Academc Forum 3 5-6 A Marov Cha Competto Model Mchael Lloyd, Ph.D. Mathematcs ad Computer Scece Abstract A brth ad death cha for two or more speces s examed aalytcally ad umercally. Descrpto of the Model Let ay S be ay fte lst. Cosder the cha where a radom term of S replaces a radom term of S. Ths was orgally Fred Worth s dea where he smulated ths maually o hs TI-83 usg lst operatos. Ths ca be thought of as a fte populato of several speces that compete wth each other for space. I wll refer to the frst selected term as the aggressor ad the secod selected term the vctm. Ths cha s smlar to the Game of Lfe whch was veted by Joh Coway 197. Ths game taes place a two-dmesoal matrx where each etry s ether populated (alve or upopulated (dead. Here are the rules for the Game of Lfe: 1. For a space that s 'populated': Each cell wth oe or o eghbors des, as f by loeless. Each cell wth four or more eghbors des, as f by overpopulato. Each cell wth two or three eghbors survves.. For a space that s 'empty' or 'upopulated', each cell wth three eghbors becomes populated. The Game of Lfe ufolds a two-dmesoal matrx where proxmty s mportat. However, the cha dscussed ths paper taes place a space that s small eough that a aggressor ca statly jump to ay other locato to clam hs vctm. Here are some obvous observatos about our model: Ths s a Marov cha sce the probablty of the ext state depeds oly o the prevous state. The absorbg states are where all but oe speces s extct. The umber of states s m where m s the umber of speces ad s the sze of the lst. The orgal problem s dffcult, so I wll stead smplfy by cosderg just two speces ad oly eepg trac of the umber of speces, ot the posto. The the state space becomes {,1,,,} where s the sze of the lst. Each state s the populato sze for the frst speces. 6
Academc Forum 3 5-6 Here s a example of a lst for 7 that was geerated usg the TI-83/4 program CHAIN. (A lst of the programs appears appedx. Each colum s a state ad t too 33 steps to reach equlbrum. Note that the sgle dvdual 1 the tal state evetually wped out the speces. 1111111111111111111111 1111111111111111111111111111111111 11111111111111111111111111 111111 1111111111111111111111111111111 1111111111111111111111111111 11111111111 Trasto ad Lmtg Probabltes It s left as a exercse for the reader to derve the followg trasto probabltes. (The proof depeds o the aggressor ad vctm beg chose depedetly. p j ( f otherwse ( + ( j ± 1 f j where, j Ths s the probablty of movg from state to state j. Note that ths s a brth ad death cha where movg from state to +1 s a brth ad movg from state to -1 s a death. The followg probablty trasto matrces ( 3 were geerated usg TI-83 program ( 1 PROBMAT. The rows ad colums are 1 umbered... For example, f 3, the the probablty of gog from 1 to s /9, stayg.5.5.5..5...5. at 1 s 5/9, ad gog from 1 to s /9. 1 1 ( 5 ( 4 1 1.1875.65.1875.16.68.16.4.5.4.5.5.5.1875.65.1875.4.5.4.16.68.16 1 1 The matrces for odd are dagoally domat. The matrces for eve are wealy dagoally domat because the p.5 ad p, -1 p-1,.5 whe /. The dagoally domace ca be prove startg wth (/ -1 ad the showg that + ( - ( -. Recall that the lmtg probablty matrx s the lmt of P p s the probablty trasto matrx. ( j P as approaches fty where 63
Academc Forum 3 5-6 Numercal evdece obtaed by qucly rasg trasto probablty matrces to the 55 th power usg a TI-83 suggests that the probablty of the speces startg wth dvduals wg s /. The followg example shows the lmtg probablty for the case where 3. 1. lm.5...5. 1 1.6.3.3.6 1 Ths shows that the bgger army s more lely to w. I ca prove ths for ay specfc value of usg the egepars of P, but I have ot bee able to prove t geeral. Note that the probablty that the sgle 1 ws s 1/7 for the umercal example gve earler ths paper. Radom Wal Vew It s possble to use the trasto probabltes to costruct a radom wal o the cosecutve tegers {,1,,,} by usg the trastoal probabltes for movg left, stayg, ad movg rght, respectvely. P[ to 1] ( P[stay at ] P[ to + 1] ( ( + ( where,1... Let T ( be the tme for the wal to reach the ed ( or 1. The T T T T ± 1 wp ( 1 / 1+ T wp ( + ( / where < < The followg system of expectatos follow: ( ( T ] + ( + ( T ] + ( T ] 1 T ] 1+ 1 T ] T ] where < < + 1 Ths smplfes to the followg trdagoal system: T 1 ] + T ] T 1] where < < ( + The followg table gves the soluto for the expected tme to ed for up to seve., the umber of elemets the space The expected values. T 1 ] 3 T 1 ]4.5 4 T 1 ]7.333, T ]9.333 5 T 1 ]1.417, T ]14.583 6 T 1 ]13.7, T ]., T 3 ]. 7 T 1 ]17.15, T ]6.133, T 3 ]3.17 64
Academc Forum 3 5-6 By the symmetry T T -, so we oly have to fd half of the expected values of T. For example, whe 3, E [ T ] 4. 5. The program ETIMES was used to compute these expected tmes. The program SIM was used to smulate the tme to ed ad cofdece tervals were computed to support the above results. The dstrbuto for the tme to ed T 1 s Geometrc(1/ whe ca be easly prove usg the trasto probabltes. I determed that the dstrbuto for T 1 s Geometrc (/9 whe 3 umercally. I do ot ow f the dstrbuto for T for 4 has a ame, but t ca be determed umercally by usg P [ T ] ( P + ( P. (P j meas the th row ad the jth colum of the matrx P. That s, add the eds of th row of the th power of P to get the cumulatve dstrbuto fucto for T. I wll brefly cosder the tme to requred to reach a partcular ed. Let L T reach be the tme to reach the left ed, ad R T reach be the tme to reach the rght ed. The T ] L ] P[reach ] + R ] P[reach ] L ] + L ], < < sce R L- by symmetry. The ozero etres of ths sparse coeffcet matrx whe solvg for L ] loo le a X. The terested reader ca solve ths lear system as was doe earler for T ]. A Bref Loo at Multple Speces Suppose that we allow speces to compete. For 1.., we have the followg a pror estmates: P[th speces ws] / The tme for oe speces to w s at least max(t, 1... These two facts follow by parttog the speces to two groups ad usg the prevous results for speces. Ideas for Further Wor Determe f the dstrbuto of T has a ame where 4. Develop a cotuous verso. Develop a -dmesoal verso le Coway s Game of Lfe. Ivestgate multple speces. Supply a proof of the lmtg probabltes for all. Refereces 65
Academc Forum 3 5-6 Coway's Game of Lfe o Wpeda (http://e.wpeda.org/w/coway s_game_of_lfe Itroducto to Stochastc Processes by Paul Hoel, Sdey Port, ad Charles Stoe 197 by Houghto Mffl Appedx of TI-83/4 Programs CHAIN ETIMES SIM PROBMAT Bography Mchael Lloyd receved hs B.S Chemcal Egeerg 1984 ad accepted a posto at Hederso State Uversty 1993 shortly after earg hs Ph.D. Mathematcs from Kasas State Uversty. He has preseted papers at meetgs of the Academy of Ecoomcs ad Face, the Amerca Mathematcal Socety, the Arasas Coferece o Teachg, the Mathematcal Assocato of Amerca, ad the Southwest Arasas Coucl of Teachers of Mathematcs. He has also bee a AP statstcs cosultat sce. 66