Spatial patterns in a nutrient-plankton model

doi:10.1016/0304-3800(78)90028-5

Ecological Modelling

Volume 4, Issue 4, May 1978, Pages 361-370

https://doi.org/10.1016/0304-3800(78)90028-5 Get rights and content

Abstract

An exact solution is given for a partial differential equation description of plankton populations interacting with nutrients in a moving water mass. Eddy diffusion and exchange with an underlying layer are included. Results are examined over one spatial dimension using different constant velocities and diffusivity values. The influence of advection relative to diffusion varies, and several system parameters alter both the amplitude and frequency of plankton pulses.

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Cited by (3)

Stability and Hopf bifurcation of a diffusive plankton model with time-delay and mixed nonlinear functional responses
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Citation Excerpt :
It is generally considered that modeling plankton systems mathematically began with Fleming’s work in the 1930s [1], where the specific model is of phytoplanktonic type. Based on the pioneer studies of Fleming, in the following few decades, more and more plankton models (including phytoplanktonic, zooplanktonic and phytoplanktonic–zooplanktonic types) were established from different aspects, and a large number of valuable results have been obtained, see [2–15] for examples. Among numerous works on plankton systems with various functional responses, some of them are spatially homogeneous, some are inhomogeneous, and the content involves, mathematically, local or global stability, long-time behavior, coexistence, bifurcation phenomenon of positive solutions for systems, etc., economically, the optimal harvest and controlling strategies of maximum sustainable yields or maximum economic benefits for aquatic organisms themselves, etc.
In this paper, we deal with a plankton reaction–diffusion model with time-delay and two different functional responses. Firstly, we consider the global stability of boundary equilibrium point. Secondly, we investigate the existence, uniqueness and stability of internal equilibrium point without time-delay. Then, we analyze the existence of Hopf bifurcation emitting from internal equilibrium point and give some characteristics on Hopf branch in detail. A new finding is presented, specifically, we find that there exist two critical values which have important effects on the occurrence of Hopf bifurcation. Finally, a few numerical examples are presented to check and illustrate the theoretical analysis, some simulation graphs, including the spatiotemporal graphs, trajectory graphs and phase portraits are depicted graphically.
Differential dispersion in planktonic food chains with constant inputs
1982, Ecological Modelling
Turbulence and phytoplankton diversity: A general model of the "paradox of plankton"
1979, Ecological Modelling
The principle of “competitive exclusion” predicts that no two species can occupy the same ecological niche at the same time and place (Hardin, 1960). Hutchinson (1953, 1961) suggested that the vast diversity of phytoplankton observed in many aquatic environments presents an apparent contradiction to this principle. Since all phytoplankton compete for the same basic resources, and since the euphotic zones of most natural waters are relatively homogeneous, such coexisting plankters appear to be simultaneously occupying the same niche. In this paper we present simulation results from a mathematical model wherein we examine the hypothesis that physical turbulence in an aquatic system can mollify interactive pressures between plankton populations and allow coexistence of species competing for the same resources. Using Bella's (1972) highly simplified model as a point of departure, we develop a new model, explicitly incorporating gross physiological mechanisms, to investigate the effects of both advective and turbulent components of water movement on the growth of three competing phytoplankton species. We observed that, in the absence of water motion, no two species were able to coexist, while under the hypothetical conditions of advection without turbulence (laminar flow), just two species were able to occur contemporaneously. Coexistence of all three species was achieved only with the addition of a random turbulent component to the model's hydrodynamic function. Moreover, this general coexistence was observed only when the major turbulent frequency approached the turnover rate of phytoplankton populations. We suggest that there is a limited region of periodicities and magnitudes for hydrodynamic energy in which phytoplankton can coexist, and that most natural aquatic environments fall within this region. We further speculate that, in general, the coupling of physical and biological processes in nature may be influenced by the relative frequency characteristics of those processes.

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Spatial patterns in a nutrient-plankton model

Abstract

Ecol. Modelling

Deep-Sea Res.

Identification method in environmental systems and its application to water pollution

Int. J. Syst. Sci.

The size of water masses containing plankton blooms

J. Mar. Res.

Some consequences of distributional heterogeneity of phytoplankton and zooplankton

Limnol. Oceanogr.