Local dynamical equivalence of certain food webs
An important challenge in theoretical ecology is to find good coarse-grained representations of complex food webs. Here, we use the approach of generalized modeling to show that it may be possible to formulate a coarse-graining algorithm that conserves the local dynamics of the model exactly. We show examples of food webs with a different number of species that have exactly identical local bifurcation diagrams. Based on these observations, we formulate a conjecture governing which populations of complex food webs can be grouped together into a single variable without changing the local dynamics. As an illustration, we use this conjecture to show that chaotic regions generically exist in the parameter space of a class of food webs with more than three trophic levels. While our conjecture is at present only applicable to relatively special cases, we believe that its applicability could be greatly extended if a more sophisticated mapping of parameters were used in the model reduction.
KeywordsModel reduction Bifurcation analysis Food web Modeling Dynamics Coarse-graining
Over the recent decades, there has been significant progress in ecological modeling. However, this progress manifests itself mostly in the description of small systems containing only few species. By contrast, to predict the dynamics of large ecosystems remains an important open challenge. In the context of marine ecosystems, it has often been pointed out that several obstacles have to be overcome in order to model even the planktonic food web, which forms their backbone, realistically.
One obstacle that is encountered in plankton modeling is the sheer diversity of marine life. The traditional approach of ecological modeling is to describe every single population by at least one differential equation. If we applied this approach to a marine food web, we would obtain a huge system, which would be prohibitively difficult to study both analytically and numerically. A promising alternative is therefore to model the food web not on the level of the population but on a coarse-grained level at which every variable describes several similar populations.
Coarse-grained descriptions have a long tradition in plankton modeling. The earliest models, the N-P-Z food chains, consisted only of three equations describing, nutrients N, phytoplankton P, and zooplankton Z (see, e.g., Steele and Henderson 1992). It is clear that models at this level of abstraction cannot capture the dynamics of the planktonic food web in any detail. However, they have the advantage that the categories N, P, and Z can be clearly defined. Many current models use more refined categories such as guilds and functional groups. In this case, it can be unclear if a given population should belong to one category or the other, or should be a category on its own (Anderson 2005). The traditional solution is to group populations that share certain biological characteristics such as feeding behavior, metabolism, or activity cycle. Some newer approaches use the tools of graph theory to identify cliques or communities that hold a similar topological position in the food web (Newman and Girvan 2004). The former approach ensures that populations that are grouped together in one category fill a similar role in the ecosystem; the latter, that they hold a similar topological position. However, does either one imply that the grouped species have a similar impact on the dynamics?
In the formulation of a coarse-grained food web model, the goal should be to group species into categories such that the dynamics of the system is not changed by the coarse-graining, at least, not qualitatively. However, to check whether this condition is satisfied does not seem to be feasible as it requires knowledge of the dynamics of the original system which we set out to determine in the first place. Nevertheless, if an approach is available to extract at least certain dynamical properties from the full model, we can check whether these dynamical properties are conserved in the coarse-grained model.
In this paper, we apply the approach of generalized modeling to explore the effect of coarse-graining on classes of simple food webs. Generalized modeling can extract certain dynamical properties, i.e., the local dynamics around steady states, very efficiently. One advantage of generalized models is that a single generalized model does not describe a single food web but the whole class of food webs, which share a similar topology. This enables us to check whether a given coarse-graining step qualitatively preserves the local dynamics in a given class of food webs. Based on these results, we conjecture that certain food webs can be coarse-grained to food chains with the same number of trophic levels. This allows us to generalize results on food chains to a class of food webs. In particular, we present evidence that parameter regions in which chaotic dynamics take place should exist in many food-chains with more than three trophic levels.
The paper is organized as follows: We start by considering some simple examples in Section 2.1. The insights gained in this section lead to the formulation of some conjecture, which is stated in Section 3. This conjecture is then applied to generalize previous results on chaotic parameter regions in food chains to a class of food webs. Finally, we summarize and discuss our findings in Section 4.
2 Two illustrative examples
Let us start by using generalized models to study two simple examples. The first example is a generalized predator–prey system with one predator. We use it to give a brief introduction to the approach of generalized modeling. Thereafter, we study a system of two populations of predators competing for one prey population. In this second example, we discuss the generalized modeling only briefly, but focus instead on the results of the modeling process: In certain cases, the local dynamics of the system is identical to the one observed in the predator–prey system. So, in order to capture the dynamics of the system, it is sufficient to describe the two populations of predators by a single variable.
2.1 The generalized predator–prey system
In conventional modeling, the first step of model analysis is often to compute the steady states of the system under consideration. In the generalized model, this is impossible with the desired degree of generality. However, there is in fact no need to compute steady states: The generalized model describes a whole class of systems, which covers the state space densely with steady states. In other words, for every steady (X *,Y *) we observe in nature, there is at least one specific model of the form of Eq. 1 that has a steady state exactly at this point. In fact, there is even a whole family of infinitely many specific models that have a steady state at (X *,Y *).
In ecology, the sensitivity of top-predator mortality to top-predator abundance m y is a well known quantity: the so-called exponent of closure. Likewise, the other parameters that appear in the predator–prey system have a well-defined ecological meaning. The parameter s x denotes the sensitivity of the population growth of the prey with respect to prey abundance. If the prey is not limited by other external factors apart from predation, the dependence should be linear (s x = 1). If, by contrast, the prey population is strongly limited by external factors, its growth rate should be insensitive to the abundance (s x = 0). Perhaps the most important parameter for the dynamics is g x, which denotes the dependence of predation on prey abundance. A Lotka-Volterra (i.e., mass action) model corresponds to g x = 1. If predator saturation is taken into account, then the predation rate is less sensitive to prey abundance, and we find 0 ≤ g x ≤ 1 depending on the strength of the saturation. Finally, the parameter g y describes the dependence of the predation rate on predator abundance. In most models, this dependence is assumed to be linear (g y = 1).
Let us now return to the stability of steady states. The Jacobian governs the time evolution of the system close to the steady state. If the real parts of all eigenvalues of the Jacobian are negative, then trajectories that start close to the steady state converge to it exponentially. In this case, the equilibrium is said to be locally asymptotically stable. If parameters are changed, the stability may be lost if the variation causes eigenvalues to acquire positive real parts. Since the Jacobian is a real matrix, this can happen in either of two different ways: A single real eigenvalue crosses the imaginary axis in the origin of the complex plane, or two eigenvalues cross the imaginary axis as a complex conjugate pair. The first case corresponds to a bifurcation of saddle-node type, such as fold, pitchfork, and transcritical bifurcations, which in general change the number and/or stability of steady states. The second case corresponds to a Hopf bifurcation, which, at least transiently, gives rise to oscillatory behavior.
Before we move on to the next section, let us recapitulate what we have achieved: Starting from a generalized model, Eq. 1, we have parameterized the Jacobian (and, therefore, the local dynamics) around every possible steady state in the class of models under consideration. Apart from the analytical computation of bifurcation points, which we have used here, the Jacobian can be analyzed in a number of ways: For instance, we can use Monte-Carlo sampling to identify the parameters to which stability is most sensitive or we can use an optimization approach to find the most stable parameter set (Steuer et al. 2006, 2007). However, in this paper, we will show that certain topologically different food webs can yield exactly the same bifurcation diagrams, which is best done analytically.
2.2 A generalized model of competing predators
While this situation is quite special, let us emphasize that it does not mean that we are talking about a single population of predators that we have split arbitrarily into two groups. Such a degenerate situation would, among other pathologies, lead to a rank-deficient Jacobian. In truth, the two populations can be quite different. What Eq. 19 implies is only that they have the same impact on the prey population and that they respond in the same way to perturbations.
To summarize: We have used the approach of generalized modeling to identify a situation in which the local bifurcation diagrams of a predator–prey system is exactly identical to the corresponding diagram of a system describing exploitative competition. While this situation is admittedly special in the context of the generalized model, it can be found in the whole family of conventional models that satisfy Eq. 19. The equivalence of bifurcation diagrams implies that the dynamics of the two models is, at least locally, identical and, therefore, the coarse-grained (predator-prey) model and the full (exploitative competition) model have similar local dynamics.
3 Equivalence of more complex food webs
Based on our results, we conjecture the following: If two or more populations in the system interact with exactly the same neighbors and their generalized parameters are identical in the sense of Eq. 19, then the populations can be modeled by a single dynamical equation with the corresponding parameters. Again, we emphasize that identical generalized parameters do not imply an identity of organisms.
In the following, we call two food webs equivalent if they satisfy the assumptions of the conjecture. In this case, we say that the food web with the higher number of populations is reducible to the food web with the lower number of populations.
Additionally, we observed that the bifurcation diagram of a given food web contains at least as many Hopf bifurcation surfaces as the food chain with the same number of trophic levels, while food webs that are not reducible to food chains generally have additional bifurcation surfaces. Two examples of such food webs that are not reducible to food chains are also shown in Fig. 4 (bottom panels).
The examples discussed above show that the food web topology has a strong impact on reducibility. We therefore have to ask what happens if we alter the topology of a reducible food web. At which point will the transition to irreducibility occur? In our general food web model, the topology is captured by a number of parameters that encode the strength of topological connections in analogy to the parameter β in the competition model from the previous section. We can tune these parameters continuously between reducible and irreducible topologies.
A bifurcation diagram of the system described above is shown in Fig. 5. In this diagram, we have chosen b 3,4 as one of the axes. For b 3,4 = 0, the food web can be reduced to the tri-trophic food chain. Therefore, the parameter plane with b 3,4 = 0 is exactly identical to a two-parameter bifurcation diagram of this food chain at the corresponding exponent of closure m y = 1.1. As we decrease b 3,4, the exact identity is lost but the topology of the bifurcation surfaces at first stays qualitatively similar. At a certain value of b 3,4 the bifurcation surface of saddle-node type folds back and disappears. From this point, the two-parameter bifurcation diagram that we find in the food web for a given value of b 3,4 is no longer equivalent to the bifurcation diagram of the tri-trophic food chain at the corresponding value of m y. However, note that the two-parameter diagram is still qualitatively similar to a two-parameter diagram of the food chain at a higher exponent of closure m y. If b 3,4 is decreased further, then a new Hopf bifurcation surface appears. The bifurcation diagram is now qualitatively different from the bifurcation diagram of the food chain. In particular, double Hopf-bifurcations, i.e., intersections of Hopf bifurcation surfaces as shown in Fig. 5 (part E), cannot appear in our tri-trophic food chain model at all.
Above, we have conjectured that local bifurcation diagrams of a class of food webs are exactly reducible to the corresponding diagrams of food chains with the same number of trophic levels. Figure 5 shows that, in the example, a larger class of food web appears to exhibit qualitatively similar local dynamics. These insights, if proved, can be used to extend certain results on food chains to classes of food webs.
One question that has been discussed in the ecological literature for quite some time is whether chaotic dynamics occur in natural food webs (May 1987; Upadhyay et al. 1998; Rai and Schaffer 2001). On the one hand, even some of the simplest ecological models exhibit chaotic dynamics (e.g., May 1974, 1976; Edwards and Yool 2000). On the other hand, chaos has rarely been observed in nature (Cushing et al. 1998; Tilman and Wedin 1991; Hanski et al. 1993; Turchin and Ellner 2000). In the past, it has been argued that chaotic dynamics could be detrimental for the survival of the participating populations and could therefore disappear in the course of natural selection. However, selection acts primarily on the level of the individual. Even on the level of the population, there is some evidence that chaos, if considered in a spatial or metapopulation context, promotes persistence (Allen et al. 1993; Solé et al. 1992; Petrovskii et al. 2004). Perhaps the most widely accepted view now is that chaos may be present in the dynamics but is difficult to detect because of environmental noise (Nychka et al. 1992; Ellner and Turchin 1995). However, another opinion is that the complexity of large ecological systems somehow prevents chaotic dynamics (Steele and Henderson 1992; Ruxton and Rohani 1998; Fussmann and Heber 2002).
Since the approach of generalized modeling focuses on local dynamics, global dynamics such as chaos is not directly accessible. However, we can draw conclusions on certain features of global dynamics by considering bifurcations of higher codimension. Unlike Hopf and saddle-node bifurcations, which are of codimension 1, bifurcations of higher codimension have more than one bifurcation condition. In comparison to codimension 1 bifurcations, the higher codimension bifurcations are often neglected since the tuning of more than one parameter is necessary to find them in experiments or numerical investigations.
An example of a codimension-2 bifurcation is the double-Hopf bifurcation, formed at the intersection of two Hopf bifurcation surfaces, which we have already encountered above. Mathematical investigations of this bifurcation have shown that there is generically a region in parameter space in which chaotic dynamics occur close to the double-Hopf bifurcation point (Kuznetsov 1989). In this parameter region chaotic dynamics can therefore, at least transiently, be observed.
In a previous paper, we have shown that double Hopf bifurcations exist in food chains with more than three trophic levels. This implies that chaotic dynamics can be found generically in these food chains at least in some parameter space. Our conjecture implies that a certain class of food webs is exactly equivalent to food chains of the same length. This means that these food webs also exhibit double-Hopf bifurcations and therefore chaotic parameter regions. Furthermore, double-Hopf bifurcations should also exist in the much larger class of food webs in which the bifurcation diagrams are qualitatively similar to the bifurcation diagrams of food chains. Finally, irreducible food webs in which the local dynamics differs qualitatively from that of food chains have, in our experience, a more complex bifurcation structure than food chains of comparable length, and therefore, the corresponding bifurcation diagrams also contain double-Hopf bifurcations.
From the evidence given above, we conclude that chaotic parameter regions should exist in a large class of complex food webs with more than three trophic levels. It is therefore unlikely that the complexity of natural food webs in itself leads to the avoidance of chaotic dynamics.
4 Remarks and discussion
In this paper, we have used the approach of generalized models to show that, in certain food webs, the local bifurcation diagrams are exactly identical to the corresponding bifurcation diagrams in other food webs and food chains. Moreover, we have conjectured that there is a universal rule that governs which food webs show this equivalence. This conjecture, if proved, can be used to generalize certain findings on food chains to classes of food webs. Perhaps more importantly, it holds the promise to yield an analytical procedure for the reduction of complex food-web models, or even dynamical biological networks in general.
A method for coarse-graining food webs based on the equivalence of local bifurcations had several advantages: It would probably preserve local dynamics exactly, and also preserve certain features of the global dynamics through conservation of local bifurcations of higher codimension. Moreover, it would probably provide hard criteria governing which populations could be coarse-grained into a single variable and which could not.
Before the present observations can be extended to a method for model reduction, more research is certainly necessary. This work should aim to understand the equivalence; prove the stated conjecture; and, in particular, explore its validity in the large class of systems in which we so far only observe a qualitative equivalence of bifurcation diagrams. Let us therefore conclude with some remarks that we believe will be conductive for this effort.
One feature of the mapping of a food web to a simpler food web is that it reduces the dimensionality of both the state space and the parameter space. In the context of our approach, the mapping in state space is not of much concern as we do not need to specify it explicitly. By contrast, the mapping in parameter space is more intriguing; We have to specify how the parameters of the original model are translated to the lower number of parameters in the reduced model.
In this paper, we have postulated the mapping of parameters ad hoc based on biological reasoning and only subsequently verified that it produced the desired result. For instance, in our discussion of the system of competing predators, we have demanded that both predators exhibit the same exponent of closure m y = m 1,y = m 2,y (Eq. 19). Only thereafter could we compare the bifurcation diagram to that of the predator–prey system. For future investigations, a different approach seems more promising: Instead of demanding the identity of certain parameters and then checking for the identity of bifurcation diagrams, we could ask which mapping of parameters, say m y(m 1,y,m 2,y) will produce a bifurcation diagram that is identical to that of the simpler system. Of course, we do not need to restrict this mapping to parameters between which a direct biological correspondence exists, but could allow the parameter of the reduced system m y also to depend on other parameters, such as g 1,y and g 2,y. While more mathematical work is certainly necessary, we believe that it should be possible to derive a rigorous procedure for the parameter mapping.
We believe that the procedure described above will reveal exact reductions for the large class of systems in which we so far only observe a qualitative similarity of bifurcation diagrams. Nevertheless, it is obvious that, for certain pairs of food webs, there cannot be any mapping of parameters that leads to identical bifurcation diagrams. For instance, we have seen that the bifurcation diagram of the four-trophic food chain contains a double-Hopf bifurcation line. Since this bifurcation is characterized by four purely imaginary eigenvalues of the Jacobian, it simply cannot be accommodated in any system with less than four state variables. Therefore, the local bifurcation diagram of the four-trophic food chain can never be mapped on that of the tri-trophic food chain.
The fact that generalized modeling reveals bifurcations of higher codimension with relative ease is advantageous as it provides examples of these bifurcations for mathematical analysis and allows us to gain insights into global features of the dynamics based on a local analysis. In the present paper, this has be utilized to provide evidence for chaos in a large class of food webs. This and other bifurcations also reveal the presence of global homoclinic and heteroclinic bifurcations. It is conceivable that future progress in normal form theory will enable us to extract even more information from the bifurcations of higher codimension.
So far, we have focused on generalized models, while in the literature, most modeling is done by conventional models in which the interactions are restricted to specific functional forms. Note, however, that it is straightforward to step back and forth between generalized and conventional representations of a system. By replacing the specific functional forms by general functions, a conventional model can be turned into a generalized model and vice versa (Gross et al. 2005). The corresponding mapping between generalized and specific model parameters can be computed analytically and, thus, provides a direct relation between the modeling approaches (van Voorn et al. 2008). For a smoother transition, it is possible to study hybrid models in which some functional relationships are specified while others are kept general (Gross and Feudel 2006).
Let us emphasize that the identity of bifurcation diagrams is therefore not restricted to the realm of generalized models. One should be able to observe the same identity also in conventional models. However, in conventional models, additional difficulties arise since the mapping is affected by the position of steady states in state space, which is often prohibitively difficult to study analytically. In the past, we have pointed out that the parameters that are identified in generalized modeling are, in a certain sense, more natural than the parameters that are commonly introduced in conventional modeling (Gross 2004; Gross and Feudel 2006). In the present paper, this enabled us to spot the identity of bifurcation diagrams even based on very simple ad hoc assumptions. While the same identity should exist also in conventional models, the corresponding mapping of the conventional parameters is probably much more involved, and hence, the identity is more difficult to spot.
To conclude, we have shown that, in generalized models, an exact equivalence between the bifurcation diagrams of different food webs can be observed, albeit in a relatively special case. Several results seem to indicate that this identity can be extended to a much larger class of systems if more refined parameter mappings are used. We hope that this insight will eventually evolve into a method for model reduction. In the context of such a method, an intermediate coarse-graining step with generalized models will probably be useful even if the desired result and starting point are conventional models.
This work was supported by the Deutsche Forschungsgemeinschaft (DFG), grant FE 359.
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