces resources available o each member of populaion 1, and hus reduces populaion growh rae. In he wo-species model, new members of populaion 2 will also reduce resources available o members of populaion 1 his is, afer all, he meaning of inerspecific compeion. The simples way o model his would be o modify he (K 1 N 1, )/K erm ino (K 1 N 1, N 2, )/K 1. However, his assumes ha each addiional member of populaion 2 will affec populaion 1 exacly as much as an addiional member of populaion 1. Tha is no necessarily he case, so we muliply N 2, in his erm by a compeiion coefficien, α 12 o express how much effec each addiional member of populaion 2 has on populaion 1, relaive o he effec of a new member of populaion 1. We modify he model for populaion 2 in a parallel way. The resuling Loka-Volerra model of wo-species compeiion is: Populaion 1: N N R N K 1 N 1, 12 N 2, 1, + 1 = 1, + α 1 1, Equaion 2 K1 Populaion 2: N N R N K 2 N 2, 21 N 1, 2, + 1 = 2, + α 2 2, Equaion 3 K2 Noe he subscrips on he compeiion coefficiens: α 12 expresses he effec of one member of populaion 2 on he growh rae of populaion 1; α 21 expresses he effec of one member of populaion 1 on he growh rae of populaion 2.

3 Inerspecific Compeiion and Compeiive Exclusion 127 In broad erms, he quesion Loka and Volerra asked was, Wha will happen o he populaion dynamics of hese wo populaions, given various values of he model parameers? Are here parameer values ha will produce a winner and a loser, one populaion ha persiss while he oher goes exinc? This would be compeiive exclusion. Will oher values resul in coexisence, in which boh compeing populaions persis indefiniely? You will look for answers o hese quesions boh analyically (algebraically) and graphically (using he spreadshee). Equilibrium Soluions One approach o answering he quesions posed above is o look for equilibrium soluions o Equaions 2 and 3. If populaion 1 is a equilibrium, hen N 1,+1 = N 1, and we can subsiue N 1, for N 1,+1 : N N R N K N α N 1, = 1, + 1 1, K 1 1, 12 2, 1 Subracing N 1, from boh sides of he equaion gives us In words, his equaion says he populaion sops growing when i is a equilibrium, which should come as no surprise. This equaion is saisfied if N 1, = 0 or if R 1 = 0, bu hese soluions are rivial. The equaion is also saisfied by he more ineresing case of K 1 N 1, α 12 N 2, = 0 If we add N 1, o boh sides and rearrange he erms, we ge N 1, α 12 N 2, Equaion 4 Noice ha his equaion is in he general form of a linear equaion, y = a + bx, and is herefore a sraigh line. We call his line a zero ne growh isocline, or ZNGI, because anywhere along i, populaion 1 has zero ne growh. In oher words, his is an equilibrium soluion for populaion 1. Jus as x and y in he general linear equaion y = a + bx can be used as coordinaes for graphing, so we can use N 1, and N 2, as coordinaes o graph Equaion 4. We can graph his isocline by finding any wo poins along i and connecing hem wih a sraigh line. Two convenien poins are where N 2, = 0 and where N 1, = 0. If N 2, = 0, hen we solve for N 1,. Equaion 4 becomes which reduces o 0 = 1 1, N 1, α 12 0 N 1, In words, if here are no members of populaion 2 in he habia, populaion 1 will sabilize a is own carrying capaciy, K 1. This seems a reasonable soluion. If we se N 1, = 0, and hen solve for N 2,. Equaion 4 becomes 0 α 12 N 2, and adding a 12 N 2, o boh sides gives us RN K N α N 1 1, 12 2, K1 Dividing boh sides by α 12 gives us α 12 N 2, N 2, / α 12

4 128 Exercise 9 N 2 K 1 /a 12 K 2 (N 1,,N 2, ) ZNGI for Pop. 1 ZNGI for Pop. 2 N 2 K 2 /a 21 K 1 Figure 1 Zero ne growh isoclines (ZNGIs) generaed by he Loka- Volerra model of wo-species compeiion. The poin (N 1,, N 2, ) represens he wo populaions a ime. In words, if here are K 1 /α 12 members of populaion 2 in he habia, here will be no resources lef over for populaion 1, and is numbers will go o zero. We can find a ZNGI and wo poins on i for populaion 2 in he same manner. N 2, = K 2 α 21 N 1, If N 1, = 0, hen N 2, = K 2 If N 2, = 0, hen N 1, = K 2 /α 21 We can draw hese isoclines on a linear graph of he wo populaions as shown in Figure 1. If we plo N 1 on he horizonal axis and N 2 on he verical, hen he soluion poins found become he inerceps of he isoclines on he axes. We can graph he populaions of he wo species a any ime by a poin on a graph. If he poin falls below and/or o he lef of a species isocline, ha populaion will coninue o increase. If he poin falls above and/or o he righ of a species isocline, ha populaion will decrease. In he case of he poin shown in Figure 1, populaion 1 will increase and populaion 2 will decrease. As ime passes, he poin will move downward (populaion 2 decreases) and o he righ (populaion 1 increases), and he poin describing he wo populaions will race some rajecory across he graph. Noice ha ime does no appear on eiher axis of his graph. Figure 1 is called a phase diagram, and he space bounded by is axes is called phase space. You will plo he rajecory of wo changing populaions hrough he phase space and from ha deermine wheher one species excludes he oher, or if hey coexis. The isoclines need no be arranged as shown in Figure 1; heir arrangemen will depend on he values of K 1, K 2, α 12, and α 21. PROCEDURES The quesions Loka and Volerra asked, and which you will answer in his exercise, are: Wha values of hese parameers will cause populaion 1 o exclude populaion 2,

5 Inerspecific Compeiion and Compeiive Exclusion 129 and vice versa? Wha parameer values will allow he wo populaions o coexis indefiniely? Wha do hese oucomes, and heir associaed parameer values, mean in ecological erms? As always, save your work frequenly o disk. INSTRUCTIONS ANNOTATION A. Se up he spreadshee. 1. Open a new spreadshee and se up iles and column headings as shown in Figure A B C D E F Loka-Volerra Model of Inerspecific Compeiion N 1 N 2 Parameers End poins R 1 --> 1.00 N 1 = 0 --> <-- N 2 = K 2 K 1 --> 1200 N 1 = K 2/a 21 --> <-- N 2 = 0 a 12 --> 0.75 N 1 --> <-- N 2 = 0 R 2 --> 1.00 N 1 = 0 --> <-- N 2 /a 12 K 2 --> 1000 a 21 --> 0.50 Time ( ) Figure 2 Ener only he ex iems for now. These are all lierals, so jus selec he appropriae cells and ype hem in. You mus leave cells B10 and C10 empy for your graphs o come ou properly. The values in cells B5 hrough C8 are he coordinaes of he endpoins of he ZNGIs for he wo species. How we go hese values will be explained in subsequen seps. 2. Se up a linear ime series from 0 o 50 in cells A11 hrough A Ener he values shown for he parameers. 4. Ener zeros in cells B5, C6, C7, and B8. See he exercise Spreadshee Hins and Tips for deails. These are in cells F4 hrough F9. Do no ener anyhing in cells B5 hrough C8 ye. These are ZNGI endpoins where each populaion is iself a zero. Cells B5 hrough C8 hold coordinaes for he endpoins of he wo ZNGIs. You mus lay ou hese endpoin cells as shown for your graphs o work properly. 5. In cells B7 and C5, ener formulae o echo he carrying capaciies of populaions 1 and 2, respecively. 6. Ener formulae o calculae he oher ZNGI endpoins. In cell B7, ener he formula =F5. In cell C5, ener he formula =F8. These are ZNGI endpoins where he compeing populaion is a zero. When you change carrying capaciies laer in he exercise, your changes will auomaically be carried over o he ZNGI endpoins. In cell B6, ener he formula =F8/F9. This corresponds o N 1, = K 2 /α 21. In cell C8, ener he formula =F5/F6. This corresponds o N 2, /α 12.

6 130 Exercise 9 7. Ener iniial populaion sizes (N 1,0 and N 2,0 ). 8. Ener formulae o calculae populaions sizes a imes = 0 hrough = 50. In cell B11, ener he value 100. In cell C11, ener he value 50. You will change hese values laer. In cell B12, ener he formula =B11+$F$4*B11*($F$5-B11-$F$6*C11)/$F$5. This corresponds o Equaion 2: N N R N K N N 1, + 1 = 1, + α 1 1, K 1 1, 12 2, 1 In cell C12, ener he formula =C11+$F$7*C11*($F$8-C11-$F$9*B11)/$F$8. This corresponds o Equaion 3: N N R N K N N 2, + 1 = 2, + α 2 2, K 2 2, 21 1, 2 Be sure o use absolue and relaive addresses as shown. 9. Copy and pase he formulae in cells B12 and C12 down heir columns hrough row 51. See Spreadshee Hins and Tips for deails on copying and pasing. B. Creae graphs. 1. Graph N 1 and N 2 (verical axis) agains ime (horizonal axis). Use an XY graph (scaerplo). Include only cells A11 hrough C51 in he block of daa o graph. Leave ou he ZNGI endpoins (cells B5 hrough C8). Use he second Char Wizard dialog box o name your series so ha hey will be labeled properly in he legend. In he dialog box (Figure 3), click he Series ab. Selec Series1 and ype Pop 1 in he box o he righ. Then selec Series 2 and ype Pop 2 in he box. Your finished graph should resemble Figure 4. Figure 3

7 Inerspecific Compeiion and Compeiive Exclusion 131 L-V Compeiion Model Populaion size (N) Time ( ) Pop1 Pop2 Figure 4 2. Graph N 2 (verical axis) agains N 1 (horizonal axis). Include cells B5 hrough C61 in he block o graph in oher words, his ime include he ZNGI endpoins, bu leave ou Time (column A). Use an XY graph (scaerplo). Your graph should resemble Figure 5. L-V Compeiion Model N N 1 Figure 5 Unforunaely, he program does no label he ZNGI endpoins for you. You will have o idenify each endpoin by is coordinaes in he spreadshee. In Figure 5, he oplef endpoin is (0, K 1 /α 12 ); he lower-lef endpoin is (0, K 2 ); he boom-righ endpoin is (K 2 /α 21, 0); and he boom-lef endpoin is (K 1, 0).

8 132 Exercise 9 QUESTIONS 1. Wha parameer values will cause species 1 o exclude species 2 from he habia? Wha do hese values mean in ecological erms? 2. Wha parameer values will reverse his oucome? Wha do hese values mean in ecological erms? 3. Wha parameer values will allow he wo species o coexis indefiniely and sably? Wha do hese values mean in ecological erms? 4. Are here parameer values under which he oucome depends on iniial populaion sizes or raes of populaion growh? Wha do hese values mean in ecological erms? LITERATURE CITED Loka, A. J The growh of mixed populaions: wo species compeing for a common food supply. Journal of he Washingon Academy of Sciences 22: