Peak-to-peak dynamics in food chain models

We show in this paper that the chaotic regimes of many food chain models often enjoy a very peculiar property, known as peak-to-peak dynamics. This means that the maximum (peak) density of the populations of any trophic level can be easily forecasted provided the last two peaks of the same population are known. Moreover, extensive simulation shows that only the last peak is needed if the forecast concerns the population at the top of the food chain and that peaks variability often increases from bottom to top. All these findings bring naturally to the conclusion that top populations should be sampled in order to have higher chances to detect peak-to-peak dynamics. The analysis is carried out by studying ditrophic food chain models with seasonally varying parameters, tritrophic food chain models with constant parameters, and more complex food chain and food web models.     

Ecological consequences of global bifurcations in some food chain models

Food chain models of ordinary differential equations (ode’s) are often used in ecology to gain insight in the dynamics of populations of species, and the interactions of these species with each other and their environment. One powerful analysis technique is bifurcation analysis, focusing on the changes in long-term (asymptotic) behaviour under parameter variation. For the detection of local bifurcations there exists standardised software, but until quite recently most software did not include any capabilities for the detection and continuation of global bifurcations. We focus here on the occurrence of global bifurcations in four food chain models, and discuss the implications of their occurrence. In two stoichiometric models (one piecewise continuous, one smooth) there exists a homoclinic bifurcation, that results in the disappearance of a limit cycle attractor. Instead, a stable positive equilibrium becomes the global attractor. The models are also capable of bistability. In two three-dimensional models a Shil’nikov homoclinic bifurcation functions as the organising centre of chaos, while tangencies of homoclinic cycle-to-cycle connections ‘cut’ the chaotic attractors, which is associated with boundary crises. In one model this leads to extinction of the top predator, while in the other model hysteresis occurs. The types of ecological events occurring because of a global bifurcation will be categorized. Global bifurcations are always catastrophic, leading to the disappearance or merging of attractors. However, there is no 1-on-1 coupling between global bifurcation type and the possible ecological consequences. This only emphasizes the importance of including global bifurcations in the analysis of food chain models.     

Persistence in three-species food chain models with group defense.

Gause-type models of a three-species food web with group defense are analyzed. Persistence criteria are derived for both the case of no mutual interference and the case when there is mutual interference of predators.     

Hopf bifurcation in three-species food chain models with group defense.

Three-species food-chain models, in which the prey population exhibits group defense, are considered. Using the carrying capacity of the environment as the bifurcation parameter, it is shown that the model without delay undergoes a sequence of Hopf bifurcations. In the model with delay it is shown that using a delay as a bifurcation parameter, a Hopf bifurcation can also occur in this case. These occurrences may be interpreted as showing that a region of local stability (survival) may exist even though the positive steady states are unstable. A computer code BIFDD is used to determine the stability of the bifurcation solutions of a delay model.     

Oscillations in plankton models with nutrient recycling

Plankton--nutrient interaction models with both instantaneous and delayed nutrient recycling are considered. The system consists of three components: autotrophic phytoplankton, herbivorous zooplankton and dissolved limiting nutrient. Local stability of the equilibria is analysed. It is shown that the positive equilibrium loses its stability when the nutrient input concentration passes through a critical value and the Hopf bifurcation occurs that induces oscillations of the populations. Numerical simulations are carried out to illustrate the obtained results.     

Relationships between individual- and population-based plankton models

Two simulations of plankton dynamics are combined: an N/P/Zmodel and an optimal foraging strategy dependent on variablevertical migration. We show that there is no optimal individualstrategy consistent with a copepod population that has two generationsand is a significant grazer on its limiting food supply. Alternativeroles for did vertical migration are considered.     

Comparing models and observations of shelf plankton

In a previous paper (Solow and Steele, J. Plankton Res., 17,1995), the differences between variability in zooplankton biomassand in copepod stage structure were demonstrated using datafrom the northern North Sea. Here, a model is used to describethe underlying demographic processes and the effects of interannualphysical variability. Comparison of output with observationscan test the theory and so help to reconcile the apparent contradictionbetween great variability and persistence in these populations.     

The role of predation in plankton models

Models of carbon and nitrogen cycles in the ocean are a majortool in elucidating short-and long-term patterns of chemicalfluxes. Variability in space and time are usually attributedto changes in ocean physics at different scales. This paperstresses the significance of the upper (predatory) closure inthese simple nutrient-plant-herbivore models. The mathematicalform used to close the system and the values given to the parametershave very marked effects on the overall response. In particularthe major differences between North Atlantic and Pacific patternsmay depend on this aspect as much as on the physical cycles.It is shown that the selection of different closure forms infive recent modelling studies corresponds to differences inthe nutrient dynamics and plankton cycles. Thus, in general,the character of the results from these models will depend onboth the form of the mortality closure and the parameter valuesused. Our ignorance in both areas is considerable.     

Chaos in three species food chains

We study the dynamics of a three species food chain using bifurcation theory to demonstrate the existence of chaotic dynamics in the neighborhood of the equilibrium where the top species in the food chain is absent. The goal of our study is to demonstrate the presence of chaos in a class of ecological models, rather than just in a specific model. This work extends earlier numerical studies of a particular system by Hastings and Powell (1991) by showing that chaos occurs in a class of ecological models. The mathematical techniques we use are based on work by Guckenheimer and Holmes (1983) on co-dimension two bifurcations. However, restrictions on the equations we study imposed by ecological assumptions require a new and somewhat different analysis.     

Chaos and order in plankton dynamics. Complex behavior of a simple model

The role of the diffusive interaction between fish-populated and fish-free habitats in a patchy environment in plankton pattern formation is studied by means of a minimal reaction-diffusion model of the nutrient-plankton-fish food chain. It is shown that such interaction can give rise to spatio-temporal plankton patterns. The fractal dimension of the patterns is shown dependent on the fish predation rate. The spatially averaged plankton dynamics depending on both fish predation rate and distance between fish-populated habitats can exhibit chaotic and regular behavior. The chaotic plankton dynamics is characteristic of a wide parameter range.     

Environmental forcing of simple plankton models

This paper examines the phenomenon of algal blooms in phytoplanktonpopulations in the context of a simple spatially homogeneousmodel. The effects of a changing environment are representedby time dependence in the model's parameters. It is argued thatthe interaction of grazing pressure and nutrient limitationgives rise to excitability in the dynamics of the model. Thisproperty allows the use of asymptotic analysis to derive a connectionbetween the nature of the environmental forcing and the productionof a bloom. Numerical experiments demonstrate the accuracy ofthis analysis and also show how. and under what conditions,its effectiveness starts to break down. Some of the biologicalimplications of these results are discussed.     

Chaos and population disappearances in simple ecological models

A class of truncated unimodal discrete-time single species models for which low or high densities result in extinction in the following generation are considered. A classification of the dynamics of these maps into five types is proven: (i) extinction in finite time for all initial densities, (ii) semistability in which all orbits tend toward the origin or a semi-stable fixed point, (iii) bistability for which the origin and an interval bounded away from the origin are attracting, (iv) chaotic semistability in which there is an interval of chaotic dynamics whose compliment lies in the origin’s basin of attraction and (v) essential extinction in which almost every (but not every) initial population density leads to extinction in finite time. Applying these results to the Logistic, Ricker and generalized Beverton-Holt maps with constant harvesting rates, two birfurcations are shown to lead to sudden population disappearances: a saddle node bifurcation corresponding to a transition from bistability to extinction and a chaotic blue sky catastrophe corresponding to a transition from bistability to essential extinction. Received: 14 February 2000 / Revised version: 15 August 2000 / Published online: 16 February 2001     

The role of higher predation in plankton population models

Zooplankton mortality in plankton population models is oftenrepresented by the so-called closure term. Recently, much attentionhas been paid to the choice of functional form used for theclosure term, primarily due to the influential paper by Steeleand Henderson (J. Plankton Res., 14, 157–172, 1992). Herewe reveal an inconsistency in the normalization of Steele andHenderson's models, and show that unforced short-term oscillations(limit cycles) can occur when a quadratic closure term is used.Furthermore, we contradict the hypothesis regarding the relationshipbetween nutrient steady-state values and the choice of closureterm: using the seven-component plankton model of Fasham (TheGlobal Carbon Cycle, Heimann,M. (ed.), pp. 457–504, 1993)with four alternative closure terms, we find the nutrient valueto depend more upon the choice of parameter values than on thechoice of closure term. However, our results agree with andstrengthen the general conclusion of Steele and Henderson'swork: that the choice of closure term can strongly influencethe dynamics of models.     

Body size and mass flow in freshwater plankton: models and tests

Observations of radiotracer (³²P) flow in size-fractionatedlake plankton were compared with the predictions of size-dependentmodels of material flow. The models assume that the metabolicactivity and turnover of the size fractions can be describedas power functions of size (Y=aM^b) One model, called size-structured,is an application of the Platt and Denman formalism: it assumesthat materials are incorporated preferentially by the smallestorganisms and flow continuously toward the larger organisms.In this model, the trophodynamic flow from a size fraction isconstrained to the next larger fraction. The tracer data indicatedthat metabolic activity could indeed be described as a continuousfunction of size, at least within the microplankton (< 75µm), but it did not support the hypothesis of a size-hierarchicaltrophodynamic flow. Another model limited the size range ofpredators to the larger (> 10 µm), size fractions whilerelaxing the size dependence constraint on the trophodynamicflow. This ‘unstructured’ model agreed better withthe ³²P dynamics and generated estimates of seston P uptakeand of the rates of biomass turnover that compared favorablywith published estimates for oligotrophic lakes. Some problemswith the fit to the data remain but might be dealt with by distinguishinga priori among functional types of planktonic organisms throughfluorescence (flow cytometry) or metabolic properties (inhibitors).     

Structural sensitivity of grazing formulations in nutrient controlled plankton models

Summary Stability and persistence properties of a family of non-spatial plankton models, each differentiated by its herbivore grazing term, are analytically compared. The dynamic persistence function in the model is shown to operate uniformly even though stability configuration characteristics of the model may be topologically distinct. The persistence threshold for each model indicates that total nutrient is a fundamental biological control. In the parameter space, all of the models studied are structurally unstable; however, an important bifurcation mechanism associated with this instability governs persistence. While, topologically, model transfigurement through parameter modulation is non-continuous, the biological populations evolve in a continuous or a lower semicontinuous manner. A basic conclusion of the paper is that fundamental problems for these marine ecological models remain unresolved since each of the models is a structurally unstable system for a fixed dynamically persistent ecology.     

Global stability in chemostat-type plankton models with delayed nutrient recycling

 In this paper, we consider chemostat-type plankton models in which plankton feeds on a limiting nutrient and the nutrient is supplied at a constant rate and is partially recycled after the death of plankton by bacterial decomposition. We use a distributed delay to describe nutrient recycling and a discrete delay to model the planktonic growth response to nutrient uptake. When one or both delays occur, by constructing Liapunov functionals, we obtain some sufficient conditions for the global attractivity of the positive equilibrium, which can be regarded as estimates of the delays for persistence of global attractivity. Received: 9 January 1996 / Revised version: 1 October 1997     

Zooplankton mortality and the dynamical behaviour of plankton population models

We investigate the dynamical behaviour of a simple plankton population model, which explicitly simulates the concentrations of nutrient, phytoplankton and zooplankton in the oceanic mixed layer. The model consists of three coupled ordinary differential equations. We use analytical and numerical techniques, focusing on the existence and nature of steady states and unforced oscillations (limit cycles) of the system. The oscillations arise from Hopf bifurcations, which are traced as each parameter in the model is varied across a realistic range. The resulting bifurcation diagrams are compared with those from our previouswork, where zooplankton mortality was simulated by a quadratic function—here we use a linear function, to represent alternative ecological assumptions. Oscillations occur across broader ranges of parameters for the linear mortality function than for the quadratic one, although the two sets of bifurcation diagrams show similar qualitative characteristics. The choice of zooplankton mortality function, or closure term, is an area of current interest in the modelling community, and we relate our results to simulations of other models.     

The food chain dynamics of the oyster,clam, and mussel in an aquaculture food chain

The food chain dynamics of the edible mussel Mytilus edulis L., the American oyster Crassostrea virginica (Gmelin) and the hard clam Mercenaria mercenaria (L.) were investigated in large experimental tanks with flowing, filtered sea water and controlled addition of phytoplankton. The feeding rate of the mussel (5.36 μg carbon removed/l/g C animal was higher than that of the oyster (3.92) and clam (3.03) but the ecological efficiencies (net production/ingested food) × 100 of the clam (23.69 %) and the oyster (18.38 %) were higher than that of the mussel (10.01 %).The food chain efficiencies (net production/available food) were lower than the ecological efficiencies, suggesting under-exploitation of the available food. The clam, although having a lower feeding rate, was more efficient in utilizing the food it filtered and so showed the highest net production.The rates (μg-at/l/g C animal) of regeneration of nutrients, especially total inorganic nitrogen (mussel, 2.1723 × 10^?3; oyster, 7.4270 × 10^?3; and clam, 8.1750 × 10^?3) along with reported high biodeposition rates of bivalves suggest that multi-species aquaculture systems would be more efficient and productive than one-species systems.