Dicuio Paper No. 9 Gregariou Behaviour of Evaive Prey Ila Ehel* Emilia Saoe** Aver Shaked*** Jue 2004 *Ila Ehel, Tel Aviv **Emilia Saoe, Napoli ***Aver Shaked, Bo Fiacial upport from the Deutche Forchuggemeichaft through SFB/TR 15 i gratefully ackowledged. Soderforchugbereich/Traregio 15 www.gey.ui-maheim.de Uiverität Maheim Freie Uiverität Berli Humboldt-Uiverität zu Berli Ludwig-Maximilia-Uiverität Müche Rheiiche Friedrich-Wilhelm-Uiverität Bo Zetrum für Europäiche Wirtchaftforchug Maheim Speaker: Prof. Korad Stahl, Ph.D. Departmet of Ecoomic Uiverity of Maheim D-68131 Maheim, Phoe: +49(0621)1812786 Fax: +49(0621)1812785
Gregariou Behaviour of Evaive Prey Ila Ehel Tel Aviv Emilia Saoe Napoli Jue 7, 2004 Aver Shaked Bo Abtract We model the formatio of a herd a a game betwee a predator ad a prey populatio. The predator receive ome iformatio about the compoitio of the herd whe he chae it, but receive o iformatio whe he chae a olitary idividual. We decribe ituatio i which the herd ad it leader are i coflict ad i which the leader bow to the herd wih but where thi i ot to the beefit of the herd. Thi verio of the paper wa compiled uder time preure for the ole beefit ofaa yet uamed (10 miute) dicuat i the firt SFB iteral coferece, Gummerbach July 2004 1
Gregariou Behaviour of Evaive Prey by I. Ehel, E. Saoe & A. Shaked 0.1 Itroductio The paper addree two quetio cocerig herd i prey populatio. The firt problem i the reao for formig a herd ad coditio for it tability. The ecod problem i about the poible coflict betwee the leader of a herd ad it member. The claical model of herd formatio i biology i a model of a edetary paive prey populatio by W.D. Hamilto [1]. It reemble a Hotellig locatio model. A fiite prey populatio i located o a circle. The prey appear at radom o the circle ad catche the prey earet to him, if there i more tha oe prey idividual cloet to him, he chooe oe of them with equal probabilitie. It i eay to ee that each prey idividual will wat to move to a poitio i which there are may prey idividual. I cotrat, our model i of evaive prey, the prey idividual ru for their life whe chaed by the predator. How doe the predator chooe a idividual to home i o whe the prey idividual differ i their ability to ecape (their peed of ruig)? The prey idividual joi a herd or hide idividually. The prey otice the herd or a olitary idividual with probabilitie that deped o the ize of the herd ad the group of olitarie. If he oticed both group he decide which group to chae. The mai idea of thi paper i that the prey receive differet iformatio about the tregth of the idividual whe he chae the herd ad whe he purue the olitarie. Whe he chae the olitarie he receive o iformatio about them, he ha to pick up oe of them at radom ad chae thi idividual. He will catch the prey with higher probability if the prey i a low idividual. Whe the predator chae the herd he receive ome iformatio durig the chae about their idividual tregth. He i therefore more likely to home i o a low idividual. The formatio of the herd i thu a outcome of the iteractio of the prey idividual ad the predator. We fid coditio for formig a herd coitig of the troget idividual i the populatio ad it tability. It i likely that the troget idividual i the herd will become it leader ad may lead the herd to cover or he may chooe to tay i the ope. We decribe ituatio i which the leader may wih to tay i the ope ad the herd refue to go alog with him. The leader may bow to the herd will ad lead it to cover, but thi outcome will ot the bet for the herd welfare: It would have bee better for them to tay i the ope. Thi model a ituatio i which a populit leader urreder to the populu ad act accordigly but it would have bee better for the public to have followed the leader origial wihe. 2
1 Herd Formatio 1.1 The Model A prey populatio Ω coit of idividual ditiguihed by their probability of beig caught by a predator whe it chae them, p 1 < p 2 <... < p. Idividual 1 i the fatet oe, he ha the lowet probability of beig caught. Each of the idividual decide whether to joi the (igle) herd or to hide a a olitary idividual. Let H Ω be the et of idividual who form the herd, ad let S be the et of olitary idividual. Note that either H or S may be empty if all idividual (or oe) joi the herd. Let h be the umber of idividual i the herd ad let be the umber of olitary idividual, with h + =. The (igle) predator urvey the terrai ad trie to locate a prey. A herd of ize h ecape hi attetio with probability q h. We aume that it i more difficult for a larger herd ot to be detected by the predator, i.e. q h+1 <q h. A olitary idividual (a herd of ize 1) i uoticed by the predator with probability q = q 1. Thu, the predator otice a olitary idividual with probability 1 q, ad he otice a particular idividual i S with probability (1 q ) /. The predator trategy pecifie what he will do if he oticed both the herd ad a olitary idividual. He may ue a pure trategy or a mixed oe: If both the herd ad olitary idividual were oticed the follow a olitary idividual with probability x. Whe the predator chae the herd he receive ome iformatio about the tregth or weake of the idividual i the herd ad he will home i o a weak idividual with a higher probability. We aume that θ i,h atifie the followig three aumptio: 1. Whe chaig P a herd, the predator will evetually home i o oe of it idividual: θ η,h =1. η H 2. The probability that a troger idividual i purued i lower tha that of a weaker idividual: If i, j H, i < j the θ i,h <θ j,h. 3. If a troger idividual joi the herd hi probability of beig purued i lower tha that of a weaker idividual who joi the herd. If i, j / H, i<jthe θ i,h{i} <θ j,h{j}. To pi thig dow we aume that the probability with which the predator home i o idividual i i: θ i,h = p i P. p η η H I cotrat, whe the predator otice olitary idividual he receive o igal about their peed ad tregth ad purue oe of thoe he oticed with equal probability. Thi ituatio defie a Herd Formig game i which each prey idividual chooe whether to joi the herd ad the predator chooe whom to purue whe 3
he oberved both the herd ad the olitarie. The payoff of each prey idividual i hi probability of urvival (it i 1 miu the probability of him beig caught). The predator payoff i hi probability of catchig a prey. We ca ow defie a equilibrium. A equilibrium i a trategy of the predator plu a partitio of the prey populatio ito a herd ad olitary idividual uch that o idividual wihe to deviate from hi trategy. Defiitio 1 Let (H, S, x) be uch that {H, S} i a partitio of the prey populatioitoaherdh ad a group of olitary idividual S, ad let x be a trategy of the predator (the probability i which he will purue a olitary idividual i the cae that he oberved both the herd ad ome olitarie). The triple (H, S, x) i a equilibrium if o prey idividual prefer to move from hi group to the other ad the predator trategy x i a bet repoe to the partitio of the prey populatio. 1.2 The Predator Deciio The predator otice the herd oly with probability (1 q h ) q, he otice oly a olitary idividual with probability q h (1 q ) ad he otice both with probability (1 q h )(1 q ). Thu, if hi trategy i to purue the herd whe he otice both, hi expected catch i: (1 q h ) q X η H θ η,h p η +(1 q h )(1 q ) X η H θ η,h p η + q h (1 q ) 1 X p σ, the firt two term decribe what happe whe the predator oticed the herd, he home i o oe of the idividual i the herd with probability θ η,h ad catche it with probability p η. If he oticed a olitary idividual but the herd ecaped hi attetio the he home i o oe of them with probability 1 ( - the total umber of olitary idividual) ad will catch it with hi characteritic probability p σ. If the predator trategy i to purue a olitary idividual whe he oticed oe, hi expected catch i: (1 q h ) q X η H θ η,h p η +(1 q h )(1 q ) 1 X p σ + q h (1 q ) 1 X p σ. The firt trategy (puruig the herd) yield a higher catch if: (1 q h ) q X θ η,h p η +(1 q h )(1 q ) X θ η,h p η + q h (1 q ) 1 X η H η H > (1 q h ) q X θ η,h p η +(1 q h )(1 q ) 1 X p σ + q h (1 q ) 1 X p σ. η H p σ 4
or, (aumig that the predator ha a poitive probability of oticig a igle olitary idividual,ad hece larger herd 1 >q): X θ η,h p η > 1 X p σ. (1) η H The predator imply compare where hi expected catch i larger ad purue that group whe he oticed both group. 1.3 Herd Formatio ad Regular Herd Aume that the predator play a mixed trategy x, i.e. if he otice both the herd ad a olitary idividual he purue a olitary idividual with probability x. Oce a herd H i formed a idividual i H ha the followig probability of beig purued ad caught: [(1 q h ) q +(1 x)(1 q h )(1 q )] θ i,h p i. The firt part decribe the probability that the predator will purue the herd, either he oberve oly the herd or ele he oberve both the herd ad the olitarie but he chooe to purue the herd (with probability 1 x). The ecod part decribe the probability that the predator will home i o idividual i ad that he will be caught. If a prey idividual deert the herd ad joi the olitarie, the herd become maller, it hrik to H 0 = H {i}, ad the et of olitarie ha expad to S 0 = S {i}. Hi probability of beig purued ad caught ow that he ha joied the olitarie i: qh 1 1 q +1 + x (1 q h 1 ) 1 q +1 1 +1 p i. Therefore, idividual i will tay i the herd iff: [(1 q h ) q +(1 x)(1 q h )(1 q )] θ i,h p i < q h 1 1 q +1 + x (1 q h 1 ) 1 q +1 1 +1 p i. Or: 1 q +1 [(1 x) q h 1 + x] θ i,h < ( +1)(1 q h )[xq +1 x] = τ h, (2) The right had ide of thi coditio i idepedet of i, it deped oly o the ize of the two group, the herd ad the olitarie. It follow (a i how ilemma2)thatiftheweaket(lowet)idividualih wihe to tay i the herd the all other idividual i the herd alo prefer tayig to beig olitary. Similarly, a olitary idividual i prefer ot to joi the herd iff: θ i,h {i} > (1 q )[(1 x) q h + x] (1 q h+1 )[xq 1 +1 x] = τ h+1, 1 (3) 5
Lemma 2 (i) If the weaket idividual i the herd prefer tayig i the herd to hidig olitarily the all other idividual i the herd prefer to tay i the herd. (ii) If the troget olitary idividual prefer to tay olitary tha all other olitarie prefer to hide olitarily rather tha joi the herd Proof. (i) Let idividual i be the weaket i the herd H: p i p η for all η H. Let idividual i prefer to tay i the herd rather tha deert it. By iequality 2: θ i,h <τ h,. The probability of the predator homig i o a troger idividual (η) i lower tha homig i o i : θ η,h θ i,h. Hece: θ η,h <τ h, ad idividual η prefer to tay i the herd. (ii) Let idividual i be the troget olitary idividual, if he prefer to remai olitary the by iequality 3: θ i,h {i} >τ h+1, 1. If ay other idividual (η)weaker tha idividual i joi the herd (itead of idividual i) the hi probability of beig purued i higher tha the correpodig probability of idividual i : θ η,h {η} >θ i,h {i}. Thu, θ η,h {η} >τ h+1, 1 ad idividual η prefer to remai olitary. What ort of herd ca be formed i equilibrium? We will be particularly itereted i regular herd, which coit of the troget idividual. Defiitio 3 AherdH i a regular herd if there exit a iteger k, 0 k.t. H = {1, 2, 3,...k}. Note that the empty herd ad the grad coalitio, the fully gregariou behaviour, are regular. Are there equilibria with irregular herd? Lemma 2 itroduce ome regularity ito the partitio of the prey populatio: If a certai idividual prefer to tay i the herd the all troger idividual i the herd prefer to tay there, ad if a idividual prefer ot to joi the herd the all weaker olitary idividual prefer to remai olitary. However, thi regularity doe ot guaratee that a equilibrium herd will ecearily be regular. The problem lie with fuctio q h. We required that q h, the probability that a herd of ize h will ot be oticed by the predator, be mootoic decreaig with h. We have made o further aumptio about how it varie with h. Extreme cae of the fuctio, whe a herd of ize h alway ecape the attetio of the predator, while a herd of ize h +1 i alway oticed by it (q h =1,q h+1 =0) may lead to a irregular herd. The herd may coit of ome idividual ad ecape the attetio of the predator. The herd may be irregular, i.e. there exit a olitary idividual who i troger tha ome idividual i the herd. He may ot wih to joi the herd for by doig o the herd will be oticed by the predator with certaity ad hi peroal probability of ecapig will be lower tha a a olitary idividual (we kip the detail of thi example). Are there equilibria with regular herd? The exitece of a regular herd hige o the formatio of a mall herd of 2 idividual. The herd, i cotrat to olitary aimal ru i oe directio thu edig a mixed igal to 6
the predator about the tregth of the idividual member of the herd. Formig a herd require cooperatio of it idividual. Startig from a ituatio where all are olitary a herd of 2 caot be formed ule both idividual prefer it. Formally, we require omewhat more tha a igle deviatio to verify that the all olitary tate i ot a equilibrium. We eed alo aume that the type are uiformly ditributed i the prey populatio. I particular we aume that the firt ad ecod troget idividual i the populatio are cloer to each other (i their peed, their probability of beig caught) tha ay other two type. Aumptio: For each idividual k 2 i the prey populatio p 2 p k, p 1 + p 2 p k 1 + p k or alteratively: p 2 p k. p 1 p k 1 Uig thee aumptio we ca how that there exit a equilibrium with aregularherd. Lemma 4 (i) The predator trict bet repoe to ay regular herd R k,k= 2, 3,..., 1 i to follow the olitarie, x =1. (ii) All of the predator trategie are bet repoe to R = Ω. Proof. (i) Coider a regular herd R k,k=2,..., 1. By (1), the predator purue the herd if: X θ η,h p η > 1 X p σ. η H But if the herd i regular the all p η,η H are maller tha ay p σ,σ S. Hece the above iequality i violated ad the predator prefer to purue the olitarie, i.e. x =1. (ii) Whe the grad regular herd R = Ω ha formed, the predator will ever oberve a olitary idividual ice it doe ot exit. Hece ay trategy x i a bet repoe to the partitio (Ω, Φ). Lemma 5 There exit a equilibrium with a regular herd. Proof. From Lemma 4, the predator bet repoe to ay regular herd i to chooe x =1. Let R k = {1, 2,..k} deote the herd of the k troget idividual. Coider the Grad Coalitio: R. If the weaket idividual i the populatio prefer to tay i the herd the (R, Φ, 1) i a equilibrium. If, however, idividual wihe to deert the herd, the we coider the herd R 1. If it weaket idividual prefer to remai i the herd the (R 1, {}, 1) i a equilibrium. Ele we coider R 2 ad o o. Thi proce cotiue util we fid a equilibrium with a regular herd k, or ele we reach the herd R 2 of the two 7
troget idividual ad idividual 2 wihe to leave the herd. We thu reach the all olitary ituatio ad we kow that the herd R 2 i ot table. But from our lat aumptio thi implie that the weak idividual i ay herd of 2 idividual will wih to deert the coalitio. Hece the all olitary ituatio with the empty (regular) herd i a equilibrium. 1.4 Stability of the Fully Gregariou Behaviour Uder what coditio i the fully gregariou behaviour a equilibrium, ad whe i it table? Aume that the etire populatio joied the herd. The predator will ever oberve a olitary idividual ad o all trategie are bet repoe to H = Ω. Aume that the predator play x =1. The herd coitig of the etire prey populatio i a equilibrium if the weaket idividual i the populatio () wihe to tay i the herd (part (i) of lemma 2). By iequality 2 thi occur whe: (1 q) θ,ω < (1 q ) = τ,0 (4) Thi ha bee obtaied from iequality 2 by ettig H = Ω,h=, =0,x=1. Thu if coditio 4 i atified the (Ω, Φ, 1) i a equilibrium. Ufortuately thi i ot a table equilibrium: The predator i idifferet betwee all hi trategie ad may drift away from x =1. If he play x<1 with a ufficietly low x, ome weak idividual may deert the herd. I the herd they are likely to be idetified a the lowet idividual ad be purued by the predator, while if they hide a olitary idividual ad x i ufficietly low the predator i le likely to purue the olitarie ad they may be better off deertig the herd. Thi may tart a owball effect, for the weaket idividual of the ow hruk herd may fid it beeficial to deert the herd ad thu move the ituatio further away from the equilibrium. However, mall deviatio from the equilibrium trategy x =1may ecourage oly the weaket idividual () to deert the herd, but oce he deerted the predator trict bet repoe i to play x =1(lemma 4), it i the bet for idividual to retur to the herd. Thu there i a x, 0 < x <1 uch that {(Ω, Φ, 1), {(R 1, {},x)} x <x<1} i a table equilibrium compoet. Startig from the equilibrium (Ω, Φ, 1), the predator may chage hi trategy withi [ x, 1], the weaket idividual may deert the herd, the predator will the witch back to x =1ad the exiled idividual will retur to the herd. The trategie iterval i which the predator may drift may be icreaed to allow more of the weaker idividual to deert the herd. oce they left the herd the predator revert to hi bet repoe x =1ad force the deerter to retur to the herd. 8
2 The Herd ad it Leader It i atural for the fatet idividual i the herd to be it leader, determiig whe ad what path the herd take to ecape the predator. I thi ectio we would like to propoe a model of the iteractio ad the coflict betwee the herd ad it leader. 2.1 The Model The model i thi ectio differ from the lat model i a few detail. A before, the prey idividual are raked by their peed. However, we aume that everythig happe i a ope avaah ad that ay herd tadig i the ope or ay idividual ruig for helter will be oticed by the predator with probability 1. We aume that the grad herd H = Ω ha bee formed. The herd or ay idividual may ru for helter. Whe purued by the predator while ruig for helter idividual i 0 probability of beig caught i αp i, 0 α 1, the cotat α repreet how cloe the helter i, mall α correpod to a cloe helter. We ditiguih betwee ruig a a herd ad ruig idividually: Whe ruig a a herd the predator will home i o oe idividual with probability θ i,ω a i the previou model, while whe ruig idividually the predator will home i o oe idividual with equal probabilitie. If all reached helter without beig caught, a heltered idividual ha probability q of ecapig the predator attetio. However, oce a heltered idividual i oticed by the predator there i o ecape ad the idividual i caught. 2.2 Stability of the Grad Herd For the grad herd to be table we eed two coditio, the firtithatthe weaket idividual () doe ot wih to deert the herd, ad the ecod i that the predator follow a igle idividual whe he otice it. The coditio for the weaket idividual to remai i the herd i: θ,ω p <αp +(1 αp )(1 q) thi aume that the predator will follow the igle ruig idividual oce it oberved it. However, whe the grad herd formed the predator ever oberve a olitary idividual ad he i therefore idifferet betwee all hi trategie. The coditio for the predator to follow the igle deertig idividual i: 1 X αp +(1 αp )(1 q) > θ i,r 1 p i. Combiig thee two coditio: 1 X θ,ω p < 1 q + αqp > θ i,r 1 p i. 9
Thi eure that the grad herd together with the predator trategy x =1, i table i thi model. Like i the previou ectio, the predator may drift ad play a earby trategy, thi may caue the weaket idividual to abado the herd, but the the trict bet repoe of the predator i to play x =1(follow the olitary ruer), thi will force the idividual back to the herd. Thi double coditio ca hold whe the idividual are ufficietly fat (the probabilitie p i are mall), ad q, the probability of ot oticig a heltered idividual, i alo mall. 2.3 Coflict betwee the Herd ad it Leader Whe the herd tad i the ope the predator probability of catchig a prey i θ i,ω p i, whe the herd ru for helter the predator probability of catchig a prey i: " # α θ i,ω p i + 1 α θ i,ω p i (1 q ). Thu, ruig for helter i better for the populatio, it expect to loe fewer idividual,if " # α θ i,ω p i + 1 α θ i,ω p i (1 q ) < θ i,ω p i or: P (1 α) θ i,ω p i 1 q < P 1 α. (5) θ i,ω p i What would the leader, the fatet idividual (1), wih to do? If he tay i the ope, ad the herd follow him, hi probability of beig caught i θ 1,Ω p 1. If he lead the herd to helter hi probability of beig caught i: " # αθ 1,Ω p 1 + 1 α θ i,ω p i The leader prefer to tay i the ope if: " # θ 1,Ω p 1 <αθ 1,Ω p 1 + 1 α θ i,ω p i (1 q ). (1 q ), 10
or: (1 α) θ 1,Ω p 1 P 1 α < 1 q. (6) θ i,ω p i Comparig iequalitie 5,6 we ca fid the coditio for a coflict betwee theleaderadtheherd:theleaderwattotayitheopewhilethewelfare of the group require that it look for helter P (1 α) θ i,ω p i (1 α) θ 1,Ω p 1 P 1 α < 1 q < P θ i,ω p i 1 α, θ i,ω p i ideed the R.H.S. of the lat iequality i greater tha it L.H.S. θ 1,Ω p 1 < P θ i,ω p i, ice p 1,θ 1,Ω are the mallet probabilitie i the um. Thi guaratee a rage of q 0 for which there i a coflict betwee the leader ad the herd. The revere coflict caot happe: If it i better for the herd to tay i the ope (iequality 5 i violated) the it follow that the leader would alo wih to tay i the ope (iequality 6). 2.4 Total Itability of the Herd. Diatifactio amog member of the herd may lead to deertio ad the breakdow of the herd. The leader of the herd ad it troget member do ot care much whe ome weak idividual deert the herd, but oce more idividual abado the group they will begi to otice it ad uffer from the decreaed ize of the herd. Aume that the predator ca idetify the weaket member of the herd with little error, thi will be the cae whe θ i,h = p β i η H pβ η for a ufficietly large β. Oce a weak pero leave the herd, the ext type i lie become the weaket i the herd ad may wih to deert it. We decribe the coditio for thi owball effect to happe. The coditio i that the predator follow the herd whatever it ize. kx θ i,rk p i > α k i=k+1 p i + " 1 α k i=k+1 p i # 1 q k, k =2, 3,.., 1. (7) Thee coditio may be atified i the followig cae: Aume that the prey populatio i early homogeeou, let p i = p for all i. The above coditio ca be writte a: p>αp+(1 αp) 1 q k, k =2, 3,.., 1 or: p>1 q k + αq k p, k =2, 3,.., 1 11
thi will hold for all k i the rage if: p>1 q 2 + αq 2 p, or: (1 α) p 1 αp > 1 q 2. (8) If thi lat iequality hold the oe ca fid a rage of probabilitie p 1 <p 2 <... < p all cloe to p.t. iequalitie 7 hold for all k i the rage ad the herd i totally utable. 2.5 The Populit Leader Whe the grad herd i totally utable the leader ha o choice but to urreder to the populu ad lead them to helter. I thi ectio we will aume that the predator make little mitake i idetifyig the weaket idividual i a herd ad that the prey populatio i early homogeeou. We will, i fact, prove our reult uder the aumptio that the predator make o mitake ad that the prey populatio i homogeeou (p i = p). Sice (early) all our reult deped o iequalitie, they ca be how to hold for populatio cloe to homogeeou populatio ad to predator makig mall mitake. Lemma 6 Let the herd H = Ω be totally Utable, the it i better for the leader to lead it herd to helter tha try to keep it i the ope if. 1 q < p 1 αp Proof. Whe the herd i totally utable, the leader will be left o hi ow, he may chooe to ru for cover himelf, ow each prey idividual ru for cover o hi ow ad the probability of the leader to be caught i: αp 1 + " 1 α # 1 q p i. = αp +[1 αp] 1 q (the prey chooe a prey with equal probabilitie ad chae him, if he failed he may idetify a heltered idividual ad catch him) If the leader take the iitiative ad lead the herd to cover, the herd will follow him with the poible exceptio of idividual (he may deert the herd, but the the predator will follow him ad idividual 1 will be better off tayig with the herd). The probability of the leader to be caught whe leadig the herd to helter i: " # 1 q αθ 1,Ω p 1 + 1 α θ i,ω p i =(1 αp) 1 q. 12 (9)
The latter probability i clearly le the the firt oe, o the leader will chooe to lead hi herd to helter rather the loe the herd. The abadoed leader ha aother optio, he may ot ru for cover but ru away from the prey i the ope, hi probability of beig caught i the: p 1 q to cover if: > (1 αp). or: p 1 αp > 1 q. p. He will prefer to lead the herd The herd ha three optio: It ca tay with the leader i the ope, it ca dipere ad ru to helter idividually, or it ca be lead by the leader to helter. I which of thee ituatio i the probability of a lo to the herd miimized? If the herd tay i the ope, the probability of a lo i: P θ i,ω p i = p If it ru to helter a a herd: 1 q + αq P θ i,ω p i =1 q + αq p If it dipere ad ru to helter idividually; 1 q + αq 1 q + αq p. P p i =1 Note that ruig to helter idividually alway ivolve lower lo to the herd, except i the homogeeou cae whe p i p ad the loe are equal. The herd welfare i highet whe it tay i the ope whe: p<1 q + αq p, or: p (1 α) 1 αp < 1 q. (10) Let u ow combie the coditio for 1. Total Itability of the herd (iequality 8) 2. The leader prefer to lead the herd to helter whe tayig i the ope will completely dipere the herd (Lemma 6, iequality 9) 3. The total welfare of the herd i higher whe it tay i the ope (iequality 10) p 1 αp > 1 (1 α) p q > 1 αp > 1 q 2 It i traightforward to ee that oe ca fid value of p, q which will atify thee iequalitie. It i the poible to defie a prey populatio p 1 <p 2 < p... < p ad a homig i fuctio θ i,h = β i uch that the followig 13 η H pβ η
hold: The herd will dipere if aked to tay i the ope, o the populit leader lead it to cover, but the the welfare of the herd i miimized, it would have bee bet for the herd to tay i the ope. Referece [1] W. Hamilto. Geometry for the elfih herd. J. Theoretical Biology, 31(2):291 311, 1971. 14