Generalized and specific modeling approaches require different computation techniques to locate bifurcations in parameter space. In Chapter 2 an innovative technique to locate bifurcations in generalized models is introduced, that allows for an efficient computation of three dimensional bifurcation diagrams. This technique is applied on two rarely investigated types of predatorprey models. One model focuses on stoichiometric constraints on the primary production and the conversion efficiency. These constraints cause dependencies that are not considered in classical predator-prey models. The other model describes how a disease spreads upon a predator population and how these dynamics influence the population interactions. The predator population is thereby structured in susceptible, infected and recovered predators. To find generic effects we use the approach of generalized modelling. The resulting bifurcation diagrams are partly combined with bifurcation diagrams of specific modeling approaches to demonstrate the interplay of generalized and specific modelling.
Although probably all natural interacting populations are influenced by limitation of nutrients and diseases, the related dependencies and the distinction between infected and uninfected are rarely considered in theoretical predator-prey models. The generalized analysis shows that these aspects of ecology qualitatively change predator-prey dynamics. In the following we briethly discuss the main results and give suggestions of further investications.
We begin with the investagtion of the generalized stoichiometric predatorprey model in Chapter 3. First, it turns out that for the classical assumption of constant efficiency the stability of equilibria depends only on two generalized parameters, the intra-specific competition and the predator sensitivity to prey.The technique for the computation of bifurcation surfaces introduced in Chapter 2 is used to show how this two-dimensional bifurcation diagram evolves when the conversion efficiency becomes variable. The additional dimension is spanned by the food quality parameter that is related to the variability of the conversion efficiency.
The analysis shows that a variable conversion efficiency has major effects on the stability and dynamics of the system. In addition to the Hopf bifurcation, a surface of tangent bifurcations and a line of codimension-2 Takens-Bogdanov bifurcation appear. One the one hand, the latter indicates that homoclinic bifurcations are also generic in stoichiometric models. On the other hand, the Takens-Bogdanov bifurcation marks the end of the Hopf bifurcation surface. Therefore, it leads to the disappearance of the paradox of enrichment for low food quality.
The computation of three-dimensional bifurcation diagrams allows for a fast overview to get a qualitative understanding of the (de)stabilizing properties of the six system parameters. By contrast to the strong influence of the variable efficiency, it shows that stoichiometric constraints on the primary production have qualitatively rather low effects. They cause only shift of the observed bifurcation scenario. In this way we identify the variable efficiency as a key process that remarkably changes the dynamics of classical predator-prey systems.
These general properties are used to understand and predict the differences of specific stoichiometric models. First, we considered in Sec. 3.5 a variable and a simplified constant conversion efficiency model by Kooijman et al. (2004). The observation that a homoclinic and a tangent bifurcation appear in the variable efficiency model but not in the constant effiency model is in agreement with the results from the generalized analysis. Also a stoichiometric model proposed by Loladze and Kuang (2000) with unsmooth processes considering two limiting nutrients shows the appearance of tangent and homoclinic bifurcations. Further, the generalized analysis coincides with the disappearance of the paradox of enrichment in the model by Loladze and Kuang (2000). From the generalized point of view this corresponds to the fact that the unsmooth conversion process in this model allows only two discrete values for a parameter of the generalized model. For one parameter value a Hopf bifurcation exist and for the other no Hopf bifurcation can occure. In order to provide an example that allows to analyze the transition between both extreme parameter values we construct a smooth analogon model. This model shows that a Takens-Bogdanov bifurcation is responsible for the disappearance of the Hopf bifurcation, as predicted by the generalized analysis. Further, the results of the generalized analysis correctly predict a shift of the bifurcation scenario when additionally mass conservation is assumed in Sec. 3.5.4. These examples show that the generalized modeling can be used in combination to specific models with identify properties that are generic for the model class.
The comparison to specific examples is done qualitatively but also quantitatively. We show that it is possible to translate specific bifurcation diagrams into generalized parameters and combine these projections with a generalized bifurcation diagram. This is done by fitting the specific coupling of certain generalized parameters. These combined bifurcation diagrams show in an exemplified way how specific and generalized parameters are connected. Further, it illustrates how multiple intersecting steady states that require different normalizations share one generalized diagram.
We show a counterintuitive stabilizing effect of intra-specific competition appearing likewise in constant and variable efficiency models. Instead of the related paradox of enrichment, this effect does not depend on the specific functional response under consideration. Further, a comparison to the observed paradox of nutrient enrichment in (Loladze and Kuang, 2000) shows that both, the paradox of enrichment and the paradox of nutrient enrichment, are combined in the paradox of competition observed in the generalized model. This illustrates the generic nature of the observed paradox of competition.
In the generalized eco-epidemic model in Chapter 4 where we consider a disease in the predator population, the visualization technique is used to locate bifurcations of higher codimension that give information about the appearance of complex dynamics. By the localization of a double-Hopf bifurcation in the generalized eco-epidemic model in Sec. 4.4.2, we show that chaotic parameter regions generally exist when the predator is infected by a disease.
This generalized analysis is used in Sec. 4.5 to construct a specific ecoepidemic model which is investigated to study the dynamics close to the double-Hopf bifurcation. Thereby, we find additional period-doubling and Neimark-Sacker bifurcations. We identify two routes into chaos, both involve a transition from quasiperiodicity. Most importantly, we demonstrate that the chaotic parameter regions are extended. In this way our analysis shows that, in the class of eco-epidemic models under consideration, chaos is generic and likely to occur. In other words, we show that diseases in predator populations can generally lead to chaotic dynamics.
More generally, the analysis in Chapter 4 shows that the localization of organizing centers by three-dimensional bifurcation diagrams reveals the regions of most interesting dynamics. Moreover, it provides plenty of examples for these situations since the generalized model represents a whole classes of models. From a technical perspective, the faithful representation of the Takens- Bogdanov bifurcation, the intersection with the Gavrilov-Guckenheimer bifurcation and most importantly, the complicated Whitney-umbrella structure of the Hopf bifurcation in Sec.4.5 (Fig.4.2) represents a masterpiece of the adaptive triangulation algorithm presented in this thesis.
In principle, the formulations of the models could be much more general than in Chapter 3 and Chapter 4. The most general formulation of a predator-prey system is dX_i/dt = F_i(X_1,X_2), i = 1,2. Obviously, this formulation hardly allows any conclusions about the involved processes. Instead, the models proposed in this thesis adopt some processes from conventional modeling approaches like the logistic growth in the eco-epidemic model or the linear death terms in the predator populations. This clearly reduces the degree of generality of the model but likewise focuses the analysis on the considered processes. Another advantage of this semi general formulation is that the results are, as it shows in the presented thesis, directly transferable to specific model examples.
In summary, the presented thesis has contributed to our understanding of stoichiometric influences on predator-prey interactions and how diseases in predator populations can influence predator-prey dynamics. On one hand, the models under consideration can be further generalized in order to see how these effects act jointly with other model modifications like nonlinear death terms. On the other hand, one could further specify and modify the eco-epidemic model to account for a specific problem. For example, one could adapt the model to analyse the dynamics of a specific disease of cats and the interaction with the rabbit population on an island (cf. Introduction). Also the method for the computation of bifurcations in generalized models could be extended in several ways. As an outlook, three possible extensions are discussed more in detail.
First, the computation of the bifurcation surfaces can potentially be extended in order to compute hypersurfaces. The proposed method for the computation of bifurcation surfaces provides a fast and efficient computation of three-dimensional bifurcation diagrams. The resolution of computed bifurcation points is locally adapted to the complexity of the surface. Once such a representation of the bifurcation surfaces is found, it is possible to trace these points while varying a fourth parameter. If necessary, additional bifurcation points can be computed in order to maintain or adapt the local resolution. In this way, one would obtain four-dimensional hypersurfaces that can be visualized in a three-dimensional diagram where the fourth parameter can be changed interactively. This technique would allow to investigate the evolution of the bifurcation landscape. Moreover, it is possible to iterate this step for additional parameters. In this way, one can explore step by step the whole bifurcation manifold. This information could be used to evaluate the (de)stabilizing effect of a parameter variation in terms of how the variation changes distance to the bifurcation surfaces in the direction of all parameters under consideration.
Second, the generalized stoichiometric predator prey model in Chapter 3 can be used as a building block in a generalized stoichiometric food web. We have shown that the variable food quality greatly affects the dynamics of simple predator prey models. In larger population models, the variable food quality applies mainly to the autotrophs at the bottom of the food chains or webs. However, as we have discussed in Sec. 3.2, the resulting variable efficiency function depends in general on all other populations. Therefore, a specific modeling approach would become very complicated with the number of considered species. A generalized modeling approach could instead be used to overcome this difficulty. In generalized food webs, the large number of parameters makes a stability analysis in terms of three-dimensional bifurcation diagrams inappropriate. Instead, a simple numerical correlation between parameter values and the stability of the steady state can reveal how the parameter influence the stability of the steady state statistically. In this way, the influence of a variable food quality on the stability of complex food webs could be studied from a very general perspective.
Third, the specific model in Chapter 4, Sec. 4.5, could be used to study stabilizing effects of diseases in conjunction with chaotic dynamics. In Sec. 4.2 we discussed that chaotic dynamics can prevent synchronization effects in patchy populations and therefore reduce the possibility of global extinction events. An interesting theoretical investigation beyond the scope of this work is to model population patches using the example model in Sec. 4.5 and adding a weak coupling due to migration between the patches. Theoretically it should be possible to observe predator-prey oscillations when the disease has no influence on the vital dynamics since the predator prey interactions are not affected by the disease. An adequate coupling should cause synchronization between the patches. Such an experiment is less artificial than it might sound. For example, the snowhare-lynx cycles from different regions in Canada show synchronization over millions of square kilometers. An onset of chaotic dynamics in the model due to an increase of the influence of the vital dynamics could perturb the synchronization (Allen et al., 1993; Ruxton, 1994; Earn et al., 1998). In this way, a disease that reduces the vitality of the predator population could prevent both the prey and the predator population from global extinction.
To this end, whenever a system is too complex for an comprehensive description like our environment, simple specific and generalized models help to understand the system properties from an elementary perspective. As the presented thesis shows, both modeling approaches can fruitfully act in concert.
This is the symposium that concludes my project