Yield and dynamics of tritrophic food chains

Strong relationships between yield and dynamic behavior of tritrophic food chains are pointed out by analyzing the classical Rosenzweig-MacArthur model. On the one hand, food chains are subdivided into undersupplied and oversupplied categories, the first being those in which a marginal increase of nutrient supply to the bottom produces a marginal increase of mean yield at the top. On the other hand, a detailed bifurcation analysis proves that dynamic complexity first increases with nutrient supply (from stationary to a low-frequency cyclic regime and, finally, to chaos) and then decreases (from chaos to a high-frequency cyclic regime). A careful comparison of the two analyses supports the conclusion that food chains cycling at high frequency are oversupplied, while all others are undersupplied. A straightforward consequence of this result is that maximization of food yield requires a chaotic regime. This regime turns out to be very often on the edge of a potential catastrophic collapse of the top component of the food chain. In other words, optimality implies very complex and dangerous dynamics, as intuitively understood long ago for ditrophic food chains by Rosenzweig in his famous article on the paradox of enrichment.     

Interplay between the paradox of enrichment and nutrient cycling in food webs

Nutrient cycling is fundamental to ecosystem functioning. Despite recent major advances in the understanding of complex food web dynamics, food web models have so far generally ignored nutrient cycling. However, nutrient cycling is expected to strongly impact food web stability and functioning. To make up for this gap, we built an allometric and size structured food web model including nutrient cycling. By releasing mineral nutrients, recycling increases the availability of limiting resources for primary producers and links each trophic level to the bottom of food webs. We found that nutrient cycling can provide a significant part of the total nutrient supply of the food web, leading to a strong enrichment effect that promotes species persistence in nutrient poor ecosystems but leads to a paradox of enrichment at high nutrient inputs. The presence of recycling loops linking each trophic level to the basal resources weakly affects species biomass temporal variability in the food web. Recycling loops tend to slightly dampen the destabilising effect of nutrient enrichment on consumer temporal variability while they have opposite effects for primary producers. By considering nutrient cycling, this new model improves our understanding of the response of food webs to nutrient availability and opens perspectives to better link studies on food web dynamics and ecosystem functioning.     

Dynamics of a plant-herbivore-predator system with plant-toxicity

A system of ordinary differential equations is considered that models the interactions of two plant species populations, an herbivore population, and a predator population. We use a toxin-determined functional response to describe the interactions between plant species and herbivores and use a Holling Type II functional response to model the interactions between herbivores and predators. In order to study how the predators impact the succession of vegetation, we derive invasion conditions under which a plant species can invade into an environment in which another plant species is co-existing with a herbivore population with or without a predator population. These conditions provide threshold quantities for several parameters that may play a key role in the dynamics of the system. Numerical simulations are conducted to reinforce the analytical results. This model can be applied to a boreal ecosystem trophic chain to examine the possible cascading effects of predator-control actions when plant species differ in their levels of toxic defense.     

Bistability and limit cycles in generalist predator–prey dynamics

Since generalist predators feed on a variety of prey species they tend to persist in an ecosystem even if one particular prey species is absent. Predation by generalist predators is typically characterized by a sigmoidal functional response, so that predation pressure for a given prey species is small when the density of that prey is low. Many mathematical models have included a sigmoidal functional response into predator–prey equations and found the dynamics to be more stable than for a Holling type II functional response. However, almost none of these models considers alternative food sources for the generalist predator. In particular, in these models, the generalist predator goes extinct in the absence of the one focal prey. We model the dynamics of a generalist predator with a sigmoidal functional response on one dynamic prey and fixed alternative food source. We find that the system can exhibit up to six steady states, bistability, limit cycles and several global bifurcations.     

Modelling Holling type II functional response in deterministic and stochastic food chain models with mass conservation

The Rosenzweig-MacArthur predator-prey model is the building block in modeling food chain, food webs and ecosystems. There are a number of hidden assumptions involved in the derivation. For instance the prey population growth is logistic without predation but also with predation. In order to reveal these we will start with modelling a resource-predator-prey system in a closed spatially homogeneous environment. This allows us to keep track of the nutrient flow. With an instantaneous remineralisation of the products excreted in the environment by the populations and dead body mass there is conservation of mass. This allows for a model dimension reduction and yields the mass balance predator-prey model. When furthermore the searching and handling processes are much faster that the population changing rates, the trophic interaction is described by a Holling type II functional response, also assumed in the Rosenzweig-MacArthur model. The derivation uses an extended deterministic model with number of searching and handling predators as model variables where the ratio of the predator/prey body masses is used as a mechanistic time-scale parameter. This extended model is also used as a starting point for the derivation of a stochastic model. We will investigate the stochastic effects of random switching between searching and handling of the predators and predator dying. Prey growth by consumption of ambient resources is still deterministic and therefore the stochastic model is hybrid. The transient dynamics is studied by numerical Monte Carlo simulations and also the quasi-equilibrium distribution for the population quantities is calculated. The body mass of the prey individual is the scaling parameter in the stochastic model formulation. This allows for a quantification of the mean-field approximation criterion for the justification of replacement of the stochastic by a deterministic model.     

Patchiness in a minimal nutrient — phytoplankton model

We present a minimal two-component model that can exhibit various types of spatial patterns including patchiness. The model, comprising nutrients and phytoplankton, includes the effect of nutrient uptake by phytoplankton as a Holling type II functional response, and also includes the effect of zooplankton grazing on phytoplankton as a Holling type II non-dynamical term. The mean-field model without the diffusion and advection terms shows both bistability and limit-cycle oscillations as a few parameters such as the input rate of nutrients and the maximum feeding rate of zooplankton are changed. If the parameter values are chosen from the limit-cycle oscillation region, the corresponding reaction-advection-diffusion equations show spatial pattern formations by the combined effects of advection and diffusion by turbulent stirring and mixing, and biological interactions. As the nutrient input is increased, the system behaviour changes from the extinction of the entire phytoplankton to the formation of filamentous patterns, patchiness patterns and homogeneous distributions. These observations suggest that the spatial pattern of phytoplankton can function as an indicator to evaluate the eutrophication level in aquatic ecosystems.     

Increased searching and handling effort in tall swards lead to a Type IV functional response in small grazing herbivores

Understanding the functional response of species is important in comprehending the species’ population dynamics and the functioning of multi-species assemblages. A Type II functional response, where instantaneous intake rate increases asymptotically with sward biomass, is thought to be common in grazers. However, at tall, dense swards, food intake might decline due to mechanical limitations or if animals selectively forage on the most nutritious parts of a sward, leading to a Type IV functional response, especially for smaller herbivores. We tested the predictions that bite mass, cropping time, swallowing time and searching time increase, and bite rate decreases with increasing grass biomass for different-sized Canada geese (Branta canadensis) foraging on grass swards. Bite mass indeed showed an increasing asymptotic relationship with grass biomass. At high biomass, difficulties in handling long leaves and in locating bites were responsible for increasing cropping, swallowing, and searching times. Constant bite mass and decreasing bite rate caused the intake rate to decrease at high sward biomass after reaching an optimum, leading to a Type IV functional response. Grazer body mass affected maximum bite mass and intake rate, but did not change the shape of the functional response. As grass nutrient contents are usually highest in short swards, this Type IV functional response in geese leads to an intake rate that is maximised in these swards. The lower grass biomass at which intake rate was maximised allows resource partitioning between different-sized grazers. We argue that this Type IV functional response is of more importance than previously thought.     

The role of prey taxis in biological control: a spatial theoretical model

We study a reaction-diffusion-advection model for the dynamics of populations under biological control. A control agent is assumed to be a predator species that has the ability to perceive the heterogeneity of pest distribution. The advection term represents the predator density movement according to a basic prey taxis assumption: acceleration of predators is proportional to the prey density gradient. The prey population reproduces logistically, and the local population interactions follow the Holling Type II trophic function. On the scale of the population, our spatially explicit approach subdivides the predation process into random movement represented by diffusion, directed movement described by prey taxis, local prey encounters, and consumption modeled by the trophic function. Thus, our model allows studying the effects of large-scale predator spatial activity on population dynamics. We show under which conditions spatial patterns are generated by prey taxis and how this affects the predator ability to maintain the pest population below some economic threshold. In particular, intermediate taxis activity can stabilize predator-pest populations at a very low level of pest density, ensuring successful biological control. However, very intensive prey taxis destroys the stability, leading to chaotic dynamics with pronounced outbreaks of pest density.     

Positive complexity-stability relations in food web models without foraging adaptation

May's [1972. Will a large complex system be stable? Nature 238, 413-414] local stability analysis of random food web models showed that increasing network complexity leads to decreasing stability, a result that is contradictory to earlier empirical findings. Since this seminal work, research of complexity-stability relations became one of the most challenging issues in theoretical ecology. We investigate conditions for positive complexity-stability relations in the niche, cascade, nested hierarchy, and random models by evaluating the network robustness, i.e., the fraction of surviving species after population dynamics. We find that positive relations between robustness and complexity can be obtained when resources are large, Holling II functional response is used and interaction strengths are weighted with the number of prey species, in order to take foraging efforts into account. In order to obtain these results, no foraging dynamics needs to be included. However, the niche model does not show positive complexity-stability relations under these conditions. By comparing to empirical food web data, we show that the niche model has unrealistic distributions of predator numbers. When this distribution is randomized, positive complexity-stability relations can be found also in the niche model.     

Which functional responses preclude extinction in ecological population-dynamic models?

The growth and loss terms of interacting populations, called functional responses, are known to have a significant impact on the extinction dynamics of ecological models. We are able to construct models that preclude extinction for any parameter value, simply through the use of particular combinations of functional responses. These structural coexistence (SC) models have functional responses where the per capita growth terms remain positive (non-vanishing), while the per capita loss terms tend to zero (vanishing) as the relevant population tends to zero. Any of the commonly used functional responses, such as Holling Types I, II, and III, lead to non-vanishing growth terms for nutrient uptake, while any type of nonlinearity such as Ivlev or density dependent mortality of the population leads to vanishing loss terms. In order for herbivore/carnivore feeding terms to simultaneously be a vanishing loss term for the prey and a non-vanishing growth term for the predator, the exponent on the predator population must be exactly one, whilst the exponent on the prey population must be greater than one (such as a Holling Type III response). Any SC system with at least one autotroph and (possibly many) heterotrophs will always possess an internal equilibrium point. We show that the inclusion of linear mortality terms are, however, sufficient to restore the possibility of population extinctions. This allows for the formulation of ‘mixed’ systems, where some populations are guaranteed to coexist, whilst others are subject to the possibility of extinction. SC models have use in studies of, for example, biogeochemical cycling or the plankton base of fisheries models, where extinction is not desirable or relevant.     

Rhizosphere engineering for sustainable crop production: entropy-based insights

There is a growing interest in exploring interactions at root–soil interface in natural and agricultural ecosystems, but an entropy-based understanding of these dynamic rhizosphere processes is lacking. We have developed a new conceptual model of rhizosphere regulation by localized nutrient supply using thermodynamic entropy. Increased nutrient-use efficiency is achieved by rhizosphere management based on self-organization and minimized entropy via equilibrium attractors comprising (i) optimized root strategies for nutrient acquisition and (ii) improved information exchange related to root–soil–microbe interactions. The cascading effects through different hierarchical levels amplify the underlying processes in plant–soil system. We propose a strategy for manipulating rhizosphere dynamics and improving nutrient-use efficiency by localized nutrient supply with minimization of entropy to underpin sustainable food/feed/fiber production.     

Remarks on antipredator behavior and food chain dynamics

When consumers feeding on a resource spend time in avoiding high risks of predation, the predator functional response declines with predator density. While this is well established, less attention has been paid to the dependence of the consumer functional response on predator density. Here we show how the separation of behavioral and ecological timescales allows one to determine both responses starting from an explicit behavioral model. Within the general set-up considered in this paper, the two functional responses can tend toward Holling type II responses when consumers react only weakly to predation. Thus, the main characteristics of the standard Rosenzweig-MacArthur tritrophic food chain (logistic resource and Holling type II consumer and predator) remain valid also when consumers have weak antipredator behavior. Moreover, through numerical analysis, we show that in a particular but interesting case pronounced antipredator behaviors stabilize the system.     

Bottom‐up and top‐down effects of browning and warming on shallow lake food webs

Productivity and trophic structure of aquatic ecosystems result from a complex interplay of bottom‐up and top‐down forces that operate across benthic and pelagic food web compartments. Projected global changes urge the question how this interplay will be affected by browning (increasing input of terrestrial dissolved organic matter), nutrient enrichment and warming. We explored this with a process‐based model of a shallow lake food web consisting of benthic and pelagic components (abiotic resources, primary producers, grazers, carnivores), and compared model expectations with the results of a browning and warming experiment in nutrient‐poor ponds harboring a boreal lake community. Under low nutrient conditions, the model makes three major predictions. (a) Browning reduces light and increases nutrient supply; this decreases benthic and increases pelagic production, gradually shifting productivity from the benthic to the pelagic habitat. (b) Because of active habitat choice, fish exert top‐down control on grazers and benefit primary producers primarily in the more productive of the two habitats. (c) Warming relaxes top‐down control of grazers by fish and decreases primary producer biomass, but effects of warming are generally small compared to effects of browning and nutrient supply. Experimental results were consistent with most model predictions for browning: light penetration, benthic algal production, and zoobenthos biomass decreased, and pelagic nutrients and pelagic algal production increased with browning. Also consistent with expectations, warming had negative effects on benthic and pelagic algal biomass and weak effects on algal production and zoobenthos and zooplankton biomass. Inconsistent with expectations, browning had no effect on zooplankton and warming effects on fish depended on browning. The model is applicable also to nutrient‐rich systems, and we propose that it is a useful tool for the exploration of the consequences of different climate change scenarios for productivity and food web dynamics in shallow lakes, the worldwide most common lake type.     

Comparison of Coding Capabilities of Type I and Type II Neurons

We consider the dependence of information transfer by neurons on the Type I vs. Type II classification of their dynamics. Our computational study is based on Type I and II implementations of the Morris-Lecar model. It mainly concerns neurons, such as those in the auditory or electrosensory system, which encode band-limited amplitude modulations of a periodic carrier signal, and which fire at random cycles yet preferred phases of this carrier. We first show that the Morris-Lecar model with additive broadband noise ("synaptic noise") can exhibit such firing patterns with either Type I or II dynamics, with or without amplitude modulations of the carrier. We then compare the encoding of band-limited random amplitude modulations for both dynamical types. The comparison relies on a parameter calibration that closely matches firing rates for both models across a range of parameters. In the absence of synaptic noise, Type I performs slightly better than Type II, and its performance is optimal for perithreshold signals. However, Type II performs well over a slightly larger range of inputs, and this range lies mostly in the subthreshold region. Further, Type II performs marginally better than Type I when synaptic noise, which yields more realistic baseline firing patterns, is present in both models. These results are discussed in terms of the tuning and phase locking properties of the models with deterministic and stochastic inputs.     

Patchy agglomeration as a transition from monospecies to recurrent plankton blooms

We propose a model for explaining both red tides and recurring phytoplankton blooms. Three assumptions are made, namely the presence of toxin producing phytoplankton, the satiation phenomenon in zooplankton's feeding, modelled by a Holling type II response, and phytoplankton aggregation leading to formation of patches. The dynamics of the plankton population is shown to depend on the fraction of the phytoplankton population that aggregates to form colonies and on the number of the latter.     

Dwarf alleles differentially affect barley root traits influencing nitrogen acquisition under low nutrient supply

Sustainable food production depends critically on the development of crop genotypes that exhibit high yield under reduced nutrient inputs. Rooting traits have been widely advocated as being able to influence optimal plant performance, while breeding-based improvements in yield of spring barley suggest that this species is a good model crop. To date, however, molecular genetics knowledge has not delivered realistic plant ideotypes, while agronomic trials have been unable to identify superior traits. This study explores an intermediate experimental system in which root traits and their effect on plant performance can be quantified. As a test case, four modern semi-dwarf barley varieties, which possess either the ari-e.GP or the sdw1 dwarf allele, were compared with the long-stemmed old variety Kenia under two levels of nutrient supply. The two semi-dwarf types differed from Kenia, exhibiting smaller stem mass and total plant nitrogen (N), and improved partitioning of mass and N to grain. Amongst the semi-dwarfs, the two ari-e.GP genotypes performed better than the two sdw1 genotypes under standard and reduced nutrient supply, particularly in root mass, root investment efficiency, N acquisition, and remobilization of N and mass to grain. However, lack of between-genotype variation in yield and N use efficiency indicated limited potential for exploiting genetic variation in existing varieties to improve barley performance under reduced nutrient inputs. Experimental approaches to test the expression of desirable root and shoot traits are scrutinized, and the potential evaluated for developing a spring barley ideotype for low nutrient conditions.     

When is evolutionary branching in predator-prey systems possible with an explicit carrying capacity?

In this study we use the theory of adaptive dynamics firstly to explore the differences in evolutionary behaviour of a generalist predator (or more specifically an omnivorous or intraguild predator) in a predator-prey model, with a Holling Type II functional response, when two distinct forms for the carrying capacity are used. The first of these involves the carrying capacity as an emergent property, whilst in the second it appears explicitly in the dynamics. The resultant effect this has on the intraspecific competition in each case is compared. Taking an identical trade-off in each case, we find that only with an emergent carrying capacity is evolutionary branching possible. Our study then concentrates solely on the case where the carrying capacity appears explicitly. Using the same model as above, but choosing alternate trade-offs, we find branching can occur with an explicit carrying capacity. Our investigation finishes by taking a more general functional response in an attempt to derive a condition for when branching can or cannot occur. For a predator-prey model, branching cannot occur if the functional response can be separated into two components, one a function of the population densities, X and Z, and the other a function of the evolving parameter z (traded off against the intrinsic growth rate), i.e. if F(z,X,Z) = F(1)(z)F(2)(X,Z). This search for evolutionary branching is motivated by its possible role in speciation.     

The impact of nonlinear functional responses on the long-term evolution of food web structure

We investigate the long-term web structure emerging in evolutionary food web models when different types of functional responses are used. We find that large and complex webs with several trophic layers arise only if the population dynamics is such that it allows predators to focus on their best prey species. This can be achieved using modified Lotka-Volterra or Holling/Beddington functional responses with effective couplings that depend on the predator's efficiency at exploiting the prey, or a ratio-dependent functional response with adaptive foraging. In contrast, if standard Lotka-Volterra or Holling/Beddington functional responses are used, long-term evolution generates webs with almost all species being basal, and with additionally many links between these species. Interestingly, in all cases studied, a large proportion of weak links result naturally from the evolution of the food webs.     

Plant-herbivore interactions mediated by plant toxicity

We explore the impact of plant toxicity on the dynamics of a plant-herbivore interaction, such as that of a mammalian browser and its plant forage species, by studying a mathematical model that includes a toxin-determined functional response. In this functional response, the traditional Holling Type 2 response is modified to include the negative effect of toxin on herbivore growth, which can overwhelm the positive effect of biomass ingestion at sufficiently high plant toxicant concentrations. Two types of consumption decisions of the herbivore are considered. One of these (Case 1) incorporates the adaptation of the herbivore to control its rate of consumption of plant items when that is likely to lead to levels of toxicity that more than offset the marginal gain to the herbivore of consuming more plant biomass, while the other (Case 2) simply assumes that, although the herbivore’s rate of ingestion of plant biomass is negatively affected by increasing ingestion of toxicant relative to the load it can safely deal with, the herbivore is not able to prevent detrimental or even lethal levels of toxicant intake. A primary result of this work is that these differences in behavior lead to dramatically different outcomes, summarized in bifurcation diagrams. In Case 2, a wide variety of dynamics may occur due to the interplay of Holling Type 2 dynamics and the effect of the plant toxicant. These dynamics include the occurrence of bistability, in which both a periodic solution and the herbivore-extinction equilibrium are attractors, as well the possibility of a homoclinic bifurcation. Whether the herbivore goes to extinction in the bistable case depends on initial conditions of herbivore and plant biomasses. For relatively low herbivore resource acquisition rates, the toxicant effect increases the likelihood of ‘paradox of enrichment’ type limit cycle oscillations, but at higher resource acquisition rates, the toxicant may decrease the likelihood of these cycles.     

Diet segregation between two sympatric 'small'Barbus spp: an experimental study of mechanisms

Gut contents of two co‐occurring species of 'small' diploid barbs (<10  L _F cm) in Lake Tana revealed that zooplankton is the major diet component for B. tanapelagius (75% based on volume), but less prominent in B. humilis (40%). Functional response experiments in the laboratory were conducted to elucidate the mechanisms causing this difference. The type of functional response by the two 'small' barbs under different microcrustacean zooplankton densities (10, 20, 40, 60 and 80 ind.l⁻¹) was examined. The functional response of B. tanapelagius to increasing prey densities corroborates with Holling Type II model, whereas B. humilis exhibits a Type III functional response. Predation rate is higher for B. tanapelagius at low zooplankton density (<40 ind.l⁻¹) and equals the level of B. humilis at higher densities (>40 ind.l⁻¹). This suggests that at lower zooplankton densities B. humilis is a less efficient forager on zooplankton prey items than B. tanapelagius . In Lake Tana average zooplankton density is relatively low (<35 ind.l⁻¹). Under these food conditions, B. humilis is forced to feed on other food items (e.g. benthic invertebrates), whereas B. tanapelagius primarily feeds on zooplankton. The feeding potentials of the two 'small' barbs, as deduced from their morphology explain their different performances and their segregation in space and food resources.