Consequences of population models for the dynamics of food chains

Maintenance affects the stability of a two-tiered microbial 'food chain'?

Microbial 'food chains' are fundamentally different from canonical food chains in the sense that the waste products of the organisms on one trophic level are consumed by organisms of the next trophic level rather than the organisms themselves. In the present paper we introduce a generalised model of a two-tiered microbial 'food chain' with feedback inhibition, after applying an appropriate dimensionless transformation, and investigate its stability analytically. We then parameterised the model with consensus values for syntrophic propionate degradation compiled by the IWA Task Group for Mathematical Modelling of Anaerobic Digestion Processes. Consumption of energy for all processes other than growth is called maintenance. In the absence of maintenance and decay the microbial 'food chain' is intrinsically stable, but when decay is included in the description this is not necessarily the case. We point out that this is in analogy to canonical food chains where introduction of maintenance in the description of a stable (equilibrium or limit cycle) predator-prey system generates chaos.

Direct observation of homoclinic orbits in human heart rate variability

Homoclinic trajectories of the interbeat intervals between contractions of ventricles of the human heart are identified. The interbeat intervals are extracted from 24-h Holter ECG recordings. Three such recordings are discussed in detail. Mappings of the measured consecutive interbeat intervals are constructed. In the second and in some cases in the fourth iterate of the map of interbeat intervals homoclinic trajectories associated with a hyperbolic saddle are found. The homoclinic trajectories are often persistent for many interbeat intervals, sometimes spanning many thousands of heartbeats. Several features typical for homoclinic trajectories found in other systems were identified, including a signature of the gluing bifurcation. The homoclinic trajectories are present both in recordings of heart rate variability obtained from patients with an increased number of arrhythmias and in cases in which the sinus rhythm is dominant. The results presented are a strong indication of the importance of deterministic nonlinear instabilities in human heart rate variability.

Numerical bifurcation analysis of a tri-trophic food web with omnivory

We study the consequences of omnivory on the dynamic behaviour of a three species food web under chemostat conditions. The food web consists of a prey consuming a nutrient, a predator consuming a prey and an omnivore which preys on the predator and the prey. For each trophic level an ordinary differential equation describes the biomass density in the reactor. The hyperbolic functional response for single and multi prey species figures in the description of the trophic interactions. There are two limiting cases where the omnivore is a specialist; a food chain where the omnivore does not consume the prey and competition where the omnivore does not prey on the predator. We use bifurcation analysis to study the long-term dynamic behaviour for various degrees of omnivory. Attractors can be equilibria, limit cycles or chaotic behaviour depending on the control parameters of the chemostat. Often multiple attractor occur. In this paper we will discuss community assembly. That is, we analyze how the trophic structure of the food web evolves following invasion where a new invader is introduced one at the time. Generally, with an invasion, the invader settles itself and persists with all other species, however, the invader may also replace another species. We will show that the food web model has a global bifurcation, being a heteroclinic connection from a saddle equilibrium to a limit cycle of saddle type. This global bifurcation separates regions in the bifurcation diagram with different attractors to which the system evolves after invasion. To investigate the consequences of omnivory we will focus on invasion of the omnivore. This simplifies the analysis considerably, for the end-point of the assembly sequence is then unique. A weak interaction of the omnivore with the prey combined with a stronger interaction with the predator seems advantageous.

Quantitative aspects of metabolic organization: a discussion of concepts

Metabolic organization of individual organisms follows simple quantitative rules that can be understood from basic physical chemical principles. Dynamic energy budget (DEB) theory identifies these rules, which quantify how individuals acquire and use energy and nutrients. The theory provides constraints on the metabolic organization of subcellular processes. Together with rules for interaction between individuals, it also provides a basis to understand population and ecosystem dynamics. The theory, therefore, links various levels of biological organization. It applies to all species of organisms and offers explanations for body-size scaling relationships of natural history parameters that are otherwise difficult to understand. A considerable number of popular empirical models turn out to be special cases of the DEB model, or very close numerical approximations. Strong and weak homeostasis and the partitionability of reserve kinetics are cornerstones of the theory and essential for understanding the evolution of metabolic organization.

Multiple attractors and boundary crises in a tri-trophic food chain

The asymptotic behaviour of a model of a tri-trophic food chain in the chemostat is analysed in detail. The Monod growth model is used for all trophic levels, yielding a non-linear dynamical system of four ordinary differential equations. Mass conservation makes it possible to reduce the dimension by 1 for the study of the asymptotic dynamic behaviour. The intersections of the orbits with a Poincaré plane, after the transient has died out, yield a two-dimensional Poincaré next-return map. When chaotic behaviour occurs, all image points of this next-return map appear to lie close to a single curve in the intersection plane. This motivated the study of a one-dimensional bi-modal, non-invertible map of which the graph resembles this curve. We will show that the bifurcation structure of the food chain model can be understood in terms of the local and global bifurcations of this one-dimensional map. Homoclinic and heteroclinic connecting orbits and their global bifurcations are discussed also by relating them to their counterparts for a two-dimensional map which is invertible like the next-return map. In the global bifurcations two homoclinic or two heteroclinic orbits collide and disappear. In the food chain model two attractors coexist; a stable limit cycle where the top-predator is absent and an interior attractor. In addition there is a saddle cycle. The stable manifold of this limit cycle forms the basin boundary of the interior attractor. We will show that this boundary has a complicated structure when there are heteroclinic orbits from a saddle equilibrium to this saddle limit cycle. A homoclinic bifurcation to a saddle limit cycle will be associated with a boundary crisis where the chaotic attractor disappears suddenly when a bifurcation parameter is varied. Thus, similar to a tangent local bifurcation for equilibria or limit cycles, this homoclinic global bifurcation marks a region in the parameter space where the top-predator goes extinct. The 'Paradox of Enrichment' says that increasing the concentration of nutrient input can cause destabilization of the otherwise stable interior equilibrium of a bi-trophic food chain. For a tri-trophic food chain enrichment of the environment can even lead to extinction of the highest trophic level.

The role of intracellular components in food chain dynamics

The dynamics of a simple food web, including multiple substrates and predator-prey interactions, is studied. An individual-based model is presented that describes the intracellular composition of the biomass of each population with two components: reserves and structural biomass. The model describes the simultaneous uptake of multiple substrates via specific carriers and their assimilation into reserve energy via multiple assimilation pathways. The available energy is used for maintenance and growth. Parameters are estimated by curve-fitting data from the literature under the condition that the elemental balances and the enthalpy balances are met. The proposed model provides an adequate description of the macromolecular composition of biomass. The model is not too complicated to be of use in the study of food webs. The consequences of the presence of intracellular components in a food web on its long-term dynamics are investigated with bifurcation analysis.

Chaotic dynamics of a food web in a chemostat

We analyze a mathematical model of a simple food web consisting of one predator and two prey populations in a chemostat. Monod's model is employed for the dependence of the specific growth rates of the two prey populations on the concentration of the rate-limiting substrate and a generalization of Monod's model for the dependence of the specific growth rate of the predator on the concentrations of the prey populations. We use numerical bifurcation techniques to determine the effect of the operating conditions of the chemostat on the dynamics of the system and construct its operating diagram. Chaotic behavior resulting from successive period doublings is observed. Multistability phenomena of coexistence of steady and periodic states at the same operating conditions are also found.

Resistance of a food chain to invasion by a top predator

We study the invasion of a top predator into a food chain in a chemostat. For each trophic level, a bioenergetic model is used in which maintenance and energy reserves are taken into account. Bifurcation analysis is performed on the set of nonlinear ordinary differential equations which describe the dynamic behaviour of the food chain. In this paper, we analyse how the ability of a top predator to invade the food chain depends on the values of two control parameters: the dilution rate and the concentration of the substrate in the input. We investigate invasion by studying the long-term behaviour after introduction of a small amount of top predator. To that end we look at the stability of the boundary attractors; equilibria, limit cycles as well as chaotic attractors using bifurcation analysis. It will be shown that the invasibility criterion is the positiveness of the Lyapunov exponent associated with the change of the biomass of the top predator. It appears that the region in the control parameter space where a predator can invade increases with its growth rate. The resulting system becomes more resistant to further invasion when the top predator grows faster. This implies that short food chains with moderate growth rate of the top predator are liable to be invaded by fast growing invaders which consume the top predator. There may be, however, biological constraints on the top predator's growth rate. Predators are generally larger than prey while larger organisms commonly grow slower. As a result, the growth rate generally decreases with the trophic level. This may enable short food chains to be resistant to invaders. We will relate these results to ecological community assembly and the debate on the length of food chains in nature.