Generalized modeling of ecological population dynamics

Transcription

1 Theor Ecol (2011) 4: DOI /s ORIGINAL PAPER Generalize moeling of ecological population ynamics Justin D. Yeakel Dirk Stiefs Mark Novak Thilo Gross Receive: 1 October 2010 / Accepte: 13 January 2011 / Publishe online: 12 February 2011 The Author(s) This article is publishe with open access at Springerlink.com Abstract Over the past 7 years, several authors have use the approach of generalize moeling to stuy the ynamics of foo chains an foo webs. Generalize moels come close to the efficiency of ranom matrix moels, while being as irectly interpretable as conventional ifferential-equation-base moels. Here, we present a peagogical introuction to the approach of generalize moeling. This introuction places more emphasis on the unerlying concepts of generalize moeling than previous publications. Moreover, we propose a shortcut that can significantly accelerate the formulation of generalize moels an introuce an iterative proceure that can be use to refine existing generalize moels by integrating new biological insights. J. D. Yeakel Department of Ecology an Evolutionary Biology, University of California, 1156 High Street, Santa Cruz, CA 95064, USA jyeakel@gmail.com D. Stiefs Max-Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, Dresen, Germany stiefs@pks.mpg.e M. Novak Long Marine Lab, 100 Shaffer Roa, Santa Cruz, CA 95060, USA mnovak1@ucsc.eu T. Gross (B) Center for Dynamics, Dresen an Max-Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, Dresen, Germany thilo.gross@physics.org, gross@pks.mpg.e Keywors Omnivory Generalize moeling Bifurcation Foo chain Foo web Preator prey system Intraguil preation Introuction Ecological systems are fascinating because of their complexity. Not only o ecological communities harbor a multitue of ifferent species, but even the interaction of just two iniviuals can be amazingly complex. For unerstaning ecological ynamics, this complexity poses a consierable challenge. In conventional mathematical moels, the ynamics of a system of interacting species are escribe by a specific set of orinary ifferential equations (ODEs). Because these equations are formulate on the level of the population, all complexities arising in the interaction of iniviuals must be cast into specific functional forms. Inee, several important works in theoretical ecology present erivations of functional forms that inclue certain types of iniviual-level effects (Holling 1959; Rosenzweig 1971; Berryman 1981; Getz 1984; Fryxell et al. 2007). Although these allow for a much more realistic representation than, say, simple mass-action moels, they cannot come close to capturing all the complexities existing in the real system. Even if etaile knowlege of the interactions among iniviuals were available an coul be turne into mathematical expressions, these woul arguably be too complex to be conucive to a mathematical analysis. In this light, the functional forms that are commonly use in moels can be seen as a compromise, reflecting the aim of biological realism,

2 180 Theor Ecol (2011) 4: the nee to keep equations simple, an often the lack of etaile information. Because of the many unknowns that exist in ecology, it is esirable to obtain results that are inepenent of the specific functional forms use in the moel. This has been achieve by a number of stuies that employe general moels, in which at least some functional forms were not specifie (Garner an Ashby 1970; May1972; DeAngelis et al. 1975; Muroch an Oaten 1975; Levin 1977; Muroch 1977; Wollkin et al. 1982). These works consiere not specific moels, but rather classes of moels comprising simple, commonly use, functions, as well as the whole range of more complex alternatives. That ecological systems can be analyze without restricting the interactions between populations to specific functional forms is in itself not surprising in every mathematical analysis the objects that are analyze can be treate as unknown. The results of the analysis will then epen on certain properties of the unknown objects. In a general ecological moel we thus obtain results that link ynamical properties of the moel, e.g., the presence of preator prey oscillations to properties of the (unknown) functions escribing certain processes, e.g., the slope of the functional response evaluate at a certain point. Accoringly, the analysis of general moels reveals the ecisive properties of the functional forms that have a istinctive impact on the ynamics. Whether such results are ecologically meaningful epens crucially on our ability to attach an ecological interpretation to the ecisive properties that are ientifie. In the present paper, we specifically consier the approach calle generalize moeling. This approach constitutes a proceure by which the local ynamics in moels can be analyze in such a way that the results are almost always interpretable in the context of the application. Generalize moeling was originally evelope for stuying foo chains (Gross an Feuel 2004; Gross et al. 2004, 2005) an was only later propose as a general approach to nonlinear ynamical systems (Gross an Feuel 2006). Subsequently, generalize moeling was use in systems biology, where it is sometimes calle structural-kinetic moeling (Steuer et al. 2006, 2007; Zumsane an Gross 2010; Reznik an Segrè 2010) an is covere in recent reviews (Steuer 2007; Sweetlove et al. 2008; Jamshii an Palsson 2008; Steuer an Junker 2009; Roriguez an Infante 2009; Schallau an Junker 2010). In ecology, generalize moels have been employe in several recent stuies (Baurmann et al. 2007; van Voorn et al. 2007; Gross et al. 2009; Gross an Feuel 2009; Stiefsetal.2010), for instance for exploring the effects of foo quality on proucer-grazer systems (Stiefs et al. 2010) an for ientifying stabilizing factors in large foo webs (Gross et al. 2009). The latter work emonstrate that the approach of generalize moeling can be applie to large systems comprising 50 ifferent species an billions of foo web topologies. In the present paper, we present a peagogical introuction to generalize moeling an explain the unerlying iea on a eeper level than previous publications. Furthermore, we propose some new techniques that consierably facilitate the formulation an analysis of generalize moels. The approach is explaine using a series of ecological examples of increasing complexity, incluing a simple moel of omnivory that has so far not been analyze by generalize moeling. We start out in Local analysis of ynamical systems with a brief introuction to funamental concepts of ynamical systems theory. Reaers who are familiar with bifurcations may wish to move irectly to Density-epenent growth of a single species, where we introuce generalize moeling by consiering the example of a single population. In contrast to previous generalize analyses of this system, we use a shortcut that accelerates the formulation of generalize moels. An alternative erivation is use in Section Preator prey ynamics, where we apply generalize moeling to a preator prey system. Our final example, shown in Intraguil preation, is a simple omnivory scenario involving three species. This example alreay contains all of the ifficulties that are encountere in the analysis of larger foo webs. Local analysis of ynamical systems Generalize moeling buils on the tools of nonlinear ynamics an ynamical systems theory. Specifically, information is typically extracte from generalize moels by a local bifurcation analysis. Mathematically speaking, a bifurcation is a qualitative transition in the long-term ynamics of the system, such as the transition from stationary (equilibrium) to oscillatory (cyclic) long-term ynamics. The corresponing critical parameter value at which the transition occurs is calle the bifurcation point. In this section, we review the basic proceure for locating bifurcation points in systems of couple ODEs. This analysis is central to the exploration of both generalize an conventional moels an is also covere in many excellent text books, for

3 Theor Ecol (2011) 4: instance (Kuznetsov 2004; Guckenheimer an Holmes 2002). In the following, we consier systems of N couple equations t x i = f i (x) (1) where x = (x 1,...,x N ) is a vector of variables an f (x) is a vector-value function. In population ynamics, each x i typically correspons to a population, representing the abunance, biomass, or biomass ensity. The simplest form of long-term behavior that can be observe in systems of ODEs is stationarity. In a steay state x the right-han sie of the equations of motion vanishes, t x i = 0 (2) for all i. Therefore, a system that is place in a steay state will remain at rest for all time. Stationarity alone oes not imply that a state is a stable equilibrium. A system that is perturbe slightly from the steay state may either return to the steay state asymptotically in time or epart from the steay state entirely. For eciing whether a steay state is stable against small perturbations, we consier the local linearization of the system aroun the steay state, which is given by the corresponing Jacobian J, an N N matrix with J ij = f i (x) x (3) j where inicates that the erivative is evaluate in the steay state. Because the Jacobian is a real matrix, its eigenvalues are either real or form complex conjugate eigenvalue pairs. A given steay state is stable if all eigenvalues of the corresponing Jacobian J have negative real parts. When the function f (x) is change continuously, for instance by a graual change of parameters on which f (x) epens, the eigenvalues of the corresponing Jacobian change continuously as well. Local bifurcations occur when a change in parameters causes one or more eigenvalues to cross the imaginary axis of the complex plane. In general, this happens in either of two scenarios: In the first scenario, a real eigenvalue crosses the imaginary axis, causing a salenoe bifurcation. In this bifurcation, two steay states collie an annihilate each other. If the system was resiing in one of the steay states before the transition, the variables typically change rapily while the system approaches some other attractor. In ecology crossing a sale-noe bifurcation backwars can, for instance, mark the onset of a strong Allee effect. In this case, one of the two steay states emerging from the bifurcation is a stable equilibrium, whereas the other is an unstable sale, which marks the tipping point between longterm persistence an extinction. In the secon scenario, a complex conjugate pair of eigenvalues crosses the imaginary axis, causing a Hopf bifurcation. In this bifurcation, the steay state becomes unstable an either a stable limit cycle emerges (supercritical Hopf) or an unstable limit cycle vanishes (subcritical Hopf). The supercritical Hopf bifurcation marks a smooth transition from stationary to oscillatory ynamics. A famous example of this bifurcation in biology is foun in the Rosenzweig MacArthur moel (Rosenzweig an MacArthur 1963), where enrichment leas to estabilization of a steay state in a supercritical Hopf bifurcation. By contrast, the subcritical Hopf bifurcation is a catastrophic bifurcation after which the system eparts rapily from the neighborhoo of the steay state. In aition to the generic local bifurcation scenarios, iscusse above, egenerate bifurcations can be observe if certain symmetries exist in the system. In many ecological moels, one such symmetry is relate to the unconitional existence of a steay state at zero population ensities. If a change of parameters causes another steay state to meet this extinct state, then the system generally unergoes a transcritical bifurcation in which the steay states cross an exchange their stability. The transcritical bifurcation is a egenerate form of the sale-noe bifurcation an is, like the sale-noe bifurcation, characterize by the existence of a zero eigenvalue of the Jacobian. Although we assume that the steay state X uner consieration is positive we shows in Section Preator prey ynamics that the generalize analysis can inclue transcritical bifurcations as limit cases. Density-epenent growth of a single species In this section, we emonstrate how the approach of generalize moeling can be use to fin local bifurcations in general ecological moels. We start with the simplest example: the growth of a single-population X. A generalize moel escribing this type of system can be written as X = S(X) D(X) (4) t

4 182 Theor Ecol (2011) 4: where X enotes the biomass or abunance of population X, S(X) moels the intrinsic gain by reprouction, an D(X) escribes the loss ue to mortality. In the following we o not restrict the functions S(X) an D(X) to specific functional forms. We consier all positive steay states in the whole class of systems escribe by Eq. 4 an ask which of those states are stable equilibria. For this purpose we enote an arbitrary positive steay state of the system as X. We emphasize that X is not a placeholer for any specific steay state that will later be replace by numerical values, but shoul rather be consiere a formal surrogate for every positive steay state that exists in the class of systems. For fining the ecisive factors governing the stability of X, we compute the Jacobian J = S X D X. (5) Because evaluate in the steay state, the two terms appearing on the right-han sie of this equation are no longer functions but constant quantities. We coul therefore formally consier these terms as unknown parameters. While mathematically soun, parameterizing the Jacobian in this way leas to parameters that are har to interpret in the context of the moel an are therefore not conucive to an ecological analysis. We therefore take a slightly ifferent approach an use the ientity F X = F log F X log X, (6) where F is an arbitrary positive function an we abbreviate F(X ) by F. The ientity, Eq. 6, hols for all F > 0 an X > 0; its erivation is shown in Appenix 1. Substituting the ientity into the Jacobian, we obtain J = S X s x D X x. (7) where s x := log S log X, (8) x := log D log X. (9) We note that S / X an D / X enote per-capita gain an loss rates, respectively. Because the gain an loss have to balance in the steay state we can efine α := S X = D X. (10) The parameter α can be interprete as a characteristic timescale of the population ynamics. If X measures abunance then this timescale is the per-capita mortality rate or in other wors, the inverse of an iniviual s life expectancy. If X is efine as a biomass then α enotes the biomass turnover rate. Using α the Jacobian can be written as J = α (s x x ). (11) Let us now iscuss the interpretation of the parameters s x an x. For this purpose, note that these parameters are efine as logarithmic erivatives of the original functions. Such parameters are also calle elasticities, because they provie a nonlinear measure for the sensitivity of the function to variations in the argument. For any power-law ax p the corresponing elasticity is p. For instance, all constant functions have an elasticity of 0, all linear functions an elasticity of 1, an all quaratic functions have an elasticity of 2. This also extens to ecreasing functions such as a/ X for which the corresponing elasticity is 1. For more complex functions, the value of the elasticity can epen on the location of the steay state. However, even in this case the interpretation of the elasticity is intuitive. For instance, the Holling type-ii functional response is linear for low prey ensity an saturates for high prey ensity (Holling 1959). The corresponing elasticity is approximately 1 in the linear regime, but asymptotically ecreases to 0 as the preation rate approaches saturation. A similar comparison for the Holling type-iii function is shown in Fig. 1. Elasticities are use in several scientific isciplines because they are irectly interpretable an can be easily estimate from ata (Fell an Sauro 1985). In particular, we emphasize that elasticities are efine in the state that is observe in the system uner consieration, an thus o not require reference to unnatural situations, such as half-maximum values or rates at saturation that often cannot be observe irectly. We note that in previous publications the elasticities have sometimes been calle exponent parameters an have been obtaine by a normalization proceure. In comparison to this previous proceure, the application of Eq. 6, propose here, provies a significant shortcut. We now return to the iscussion of the example system. So far, we have manage to express the Jacobian etermining the stability of all steay states by the three parameters α, s x,an x. Because this simple example contains only one variable, the Jacobian is a 1-by-1 matrix. Therefore, the Jacobian has only one eigenvalue which is irectly λ = α (s x x ). (12)

5 Theor Ecol (2011) 4: a b Fig. 1 Illustration of elasticity. a In the specific example system, a reprouction rate, S(X) of the form of a Holling type-iii functional response, ax 2 /(k 2 + X 2 ), is assume. This function starts out quaratically at low values of the population ensity X, but saturates as X increases. b The corresponing elasticity, s x, is close to two near the quaratic regime χ = X /k 0, but approaches zero as saturation sets in The steay state uner consieration is stable if λ<0, or equivalently s x < x. (13) In wors: In every system of the form of Eq. 4, a given steay state is stable whenever the elasticity of the mortality in the steay state excees the elasticity of reprouction. A change in stability occurs when the elasticities of gain an loss become equal, s x = x. (14) If this occurs the eigenvalue of the Jacobian vanishes an the system unergoes a sale-noe bifurcation. For gaining a eeper unerstaning of how the generalize analysis relates to conventional moels, it is useful to consier a specific example. We emphasize that this step is not part of the analysis of the generalize moel, but is presente here solely for the purpose of illustration. One moel that immeiately comes to min is logistic growth, which can formally be written as a linear reprouction an quaratic mortality. However, base on our iscussion above, it is immeiately apparent that linear reprouction must correspon to s x = 1 an quaratic mortality to x = 2. Without further analysis we can therefore say that steay states foun for a single population uner logistic growth must always be stable regarless of the other parameters. A more interesting example is obtaine when one assumes a reprouction rate following a Holling type- III kinetic an linear mortality, t X = ax2 bx, (15) k 2 + X2 where a is the growth rate at saturation, k is the halfsaturation value of growth, an b is the mortality rate. This example system can be investigate by explicit computation of steay states an subsequent stability an bifurcation analysis. This proceure is shown in most textbooks on mathematical ecology an is hence omitte here. For the present example the conventional analysis reveals that, for high k only a trivial equilibrium at zero population ensity exists, so that the population becomes extinct eterministically (Fig. 2a). As k is reuce, a sale-noe bifurcation occurs, which marks the onset of a strong Allee effect. In the bifurcation, a stable nontrivial equilibrium an an unstable sale point are create. Beyon the bifurcation a population can persist if its initial abunance is above the sale point. In this case, the population asymptotically approaches the stable equilibrium. By contrast, a population which is initially below the sale point eclines further an approaches the trivial (extinct) equilibrium. For comparing the results from the specific analysis to the generalize moel, we compute the elasticities that characterize the steay states foun in the specific moel. Because the mortality rate is assume to be linear, we know x = 1. The elasticity of the growth function can be foun by applying Eq. 6 to the known growth function of the specific moel. This yiels s x = χ 2 (16) where χ = X /k. A etaile erivation of this relationship using a normalization proceure instea of the shortcut, Eq. 6, is given in Gross et al. (2004). Equation 16 shows that the elasticity of growth is s x 2 for X k, but approaches s x = 0 in the limit X k (Fig. 1). In Fig. 2a, we have color coe the growth elasticity of steay states visite by the system as k is change. We note that the sale-noe bifurcation occurs at s x = x = 1, conforming to our expectation from the generalize moel. Moreover, in the unstable sales we fin

6 184 Theor Ecol (2011) 4: a b Fig. 2 Comparison of generalize an conventional moeling. a Bifurcation iagram of a specific example (Eq. 15). The lines correspon to the locations of steay states, which are stable equilibria (soli) or sales (ashe). The color encoes the elasticity of growth, s x, in the respective steay states. The figure confirms the our expectation from the generalize moel that steay states are stable whenever s x < x,where x = 1 in the specific example. The two steay states vanish in a sale-noe bifurcation, which occurs at s x = x. Parameters: a = b = 1. b General results. The corresponence between generalize an specific moel can be seen explicitly by mapping the steay states from the specific moel into the generalize parameter plane. In this plane, the stable an unstable states are separate by the stability bounary (black line, Eq. 14) at which the sale-noe bifurcation occurs s x > x, whereas the stable equilibria are characterize by s x < x, which is in agreement with Eq. 13. We can now map the steay states foun in the specific moel into the generalize parameter plane spanne by the elasticities s x an x (Fig. 2b). Because x = 1 in the specific example, irrespective of X,all steay states en up on a single line in the generalize iagram. Other areas of the bifurcation iagram, not visite by the specific example, correspon to other moels that assume other functional forms for the mortality. In this iagram the two colliing branches of stable an unstable steay states are mappe into the corresponing stable an unstable region of the generalize parameter space, respectively. Therefore the two branches appear on ifferent sies of the bifurcation. However, from the bifurcation conition, Eq. 14, we know that this bifurcation must occur as the iagonal line in the iagram is crosse. The comparison of the two bifurcation iagrams in Fig. 2 highlights the ifferences between generalize an conventional moeling. In the conventional moel ifferent numbers of steay states are foun epening on the specific functional forms that are assume. Moreover, for a given set of parameter values multiple steay states can coexist that iffer in their stability properties. Because the generalize moel comprises a whole class of specific moels a single set of generalize parameters correspons to an infinite number of ifferent steay states, foun in ifferent specific moels. However, the solution branches of this family of moels have been unfole such that all steay states corresponing to the same set of generalize parameters must have the same stability properties. It is apparent that for a given specific example, the conventional analysis reveals more etaile insights than the generalize analysis. For instance the presence of the strong Allee effect that is irectly evient in the conventional bifurcation iagram, Fig. 2a, can only be inferre inirectly from the presence of the salenoe bifurcation in the generalize analysis, Fig. 2b. However, the conventional analysis provies insights only into the ynamics of the specific moel uner consieration, whereas the generalize analysis reveals the stability bounary (black line in Fig. 2b) that is vali for the whole class of moels an is hence robust against uncertainties in the specific moel. A major avantage of the generalize moel is that results are obtaine without explicit computation of steay states. In the conventional moel that we iscusse in this section, steay states can be compute analytically. However, even for slightly more complex moels this computation becomes infeasible as it involves (uner the best circumstances) factorization of large polynomials. Also the numerical computation of steay states poses a serious challenge for which no algorithm with guarantee convergence is known. Because generalize moeling avois the explicit computation of steay states, the approach can be scale to much larger networks. The aitional complications that arise in the generalize moeling of larger systems an their resolution are the subject of the subsequent sections.

7 Theor Ecol (2011) 4: Preator prey ynamics In our secon example, we consier a slightly more complex system where intra- an interspecific interactions are consiere. Departing from the single-species moel, we introuce a preator Y whose growth is entirely epenent on X. This leas to the generalize moel X = S(X) D(X) F(X, Y), (17) t Y = γ F(X, Y) M(Y), (18) t where S(X) an D(X) escribe the reprouction an mortality of the prey X, the function F(X, Y) moels the interaction of X with the preator Y, M(Y) is the mortality of Y, an γ is a constant conversion efficiency. In principle this moel can be analyze by the proceure propose in the previous section. However, for gaining a ifferent perspective we use the alternative proceure for eriving Jacobians that was use in previous papers on generalize moeling, such as Gross an Feuel (2006). This alternative proceure starts by introucing a set of normalize variables x = X X (19) y = Y Y (20) an normalize functions, inicate by lower-case letters, such as s(x) = S(X) S = S(xX ) S, (21) where we use the abbreviate notation S := S(X ). Using these efinitions we can rewrite Eqs. 17, 18 as t x = S D F s(x) (x) f (x, y), (22) X X X t y = γ F M f (x, y) m(y). (23) Y Y By normalizing the system we have mappe the previously unknown steay state (X, Y ) to a known location (x, y ) = (1, 1). Linearizing the system aroun this steay state, we obtain the Jacobian J = S X s x D X x F X f x γ F Y f x F X f y γ F Y f y M, Y m y (24) where we again use roman inices to inicate partial erivatives, s x = s(x)/ x. As suggeste by this notation the partial erivatives of the normalize functions are logarithmic erivatives (i.e., elasticities) of the original functions. For instance s x := log S log X (25) We now continue in analogy to the previous section. We absorb the steay-state abunances X, Y, rates S, D, F, M, an the constant γ into a set of scale parameters. In oing so, we have to take care to satisfy the emans of stationarity for every variable. Let us first consier the preator. The corresponing equation of motion, Eq. 18, implies γ F M = 0. (26) Analogously to the example from the previous section, we can therefore efine α y := γ F Y = M Y. (27) which automatically satisfies Eq. 26 but oes not restrict the per-capita rates otherwise. As in the previous example, the α y can be interprete as a characteristic time scale, now escribing the preator population. Equation 27 allows us to replace all occurrences of the unknown constants from the elements of the Jacobian referring to the preator, i.e., the bottom row in Eq. 24. This is possible because the equation of motion for the preator, Eq. 18, contains only two terms an can therefore be characterize by two rates, the reprouction rate an the mortality rate. By consiering steay states we impose one constraint, Eq. (26). Thus, only one egree of freeom remains, which can be capture by one parameter α y. An aitional complication is encountere for the prey. Because we moele preation an mortality from intraspecific competition as inepenent loss terms, the corresponing equation of motion, Eq. 17 contains three terms. Demaning stationarity yiels the constraint S D F = 0. (28)

8 186 Theor Ecol (2011) 4: The presence of three terms, subject to one constraint, implies that two generalize parameters have to be efine to replace all occurrences of the unknown rates. From the point of view of mathematics, many alternative ways of efining these parameters exist. However, from an ecological point of view these ways iffer in the interpretability of the parameters they introuce. Analyzing generalize moels we foun it almost always avantageous to introuce (a) one parameter capturing the characteristic timescale of the corresponing variable, i.e., the turnover rate, (b) a set of branching parameters capturing the relative contribution of the iniviual loss terms to the total turnover, (c) a set of branching parameters capturing the relative contribution of iniviual gain terms to the total turnover. By efinition, the turnover rate (step a) equals the sum of all gains an the sum of all losses. In the present example, we thus efine α x := S X = D X + F X, (29) where all gains appear on the left sie of the equals sign an all losses appear on the right. In step (b), we efine the parameter β := 1 α x D X, (30) an its complement β := 1 α x F X, (31) that capture the relative contribution of losses from preation an intraspecific competition to the total turnover rate. Because the losses have to a up to the total turnover, the parameters have to obey β + β = 1. Therefore, only one of the two parameters, say β, can be consiere as an inepenent parameter. Because there is only one gain term, step (c) in the outline above is not necessary in the present example. As we alreay argue above, we obtaine two inepenent parameters escribing the biomass flow in the prey population: the per-capita turnover rate, α, an the relative contribution of intraspecific competition to the total turnover rate β, i.e., the fraction of losses cause by competition. In general, the same strategy for efining branching parameters can be applie to equations containing any number of terms. For each variable, firstly efine a parameter α, which enotes the total turnover rate, separating gain an loss terms, an ientifying the characteristic timescale of a species. Branching parameters are then assigne to any number of terms that efine the relative contribution of the iniviual gains an losses to the total turnover within a system. Returning to the Jacobian of the preator prey system, we substitute the scale an branching parameters into Eq. 24, which yiels ( ( ) J αx sx β = x β f x α x β f y α y f x α y ( fy m y ) ). (32) In contrast to the system from the previous section, the Jacobian is now a 2-by-2 matrix. For this Jacobian, the eigenvalues can still be compute analytically. However, analytical eigenvalue computation is teious alreay for systems with three variables, an in general impossible for systems with more than four variables. Nevertheless, analytical results can be obtaine even for larger systems by eriving test functions that irectly test for bifurcations, without an intermeiate computation of eigenvalues. Sale-noe bifurcations occur when a single real eigenvalue crosses the imaginary axis (Kuznetsov 2004). Therefore, a zero eigenvalue must be present in a sale-noe bifurcation. This implies that the prouct of all eigenvalues must vanish in this bifurcation. Because the prouct of all eigenvalues equals the eterminant of a matrix, we can locate sale-noe bifurcations by emaning that the eterminant of the Jacobian, et J, vanishes. For the present example this yiels the conition s x = (β x + β f x )( f y m y ) β f x f y. (33) f y m y For fining the Hopf bifurcations, we note that the trace of a matrix (the sum of iagonal elements) is ientical to the sum of the eigenvalues (Kuznetsov 2004). For a two-imensional system this implies that the trace of the Jacobian, tr J, must vanish in a Hopf bifurcation, because there is only one purely symmetric eigenvalue pair, which as up to zero. For etecting Hopf bifurcations we have to aitionally eman that the et J > 0, because tr J = 0 is also satisfie if there is a real symmetric pair of eigenvalues, which is not characteristic of the Hopf bifurcation. In the preator prey moel, the Hopf bifurcation is foun at s x = β x + β f x r( f y m y ), (34) where r = α y /α x is the turnover rate of the preator measure in multiples of the turnover rate of the prey. For systems with more than two variables, the test function for the Hopf bifurcation can be erive by a proceure that is escribe in Gross an Feuel (2004). The results of the bifurcation analysis suggest that high values of f x exert a stabilizing influence on the system. Previous stuies (Gross et al. 2004; Stiefsetal.

9 Theor Ecol (2011) 4: ) showe that this parameter is relevant for enrichment scenarios. In many previously propose moels, preator saturation increases when resources are ae, leaing to a ecrease of f x an therefore to instability. Ientification of f x as a crucial parameter for stability in the generalize moel enables us to ask what functional responses woul lea to an intermeiate stabilizing effect of enrichment that is sometimes observe in nature. The iscussion in Gross et al. (2004) showe that reasonable functional responses can be foun that exhibit such an intermeiate stabilization, but are very har to istinguish from, say Holling type- II kinetics, if they were encountere in nature. To illustrate the ifferences between generalize an conventional moeling, we again compare the generalize moel with a specific example. For this purpose we focus on the Rosenzweig MacArthur moel. In this moel, the prey exhibits logistic growth in absence of the preator, the preator prey interaction is moele by a Holling type-ii functional response, an the mortality of the preator is assume to be ensity inepenent. This leas to ( t X = rx 1 X ) axy k b + X, t Y = γ axy my, (35) b + X where r is the intrinsic growth rate of X, k is the carrying capacity of X, a is the preation rate at saturation, b is the half-saturation value of the preation rate, γ is the biomass conversion efficiency, an m is the mortality rate of Y. The results of a conventional bifurcation analysis are shown in Fig. 3a. If the carrying capacity k is too small then the preator population cannot invae the system. As the carrying capacity is increase a transcritical bifurcation occurs in which a stable equilibrium appears, such that the preator prey system can resie in stationarity. If the carrying capacity is increase further, a supercritical Hopf bifurcation occurs, in which the equilibrium is estabilize. Subsequently, the system resies on a stable limit cycle, which emerges from the Hopf bifurcation. On this cycle, pronounce preator prey oscillations can be observe, which become larger as the carrying capacity is further increase. One can imagine that if an aitional parameter is change then critical values of the carrying capacity at which the bifurcations occur change as well. This can be visualize in two-parameter bifurcation iagrams, which we have alreay use for the generalize moel in Fig. 2b. In such iagrams, Hopf an sale-noe bifurcation points form lines in the two-imensional parameter space. For the specific example of the Rosenzweig MacArthur system, a twoparameter bifurcation iagram is shown in Fig. 3b. This iagram illustrates that increasing the mortality rate m of the preator, shifts both the transcritical bifurcation point an the Hopf bifurcation point to higher values of the carrying capacity. For comparing the specific example to the generalize moel, we compute the generalize parameters that are observe in the steay states of the specific moel. Above, we have alreay note that logistic growth can be unerstoo as a combination of linear reprouction an quaratic mortality, which correspons to s x = 1 an x = 2. Furthermore, the assumptions of ensity inepenent mortality an linear epenence of the preation rate on the preator imply m y = f y = 1. The elasticity f x of the preation rate with respect to prey was erive in Gross et al. (2004) an is f x = χ, (36) where χ = X /b. Accoringly, f x = 1 in the limit of vanishing prey ensity an f x = 0 in the limit of infinite prey. Note that in the Rosenzweig MacArthur moel, the preator population tightly controls the prey population. Once the preator can invae, any further increase in carrying capacity only increases the stationary population of the preator, while the stationary population size of the prey remains invariant. Apart from the parameters β an f x, shown in Fig. 3c, the only other parameter that is not fixe to a specific value is the relative turnover of the preator r = α y /α x. This parameter cannot affect the transcritical bifurcation, because turnover rates by construction cannot appear in test functions of transcritical or sale-noe bifurcations. By contrast, turnover rates in general affect Hopf bifurcations. However, in the present example the epenence of the Hopf bifurcation test function, Eq. 34 on r isappears if ensity inepenent mortality an linear epenence of the preation rate on the preator population are assume. Therefore, the parameter has no influence on the bifurcation surfaces. We note that this is a special property of the Rosenzweig MacArthur system an not a generic feature of the larger class of systems escribe by the generalize moel. As argue in van Voorn et al. (2007) an Gross an Feuel (2009) one can expect that typically mortality is slightly super-linear because of overcrowing, iseases an other limiting resources, whereas preation may be sublinear ue to preator interference. In this case, large values of r can have a stabilizing effect.

10 188 Theor Ecol (2011) 4: a b c Fig. 3 Comparison of a generalize moel an a specific example for preator prey interaction. a Bifurcation iagram of the Rosenzweig MacArthur moel. The preator Y can invae the system when the carrying capacity k of the prey excees a threshol, corresponing to a transcritical bifurcation (TC). Increasing the carrying capacity further eventually leas to estabilization in a Hopf bifurcation (H). Lines mark stable (soli) an unstable (ashe) steay states an the upper an lower turning points of a stable limit cycle (otte). Parameters: r = 1, a = 2, γ = 0.5, b = 1, m = 0.5. b A two-parameter bifurcation iagram of the Rosenzweig MacArthur moel as a function of the mortality of Y, m, an the carrying capacity of X, k. Stable equilibria are confine to the narrow region between the TC (blue) anthe H(re) bifurcation points. The black lines inicate the steay states foun in the section of this iagram shown in (a). c Atwoparameter bifurcation iagram of the generalize preator prey moel as a function of the proportional mortality of X ue to intraspecific competition (β) an the elasticity of the preation rate with respect to the prey ( f x ). Bifurcations an labels are as above. The black line plots the trajectory of the Rosenzweig MacArthur system as a specific example of the class of moels. A three-parameter bifurcation iagram of the generalize preator prey moel. The bifurcation points now form surfaces. The black lines inicate the steay states from (a), while the grey plane inicates all steay states that can be reache in the Rosenzweig MacArthur moel if k an m are varie as in (b) We can now map the steay states to the specific system into the generalize parameter space. A twoparameter bifurcation iagram of the generalize moel is shown in Fig. 3c. In this iagram, bifurcations of sale-noe type occur only on the bounary of the parameter space, where the branching parameter β vanishes. This parameter value inicates that none of the biomass loss of the prey occurs because of preation. Even without comparing to the specific example we can conclue that this bifurcation must be a transcritical bifurcation in which the preator enters the system. To illustrate this we map aitionally the twoimensional bifurcation iagram (Fig. 3b) into the generalize parameter space. This mapping is visualize in a three-imensional bifurcation iagram shown in Fig. 3. Such three-imensional iagrams can be generate from analytical test functions using the metho escribe in Stiefs et al. (2008) an have been use in a number of previous stuies (Gross an Feuel 2004; Gross et al. 2004, 2005, 2009; Stiefsetal.2010; Gross an Feuel 2009). As in the two-parameter iagrams, every point in the iagram represents a family of steay states. The parameter volume is ivie by bifurcation surfaces, which separate steay states with qualitatively ifferent local ynamics. Specifically, all steay states locate between the two bifurcation surfaces are stable, whereas the steay states below the Hopf bifurcation surface are unstable.

11 Theor Ecol (2011) 4: In the present example, we were able to show all relevant parameters in a single three-parameter bifurcation iagram. Let us remark that this is in general not possible as a larger number of parameters is often necessary to capture the ynamics of a system at the esire egree of generality. Even if a generalize moel contains only five parameters, the three-imensional slice that can be visualize in a single three-parameter iagram is relatively small when compare with the five-imensional space. Nevertheless, plotting threeparameter bifurcation iagrams can be very valuable because a three-imensional iagram is often sufficient to locate bifurcations of higher coimension. Such bifurcations are forme at the point in parameter space where ifferent bifurcation surfaces meet or intersect. The presence of such bifurcations can reveal aitional insights into global properties of the ynamics. For instance in Gross et al. (2005) the presence of a certain bifurcation of higher coimension in generalize moels was use to show that chaotic ynamics generically exist in long foo chains. An extensive iscussion of bifurcations of higher coimension an their ynamical implications is presente in Kuznetsov (2004). For obtaining a general overview of the ynamics of larger systems containing hunres or thousans of parameters, bifurcation iagrams are not suitable. However, these systems can be analyze by statistical sampling techniques escribe in the following section. Intraguil preation As the final example in the present paper, we consier the effect of omnivory on a small foo web. Omnivory is efine by an organism s ability to consume prey that inhabit multiple trophic levels. It has been the subject of much recent interest because it is notable for its pervasiveness within well-stuie ecosystems (Polis 1991), as well as its relatively complex ynamics (McCann an Hastings 1997; Kuijper et al. 2003; Tanabe an Namba 2008). Omnivory has been historically viewe as a paraoxical interaction. Initially, the presence of omnivory was thought to be entirely estabilizing, an, as a consequence, rarely observe in nature (Pimm an Lawton 1978). However, further explorations of ecological networks have reporte omnivory to be a common architectural component within larger foo webs (Bascompte et al. 2005; Stouffer an Bascompte 2010). Furthermore, theoretical investigations have reveale parameter regions that lea to both stabilizing an estabilizing ynamics in simple moels (Holt an Polis 1997; McCann an Hastings 1997; Kuijper et al. 2003; Tanabe an Namba 2008; Namba et al. 2008; Very an Amarasekare 2010). These theoretical arguments are limite by the fact that such moels are either constraine to specific functional forms or report ynamics across parameter ranges that may not be biologically significant. A generalization of the entire class of simple omnivory moels is poise to eluciate uner which conitions stable or unstable ynamics are boun to occur, regarless of the functional relationships among or between species in the moel. A specific case of omnivory is intraguil preation (IGP), which in its simplest incarnation appears in a three-species system containing a consumer an resource pair (as in the prior example), an an omnivore that preates upon both the consumer an resource. Traitional analyses of IGP moels reache the following: (1) the coexistence of all species in the system is contingent on the greater competitive abilities of Y, relative to the omnivore Z, to consume the share resource (Holt an Polis 1997; McCann an Hastings 1997); (2) enrichment estabilizes the system (Holt an Polis 1997; Diehl an Feissel 2001); (3) if the gain of the omnivore by preation on the consumer excees the negative competitive effects of the consumer, then the consumer facilitates a larger population of the omnivore than can be maintaine in its absence (Diehl an Feissel 2001). In the present paper, we consier the generalize moel X = S(X) D(X) F(X, Y) G(X, Y, Z ) t Y = γ F(X, Y) H(X, Y, Z ) M(Y) t Z = K(X, Y, Z ) M(Z ). (37) t In aition to the terms alreay presente in the preator prey moel from Preator prey ynamics, we inclue the functions G an H, which enote the loss of the resource an consumer from preation by the omnivore an the function K enoting the gain of the omnivore that arises from this preation. Note that we moele the two ifferent preatory losses of X as separate terms G an H because these losses can be assume to arise inepenently of each other. By contrast the gain of the omnivore erives from preation on two ifferent prey species an is moele as a single term K because finite hanling time, saturation effects, an possibly active prey-switching behavior prevent the preator from feeing on both sources inepenently of each other.

12 190 Theor Ecol (2011) 4: By following the proceure escribe in the previous sections, we construct the Jacobian α x (s x δ x δ(β x f x β x g x )) α x δ(β x f y + β x g y ) α x β x δg z J = α y ( f x β y h x ) α y ( f y β y h y β y m y ) α y β y h z (38) α z k x α z k y α z (k z m z ) where the elasticities are efine as in the previous sections an the scale parameters are α x = S X = D X + F X + G X, α y = γ F Y = H Y + M Y, α z = K Z = M Z, (39) the branching parameters are δ = D D + F + G, β x = F F + G, β y = H H + M, (40) an δ = 1 δ, β x = 1 β x,an β y = 1 β y. Let us remark that the branching parameters in the moel were efine such that the parameter δ separates the preatory losses of the resource from the competition term. This was one to reflect our opinion that these losses are qualitatively ifferent. An alternative proceure woul have been to use three branching parameters, β, β f, β g, to enote irectly the ifferent proportions the three losses contribute to the total percapita loss rate of X. In this case, we woul have to eman β + β f + β g = 1 for consistency, such that only two of the parameters coul be varie inepenently. In principle, the Jacobian of the omnivory moel coul be analyze straight away. However, more insights can be gaine by builing more biological knowlege into the moel. In the following, we integrate this knowlege into the Jacobian erive above, by a refinement proceure that can be use to iteratively integrate new information into the generalize moel when such information becomes available. In the present example, we want to integrate the observation that the ifferent elasticities associate with functions escribing preation by the omnivore cannot be unrelate. Above, we alreay argue that the nonlinearity in the preator functional response arises mainly from preator saturation. If the availability of a given prey species is increase then preator saturation increases an consequently preation on other prey populations ecreases. Thus saturation governs both the nonlinearity of a given preator prey interaction an the response of the preation rate to changes in the abunance in a another prey species. For making these epenencies explicit in our moel we first note that saturation epens on the total amount of prey available to the preator (see Holling 1959 for a etaile iscussion). For simplicity, we assume that this total amount, T, is a weighte sum of sizes of the two prey populations, such that T(X, Y) = T x X + T y Y. (41) We enote the relative proportions that both types of prey (here, consumer an resource) contribute to the iet of the preator (omnivore) as t x = T x X T(X, Y), t y = T yy T(X, Y), (42) such that t x + t y = 1. If a species contributes a given proportion to the iet of the omnivore, it is reasonable to assume that the same species carries an equal portion of the losses inflicte by the omnivore, such that G(X, Y, Z ) t x K(T(X, Y), Z ) an H(X, Y, Z ) t y K(T(X, Y), Z ). By consiering these assumptions in the steay state uner consieration an applying the ientity Eq. 6 we fin g x = k t t x + t y, g y = k t t y t y, h x = k t t x t x, h y = k t t y + t x, k x = k t t x, k y = k t t y. (43) An exemplary erivation of one of the relations is shown in Appenix 2. The new parameter k t appearing above is the elasticity of the omnivore s gain with respect to the total amount of available prey, i.e., the saturation of the omnivore. This parameter can be interprete completely analogously to the parameter f x in the preator prey system. Taking aitional biological insights into account has le to relationships that can be irectly substitute into

13 Theor Ecol (2011) 4: the previously erive Jacobian. Doing so removes six parameters from the generalize moel at the cost of introucing two new ones. The substitution makes the moel less general an more specific, allowing us to extract more conclusions on a narrower range of moels. By this proceure new insights on a given system can be integrate iteratively without re-engineering the moel from scratch. We believe that such refinements will be valuable for future foo web moels possibly containing hunres of species. Let us remark that iterative refinement is not contingent on the availability of a specific, i.e., nongeneral, equation. Instea of the specific relationship in Eq. 41, we coul have also use the general relationship T(X, Y) = C x (X) + C y (Y), wherec x an C y are general functions. Even substituting this general relationship into the moel leas to a reuction of parameters of the moel. Furthermore, the functions C x an C y can be use to introuce active prey switching. This has been one for instance in the foo web moels propose in Gross an Feuel (2006) an Gross et al. (2009). Using the techniques escribe above, the local bifurcations of the IGP moel can be calculate analytically. However, because the number of parameters is relatively large, even three-parameter iagrams reveal only a very limite insight into the ynamics of the system. We therefore use an alternative approach an explore the parameter space by a numerical sampling proceure. Because all parameters in the moel have clear interpretation, we can assign a range of realistic values to each of the parameters (see Table 1). We generate an ensemble of parameter sets by ranomly assigning each parameter a value rawn from the respective range. The stability of the steay state corresponing to a sample parameter set is then etermine by numerical computation of the eigenvalues of the corresponing Jacobian. Because of the numerical efficiency of eigenvalue computation, ensembles of millions or billions of sample parameter sets can be evaluate in reasonable computational time. Base on such large ensembles, a soun statistical analysis of moels containing hunres or thousans of parameters is feasible. An example of such an analysis in a 50- species moel was presente in Gross et al. (2009). We refer the reaer to this paper for an illustration of the ecological insights that can be gaine from correlation analysis in generalize moels. To assess the epenence of the stability of the IGP moel on the parameters, we generate 10 8 ranom parameter sets. Both scale an elasticity parameter values were rawn inepenently from uniform istributions. Subsequently, each parameter set was assigne a stability value of 1 if it is foun to correspon to a stable steay state an 0 if it correspons to an unstable steay state. The epenence of system stability on iniviual parameter values was then quantifie by computing the correlation coefficient between a given parameter an the stability value over the whole ensemble. Strong positive correlations inicate that large values of the respective parameter promote stability, while strong negative correlations inicate that large values of the parameter reuce stability. The results of the numerical analysis (Fig. 4) show that the proportional loss of the resource ue to intraspecific competition, δ, correlate with stability. This is not surprising because of the known stabilizing effect of super-linear mortality. To a lesser extent high values of β x promote stability. This shows that stability is enhance if the preatory losses of the resource occur Table 1 Values an ranges of the parameter sampling assume to compute Fig. 4 The timescales, α x, α y an α z, areassumetoscale allometrically. The parameter r escribes the timescale separation. The elasticities, s x, x,m y, f y, f x,k t an k z,are assume to be one (linear), two (quaratic) or between 0 an 1 (saturation, see text). Only the elasticity m z is assume to be slightly super linear. The branching parameters, δ,β x an β y,are per efinition between 0 an 1 Parameter Value or range Meaning α x 1 Turnover rate of resource α y r Turnover rate of consumer α z r 2 Turnover rate of omnivore r 0 to 1 Allometric factor s x 1 Elasticity of resource prouction x 2 Elasticity of intraspecific competition in resource m y 1 Elasticity of consumer mortality m z 1 Elasticity of omnivore mortality f y 1 Elasticity of consumption with respect to consumer f x 0 to 1 Elasticity of consumption with respect to resource k t 0 to 1 Elasticity of preation with respect to prey/resource k z 1 Elasticity of preation with respect to omnivore δ 0 to 1 Proportion of losses of the resource ue to mortality β x 0 to 1 Proportion of resource consumption ue to consumer β y 0 to 1 Proportion of losses of consumer ue to preation t x 0 to 1 Proportion of resource in omnivore iet t y 1 t x Proportion of consumer in omnivore iet