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MaMPFi: Mathematical Modelling of Plankton-Fish Dynamics

Temporal plankton dynamics
Spatial plankton dynamics
Spatio-temporal plankton dynamics
Viral infections of phytoplankton


Cooperation (in alphabetical order)

Frank M. Hilker
Institute of Mathematics, School of Mathematics/Computer Science, and Institute of Environmental Systems Research, Osnabrück University, Germany

Alex James
Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand

Diomar Cristina Mistro and Luiz Alberto Díaz Rodrigues
Departamento de Matemática, Universidade Federal de Santa Maria RS , Brasil

Sergei V. Petrovskii
Department of Mathematics, University of Leicester, Leicester, UK

Jean-Christophe Poggiale
Aix-Marseille Université, Institut Méditerranéen d'Océanologie, Marseille, France

Michael Sieber
Department for Evolutionary Theory, Max Planck Institute for Evolutionary Biology, Plön, Germany

Ivo Siekmann
School of Applied Mathematics, Faculty of Engineering and Technology, Liverpool John Moores University, UK

Ezio Venturino
Dipartimento di Matematica, Università degli Studi di Torino, Italia

The history of modelling plankton dynamics is already quite long and has been initiated by fishery science in the early 20th century. The main aim of modelling population dynamics is to improve the understanding of the functioning of food chains and webs and their dependence on internal and external conditions. Hence, mathematical models of biological population dynamics have not only to account for growth and interactions but also for spatial processes like random or directed and joint or relative motion of species as well as the variability of the environment. Early attempts began with physico-chemical diffusion, exponential growth and Lotka-Volterra type interactions. These approaches have been continuously refined to more realistic descriptions of the development of natural populations. The aim of this page it to give an extensive introduction to the subject and the bibliography. The fascinating variety of spatio-temporal patterns in such systems and the governing mechanisms of their generation and further dynamics are decribed and related to plankton.


The exploration of pattern formation mechanisms in nonlinear complex systems is one of the central problems of natural, social, and technological sciences. The development of the theory of self-organized temporal, spatial or functional structuring of nonlinear systems far from equilibrium has been one of the milestones of structure research (Haken, 1977; Nicolis & Prigogine, 1977). The occurrence of multiple steady states and transitions from one to another after critical fluctuations, the phenomena of excitability, oscillations, waves and, in general, the emergence of macroscopic order from microscopic interactions in various nonlinear nonequilibrium systems in nature and society has required and stimulated many theoretical and, if possible, experimental studies. Mathematical modelling has turned out to be one of the useful methods to improve the understanding of such structure generating mechanisms.

In the 17th century, the Dutch pioneer microscopist Anton van Leeuwenhoek was probably the first human being to see minute creatures, which he called animalcules, in pond water (Hallegraef, 1988). The German Victor Hensen who organized Germany's first big oceanographic expedition in 1889 (Hensen, 1992; Porep, 1970) introduced the term plankton (due to the Greek planktos = made to wander).

Phytoplankton are microscopic plants that drive all marine ecological communities and the life within them. Due to their photosynthetic growth, the world's phytoplankton generate half of the oxygen that the mankind needs for maintaining life and it absorbs half of the carbon dioxide that may be contributing to global warming. It is not only oxygen and carbon dioxide but there are also other substances and gases that are recycled by phytoplankton, e.g. phosphorus, nitrogen and sulphur compounds (Bain, 1968; Ritschard, 1992; Duinker & Wefer, 1994; Malin, 1997). Hence, the phytoplankton is one of the main factors controlling the further development of the world's climate and there is a vast literature supporting that, cf. Charlson et al. (1987) and Williamson & Gribbin (1991).

Zooplankton are the animals in plankton. In marine zooplankton both herbivores and predators occur, herbivores graze on phytoplankton and are eaten by zooplankton predators. Together, phyto- and zooplankton form the basis for all food chains and webs in the sea. In its turn, the abundance of the plankton species is affected by a number of environmental factors such as water temperature, salinity, sunlight intensity, biogen availibility etc. (Raymont, 1980; Sommer, 1994). Temporal variability of the species composition is caused by seasonal changes and trophical prey--predator interactions between phyto- and zooplankton. The latter have first been introduced by Lotka (1925) and Volterra (1926).

Because of its apparent importance, the dynamics of plankton systems have been under continuous investigation during more than hundred years. It should be noted that, practically from the very beginning, regular plankton studies have combined field observations, laboratory experiments and mathematical modelling. It was in the 19th century that fisheries stimulated the interest in plankton dynamics because strong positive correlations between zooplankton and fish abundance were found. The already mentioned German plankton expedition of 1889 was mainly motivated by fisheries interests. At the same time, fishery science began to develop. In the beginning of the 20th century, first mathematical models were developed in order to understand and to predict fish stock dynamics and its correlations with biological and physical factors and human interventions, cf. Cushing (1975), Gulland (1977) and Steele (1977).

Temporal plankton dynamics

The mathematical modelling of phytoplankton productivity has its roots in the work by Fleming (1939), Ivlev (1945), Riley (1946), Odum (1953) and others. A review of the developments has been given by Droop (1983). Recently, a collection of mostly used models has been presented by Behrenfeld and Falkowski (1997).

The control of phytoplankton blooming by zooplankton grazing has been modelled first by Fleming (1939), using a single ordinary differential equation for the temporal dynamics of phytoplankton biomass. Other approaches have been the construction of data fitted functions (Riley, 1963) and the application of standard Lotka-Volterra equations to describe the prey-predator relation of phytoplankton and zooplankton (Segel & Jackson, 1972; Dubois, 1975; Levin & Segel, 1976; Mimura & Murray, 1978). More realistic descriptions of zooplankton grazing with functional responses to phytoplankton abundance have been introduced by Ivlev (1945) with a certain modification by Mayzaud and Poulet (1978). Holling-type response terms (Holling, 1959) which are also known from Monod or Michaelis-Menten saturation models of enzyme kinetics (Michaelis & Menten, 1913; Monod & Jacob, 1961) are just as much in use (cf. Steele & Henderson 1981, 1992a,b; Scheffer, 1991a,b, 1998; Pascual, 1993; Malchow, 1993; Truscott & Brindley, 1994a,b). Observed temporal structures are the well-known stable prey-predator oscillations as well as the oscillatory or monotonous relaxation to one of the possibly multiple steady states. Excitable systems are of special interest because their long-lasting relaxation to the steady state after a supercritical external perturbation like a sudden temperature increase and nutrient inflow is very suitable to model red or brown tides (Beltrami, 1989, 1996; Truscott & Brindley, 1994a,b). Models with more than three interacting species might be driven to quasiperiodic and chaotic population oscillations (Scheffer, 1991b, 1998). Recently, it has been shown that resource competition models can generate oscillations and chaos when plankton species compete for three or more recources. Furthermore, these oscillations and chaotic variations in species abundances stabilize the coexistence of many species on a few resources (Huisman & Weissing, 1999).

Another interesting group of studies is devoted to externally forced systems. This ideally periodic forcing appears rather natural due to daily, seasonal or annual cycles of photosynthetically active radiation, temperature, nutrient availability etc. (Evans & Parslow, 1985; Truscott, 1995; Popova et al., 1997; Ryabchenko et al., 1997). Natural forcings are of course superposed by a certain environmental noise. A number of forced models for parts of or the complete food chain from nutrients, phytoplankton and zooplankton to planktivorous fish have been investigated and many different routes to chaotic dynamics have been demonstrated (Kuznetsov, Muratori & Rinaldi, 1992; Ascioti et al., 1993; Doveri et al., 1993; Rinaldi & Muratori, 1993; Scheffer et al., 1997; Scheffer, 1998; Steffen & Malchow, 1996a,b; Steffen, Medvinsky & Malchow, 1998).

Spatial plankton dynamics

The spatial spread of plankton is most simply described as passive diffusion within the turbulent water on all spatial scales. This is particularly right for the mesoscale of 100 metres to 100 kilometres where patchiness of plankton is usually observed (Daly & Smith, 1993; Powell & Okubo, 1994; Mann & Lazier, 1996). Here, one has only to account for the difference between the horizontal and the much smaller vertical turbulent exchange coefficients. A well-studied stripy plankton pattern is due to the trapping of populations of sinking microorganisms in Langmuir circulation cells (Stommel, 1948; Leibovich, 1993). Other physically determined plankton distributions like steep density gradients due to local temperature differences, nutrient upwelling, turbulent mixing or internal waves have been reported too (Yoder et al., 1994; Franks, 1997; Abraham, 1998).
On a small spatial scale of some ten's of centimetres and under relative physical uniformity also differences in individual "molecular" diffusivities and the ability of locomotion might create finer spatial structures, e.g. due to bioconvection and gyrotaxis (Platt, 1961; Winet & Jahn, 1972; Pedley & Kessler, 1992; Timm & Okubo, 1994). Till now not for plankton but for certain bacteria, the mechanism of diffusion-limited aggregation (Witten & Sander, 1981) has been proposed and experimentally proven for the spatial fingering of colonies (Matsushita & Fujikawa, 1990; Ben-Jacob et al., 1992).


Spatio-temporal plankton dynamics

The interplay of phytoplankton and zooplankton growth, interactions and transport yields the whole variety of spatio-temporal population structures, in particular the phenomenon of plankton patchiness (cf. Fasham, 1978; Okubo, 1980). The mathematical modelling requires the use of reaction-diffusion and perhaps advection equations. A good introduction to the latter field has been provided by Holmes et al. (1994).
Kierstead, Slobodkin (1953) and Skellam (1951) were perhaps the first to think of the critical size problem for plankton patches, presenting their nowadays called KISS model with the coupling of exponential growth and diffusion of a single population. Of course, their patches are unstable because this coupling leads to an explosive spatial spread of the initial patch of species with surprisingly the same diffusive front speed as the asymptotic speed of a logistically growing population (Luther, 1906; Fisher, 1937; Kolmogorov, Petrovskii & Piskunov, 1937).
Populations with an Allee effect (Allee, 1931; Allee et al., 1949), i.e. the existence of a minimum viable number of species of a population yields two stable population states - extinction and survival at its carrying capacity, show a spatial critical size as well (Schlögl, 1972; Nitzan, Ortoleva & Ross, 1974; Ebeling & Schimansky-Geier, 1980; Malchow & Schimansky-Geier, 1985). Population patches greater than the critical size will survive, the others will go extinct. However, bistability and the emergence of a critical spatial size do not necessarily require an Allee effect, also logistically growing preys with a parametrized predator of type II or III functional response can exibit two stable steady states and the related hysteresis loops (cf. Ludwig, Jones & Holling, 1978; Wissel, 1989).

The consideration of dynamic predation leads to the full spectrum of spatial and spatio-temporal patterns like regular and irregular oscillations, propagating fronts, target patterns and spiral waves, pulses as well as stationary spatial patterns. Many of these structures were first known from oscillating chemical reactions (cf. Field & Burger, 1985) but have never been observed in natural plankton populations. However, spirals have been seen in the ocean as rotary motions of plankton patches on a kilometer scale (Wyatt, 1973). Furthermore, they have been found important in parasitoid-host systems (Boerlijst, Lamers & Hogeweg, 1993). For other motile microorganisms, travelling waves like targets or spirals have been found in the cellular slime mold Dictyostelium discoideum (Gerisch, 1968, 1971; Keller & Segel, 1970, 1971a,b; Segel & Stoeckly, 1972; Segel, 1977; Newell, 1983; Alt & Hoffmann, 1990; Siegert & Weijer, 1991; Steinbock, Hashimoto & Müller, 1991; Ivanitskii, Medvinskii & Tsyganov, 1994; Vasiev, Hogeweg & Panfilov, 1994; Höfer, Sherratt & Maini, 1995). These amoebae are chemotactic species, i.e. they move actively up the gradient of a chemical attractant and aggregate. Chemotaxis is a kind of density-dependent cross-diffusion and it is an interesting open question whether there is preytaxis in plankton or not. Bacteria like Escherichia coli or Bacillus subtilis also show a number of complex colony growth patterns (Shapiro & Hsu, 1989; Shapiro & Trubatch, 1991), different to the already mentioned diffusion-limited aggregation patterns. Their emergence requires as well cooperativity and active motion of the species which has also been modelled as density-dependent diffusion and predation (Kawasaki, Mochizuki & Shigesada, 1995; Kawasaki et al., 1997).


Since the classic paper by Turing (1952) on the role of nonequilibrium reaction-diffusion patterns in biomorphogenesis, dissipative mechanisms of spontaneous spatial and spatio-temporal pattern formation in a homogeneous environment are of continuous interest in theoretical biology and ecology. Turing showed that the nonlinear interaction of at least two agents with considerably different diffusion coefficients can give rise to spatial structure. Segel and Jackson (1972) were the first to apply Turing's idea to a problem in population dynamics: The dissipative instability in the prey-predator interaction of phytoplankton and herbivorous copepods with higher herbivore motility. Levin and Segel (1976) suggested this scenario of spatial pattern formation for a possible origin of planktonic patchiness.

Recently, local bistability, predator-prey limit-cycle oscillations, plankton front propagation and the generation and drift of planktonic Turing patches were found in a minimal phytoplankton-zooplankton interaction model (Malchow, 1993, 1994) that was originally formulated by Scheffer (1991a), accounting for the effects of nutrients and planktivorous fish on alternative local equilibria of the plankton community. The emergence of diffusion-induced spatio-temporal chaos has been found by Pascual (1993) along a linear nutrient gradient in the same model without fish predation. Chaotic oscillations behind propagating prey-predator fronts (cf. Sherratt, Lewis & Fowler, 1995; Sherratt, Eagan & Lewis, 1997) of phytoplankton and zooplankton have been obtained as well (Petrovskii & Malchow, 1999).

Conditions for the emergence of three-dimensional spatial and spatio-temporal patterns after differential-flow-induced instabilities (Rovinsky & Menzinger, 1992) of spatially uniform populations were derived by Malchow (1996, 1998) and illustrated by patterns in Scheffer's model. Instabilities of the spatially uniform distribution can appear if phytoplankton and zooplankton move with different velocities but regardless of which one is faster. This mechanism of generating patchy patterns is much more general than the Turing mechanism which depends on strong conditions on the diffusion coefficients and one can imagine a wide range of applications in population dynamics.

Many mechanisms of the spatio-temporal variability of natural plankton populations are not known yet. Pronounced physical patterns like thermoclines, upwelling, fronts and eddies often set the frame for the biological processes. However, under conditions of relative physical uniformity, the temporal and spatio-temporal variability can be a consequence of the coupled nonlinear biological and chemical dynamics (Levin & Segel, 1976; Steele & Henderson, 1992). Daly and Smith (1993) concluded "... that biological processes may be more important at smaller scales where behaviour such as vertical migration and predation may control the plankton production, whereas physical processes may be more important at larger scales in structuring biological communities ...". O'Brien and Wroblewski (1973) introduced a dimensionless parameter, containing the characteristic water speed and the maximum specific biological growth rate, to distinguish parameter regions of biological and physical dominance, cf. also the forthcoming papers by Wroblewski, O'Brien and Platt (1975) and by Wroblewski and O'Brien (1976).

The effect of external hydrodynamical forcing on the appearance and stability of nonequilibrium spatio-temporal patterns has also been studied in Scheffer's model (Malchow & Shigesada, 1994), making use of the separation of the different time scales of biological and physical processes. A channel under tidal forcing served as a hydrodynamical model system with a relatively high detention time of matter. Examples were provided on different time scales: The simple physical transport and deformation of a spatially nonuniform initial plankton distribution as well as the biologically determined formation of a localized spatial maximum of phytoplankton biomass.

A general class of planktonic prey-predator systems is considered, however, the model by Scheffer (1991a) is used as an example. The fish is not considered as parameter anymore but it is treated as localized in a school, cruising and feeding according to defined rules. The process of aggregation of individual fishes and the persistence of schools under environmental or social constraints has already been studied by many other authors (Radakov, 1973; Cushing, 1975; Steele, 1977; Blake, 1983; Okubo, 1986; Grünbaum & Okubo, 1994; Huth & Wissel, 1994; Reuter & Breckling, 1994; Gueron et al., 1996, Niwa, 1996; Romey, 1996; Flierl et al., 1999, Stöcker, 1999) and is not considered.


Viral infections of phytoplankton

Viruses are evidently the most abundant entities in the sea and the question may arise whether they control ocean life. However, there is much less known about marine viruses and their role in aquatic ecosystems and the species that they infect, than about plankton patchiness and blooming, for reviews, cf. Fuhrman (1999). A number of studies (Wommack et al., 2000; Bergh et al., 1989; Tarutani et al., 2000; Suttle et al., 1990; Wilhelm & Suttle, 1999) shows the presence of pathogenic viruses in phytoplankton communities. Fuhrman (1999) has reviewed the nature of marine viruses and their ecological as well as biogeological effects. Suttle et al. (1990) have shown by using electron microscopy that the viral disease can infect bacteria and phytoplankton in coastal water. Parasites may modify the behaviour of the infected members of the prey population. Virus-like particles are described for many eukaryotic algae (van Etten et al., 1991; Reiser, 1993), cyanobacteria (Suttle et al., 1993) and natural phytoplankton communities (Peduzzi & Weinbauer, 1993). There is some evidence that viral infection might accelerate the termination of phytoplankton blooms (Jacquet et al., 2002; Gastrich et al., 2004). Viruses are held responsible for the collapse of Emiliania huxleyi blooms in mesocosms (Bratbak et al., 1995) and in the North Sea (Brussard et al., 1996) and are shown to induce lysis of Chrysochromulina (Suttle & Chan, 1993). Because most viruses are strain-specific, they can increase genetic diversity (Nagasaki & Yamaguchi, 1997). Nevertheless, despite the increasing number of reports, the role of viral infection in the phytoplankton population is still far from understood.

Viral infections of phytoplankton cells can be lysogenic or lytic. The understanding of the importance of lysogeny is just at the beginning (Wilcox & Fuhrman, 1994; Jiang & Paul, 1998; McDaniel et al., 2002; Ortmann et al., 2002). Contrary to lytic infections with destruction and without reproduction of the host cell, lysogenic infections are a strategy whereby viruses integrate their genome into the host's genome. As the host reproduces and duplicates its genome, the viral genome reproduces, too.

Mathematical models of the dynamics of virally infected phytoplankton populations are rare as well, the already classical publication is by Beltrami and Carroll (1994). More recent work is of Chattopadhyay et al. (2002-2004). The latter deal with lytic infections and mass action incidence functions. Malchow et al. (2004,2005) observed oscillations and waves in a phytoplankton-zooplankton system with Holling-type II and III grazing under lysogenic viral infection and frequency-dependent transmission (Nold, 1980; McCallum et al., 2001).

We focus on modelling the influence of lysogenic and lytic infections on the local and spatio-temporal dynamics of interacting phytoplankton and zooplankton with Holling-type II and III grazing. Furthermore, the impact of multiplicative noise (Allen, 2003; Anishenko et al., 2003) is investigated.