POPULATIO MODELS Michael B. Bonsall Deparmen of Zoology, Universiy of Oxford, Oxford, UK Keywords: compeiion models; densiy-dependence; ecology; populaion dynamics; predaion models; sochasic models Conens. Inroducion. Coninuous-Time Populaion Models.. Pure Birh Processes.. Pure Deah Processes.3. Birh-deah Processes.4. Logisic Model.5. Compeiive Ineracions.5.. Inerspecific Compeiion.5.. Mechanisic Inerspecific Compeiion.6. Predaor-Prey Ineracions.7. Age Srucure 3. Discree-Time Populaion Models 3.. Single Species Ineracions 3.. Predaor-Prey Ineracions 3.3. Age Srucure 4. Sochasic Populaion Models 4.. Birh-deah Process 4.. Predaion and Compeiion 5. Fuure Developmens Glossary Bibliography Biographical Skech Summary This chaper presens an overview of he mahemaical models associaed wih populaion ecology. Since he incepion of he academic subjec of ecology, mahemaical models have been used o predic and undersand changes in populaions. Dealing wih numbers of individuals of a species or ineracions beween species ha change over ime is a dynamical sysem. Such dynamical ecological sysems include he ineracion beween compeiors for limiing food, he effecs of muualiss or he rophic effecs of predaor-prey ineracions. This chaper begins wih a broad inroducion o populaion ecology and shows how mahemaical models have been developed o explore a range of differen ecological processes in coninuous and discree ime. Towards he end of he chaper, brief consideraion is given o sochasic formulaions of hese ecological processes. The chaper concludes wih brief perspecives on he fuure role of mahemaics in populaion ecology.
. Inroducion Populaion ecology is he sudy of he changes in abundance and disribuion of species over ime and hrough space. I is a quaniaive subjec and relies heavily on he language of mahemaics o formalize conceps. Four key parameers: birhs (B), deahs (D), immigraion (I)and emigraion (E)provide he fundamenal basis for undersanding he paerns of emporal abundance and spaial disribuion of organisms. The changes in numbers () of a species hrough ime and space can be expressed simply as: = B D+ I E. () This expression () saes ha an organism will increase in numbers hrough birhs and/or immigraion, and decrease hrough deah and/or emigraion. The overall objecive of his chaper is o inroduce how mahemaical models and echniques can be applied o problems in populaion ecology. More specifically, in his chaper, he populaion processes associaed wih birhs and deahs are considered. Three major ypes of populaion models are presened: coninuous-ime models, discree-ime models and sochasic models. A comprehensive discussion of heir role in undersanding he paerns and processes associaed wih single species, compeiive and predaor-prey ineracions is presened. The chaper concludes wih a consideraion of he prospecs for models in populaion ecology and heir general fuure role in biology. The formulaion of mahemaical models provides a logical framework for developing, esing and criicizing ecological hypoheses. Mahemaical models of ecological heory can be formulaed in one of wo ways. Firsly as deerminisic models (i.e. using difference or differenial equaions) such as he coninuous-ime Loka-Volerra model of inerspecific compeiion or he discree-ime icholson-bailey predaor-prey model, or secondly, as sochasic models in which he occurrence of evens is considered o be probabilisic even hough he underlying raes remain consan. Alhough he use of mahemaics in biology has ofen been conroversial wih biological realism ofen being ignored for he sake of mahemaical generaliies, mahemaical models have provided biology (e.g. populaion ecology and populaion geneics) wih a concepual paradigm hrough which he occurrence of key paerns and processes can be expressed and explored.. Coninuous-Time Populaion Models.. Pure Birh Processes The simples ecological scenario o envisage is a pure birh process. This occurs if, over a shor ime period, a populaion increases wihou he consrains of crowding, compeiion or coness. If i is assumed ha wihin his ime frame deah does no occur and he birh rae (λ) is he same for all individuals (geneic homogeneiy) hen he growh rae of he populaion may be considered in erms of an ordinary differenial equaion. If () denoes he populaion size a ime, in small ime inervals of lengh τ, he increase in he populaion size from ime o + τ is: ( + τ ) = ( ) + λτ ( ). ()
Subracing () from each side of his expression () and dividing hrough by τ gives: ( + τ ) ( ) = λ (). (3) τ Leing τ 0 hen yields he ordinary differenial equaion for a pure birh process: d() = λ () (4) such ha he rae of change of wih respec o is he produc of he birh rae (λ) and he curren populaion size ( ()). Eq. (4) can be inegraed o give (wih appropriae iniial condiions) he soluion: ( ) = (0) exp( λ ). (5) Given he iniial assumpions of no deah and consan birh rae, a pure birh process gives rise o an exponenial increase in populaion size. A saisical es for a pure birh process would be o plo ln( ( )) versus and check for an approximae posiive linear relaionship wih slope λ and inercep ln( (0))... Pure Deah Processes In conras o he pure birh process, a similar ordinary differenial equaion can be derived for a pure deah process. This ype of ecological process ceners on how he longeviy and survival of an organism affecs changes in populaion numbers hrough ime. Under a pure deah process i is assumed ha individuals do no give birh, hey do no suffer he consrains of crowding, compeiion or coness and he deah rae (µ) is he same for all individuals. If () denoes he populaion size a ime, hen in small ime inervals of lengh τ,he decrease in populaion size from ime o + τ : ( + τ ) = ( ) τ µ ( ). (6) The ordinary differenial Eq. (following he scheme for he pure birh process (3)) for a pure deah process is: d() = μ () (7) and has soluion: ( ) = (0) exp( μ. ) (8) Again given he assumpions, under a pure deah process a populaion would decline exponenially. A simple es of his ype of process would be o regress ln( ( )) on and check for a negaive linear relaionship wih slope µ and inercep ln( (0)).
.3. Birh-deah Processes Given ha populaions change as a resul of birh and deah processes, he pure, individual, processes of birh and deah can be combined o formulae a birh-deah process ha can be expressed as a coninuous-ime model for he changes in populaion numbers. Deriving he ordinary differenial equaion for he birh-deah process proceeds in he same way as for a pure birh and pure deah process. If i is assumed ha individuals give birh a a rae λ and die a a rae µ, and do no suffer he consrains of crowding or compeiion, he deerminisic ordinary differenial equaion is: d() = ( λ μ) ( ) (9) wih soluion: ( ) = (0) exp(( λ μ) ). (0) The ne rae of change ( λ μ) may be posiive or negaive depending on wheher λ > µ or vice versa. This can give rise o eiher exponenial increase or decrease. This is a simple represenaion of how changes in populaion numbers may occur where he rae of birh and deah are independen of populaion size. In realiy, populaion growh mus be resriced by he availabiliy of limiing resources. When hese resources are fully uilized he populaion can grow no furher: his upper bound or carrying capaciy on populaion growh is deermined by he environmen. As he oal abundance of an organism will only increase hrough birhs and immigraion, or decrease hrough deahs and emigraion, we seek general expressions ha link hese processes o populaion size and densiy. For insance, if raes are funcions of populaion numbers, he birh and deah processes can be expressed as B( )and D( ), respecively. These simple expressions allow virually any funcional form of birh or deah o be modeled and in he limiing case when B ( ) = λ and D ( ) = µ, hen he birh-deah process described above (Eqs. (9) and (0)) is recovered. If () denoes he populaion size a ime, hen in small inervals of ime of lengh τ, he increase in populaion size from ime o + τ is: ( + τ ) = ( ) + ( B ( ) D ( )) τ () Subracing ( ) and dividing hrough by τ and allowing τ 0gives: d() = B( ) D( ) () Soluions o his ordinary differenial equaion are criically dependen on he funcional forms of B( )and D( ). Alhough analyical soluions may be difficul o obain, numerical soluions are always feasible wih an appropriae numerical inegraion
algorihm. Simple funcional forms for he dependence of birhs and deahs on populaion numbers illusrae how he processes affec changes in populaion growh and abundance. For example, if in a birh-deah process B( ) = λ and D ( ) = μ ( ), hen Eq. () is: d() = [ λ μ( )]. (3) One common quesion asked of populaion models is wheher he ecological processes being modeled have any sable seady saes. Tha is, over ime will he populaion remain a a consan size or will i increase or decrease. A populaion will remain in a seady sae for some value of if boh birhs and deah raes are equal. A deerminisic seady sae ( d () ) for he birh-deah model (Eq. (3)), deermined by seing = 0 and solving for, is given by: λ μ = (4) μ This seady sae is considered o be sable if afer experiencing a small perurbaion, he populaion reurns o his seady sae value ( ). We can derive a formal es for his hrough local (as opposed o global) sabiliy analysis: i is assumed ha he populaion is displaced, hrough a small perurbaion, from is seady sae value and he subsequen populaion behavior is hen moniored. Considering a small perurbaion from a seady sae ( ) n= + (5) where n is he resuling perurbed value of, we can expand abou he seady sae for he birh and deah processes hrough he use of a Taylor series expansion. This yields: db B( ) = B( ) + n + o( n ) d = and (6) dd D ( ) = D ( ) + n + on ( ). (7) d = Subsiuing hese Eqs. ((6) and (7)) ino he original birh-deah process (Eq. ()) gives he rae of change of he populaion following a perurbaion as: dn() db dd = B( ) D( ) + n + o( n ). (8) d d =
A he seady sae, birhs and deahs are equal and following a small perurbaion, erms of n and higher erms of he Taylor expansion are assumed o be negligible. Afer a small displacemen, he populaion is hen expeced, a leas o a firs approximaion, o follow: dn() db dd = n d d = (9) wih soluion db dd n () = n(0) exp. (0) d d = This soluion (Eq. (0)) implies ha he populaion has seady sae dynamics and ha small perurbaions decay away exponenially if db dd < 0 d d = This occurs if B() > D() whenever < and B( ) <D( ) whenever >..4. Logisic Model In many populaions, he uilizaion of an available resource will evenually limi he long-erm increase of a populaion. A developmen of he birh-deah process (Eq. ()) in which he ne growh rae per individual is a funcion (f ) of oal populaion size () could ake he form: () d() = f ( ). () df If is large hen ( ) < 0 since he larger he populaion grows, he greaer he d inhibiory effec on furher growh. The simples assumpion is o allow f ( ) o be linear: f ( )= r s where r and s are posiive consans represening he birh and deah raes, respecively. This is he Verhuls-Pearl logisic equaion. I predics ha, in he absence of ineracions wih oher species, a populaion will grow o a carrying capaciy ( ), where = r/ s, he raio of growh rae o deah rae. The logisic model is a useful descripion of how populaions grow in he presence of a limiing resource. Species, however, do no ac in isolaion bu are embedded in webs of compeiive and rophic ineracions. ex, we explore he role of populaion models in undersanding he effecs of inerspecific (beween-species) compeiion before developing he heme furher o examine predaor-prey heory..5. Compeiive Ineracions
.5.. Inerspecific Compeiion Species are likely o compee for limiing resources such as food or erriory. The mahemaical heory of hese inerspecific compeiive ineracions is well esablished. Alhough correlaion does no necessarily imply causaion, (for example, a negaive correlaion beween populaions of wo organisms does no necessarily imply inerspecific compeiion), simple mahemaical models of compeiion do assume ha a species growh rae is inhibied hrough boh inraspecific (wihin-species) and inerspecific (beween species) processes. If and denoe he numbers of individuals of species and, respecively, hen he single species logisic model (Eq. ()) can be exended o a wo species form: d d = ( r s s ) (3) l = ( r s s ) (4) where r i is he growh rae of species i, s ii is he inhibiory effec of species i on iself (i.e. he inraspecific effecs) and s ij is he effec of species j on species i (i.e. inerspecific effecs). To deermine wheher one species wins in compeiion or wheher here is d d coexisence, requires an equilibrium soluions o Eq. (3) and Eq. (4): = = 0 a = and =. Using his assumpion, allows Eqs. (3) and (4) o be expressed in he form: l 0 = ( r s s ) (5) 0 = ( r s s ) (6) A equilibrium, r s r s = s s s s s r s r s = s s s s s (7) (8) r s From Eq. (3), = 0 when = and = 0 when =. From Eq. (4), = when = and = 0 when =. 0 r s r s r s
r r s s, Four ypes of populaion behavior are possible depending on wheher (i) > (ii) r r s s r r s s r r s s < (iii) > and (iv) < These invasion condiions can be represened graphically. r s If r s s r s s r s r s r s r s r s r s r s > and > species wins ou. If < and < species wins ou. If s r s s r s < > a sable equilibrium exiss and he species coexis. If < > here is an unsable equilibrium such ha if species are perurbed from he equilibrium, one of he species will ulimaely become exinc. Figure. Condiions for coexisence and compeiive exclusion. (a) Exclusion of by as is he sronger compeior. (b) Exclusion of by as is he sronger compeior. (c) Unsable equilibrium: oucome of compeiion dependen on iniial condiions. (d) Coexisence of and. Blue line is he growh isocline for and - - - red line is he growh isocline for. TO ACCESS ALL THE 5 PAGES OF THIS CHAPTER, Visi: hp://www.eolss.ne/eolss-sampleallchaper.aspx Bibliography Caswell, H. 00 Marix populaion models. Sinauer, Sunderland, Massachuses. [A user-friendly inroducion o he heory, echniques and applicaion of marix models].
Gurney, W.S.C. and isbe, R.M.998 Ecological dynamics. Oxford Universiy Press, Oxford. [An inroducion o he echniques and applicaion of populaion models]. Hassell, M.P. 000 Spaial and emporal dynamics of hos-parasioid ineracions. Oxford Universiy Press, Oxford.[A deailed monograph on he applicaion of discree and coninuous ime populaion models o predaor-prey ineracions]. May, R. M. 974 Sabiliy and complexiy in model ecosysems. Monographsin Populaion Biology, o 6, Princeon Universiy Press, Princeon. [A deailed monograph on he applicaion of mahemaical models o undersand ecological conceps]. Mueller, L. D. and Joshi, A. 000 Sabiliy in model populaions. Monographs in Populaion Biology, o 3, Princeon Universiy Press, Princeon.[A conemporary reamen of he applicaion and analysis of single-species populaion dynamics]. Pielou, E. C. 977 Mahemaical Ecology. John Wiley and Sons, London. [An excellen inroducion o he applicaion of mahemaics o populaion ecology]. Renshaw, E.99. Modelling biological populaions in space and ime. Cambridge Universiy Press, Cambridge. [A clear and concise overview of he mahemaical deails of boh deerminisic and sochasic populaion models]. Biographical Skech Michael Bonsall, is a Royal Sociey Universiy Research Fellow and Head of Ecology in he Deparmen of Zoology a he Universiy of Oxford. Michael graduaed from Imperial College London. He compleed his PhD on heoreical and empirical insec populaion dynamics and compleed posdocoral posiions a Silwood Park, Imperial College, London. He joined he Zoology Deparmen in Oxford in April 005 and is he Biology Fellow and Tuor a S. Peer's College. Michael s overall research focuses on mahemaical biology and aims, broadly, o undersand how species coexis in ime and space. This is approached using mahemaical models, observaions and experimens. He has gran-funded projecs exploring he role of insec on predaor-prey communiies and he evoluion of insec resisance o microbial insecicides, and works wih a range of inernaional collaboraors.