File: Models_of_Biological_Ineracion.doc Dae: 2 April 2011 Models of Biological Ineracion Among Species or Populaions 1. Models of biological ineracion The bio-economic models examined so far in his course have ypically considered jus one populaion of a single biological species ineracing wih is naural environmen (he biological componen) and human harvesing of ha populaion (he economic componen). There are wo circumsances in which i is sensible o develop a bio-economic model in his way. Firs, if he biological growh behaviour of a single populaion of ineres is independen of ha of any oher populaion or species. In ha case, here is no useful informaion o be brough ino he modelling exercise by joinly sudying he biological growh of his species and any oher. I is mos unlikely ha any populaion will be sricly independen of all ohers, bu i may be approximaely so in some cases. As a pracical maer, his migh jusify is analysis in isolaion. Second, we may accep ha he behaviour of he populaion of ineres is affeced in many ways by many oher populaions or species, bu wih no single relaionship being paricularly dominan. In ha case, a researcher migh choose o regard all oher populaions/species as consiuing (par of) ha populaion s environmen. The modeller would hen give greaer aenion o modelling ha environmen perhaps by reaing i as being sochasic or uncerain in some way bu would no develop models of ineracion beween specific species or populaions. Bu here are circumsances where proceeding in his way is no appropriae. Of mos imporance are hose cases where paricular species inerac in imporan ways. These relaionships may be as predaors and prey, as wih big cas and herbivore mammals; hey may involve parasiism in which one species inhibis he growh of anoher; hey may exhibi muualism (or symbiosis) where each species requires he oher for is survival (such as some baceria and hoss); or hey may involve one of several oher ypes of relaionship. 1 Wherever he relaionship beween wo or more populaions is significan, he researcher ino he behaviour of one of hose populaions is hrowing away imporan and relevan informaion by ignoring ha relaionship. In his se of noes we invesigae some models of biological ineracion beween populaions, and hen briefly show how such models can be used o shed ligh on he issues of biodiversiy and susainabiliy. 2 I will be helpful o begin by oulining a classificaion of models of biological ineracion developed by Shone (1997). We shall also invesigae some of he examples he considers. Suppose ha here are wo populaions of differen species, labelled F and P. Le f denoe he ne conribuion of a represenaive individual in F o he magniude of populaion F, and le p denoe he ne conribuion of one individual in P o he size of populaion P. We specify f and p in he following, relaively general, ways. 1
f α βf χ (1) P p δ εp φ (2) F In Equaions 1 and 2, he parameers and are populaion-specific naural growh coefficiens. They correspond o he parameer ha we called he inrinsic growh rae of a fish populaion in he logisic growh model of a single populaion examined earlier. Parameers and are populaion-specific crowding (or self-limiing) coefficiens, which can be inerpreed in he following ways: If and are boh negaive, boh species are subjec o (inra populaion) crowding effecs, and so here will be limis o which he populaion size could grow, even in he absence of any limis imposed by relaionships wih oher species. If and are boh posiive, he feriliy of each populaion increases he larger is he size of ha populaion. There is a kind of muualism wihin a populaion. Mixed cases are, of course, also possible. The parameers and relae o he ineracion beween differen species or populaions. In paricular:, boh negaive implies iner species compeiion, boh posiive implies muually beneficial ineracion (muualism or cooperaion), of opposie sign implies a predaor- prey relaionship (he populaion wih he posiive parameer being he predaor, and wih he negaive being he prey) In he absence of migraion, he growh of each populaion is given by he produc of he ne conribuion of a represenaive individual and he number of individuals in he populaion. Tha is, in coninuous ime noaion F ff P pp and in discree ime noaion F F f F 1 P P 1 p 1 1 1 P 1 (In large par, we focus on using coninuous ime noaion in hese noes.) By imposing resricions on his general specificaion, various paricular models of biological ineracion are generaed. We consider four of hese. For reference purposes, we lis hese ogeher wih he parameer values being used in our modelling examples in he able below. 2
Model Compeiion, wih no self-limiaion + + 0 0 F a bp F P c df P a = 4 c = 3 b = 3 d =1 Predaor-prey wih no self-limiaion + 0 0 + F a bp F P c df P Predaor-prey model wih populaionspecific crowding for prey F a bp uf F P c df P a = 1 c = 0.1 b = 1 d =0.1 Predaor-prey model wih populaionspecific crowding for prey and predaors F a bp uf F P c df vp P + 0 + a = 1 c = 0.1 u = 0.1 b = 1 d =0.1 + 0 + a = 1 c = 0.1 u = 0.1 v = 1/8 b = 1 d =0.1 1.1(a) Compeiion beween species wih no crowding or self-limiaion This model is obained from he general specificaion (1 and 2) by imposing he resricions { > 0, = 0, < 0} and { > 0, = 0, < 0}. The populaion growh models may hen be wrien as F a bp F (a, b > 0) (3) P c df P (c, d > 0) (4) Wha do hese equaions ell us? Firs, boh inrinsic growh rae coefficiens (here a and c) are posiive, so in isolaion each populaion becomes larger over ime. Indeed, he absence of any self-limiaion (sricly speaking, he propery ha = 0 and = 0) means ha in isolaion he populaions would grow wihou bounds. However, he populaions are no in isolaion; he negaive coefficiens in he erms -bp and -df show ha each species is in compeiion wih he oher for scarce resources. In Equaion 3, for example, he erm bp means ha he ne conribuion o he F-populaion of one individual member of F is negaively relaed o he size of he oher populaion, P. As his relaionship is rue for boh populaions, we have a species compeiion model. There are hree possible oucomes o his ineracive relaionship. I may be helpful o hink of he example of wo compeing garden plans (or perhaps a chosen plan and a weed) o visualise hese. Two of hese are equilibrium oucomes. The firs a rivial (unineresing) soluion - is ha he equilibrium sock of each is zero. A second equilibrium oucome is ha here is a posiive sock of boh, in which he compeiive force ha each exers on he oher is compleely balanced. Bu such an equilibrium will be a knife-edge, or unsable, equilibrium. 3
Any even leading one populaion o become larger han is equilibrium level will precipiae a chain reacion sequence driving he oher o zero. This poins o he hird kind of oucome: one species will become increasingly dominan, and he sock of he oher will be driven owards zero. 3 I will be helpful o give a numerical example. We assume (using Example 12.4 in Shone, 1997) ha Equaions 3 and 4 ake he paricular forms F 4 3P F (5) P 3 F P (6) By definiion, an equilibrium occurs where he wo populaion levels are simulaneously consan, ha is, F 0 and P 0. Imposing hese equilibrium condiions on 5 and 6 gives 0 4 3P F 0 3 F P which yield he pair of soluions {F = 0, P = 0} and {F = 3, P = 4/3}. These are shown in Figure 1 below (generaed using Maple), he former a he origin, he laer by he inersecion of he wo sraigh lines represening F 0 and P 0. The arrows show he direcions in which he wo populaions will move (defined by Equaions 5 and 6) from any arbirarilychosen saring poin. If sequences of hese direcional arrows are conneced ogeher, we obain dynamic ime pahs for he wo populaions. Several such pahs are shown in he diagram by he heavily-drawn curved lines. 4 I is eviden from ha diagram ha our previous conclusions are valid. A seady sae equilibrium wih posiive values of F and P does exis. However, his is an unsable knife edge equilibrium; any deviaion from his equilibrium will lead o he populaion levels diverging even furher, wih one of he species becoming increasingly dominan and he oher becoming ever-closer o zero wih he passage of ime. Moreover, almos all iniial saring poins fail o find ha equilibrium and so in he absence of deliberae managemen he equilibrium is mos unlikely o be achieved. 4
Figure 1 Equilibrium and dynamics of he compeiion wih no crowding model. 5
1.1(b) Predaor-prey model wih no populaion-specific crowding or self-limiaion Suppose we wish o examine biological ineracion beween populaions of a predaor and a prey, bu sill reain he propery ha neiher populaion is subjec o crowding or selflimiaion. This can be done by imposing he resricions { > 0, = 0, < 0} and { < 0, = 0, > 0} on he general specificaion (equaions 1 and 2). The resuling model firs developed by Loka (1925) and Volerra (1931) akes he form F a bp F (a, b > 0) P c df P (c, d > 0) In his predaor-prey model, F is he prey populaion and P is he predaor populaion. In he differenial equaion for F, he posiive inrinsic growh rae a implies ha in he absence of predaion he prey populaion would increase hrough ime. Moreover, he resricion = 0 implies he absence of any self-limiing facor, and so his growh process would be wihou bounds: he prey populaion would expand indefiniely. However, he prey populaion is consrained by is relaionship wih he predaors. The erm bfp, found by muliplying ou he erms wihin and ouside brackes, shows ha he prey populaion falls because of predaion, and ha his effec is larger he greaer is he predaor populaion. In conras, he negaive inrinsic growh coefficien for he predaor implies ha he predaor depends on he prey populaion for is exisence: in he absence of F, he populaion P would collapse o zero. However, he presence of prey acs o increase he predaor populaion; his effec is represened by he ineracion erm dfp. Wha kind of oucome would one expec in his siuaion? Inuiion suggess ha we migh find a balance beween predaor and prey populaions. To see why, noe ha an equilibrium will, by definiion, exis where boh predaor and prey populaions are consan. The prey populaion will be consan where ne recruimen (af) equals ne losses due o predaion (bfp). The predaor populaion will be consan when naural populaion loss (cp) is jus balanced by growh associaed wih presence of he prey populaion (dfp). This gives us wo equaions; knowing values of he parameers a, b, c and d, hese wo equaions could be solved for he wo unknowns, he equilibrium levels of F and P. By way of example, we ake he parameer values o be a = 1, b = 1, c = 0.1 and d = 0.1. The wo equilibrium equaions are hen P 1 and 0.1F 0.1 which yield he equilibrium soluion F * = 1 and P * = 1. 5 6
However, his inuiion and seady-sae algebra boh fail o reveal one of he imporan feaures of his model. Excep foruiously, or hrough deliberae managemen, he equilibrium oucome F * = 1 and P * = 1 will never be realised! Insead, wha will happen is ha populaions of boh F and P will coninually flucuae, cycling above and below hose seady sae levels bu no acually converging o hem. 6 (This is shown in Figure 2b below.) Moreover he ampliude of he oscillaions depends on he iniial values of he variables: differen iniial values lead o differen ampliudes. This can be observed by examining Figure 2a. 7 Figure 2a Equilibrium and dynamics in he LV predaor-prey model. 7
Figure 2b Populaion cycles in he predaor-prey model where neiher populaion is selflimiing. 1.1(c) Predaor-prey model wih populaion-specific crowding for prey The cycling, non-convergen dynamic behaviour of he model examined in he secion 1.1(b) is a mahemaical propery of he se of differenial equaions (and he associaed parameer resricions) which underlies he L-V predaor-prey model. Many biologiss regard hese dynamic properies as being inconsisen wih he observed evidence, or feel ha hey are overly resricive. Various generalisaions o he predaor-prey model have been developed. One such generalisaion involves inroducing upper limis o he populaion sizes of he prey, much as we did earlier when looking a biological models of fisheries. This can be implemened by making he parameer in he general specificaion be negaive (raher han zero). Then his model is obained from he general specificaion (1 and 2) by imposing he resricions { > 0, < 0, < 0} and { < 0, = 0, > 0}. A predaor-prey model wih logisic-like populaion specific upper size on he prey hus can be specified wih he following general srucure 8
F F a bp uf (a, b, u > 0) P c df P (c, d > 0) These equaions imply ha he prey populaion has a componen ha corresponds o a logisic form of biological growh funcion. 8, 9 To see ha hese equaions do indeed conain logisiclike growh funcions, consider he F populaion. Muliplying ou erms and rewriing, he resulan equaion gives 2 uf F F a bp uf F af uf bp F af 1 bp F af 1 bp F a F MAX where F MAX = a/u. I is eviden ha here are here wo limiing influences on he populaion of F: 1. F MAX is he maximum carrying capaciy of he populaion in he absence of he predaor populaion P; as F rises from low levels, members of he F-populaion face increasingly inense self-compeiion given he environmenal milieu in which hey are locaed. 2. The muliplicaive erm bp F implies ha he prey populaion, F, is negaively relaed o he size of he predaor populaion, P. For a numerical example we consider he following specific forms: F F F af 1 bf P F 1 F P F MAX 10 (7) P cp df P 0.1P 0.1F P (8) which involves he following parameer value assumpions: a = 1, b = 1, u = 1/10, c= 1/10, and d = 1/10. Seing F and P equal o zero, and solving hese wo equaions for F and P gives hree equilibrium soluions: F * = 0 and P * = 0; F * = 10 and P * = 0; F * = 1 and P * = 0.9. The firs of hese is he empy or 'rivial' soluion in which neiher predaors nor prey exis. The second is where predaors are enirely absen, and he prey grow o maximum populaion size, 10. The hird soluion, invesigaed furher here, is ha in which here is a join equilibrium of one prey and 0.9 predaors. (Noe ha we have no specified unis in his example, so 0.9 may, for example, be in unis of housands, in which case P = 0.9 corresponds o 900 individuals.) The dynamics of his model are shown in Figure 3a. I is eviden from looking a he direcional arrows and he examples of dynamic adjusmen pahs (shown by he heavy coninuous lines) ha F * = 1 and P * = 0.9 is an equilibrium soluion ha will evenually be achieved provided ha a leas some individuals of boh species exis; dynamic adjusmen pahs from any arbirary posiion all lead o ha equilibrium. Moreover, his is a sable equilibrium (a disurbance would only knock he sysem ou of equilibrium emporarily as dynamic adjusmens will resore he equilibrium). This is also eviden in Table 3b showing 9
he convergen oscillaory dynamics of he wo populaions over ime, saring from some arbirarily chosen iniial populaion levels. Figure 3a Equilibrium and dynamics in he predaor-prey model wih logisic self-limiing (crowding) of he prey species. 10
Figure 3b A useful exercise for you o do a his poin would be o se up a discree ime counerpar o Equaions 7 and 8 in a spreadshee, and o verify he soluion we have jus described. Try a series of alernaive saring values of F and P (ideally no oo far away from he equilibrium values) and observe wha happens. You migh also like o see wha happens as you change he parameer values of he model. If you wish o verify ha you have se up your spreadshee correcly, an example is provided in he file Ineracion.xls. 1.1. (d) Predaor-prey model wih populaion-specific crowding for prey and predaors Nex consider he case where here are upper limis o he populaion sizes of he predaors and prey. This can be implemened by making he parameers and in he general specificaion each be negaive (raher han zero). A predaor-prey model wih logisic-like populaion specific upper size limis hus can be specified wih he following general srucure 11
F P F a bp uf (a, b, u > 0) P c df vp (c, d, v > 0) These equaions imply ha each populaion has a componen ha corresponds o a logisic form of biological growh funcion. 10 We saw in he previous sub-secion ha hese equaions do indeed conain logisic-like growh funcions; his ime, consider he P populaion. Muliplying ou erms and rewriing, he resulan equaion gives 2 vp P P c vp df P cp vp df P cp 1 dp F cp 1 df P c P MAX where P MAX = c/v. We see here are wo influences on he populaion of predaors: 1. P MAX is he maximum carrying capaciy of he populaion in he absence of he populaion F; as P rises from low levels, members of he P-populaion face increasingly inense self-compeiion given he environmenal milieu in which hey are locaed. 2. The muliplicaive erm + df P implies ha he predaor populaion, P, is posiively relaed o he size of he prey populaion, F. For a numerical example we consider he following specific forms: F F F af 1 bf P F 1 F P F (7*) MAX 10 P P P cp 1 df P 0.1P 1 0.1F P P (8*) MAX 0.8 which involves he following parameer value assumpions: a = 1, b = 1, u = 1/10, c= 1/10, d = 1/10 and v = 1/8. Seing F and P equal o zero, and solving hese wo equaions for F and P gives he equilibrium soluion F * = 2/9 and P * = 44/45. 11 The dynamics of his model are shown in Figure 4. I is eviden from looking a he direcional arrows and he examples of dynamic adjusmen pahs (shown by he heavy coninuous lines) ha F * = 2/9 and P * = 44/45 is an equilibrium soluion ha will evenually be achieved (dynamic adjusmen pahs from any arbirary posiion all lead o i) and ha i is a sable equilibrium (a disurbance would only knock he sysem ou of equilibrium emporarily as dynamic adjusmens will resore he equilibrium). A useful exercise for you o do a his poin would be o se up a discree ime counerpar o Equaions 7 and 8 yourself in a spreadshee, and o verify he soluion we have jus described. Try a series of alernaive saring values of F and P (ideally no oo far away from he equilibrium values) and observe wha happens. You migh also like o see wha happens as 12
you change he parameer values of he model. If you wish o verify ha you have se up your spreadshee correcly, an example is provided in he file Ineracion.xls in he Addiional Maerials. Figure 4: Equilibrium and dynamics in he predaor-prey model wih crowding. 1.2 Economic Policy The various models of species ineracion ha we have examined in his noe can be used o generae a number of policy implicaions. One way of doing so is o add, as an exra componen o he model, a social welfare funcion (SWF). This could be specified in various ways. I migh conain F and P as argumens, and so reflec sociey s relaive valuaions over he possible combinaions of F and P populaions. Alernaively, he argumens of he SWF migh be he coss and benefis of harvesing each populaion a various raes. If he in siu sock size of F and/or P also conribues o uiliy, hose benefis should also ener as argumens of he SWF. Given a SWF, an opimisaion exercise can hen be underaken, maximising social welfare subjec o he consrains of he differenial equaions ha are hough o be appropriae for F and P, and for given iniial values of F and P. Noe ha an opimisaion analysis of his kind will only be useful for policy purposes if populaions of F and/or P can be conrolled by human inervenion. Harwick and Olewiler (1998) do an analysis of his kind for wo ineracing species, sharks and una, and demonsrae how ineresing policy inferences can be drawn from such an exercise. 13
An alernaive way of deriving policy insigh is o bring human inervenion explicily ino he model being considered. One way of doing so is o inroduce more species ino he model, including one which is bred and mainained for human benefi, ye which also ineracs wih one or more species of our model. In mos cases of ineres, his hird species will be some kind of domesicaed livesock animal or farm crop. Conrad (1999, pages 173-182) invesigaes a hree species model of grass-herbivore-predaor ineracions, and where farmers breed and mainain domesicaed cale which also compee for grass. This is a useful basis for sudying policy implicaions of farming or agriculure, and so we shall examine a slighly modified version of his model, given by Equaions 9-11. Noe ha we now change noaion slighly, using H raher han F for he herbivore. 12 Grass biomass (G) dynamics: G G gg 1 1H G (9) MAX Herbivore (H) dynamics: H H hh 1 H P G (10) Predaor (P) dynamics: P P pp 1 H P P (11) MAX Examining hese hree equaions, you will see ha he firs componen on he righ-hand side of each equaion consiues a logisic growh process, wih he qualificaion ha a fixed maximum herbivore populaion has been replaced by he erm G, indicaing ha he upper limi o which he herbivore populaion can grow is a fixed muliple of grass biomass. The remaining erms on he righ-hand side furher specify he form of biological ineracions. Grass is consumed by he herbivore populaion a he rae per individual, and so a he rae 1 H by he populaion a ime. The erm -H P shows ha herbivores are consumed by predaors a a rae given by a fixed muliple of he produc of he herbivore and he predaor populaions. In conras, he predaor populaion is posiively relaed o ha produc. Once again, given knowledge of he parameer values and iniial values for he variables G, F and P, one could idenify seady sae (equilibrium) populaions, and simulae he dynamic evoluion of hese variables hrough ime. Nex we follow Conrad by inroducing a fourh populaion, domesic cale, ino he model. This is done by assuming ha cale consume grass a he rae 2 per head. Equaion 9 is hen amended o become 13 G G gg 1 α1h α2c G (12) MAX Noe ha in Equaion 12, he variable C is no ime-subscriped. Tha is, we are reaing he cale sock as a fixed consan, predeermined by economic agens. 14
A his sage of developmen, he model has become quie rich, and can be used in various ways. The researcher could leave he various equaions of he model in general parameric form (as in 10 o 12 above) and could underake qualiaive analysis. Alernaively, paricular parameer values and iniial values of he variables G, F, P and P could be chosen and quaniaive analysis be done. In eiher case, i would be possible o address he following ypes of quesions: (1) Given paricular choices of cale sock, P, wha are he seady-sae equilibrium values of G, F and P? How do hose equilibrium values change as P is changed? (2) Using dynamic simulaion, how do he pahs of G, F and P vary over ime in response o changes in he sock of domesic cale? Do adjusmen pahs converge on a new seady sae equilibrium? (3) Wha, if any, are he limis o which cale socks can be raised before he biological sysem breaks down (wih one or more of he oher socks being driven o zero, or below some criical hreshold)? (4) Wha level of cale socks maximise some appropriaely specified social welfare funcion? Conrad akes his kind of model one sage furher by making he grass growh funcion sochasic raher han deerminisic. This can be done by replacing he deerminisic growh rae g in Equaion 12 by g, a random variable wih some suiably chosen disribuion. I should be eviden ha, provided ha he expeced value of g is g, seady sae equilibrium soluions will no be changed. However, he dynamic adjusmens pahs will now exhibi more variabiliy. Moreover, if he variance of g is sufficienly large, hese dynamic ime pahs may well breach biological hreshold poins leading o populaion collapses. In a sochasic environmen, herefore, i is likely o be he case ha human impacs on he sysem (measured in his case by he size of cale socks) will need o be smaller o avoid possible ecosysem collapse. This is one way of modelling he idea of a safe minimum sandard of conservaion ha we described earlier. Alernaively, i is a useful way of modelling susainabiliy quesions in a relaively simple bio-economic model framework. This brief accoun of a muliple species bio-economic model has been given primarily as a poiner o how you migh go abou doing his kind of modelling yourself. We ake i no furher here in hese noes. Bu he Conrad-ype model is operaionalised and simulaed in boh he Excel file and he Maple file referred o below. Using hose files, you can examine he quesions (1) o (4) ha we lised above. If you are ineresed in seeing how some of he issues here could be operaionalised, a sensible opion migh be o follow he exposiion in Conrad (1999), Chaper 8. Alernaively, we have provided in he Addiional Maerials o Chaper 17 of he Perman e al ex an Excel file (Ineracion.xls) which simulaes a sochasic version he hree species + cale model ha has jus been described, and which is used o explore he quesions raised in his subsecion. You will also find in he Addiional Maerials he Maple file which has been used o generae all he resuls (and graphics) used in his se of noes, and which you can edi yourself if you wish o experimen furher wih simulaions. (I uses version 9.5 of Maple.) 15
References: (Full deails can be found in he Bibliography in he Perman e al ex.) Conrad, 1999 Dajoz, 1977 Harwick and Olewiler (1998) Loka (1925) Shone (1977) Volerra (1931) 16
ENDNOTES: 1 Ecologiss recognise a leas four oher forms of biological ineracion (see, for example, Dajoz, 1977): compeiion, where species are in compeiion for scarce resources; cooperaion, where symbioic relaionships are chosen for common purposes, such as securiy agains predaion; commensalism, where one species benefis bu he oher neiher suffers nor benefis; and amensalism, where one species is unaffeced bu he oher suffers from a relaionship. 2 Discussions of his ype someimes refer o species ineracion and someimes o populaion ineracion. Generally, i is he former which is relevan, provided we ake care o noe ha he populaions are of differen species. In some special cases, he populaions of ineres may equae wih enire species, in which case eiher is appropriae. 3 Mahemaically, his kind of oucome is a no an equilibrium oucome in his model, as no fixed poin will ever be reached. I is worh noing ha his propery comes from he fac ha his model has no (inra-populaion) crowding or self-limiaion; hence here is no fixed poin o which i can grow. 4 Figure 1 is an example of wha is called phase plane analysis. A discree ime counerpar o he coninuous ime specificaion used for his model is given by: F 1 F 4 3P F P 1 P 3 F F The file Ineracion.xls shows how his discree ime model can be analysed by means of an Excel spreadshee. 5 There is also a second, rivial, soluion F = 0 and P = 0. 6 Sricly speaking, his is only rue in coninuous ime models. If a discree ime counerpar model is examined, cyclical behaviour will also be observed, bu i will be explosive, wih cycles of increasing ampliude (unil he sysem collapses). This can be seen by examining shee LV in he Excel file Ineracion.xls in he Addiional Maerials. 7 Figure 2 was generaed using Maple. A discree ime version can be found in he Excel file Ineracion.xls (shee = LV). 8 For he generalisaion ha we are discussing here o make sense, i is usual o specify ha he parameer c in he predaor equaion is posiive, unlike in he LV model wihou crowding effecs (where i was negaive). 17
9 Noe ha as F MAX, F /F MAX 0, and so he logisic componen disappears. Hence he linear form used in he L-V model can be regarded as a special form of logisic in which he quadraic collapses o a linear by virue of here being an infiniely large upper limi o he populaion size. 10 Noe ha as F MAX, F /F MAX 0, and so he logisic componen disappears. Hence he linear form used in he L-V model can be regarded as a special form of logisic in which he quadraic collapses o a linear by virue of here being an infiniely large upper limi o he populaion size. 11 Mahemaically here are also hree oher soluions. Can you deduce wha hese are? 12 The major variaion concerns Equaion 11 below. Unlike in our equaion, Conrad specifies he predaor populaion o be direcly relaed o he herbivore populaion, and does no conain a logisic componen in he form given by 11. 13 Conrad (1999) acually uses (in his Equaion 8.7) a discree ime counerpar o our coninuous ime specificaion 12. 18