Towards resolving the paradox of enrichment: The impact of zooplankton vertical migrations on plankton systems stability

Exploring the role of spatial and stoichiometric heterogeneity in the top-down control in eutrophic planktonic ecosystems

Understanding the impact of eutrophication on the dynamics of aquatic food webs, remains a long-term challenge in ecology. Mathematical models generally predict the destabilisation of such webs, under increasing eutrophication levels, with large oscillations of species densities that can result in their extinction. This is at odds with a number of ecological observations that show stable dynamics even for high nutrient loads. The apparent discrepancy between theory and observations is known as the Rosenzweig's 'paradox of enrichment' and various solutions to the problem have been proposed over the years. In this study, we explore the stabilisation of dynamics of a tri-trophic plankton model in a eutrophic environment which occurs as a result of interplay of space heterogeneity, ecological stoichiometry, and food taxis of predators. We build a variety of models of increasing complexity, to explore various scenarios of phytoplankton growth, zooplankton food-dependent vertical movement, and different stoichiometric limitations of zooplankton. We show that the synergy among the vertical gradient in phytoplankton growth, phytoplankton structuring in terms of their stoichiometric ratio, and food-dependent vertical movement of zooplankton, would result in a postponing of destabilisation of eutrophic systems as compared to a well-mixed system. Our approach reveals a high complexity of the bifurcation structure of the system when key model parameters, such as the degree of eutrophication and light shading, are varied. We find coexistence of limit cycles and stable equilibria and that the possibility of multiple attractors in the system can result in hysteresis phenomena when the nutrient load is manipulated. These results are relevant and should be taken into account in lake restoration programs.

Dynamics of an autonomous food chain model and existence of global attractor of the associated non-autonomous system

In this paper, we have analyzed a tri-trophic food chain model consisting of phytoplankton, zooplankton and fish population in an aquatic environment. Here, the pelagic water column is divided into two layers namely, the upper layer and the lower layer. The zooplankton population makes a diel vertical migration (DVM) from lower portion to upper portion and vice-versa to trade-off between food source and fear from predator (Fish). Here, mathematical model has been developed and analyzed in a rigorous way. Apart from routine calculations like boundedness and positivity of the solution, local stability of the equilibrium points, we performed Hopf bifurcation analysis of the interior equilibrium point of our model system in a systematic way. It is observed that the migratory behavior of zooplankton plays a crucial role in the dynamics of the model system. Both the upward and downward migration rates of DVM leads the system into Hopf bifurcation. The upward migration rate of zooplankton deteriorates the stable coexistence of all the species in the system, whereas the downward migration rate enhance the stability of the system. Further, we analyze the non-autonomous version of the system to capture seasonal effect of environmental variations. We have shown that under certain parametric restrictions periodic coexistence of all the species of our system is possible. Finally, extensive numerical simulation has been performed to support our analytical findings.

EXPLORING THE ROLE OF VERTICAL HETEROGENEITY IN THE STABILIZATION OF PLANKTONIC ECOSYSTEMS UNDER EUTROPHICATION

Understanding plankton dynamics in marine and lake ecosystems under eutrophication is currently a hot topic in the literature. Simple theoretical models predict appearance of large amplitude oscillations of species densities in nutrient-rich waters; however, such predictions do not always correspond to field observations. Recent models taking into account heterogeneity of the growth rate of phytoplankton and active food-searching behavior of zooplankton demonstrated that grazers can efficiently control phytoplankton densities at low values even for a high nutrient stock. In this paper, we extend the previous modeling findings on the role of fast-moving plankton grazers by exploring a more realistic case where the limiting nutrient is a dynamical variable. Thus, the growth of phytoplankton across the water column depends on both light attenuation and dynamical depletion of nutrients. We also consider a more realistic scenario of a depth-dependent vertical turbulent diffusion. Most of the previous results on stabilization of planktonic ecosystems still hold; however, some alternative mechanisms of bloom suppression can also be possible. In particular, we demonstrate that the foraging of zooplankton according to the ideal free distribution (IFD) of food (which was previously considered to be a crucial condition for stabilization) may be less stabilizing than random foraging of zooplankton. We also show that stable top-down control in the ecosystem would be highly dependent on values of vertical diffusion and on the nutrient concentration in deep layers.

Revisiting the Stability of Spatially Heterogeneous Predator-Prey Systems Under Eutrophication

We employ partial integro-differential equations to model trophic interaction in a spatially extended heterogeneous environment. Compared to classical reaction-diffusion models, this framework allows us to more realistically describe the situation where movement of individuals occurs on a faster time scale than on the demographic (population) time scale, and we cannot determine population growth based on local density. However, most of the results reported so far for such systems have only been verified numerically and for a particular choice of model functions, which obviously casts doubts about these findings. In this paper, we analyse a class of integro-differential predator-prey models with a highly mobile predator in a heterogeneous environment, and we reveal the main factors stabilizing such systems. In particular, we explore an ecologically relevant case of interactions in a highly eutrophic environment, where the prey carrying capacity can be formally set to 'infinity'. We investigate two main scenarios: (1) the spatial gradient of the growth rate is due to abiotic factors only, and (2) the local growth rate depends on the global density distribution across the environment (e.g. due to non-local self-shading). For an arbitrary spatial gradient of the prey growth rate, we analytically investigate the possibility of the predator-prey equilibrium in such systems and we explore the conditions of stability of this equilibrium. In particular, we demonstrate that for a Holling type I (linear) functional response, the predator can stabilize the system at low prey density even for an 'unlimited' carrying capacity. We conclude that the interplay between spatial heterogeneity in the prey growth and fast displacement of the predator across the habitat works as an efficient stabilizing mechanism. These results highlight the generality of the stabilization mechanisms we find in spatially structured predator-prey ecological systems in a heterogeneous environment.

Alternative food and external source of infection stabilize predator–prey oscillations — A conclusion drawn from an eco-epidemiological model

Das et al. [Effect of disease-selective predation on prey infected by contact and external sources, Biosystems 95(3) (2009) 188–199] proposed an eco-epidemiological model where the prey species is infected through the external source of infection and contact of the species. In this present study we have modified their model by assuming that the predator consumes both the susceptible as well as the infected prey following the modified Holling type-II functional response. Our main focusing points of this study are the role of infection rate (both internal and external), alternative food, and half-saturation constant in the predator–prey dynamics with disease in the prey population. We have shown the local stability of the boundary as well as the interior equilibrium point under certain conditions. We have also worked out the permanence of the system. Our simulation results show that the system enters into limit cycle oscillations from stable position for higher values of the contact rate. But it is also shown that the external infection rate, enrichment of the alternative food of the predator population and the half-saturation constant can prevent limit cycle oscillations and stabilize the system. Thus external disease propagation, enrichment of the alternative food resource, and the half-saturation constant are the key factors for preventing the oscillatory behavior of the species.

EFFECT OF KAIROMONE ON PREDATOR–PREY DYNAMICS — A DELAY MODEL

In most of the predator–prey systems, prey individuals make transitions between vulnerable and invulnerable states or locations. This transition is regulated by various inducible defense mechanisms. Diel vertical migration (DVM) in zooplankton is the most effective and instantaneous defense observed in zooplankton population. Zooplankton shows downward vertical migration in the daytime in the presence of predators (or predator kairomones) to avoid predation (i.e. refuge use), and it enters into the surface water again at night to graze phytoplankton. The dynamics of the planktonic ecosystem under DVM of zooplankton along with fish kairomone and the multiple delays due to migration for vulnerable and invulnerable prey and reproduction in the predator population is of considerable interest both in theoretical and experimental ecologists. By developing mathematical model, we analyze such a system. The conditions for which the system enters into Hopf-bifurcation are obtained. Moreover, the conditions for which the bifurcating branches are supercritical are also derived. Our results indicate that DVM along with the effect of kairomone and multiple delays with a certain range are responsible to enhance the stability of the system around the positive interior equilibrium point.

Towards a correct description of zooplankton feeding in models: taking into account food-mediated unsynchronized vertical migration

Complex nature of foraging behaviour of zooplankton makes it difficult to describe adequately zooplankton grazing in models with vertical space. In mean-field models (based on systems of PDEs or coupled ODEs), zooplankton feeding at a given depth is normally computed as the product of the local functional response and the zooplankton density at this depth. Such simplification is often at odds with field observations which show the absence of clear relationship between intake rates of organisms and the ambient food density. The observed discrepancy is generic and is often caused by fast non-synchronous vertical migration of organisms with different nutrition status. In this paper, we suggest a simple way of incorporating unsynchronized short-term vertical migration of zooplankton into the mean-field modelling framework. We compute grazing of zooplankton in each layer depending on feeding activity of organisms in the layer. We take into account grazing impact of animals which are in the active phase of foraging cycle at the given moment of time but neglect the impact of animals which are in the non-active phase of the cycle (e.g. digesting food). Unsynchronized vertical migration determines the vertical distribution of actively feeding animals in layers depending on vertical distribution of food. In this paper, we compare two generic plankton models: (i) a model based on 'classical' grazing approach and (ii) a model incorporating food-mediated unsynchronized vertical migration of zooplankton. We show that including unsynchronized food-mediated migration would make the behaviour of a plankton model more realistic. This would imply a significant enhancement of ecosystem's stability and some additional mechanisms of regulation of algal blooms. In the system with food-mediated unsynchronized vertical migration, the control of phytoplankton by herbivorous becomes possible even for very large concentrations of nutrients in the water (formally, when the system's carrying capacity tends to infinity).