The impact of diffusion and stirring on the dynamics of interacting populations

Viral infections of oceanic plankton blooms

Viruses are known to impact blooms of phytoplankton in the ocean, in some cases causing the bloom to crash. Here, using a population model that includes viral infection, we investigate the conditions under which the presence of a virus significantly impacts the population dynamics. A major focus is how spatial variability influences the spread of an epidemic in a stirring and mixing field. The combination of viral infection and diffusion can cause waves of the epidemic to sweep through the domain, with the epidemic lasting much longer than in the homogeneous case. Stirring by the fluid flow can greatly increase this effect causing an increase in the fraction of the bloom that is affected and in certain circumstances (high diffusion and stirring) can totally suppress the bloom. The fluid environment affects the relative spatial structure of the components of the system. High values of the concentrations of the virus and infected phytoplankton are found in thin filaments along fronts of uninfected (susceptible) phytoplankton.

How turbulence regulates biodiversity in systems with cyclic competition

Cyclic, nonhierarchical interactions among biological species represent a general mechanism by which ecosystems are able to maintain high levels of biodiversity. However, species coexistence is often possible only in spatially extended systems with a limited range of dispersal, whereas in well-mixed environments models for cyclic competition often lead to a loss of biodiversity. Here we consider the dispersal of biological species in a fluid environment, where mixing is achieved by a combination of advection and diffusion. In particular, we perform a detailed numerical analysis of a model composed of turbulent advection, diffusive transport, and cyclic interactions among biological species in two spatial dimensions and discuss the circumstances under which biodiversity is maintained when external environmental conditions, such as resource supply, are uniform in space. Cyclic interactions are represented by a model with three competitors, resembling the children's game of rock-paper-scissors, whereas the flow field is obtained from a direct numerical simulation of two-dimensional turbulence with hyperviscosity. It is shown that the space-averaged dynamics undergoes bifurcations as the relative strengths of advection and diffusion compared to biological interactions are varied.

Stirring effects in models of oceanic plankton populations

We present an overview and extend previous results on the effects of large scale oceanic transport processes on plankton population dynamics, considering different types of ecosystem models. We find that increasing stirring rate in an environment where the carrying capacity is non-uniformly distributed leads to an overall decrease of the effective carrying capacity of the system. This may lead to sharp regime shifts induced by stirring in systems with multiple steady states. In prey-predator type systems, stirring leads to resonant response of the population dynamics to fluctuations enhancing the spatial variability-patchiness-in a certain range of stirring rates. Oscillatory population models produce strongly heterogeneous patchy distribution of plankton blooms when the stirring is weak, while strong stirring may either synchronise the oscillatory dynamics, when the inhomogeneity is relatively weak, or suppress oscillations completely (oscillator death) by reducing the effective carrying capacity below the bifurcation point.

Resonant plankton patchiness induced by large-scale turbulent flow

Here we study how large-scale variability of oceanic plankton is affected by mesoscale turbulence in a spatially heterogeneous environment. We consider a phytoplankton-zooplankton (PZ) ecosystem model, with different types of zooplankton grazing functions, coupled to a turbulent flow described by the two-dimensional Navier-Stokes equations, representing large-scale horizontal transport in the ocean. We characterize the system using a dimensionless parameter, γ=T(B)/T(F), which is the ratio of the ecosystem biological time scale T(B) and the flow time scale T(F). Through numerical simulations, we examine how the PZ system depends on the time-scale ratio γ and find that the variance of both species changes significantly, with maximum phytoplankton variability at intermediate mixing rates. Through an analysis of the linearized population dynamics, we find an analytical solution based on the forced harmonic oscillator, which explains the behavior of the ecosystem, where there is resonance between the advection and the ecosystem predator-prey dynamics when the forcing time scales match the ecosystem time scales. We also examine the dependence of the power spectra on γ and find that the resonance behavior leads to different spectral slopes for phytoplankton and zooplankton, in agreement with observations.

Spatially implicit plankton population models: Transient spatial variability

Ocean plankton models are useful tools for understanding and predicting the behaviour of planktonic ecosystems. However, when the regions represented by the model grid cells are not well mixed, the population dynamics of grid cell averages may differ from those of smaller scales (such as the laboratory scale). Here, the 'mean field approximation' fails due to 'biological Reynolds fluxes' arising from nonlinearity in the fine-scale biological interactions and unresolved spatial variability. We investigate the domain-scale behaviour of two-component, 2D reaction-diffusion plankton models producing transient dynamics, with spatial variability resulting only from the initial conditions. Failure of the mean field approximation can be quite significant for sub grid-scale mixing rates applicable to practical ocean models. To improve the approximation of domain-scale dynamics, we investigate implicit spatial resolution methods such as spatial moment closure. For weak and moderate strengths of biological nonlinearity, spatial moment closure models generally yield significant improvements on the mean field approximation, especially at low mixing rates. However, they are less accurate given weaker transience and stronger nonlinearity. In the latter case, an alternative 'two-spike' approximation is accurate at low mixing rates. We argue that, after suitable extension, these methods may be useful for understanding and skillfully predicting the large-scale behaviour of marine ecosystems.

Plankton cycles disguised by turbulent advection

Mathematical models used to represent plankton dynamics often display limit-cycle behavior in a range of realistic parameter values. However, experimental data do not show evidence of plankton oscillations besides externally driven seasonal blooms, casting doubts on the validity of the models themselves. In this work we show that spatial-temporal variability, coupled with advection by mesoscale turbulence, can disguise limit-cycle behavior to the point that it cannot be detected in fixed-point measurements of plankton abundance. The results presented here have more general implications as they indicate that the behavior of ecosystem models in the presence of advection can be very different from that occurring for homogeneous conditions. Care should thus be exercised in drawing general conclusions from the analysis of homogeneous ecosystem models.