### Citations

990 |
Quantum Mechanics. Nonrelativistic Theory [in Russian
- Landau, Lifshits
- 2002
(Show Context)
Citation Context ...ows an example for such a localisation. Solving the eigenvalue problem Dahmen et al. exploited a similarity of equation (3.27) to the well studied “square well potential” problem of quantum mechanics =-=[53]-=-. They showed that depending on the dimensionless parameter x = (2/L) √ D/(µ0 + m), which characterises the growth rate in relation to the diffusivity and the habitat length, the critical flow velocit... |

493 |
The advance of advantageous genes
- Fisher
- 1937
(Show Context)
Citation Context ...patio-temporal dynamics of the population can be represented by the following equation ( ∂P(x, t) = µ0P 1 − ∂t P ) + D K ∂2P . (3.18) ∂x2 which is known as the Fisher-Kolmogorov equation after Fisher =-=[20]-=-, who considered the logistic dynamics of advantageous genes and Kolmogorov et al. [48], who investigated a general form of this problem (see also [62, 51, 108] ). Fisher [20] and Kolmogorov et al. [4... |

483 |
Light and Photosynthesis in Aquatic Ecosystems, second ed.
- Kirk
- 1994
(Show Context)
Citation Context ...he growth rate is saturated, fN(N) → 1, and we can neglect the limitation of growth by nutrients. The spatial profile of light intensity in a water column is described by Lambert-Beer’s law (see e.g. =-=[45]-=-) which states that the gradient of light intensity at depth z is proportional to the light intensity at this depth dI = −κI . (4.7) dz The coefficient κ includes both the absorption of light by water... |

205 |
On conditions for the vernal blooming of phytoplankton,
- Sverdrup
- 1953
(Show Context)
Citation Context ... However, they can be compensated by the production in the euphotic zone, if the unfavourable region is relatively small. Considering a simple mathematical model of a well mixed water column Sverdrup =-=[102]-=- defined a critical depth as the depth of a water column at which the total growth is equal to the total loss of biomass. Similar to the compensation depth, the critical depth can be reinterpreted in ... |

204 | Random dispersal in theoretical populations. - Skellam - 1951 |

177 |
Elements of Mathematical Ecology,
- Kot
- 2001
(Show Context)
Citation Context ...n as the Fisher-Kolmogorov equation after Fisher [20], who considered the logistic dynamics of advantageous genes and Kolmogorov et al. [48], who investigated a general form of this problem (see also =-=[62, 51, 108]-=- ). Fisher [20] and Kolmogorov et al. [48] have shown that if initially some part of the habitat is not occupied, P(x) = 0, then the population will propagate into this part with the constant velocity... |

157 |
Photoinhibition of photosynthesis in natural assemblages of phytoplankton.
- PLATT, GALLEGOS, et al.
- 1980
(Show Context)
Citation Context ...on by light and the nutrient. The specific form of these functions depends on many factors, for instance, strong light may photoinhibit photosynthesis and reduce the growth rate for large values of I =-=[83, 22, 86]-=-. However, usually it is suggested that f(x) → 1 as x → ∞, that is, the maximum growth rate is achieved when all resources are unlimited. For phytoplankton modelling, the most frequently used form is ... |

99 |
Periodic-Parabolic Boundary Value Problems and Positivity,
- Hess
- 1991
(Show Context)
Citation Context ...his condition is not necessary and does not hold even in the simple models considered before. General conditions of uniqueness and the existence of positive eigenvalues were obtained by Hess and Kato =-=[29]-=-, Senn and Hess [92], Cantrell and Cosner [9]. The analysis of 2D and 3D models raises even more questions. In higher dimensions the persistence of a population may depend on the geometric form of the... |

98 |
The deep chlorophyll maximum: Comparing vertical profiles ofchlorophyll a.
- CuLLEN
- 1982
(Show Context)
Citation Context ...on favours a phytoplankton build-up in deeper layers. This tension between light and nutrient limitation from two opposite sides frequently causes optimal growth conditions in subsurface layers (e.g. =-=[1, 13, 31]-=-). This fact often leads to the appearance of maxima of chlorophyll or biomass distributions at approximately 30-100 m depth. So called deep chlorophyll maxima (DCM) [1, 111, 13, 110, 31] and deep bio... |

98 |
Consequences of the Allee effect for behaviour, ecology and conservation.
- Stephens, J
- 1999
(Show Context)
Citation Context ...ad 2A. B. Ryabov and B. Blasius The role of diffusion and advection to omit the interesting aspects concerning conditional persistence in situations where the growth is influenced by an Allee effect =-=[100]-=-. Fourth, the interaction of turbulent currents with other processes can produce complex spatial structures in phytoplankton distributions [57], whereas locally oscillatory behaviour may lead to spati... |

83 |
Resource competition.
- Grover
- 1997
(Show Context)
Citation Context ...he sake of simplicity, assume that phytoplankton growth is limited only by the availability of light and a nutrient (the model can easily be extended to take into account multiple nutrient limitation =-=[26]-=-). In our approximation the dynamics of a phytoplankton population obey a reaction-diffusionadvection equation, similar to equation (3.1) considered in the previous chapter (see [85, 95, 46, 36] among... |

80 |
Dynamic model of phytoplankton growth and acclimation: responses of the balanced growth rate and the chlorophyll a:carbon ratio to light, nutrient-limitation and temperature.
- Geider, MacIntyre, et al.
- 1997
(Show Context)
Citation Context ...on by light and the nutrient. The specific form of these functions depends on many factors, for instance, strong light may photoinhibit photosynthesis and reduce the growth rate for large values of I =-=[83, 22, 86]-=-. However, usually it is suggested that f(x) → 1 as x → ∞, that is, the maximum growth rate is achieved when all resources are unlimited. For phytoplankton modelling, the most frequently used form is ... |

80 |
Response of ocean ecosystems to climate warming.
- Sarmiento, Slater, et al.
- 2004
(Show Context)
Citation Context ...lar, this can happen if D → Dmin = v 2 /4µ0 which is of importance in marine biology as climate models predict that the ongoing global warming may result in a higher stratification of the ocean water =-=[6, 91]-=-, increasing thereby the requirement on the critical (vertical) patch size for sinking phytoplankton species. 20A. B. Ryabov and B. Blasius The role of diffusion and advection Persistence in a river ... |

73 |
Emergent biogeography of microbial communities in a model ocean
- Follows, Dutkiewicz, et al.
- 2007
(Show Context)
Citation Context ...gave rise to a growing set of ecological models, which include cycling of many chemicals [117], coupling with meteorological data [42], interplay of different phytoplankton groups, and 3D simulations =-=[60, 21]-=-. Here, however, we will focus on the theoretical aspects and consider only simple conceptual models. 30A. B. Ryabov and B. Blasius The role of diffusion and advection Conservative models The nutrien... |

71 |
Oceanic diffusion diagrams,
- Okubo
- 1971
(Show Context)
Citation Context ... Below we will provide another heuristic derivation confirming the validity of this expression. Note that in large aquatic basins the horizontal turbulent mixing increases with the scale of phenomena =-=[73, 67, 66, 74]-=-. Petrovskii [77, 78] showed that this should result in the increase of the front propagation velocity. Moreover, this velocity should grow with the size of the area occupied by a population. Pseudo w... |

71 |
Diffusion and Ecological Problems
- Okubo
(Show Context)
Citation Context ...efer to detailed discussions, providing information about the main techniques that are useful in this field of research. There exist many excellent reviews about spatial population dynamics (see e.g. =-=[32, 47, 65, 72]-=-). Nevertheless to our knowledge, the interplay of advection and diffusion for a heterogeneous population, as elaborated in this text, has never been described. So, while the text will not provide muc... |

60 |
Partial differential equations in ecology: spatial interactions and population dynamics.
- HOLMES, LEWIS, et al.
- 1994
(Show Context)
Citation Context ...efer to detailed discussions, providing information about the main techniques that are useful in this field of research. There exist many excellent reviews about spatial population dynamics (see e.g. =-=[32, 47, 65, 72]-=-). Nevertheless to our knowledge, the interplay of advection and diffusion for a heterogeneous population, as elaborated in this text, has never been described. So, while the text will not provide muc... |

59 |
Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem. In: P. Pelce (Ed.) Dynamics of Curved Fronts (Boston: Academic Press). Cancer invasion of brain tissue 31
- Kolmogorov, Petrovsky, et al.
- 1988
(Show Context)
Citation Context ...( ∂P(x, t) = µ0P 1 − ∂t P ) + D K ∂2P . (3.18) ∂x2 which is known as the Fisher-Kolmogorov equation after Fisher [20], who considered the logistic dynamics of advantageous genes and Kolmogorov et al. =-=[48]-=-, who investigated a general form of this problem (see also [62, 51, 108] ). Fisher [20] and Kolmogorov et al. [48] have shown that if initially some part of the habitat is not occupied, P(x) = 0, the... |

58 |
Potential impact of climate change on marine export production,
- Bopp, Monfray, et al.
- 2001
(Show Context)
Citation Context ...lar, this can happen if D → Dmin = v 2 /4µ0 which is of importance in marine biology as climate models predict that the ongoing global warming may result in a higher stratification of the ocean water =-=[6, 91]-=-, increasing thereby the requirement on the critical (vertical) patch size for sinking phytoplankton species. 20A. B. Ryabov and B. Blasius The role of diffusion and advection Persistence in a river ... |

53 |
Summer heatwaves promote blooms of harmful cyanobacteria.
- Johnk, Huisman, et al.
- 2008
(Show Context)
Citation Context ...ers with depth and time. This approach (see also [113, 82, 115, 112]) gave rise to a growing set of ecological models, which include cycling of many chemicals [117], coupling with meteorological data =-=[42]-=-, interplay of different phytoplankton groups, and 3D simulations [60, 21]. Here, however, we will focus on the theoretical aspects and consider only simple conceptual models. 30A. B. Ryabov and B. B... |

47 | Critical depth and critical turbulence: Two different mechanisms for the development of phytoplankton blooms. - OOSTVEEN, WEISSING - 1999 |

41 | Algal games: the vertical distribution of phytoplankton in poorly mixed water columns.
- Klausmeier, Litchman
- 2001
(Show Context)
Citation Context ...trient limitation [26]). In our approximation the dynamics of a phytoplankton population obey a reaction-diffusionadvection equation, similar to equation (3.1) considered in the previous chapter (see =-=[85, 95, 46, 36]-=- among others) ∂P(z, t) = µ(N, I)P − mP − v ∂t ∂P ∂ + ∂z ∂z D∂P , (4.1) ∂z where m is the mortality (compare to equation (2.3)), v is the phytoplankton sinking velocity, and D is the diffusivity, whic... |

40 |
Phytoplankton patchiness: the role of lateral stirring and mixing
- Martin
- 2003
(Show Context)
Citation Context ...tions where the growth is influenced by an Allee effect [100]. Fourth, the interaction of turbulent currents with other processes can produce complex spatial structures in phytoplankton distributions =-=[57]-=-, whereas locally oscillatory behaviour may lead to spatio-temporal chaotic waves [79, 80]. Finally, we did not even touch such important problems as model validation and the simulation of multicompon... |

39 |
Reduced mixing generates oscillations and chaos in the oceanic deep chlorophyll maximum.
- Huisman, Thi, et al.
- 2006
(Show Context)
Citation Context ...e shown below, these complicated, selforganised dynamics can lead to new phenomena and diverse behaviour. For example, if the mixing is small, the final solution becomes non-stationary and oscillates =-=[36]-=-, whereas in the presence of an upper mixed layer the system may exhibit bistability and the solution may be sensitive to the initial conditions [118, 89]. Equation of growth To formulate a mathematic... |

38 |
How habitat edges change species interactions.
- Fagan, Cantrell, et al.
- 1999
(Show Context)
Citation Context ...y depend on the geometric form of the favourable patches [15], the form of the edges separating the patches and finally it may depend on the behaviour of individuals, moving across or along the edges =-=[19]-=-. 3.2. Persistence on an infinite habitat Travelling fronts While in the previous section we have highlighted some negative aspects of diffusion for the fate of a population, in this section we show t... |

38 |
Quantitative ecology of the plankton of the western North Atlantic.
- Riley, Stommel, et al.
- 1949
(Show Context)
Citation Context ...ons and constitutes, for example, a necessary condition for the persistence of a population in a river [99], as well as for the persistence of sinking phytoplankton species in a vertical water column =-=[87, 95, 33]-=-. Derivation of the front propagation velocity These results about the spread of a population in an advective flow can be elegantly used to derive the population spread, Eq. (3.19), in a system withou... |

34 |
Transmissometer measurement of POC.
- Bishop
- 1999
(Show Context)
Citation Context ... leads to the appearance of maxima of chlorophyll or biomass distributions at approximately 30-100 m depth. So called deep chlorophyll maxima (DCM) [1, 111, 13, 110, 31] and deep biomass maxima (DBM) =-=[49, 5]-=- are ubiquitous phenomena and can be observed in many oligotrophic regions in the ocean, marine systems, and deep lakes. Another important component of a stratified water column is an upper mixed laye... |

34 |
Resource competition in a variable environment: phytoplankton growing according to the variable-internal-stores model.
- Grover
- 1991
(Show Context)
Citation Context ...o be omitted here. First, we had to restrict to single species populations, while one of the most important and still debatable challenges concerns the competition of spatially structured populations =-=[104, 25, 98]-=- and the high diversity of phytoplankton species (see e.g. [88] and the references therein). Second, for the sake of simplicity we included only Fickian diffusion models, whereas in the ocean diffusiv... |

32 | Light limitation of phytoplankton biomass and macronutrient utilization in the Southern Ocean,
- Mitchell, Brody, et al.
- 1991
(Show Context)
Citation Context ...phic aquatic environments (as observed in many regions), where the nutrients are in ample supply and light becomes a crucial factor which determines the distribution and the dynamics of phytoplankton =-=[59, 90, 11]-=-. So in the following we assume that the nutrient dependence of the growth rate is saturated, fN(N) → 1, and we can neglect the limitation of growth by nutrients. The spatial profile of light intensit... |

31 | Island biogeography and the design of nature reserves. - Diamond, May - 1981 |

29 |
Marine snow: sinking rates and potential role in vertical flux. Deep-Sea Res.
- Shanks, Trent
- 1980
(Show Context)
Citation Context ...(4.12) ∂N(z, t) = −µ(N, I)P + rT +D ∂t ∂2N , ∂z2 where vP and vT are the sinking velocities of phytoplankton and detritus respectively. Note that usually detritus sinks much faster than phytoplankton =-=[94, 84]-=-. In this model the cycle of chemicals includes three stages: the transfer of biomass to detritus with mortality m, the remineralisation of detritus back into nutrients with remineralisation rate r, a... |

28 |
Populations persistence in river and estuaries.
- Speirs, Gurney
- 2001
(Show Context)
Citation Context ...or x > 0. In both plots P(x, 0) = 1 for x ≤ 0. crucial factor not only for an invasion process but also for the survival of a population in the presence of drift, sinking or other advective processes =-=[14, 99, 101]-=-. There is a variety of ways to derive relation (3.19). The more common and rigorous approach suggests to assume a travelling solution of the form P(x − vt) and then to prove that this solution is sta... |

27 |
The effects of spatial heterogeneity in population dynamics
- Cantrell, Cosner
- 1991
(Show Context)
Citation Context ...cterises the grazing rate. Influence of boundary conditions and spatial arrangement It is interesting to extend the analysis for more complicated spatial geometries. Seno [93] and Cantrell and Cosner =-=[7]-=- investigated the influence of a spatial sequence of favourable and unfavourable habitats and boundary conditions. Seno considered a population of organisms migrating between n patches of different qu... |

27 |
A quantitative study on the phytoplankton of the Bay of Fundy and the Gulf of Maine (including observations on hydrography, chemistry and morbidity).
- Gran, Braarud
- 1935
(Show Context)
Citation Context ...ed water column, owing to additional losses of biomass across the thermocline. The fact that a deep upper layer can prevent phytoplankton blooming was noted experimentally in 1935 by Gran and Braarud =-=[23]-=-, who investigated the conditions of phytoplankton blooming in the upper mixed layer. They reported that until there exists a deep UML, phytoplankton production cannot exceed the destruction by respir... |

27 |
The regulation of inhomogeneous population
- Gurney, Nisbet
- 1975
(Show Context)
Citation Context ...d the centre of a patch. He found that the modified critical size equals Lc = L0f (v 2 /4αD), where f(0) = 1 and the function f(v 2 /4αD) monotonically decreases with v toward zero. Gurney and Nisbet =-=[27]-=- considered a model in which the growth rate parabolically depends on the distance from the habitat centre. This approach is more realistic since it includes a gradual transition from favourable to un... |

27 | How do sinking phytoplankton species manage to persist?
- ARRAYAS, EBERT, et al.
- 2002
(Show Context)
Citation Context ...ons and constitutes, for example, a necessary condition for the persistence of a population in a river [99], as well as for the persistence of sinking phytoplankton species in a vertical water column =-=[87, 95, 33]-=-. Derivation of the front propagation velocity These results about the spread of a population in an advective flow can be elegantly used to derive the population spread, Eq. (3.19), in a system withou... |

27 | Mathematical challenges in spatial ecology.
- Neuhauser
- 2001
(Show Context)
Citation Context ...efer to detailed discussions, providing information about the main techniques that are useful in this field of research. There exist many excellent reviews about spatial population dynamics (see e.g. =-=[32, 47, 65, 72]-=-). Nevertheless to our knowledge, the interplay of advection and diffusion for a heterogeneous population, as elaborated in this text, has never been described. So, while the text will not provide muc... |

26 |
Stable isotopes resolve the drift paradox for Baetis mayflies in an artic river.
- Hersey, Pastor, et al.
- 1993
(Show Context)
Citation Context ...drift paradox” because any advection will ensure that the average location of a population will move downstream, so at first glance it seems counter-intuitive that a population can persist in a river =-=[28]-=-. In their study Speirs and Gurney used the linear model (3.24) and assumed an impenetrable boundary upstream and a totally hostile environment downstream. In this model the critical size Lv of the fa... |

23 |
Changes in turbulent mixing shift competition for light between phytoplankton species.
- Huisman, Sharples, et al.
- 2004
(Show Context)
Citation Context ...arrier. Thus, in an incompletely mixed water column, a more light limited or buoyant species obtains a competitive advantage to another species, whose favourable patch is located in subsurface layers =-=[34, 89]-=-. Arbitrary spatial dependence of the growth rate The behaviour of model (3.6) in the case where the growth rate µ(x) is an arbitrary function of the coordinates was investigated by Cantrell and Cosne... |

23 |
A theoretical study of phytoplankton growth and nutrient distribution
- Jamart, Winter, et al.
- 1977
(Show Context)
Citation Context ...d by Okubo [69, 71]. Radach and Maier-Reimer [85] suggested a mathematical model of phytoplankton growth which included light-nutrient-phytoplankton dynamics. This model was extended by Jamart et al. =-=[41]-=- who considered limitation by two nutrients, grazing and the variability of the parameters with depth and time. This approach (see also [113, 82, 115, 112]) gave rise to a growing set of ecological mo... |

23 |
Modeling the productivity of phytoplankton.
- Platt, Denman, et al.
- 1977
(Show Context)
Citation Context ...dmit only numerical investigation. However, they are more realistic and provide some understanding of the processes occurring in deep waters of many regions where surface layers are nutrient depleted =-=[85, 113, 82, 115, 112, 109, 36]-=-. Furthermore, in the tension of two opposing resource gradients the location and the size of a production layer becomes a function of the phytoplankton abundance and the initial conditions, that can ... |

21 |
Towards a resolution of 'the paradox of the plankton': A brief overview of the proposed mechanisms[J]. Ecological Complexity.
- Roy, Chattopadhyay
- 2007
(Show Context)
Citation Context ...ns, while one of the most important and still debatable challenges concerns the competition of spatially structured populations [104, 25, 98] and the high diversity of phytoplankton species (see e.g. =-=[88]-=- and the references therein). Second, for the sake of simplicity we included only Fickian diffusion models, whereas in the ocean diffusivity depends on the scale of phenomena [72]. Third, we had 2A. ... |

20 |
Spatial patterning of the spruce budworm,
- Ludwig, Aronson, et al.
- 1979
(Show Context)
Citation Context ...available (see e.g. [52], [2], [61], [72]). The stationary solution can be expressed in terms of elliptic functions and was investigated by Skellam [96], Levandowsky and White [54], and Ludwig et al. =-=[56]-=-. However, applying the invasibility criteria, we can conclude that this system possesses the same critical values as the KiSS model. Indeed, as the population density approaches zero, 10A. B. Ryabov... |

19 |
A minimal model of pattern formation in a prey–predator system
- Petrovskii, Malchow
- 1999
(Show Context)
Citation Context ... of turbulent currents with other processes can produce complex spatial structures in phytoplankton distributions [57], whereas locally oscillatory behaviour may lead to spatio-temporal chaotic waves =-=[79, 80]-=-. Finally, we did not even touch such important problems as model validation and the simulation of multicomponent natural ecosystems [60]. The review is structured as follows. In chapter two, we short... |

18 |
A Model of Phytoplankton Blooms.
- Huppert, Blasius, et al.
- 2002
(Show Context)
Citation Context ...tains a stable feasible equilibrium and the system spirals in phase space to an isolated 5 (2.7) (2.8)A. B. Ryabov and B. Blasius The role of diffusion and advection fixed point (Fig. 1C), see e.g., =-=[38, 10, 39]-=-. Furthermore, for perfect nutrient recycling (ε = 1) one can easily show that S(t) follows the simple dynamics dS dt = δ(Ni − S) . (2.9) Therefore, after some transient time, determined by the exchan... |

18 |
Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics
- Petrovskii, Malchow
- 2001
(Show Context)
Citation Context ... of turbulent currents with other processes can produce complex spatial structures in phytoplankton distributions [57], whereas locally oscillatory behaviour may lead to spatio-temporal chaotic waves =-=[79, 80]-=-. Finally, we did not even touch such important problems as model validation and the simulation of multicomponent natural ecosystems [60]. The review is structured as follows. In chapter two, we short... |

17 |
On the effects of spatial heterogeneity on the persistence of interacting species
- Cantrell, Cosner
- 1998
(Show Context)
Citation Context ...stence of a species than two smaller patches of the same total size, since two patches would have four ends, that would double the loss rate. Diamond and May [15], McMurtrie [58], Cantrell and Cosner =-=[8]-=- applied this concept of a critical patch size to the design of national parks and natural reserves of optimal size and form. Logistic growth As the growth of biomass in the KiSS model is not limited,... |

16 | Mixing and the dynamics of the deep chlorophyll maximum in Lake Tahoe
- Abbott, Denman, et al.
- 1984
(Show Context)
Citation Context ...on favours a phytoplankton build-up in deeper layers. This tension between light and nutrient limitation from two opposite sides frequently causes optimal growth conditions in subsurface layers (e.g. =-=[1, 13, 31]-=-). This fact often leads to the appearance of maxima of chlorophyll or biomass distributions at approximately 30-100 m depth. So called deep chlorophyll maxima (DCM) [1, 111, 13, 110, 31] and deep bio... |

16 |
Factors controlling the development of phytoplankton blooms in the Antarctic Ocean — a mathematical-model.
- Sakshaug, Slagstad, et al.
- 1991
(Show Context)
Citation Context ...phic aquatic environments (as observed in many regions), where the nutrients are in ample supply and light becomes a crucial factor which determines the distribution and the dynamics of phytoplankton =-=[59, 90, 11]-=-. So in the following we assume that the nutrient dependence of the growth rate is saturated, fN(N) → 1, and we can neglect the limitation of growth by nutrients. The spatial profile of light intensit... |

15 | Critical conditions for phytoplankton blooms
- Ebert, Arrayás, et al.
- 2001
(Show Context)
Citation Context ...and light-limited growth, respectively. However, this non-linear form often admits only numerical investigation. Analytical solutions are commonly possible only for a linear or algebraic form of f(x) =-=[95, 17]-=-. Boundary conditions By default, we assume that the surface and bottom are impenetrable for phytoplankton ( vP(z, t) − D ∂P )∣ ∣∣∣z=0,ZB = 0 . (4.5) ∂z To model a stratified water column, one can eit... |

15 |
The size of water masses containing plankton
- Kierstead, B
- 1953
(Show Context)
Citation Context ...e patch can cause additional losses which may even lead to local extinction. KiSS model One simple and elegant model to study this problem has been independently introduced by Kierstead and Slobodkin =-=[44]-=- and Skellam [96]. The main idea is to separate the landscape into a favourable area (which will be denoted as the species’ habitat) adjoining some hostile environment from both sides. For its simplic... |

14 |
The influence of density stratification on particle settling, dispersion and population growth.
- CONDIE, BORMANS
- 1997
(Show Context)
Citation Context ...L). A UML commonly occurs in oceans and lakes due to mechanical perturbation of the surface waters (e.g. due to wind, waves, and storms). This layer is separated from the deep layers by a thermocline =-=[12]-=-, which is defined as a relatively thin layer below a UML characterised by an strong change in temperature with depth. Mixing in a UML is much stronger than in the layers below it. As a result, the di... |

13 | Analysis of the self-shading effect on algal vertical distribution in natural waters.
- Shigesada, Okubo
- 1981
(Show Context)
Citation Context ...ons and constitutes, for example, a necessary condition for the persistence of a population in a river [99], as well as for the persistence of sinking phytoplankton species in a vertical water column =-=[87, 95, 33]-=-. Derivation of the front propagation velocity These results about the spread of a population in an advective flow can be elegantly used to derive the population spread, Eq. (3.19), in a system withou... |

12 |
Beneath the surface: Characteristics of oceanic ecosystems under weak mixing conditions – a theoretical investigation
- Beckmann, Hense
- 2007
(Show Context)
Citation Context ... this system. The sedimentation of organic matter removes the nutrient fixed in phytoplankton cells from the upper layer, which leads to the formation of deep phytoplankton maxima. Beckmann and Hense =-=[3]-=- performed numerical simulations and analytical evaluations of model (4.12), assuming that detritus sinks relatively fast, whereas the phytoplankton sinking is negligible. Fig. 13 reproduces a typical... |

12 | Persistence, spread and the drift paradox
- Pachepsky, Lutscher, et al.
- 2005
(Show Context)
Citation Context ...model has only one boundary with the hostile environment, we obtain the limit L v → L0/2 as v → 0, where L0 is the critical patch size (3.12) of the KiSS model. Extending this model, Pachepsky et al. =-=[75]-=- derived conditions for the persistence and spread of a population of organisms living and reproducing on the sediment and occasionally entering the water flow where they can drift and disperse. Local... |

11 |
Is phytoplankton growth in the Wadden Sea light or nitrogen limited?
- Colijn, Cadée
- 2003
(Show Context)
Citation Context ...phic aquatic environments (as observed in many regions), where the nutrients are in ample supply and light becomes a crucial factor which determines the distribution and the dynamics of phytoplankton =-=[59, 90, 11]-=-. So in the following we assume that the nutrient dependence of the growth rate is saturated, fN(N) → 1, and we can neglect the limitation of growth by nutrients. The spatial profile of light intensit... |

11 |
The benthic biological sub model in the European regional seas ecosystem model
- Ebenhöh, Kohlmeier, et al.
- 1995
(Show Context)
Citation Context .... In contrast, the nutrient concentration can behave just in the opposing way. The sedimentation of dead biomass (detritus), with the successive remineralisation in the deep layers or in the sediment =-=[16]-=- causes an increase of the nutrient concentration with depth [117]. Thus, while light limitation may lead to the formation of a surface phytoplankton maximum, a lower nutrient concentration favours a ... |

11 |
Deep chlorophyll-a maxima (DCMs) in Antarctic waters—I. relationships between DCMs and the physical, chemical, and optical conditions in the upper water column
- Holm-Hansen, Hewes
- 2004
(Show Context)
Citation Context ...on favours a phytoplankton build-up in deeper layers. This tension between light and nutrient limitation from two opposite sides frequently causes optimal growth conditions in subsurface layers (e.g. =-=[1, 13, 31]-=-). This fact often leads to the appearance of maxima of chlorophyll or biomass distributions at approximately 30-100 m depth. So called deep chlorophyll maxima (DCM) [1, 111, 13, 110, 31] and deep bio... |

11 |
Population dynamics of sinking phytoplankton in light-limited environments: Simulation techniques and critical parameters.
- SOMMEIJER
- 2002
(Show Context)
Citation Context ...ing infinite mixing within the UML and a small diffusivity DD in deep layers. Assuming continuity of the flux across the thermocline, we obtain the boundary condition at the bottom of a UML (see e.g. =-=[35]-=-) vP(z)| z=ZT −0 = ( )∣ ∂P ∣∣∣z=ZT+0 vP(z) − DD ∂z where ZT is the depth of the thermocline. On the other hand, to simulate the water column in a single framework, one can assume a gradual transition ... |

11 |
Development of a deep chlorophyll maximum of Heterocapsa triquetra (Ehrenb), at the entrance to the Gulf of Finland,
- Kononen, Huttunen, et al.
- 2003
(Show Context)
Citation Context ... leads to the appearance of maxima of chlorophyll or biomass distributions at approximately 30-100 m depth. So called deep chlorophyll maxima (DCM) [1, 111, 13, 110, 31] and deep biomass maxima (DBM) =-=[49, 5]-=- are ubiquitous phenomena and can be observed in many oligotrophic regions in the ocean, marine systems, and deep lakes. Another important component of a stratified water column is an upper mixed laye... |

11 |
On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions
- SENN, HESS
- 1982
(Show Context)
Citation Context ... necessary and does not hold even in the simple models considered before. General conditions of uniqueness and the existence of positive eigenvalues were obtained by Hess and Kato [29], Senn and Hess =-=[92]-=-, Cantrell and Cosner [9]. The analysis of 2D and 3D models raises even more questions. In higher dimensions the persistence of a population may depend on the geometric form of the favourable patches ... |

11 |
Simulation of three-dimensional phytoplankton dynamics: competition in light-limited environments.
- Thi, Huisman, et al.
- 2005
(Show Context)
Citation Context ..., yield an integro-differential system of equations. It is not straightforward to obtain rigorous or analytical results for such a system and even a numerical solution encounters certain difficulties =-=[35, 103]-=-. Nevertheless, without solving any equations, it is clear that the light intensity in the water column is reduced with increasing depth. Thus the light limitation forms a favourable area close to the... |

10 |
Life and death near a windy oasis
- Dahmen, Nelson, et al.
- 2000
(Show Context)
Citation Context ...or x > 0. In both plots P(x, 0) = 1 for x ≤ 0. crucial factor not only for an invasion process but also for the survival of a population in the presence of drift, sinking or other advective processes =-=[14, 99, 101]-=-. There is a variety of ways to derive relation (3.19). The more common and rigorous approach suggests to assume a travelling solution of the form P(x − vt) and then to prove that this solution is sta... |

10 |
Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics
- Ishii, Takagi
- 1982
(Show Context)
Citation Context ... model does not depend on the sinking velocity. The sinking just shifts the bulk of biomass downward, preserving, however, the total amount of biomass in the water column (Fig. 11A). Ishii and Takagi =-=[40]-=- relaxed the condition Kbg = 0 and proved some existence, stability and uniqueness results for this system. Assuming an algebraic form of the growth rate, µ(I) ∼ I α , Ebert et al. [17] have found som... |

9 | Simple models of steady deep maxima in chlorophyll and biomass
- Hodges, Rudnick
- 2003
(Show Context)
Citation Context ...umn ∂N(0, t) = 0 , N(ZB, t) = NB . (4.11) ∂z Fig. 12 shows typical final distribution of phytoplankton and nutrient given by model (4.10), supplemented by equation (4.8) for light. Hodges and Rudnick =-=[30]-=- pointed out that, independent of the functional form of the growth rate and of the light distribution (assuming that light decreases with depth), this model can reproduce a deep stationary phytoplank... |

9 | Localization and extinction of bacterial populations under inhomogeneous growth conditions
- Lin, Mann, et al.
- 2004
(Show Context)
Citation Context ...ocity is higher, the population becomes extinct. The value of the critical patch size and the critical diffusivity can be easily expressed from equation (3.34). Examining bacterial growth, Lin et al. =-=[55]-=- confirmed the results of Dahmen et al. [14] experimentally and by means of numerical simulation. Joo and Lebowitz [43] carried out computer simulations in a stochastic spatially discrete population m... |

9 |
Minimum domains for spatial patterns in a class of reation diffusion equations
- Murray, Sperb
- 1983
(Show Context)
Citation Context ...s, giving rise to a “pseudo-wave”. Advection Consider now an extension of the Fisher-Kolmogorov equation for a population which is additionally subjected to an advective flow with constant velocity v =-=[63, 64, 14, 99, 101, 4, 50]-=- ( ∂P(x, t) = µ0P 1 − ∂t P ) − v K ∂P ∂x + D∂2 P . (3.21) ∂x2 To investigate the role of advection let us again suppose initial conditions, such that only the left part of the habitat is populated, wh... |

9 |
The vertical structure of phytoplankton growth dynamics: A mathematical model
- Radach, Maier-Reimer
- 1975
(Show Context)
Citation Context ...trient limitation [26]). In our approximation the dynamics of a phytoplankton population obey a reaction-diffusionadvection equation, similar to equation (3.1) considered in the previous chapter (see =-=[85, 95, 46, 36]-=- among others) ∂P(z, t) = µ(N, I)P − mP − v ∂t ∂P ∂ + ∂z ∂z D∂P , (4.1) ∂z where m is the mortality (compare to equation (2.3)), v is the phytoplankton sinking velocity, and D is the diffusivity, whic... |

8 |
Spatial heterogeneity and critical patch size: Area effects via diffusion in closed environments
- Cantrell, Cosner
- 2001
(Show Context)
Citation Context ... the simplest combination of parameters yields √ D L0 = c , where c is a non-dimensional constant which equals π in the 1D model. The same expression holds in 2D systems [96, 44], and it can be shown =-=[32, 9]-=- that the constant c depends on the principal eigenvalue, thereby it represents the geometry of the model. Equation (3.12) can also be written in the form µ0 D ≤ D0 = L2 µ0 π 2 , (3.13) yielding the c... |

8 | Turbulent mixing and phytoplankton spring bloom development in a deep lake.
- Peeters, Straile, et al.
- 2007
(Show Context)
Citation Context ...plankton production cannot exceed the destruction by respiration and phytoplankton blooming is not possible. The concept of the maximal diffusivity is also consistent with field experiments, see e.g. =-=[106, 18, 76]-=-. 4.2. Light and Nutrient limitation In the last section we will discuss models which take into account both light and nutrient limitation of phytoplankton growth. These models are more difficult to a... |

7 |
Non-Hermitian localization and population biology,
- Nelson, Shnerb
- 1998
(Show Context)
Citation Context ...s, giving rise to a “pseudo-wave”. Advection Consider now an extension of the Fisher-Kolmogorov equation for a population which is additionally subjected to an advective flow with constant velocity v =-=[63, 64, 14, 99, 101, 4, 50]-=- ( ∂P(x, t) = µ0P 1 − ∂t P ) − v K ∂P ∂x + D∂2 P . (3.21) ∂x2 To investigate the role of advection let us again suppose initial conditions, such that only the left part of the habitat is populated, wh... |

7 |
Horizontal dispersion and critical scale for phytoplankton patches.
- Okubo
- 1978
(Show Context)
Citation Context ...d the critical (minimal) size of the favourable patch, which provides the survival of the population √ D L ≥ L0 = π . (3.12) 9 µ0A. B. Ryabov and B. Blasius The role of diffusion and advection Okubo =-=[71, 72]-=- demonstrated that this equation can be obtained by means of simple dimensional analysis. Suppose that L0 = f(D, µ0). Dimensionally, [D] = m2 /s, [µ0] = 1/s, and [L] = m. Thereby, the simplest combina... |

7 |
On the plankton front waves accelerated by marine turbulence
- Petrovskii
- 1999
(Show Context)
Citation Context ...r heuristic derivation confirming the validity of this expression. Note that in large aquatic basins the horizontal turbulent mixing increases with the scale of phenomena [73, 67, 66, 74]. Petrovskii =-=[77, 78]-=- showed that this should result in the increase of the front propagation velocity. Moreover, this velocity should grow with the size of the area occupied by a population. Pseudo waves We should stress... |

6 |
Bounding biomass in the Fisher equation
- Birch, Tsand, et al.
(Show Context)
Citation Context ...s, giving rise to a “pseudo-wave”. Advection Consider now an extension of the Fisher-Kolmogorov equation for a population which is additionally subjected to an advective flow with constant velocity v =-=[63, 64, 14, 99, 101, 4, 50]-=- ( ∂P(x, t) = µ0P 1 − ∂t P ) − v K ∂P ∂x + D∂2 P . (3.21) ∂x2 To investigate the role of advection let us again suppose initial conditions, such that only the left part of the habitat is populated, wh... |

6 |
Chaos in a periodically forced chemostat with algal mortality
- Clodong, Blasius
(Show Context)
Citation Context ...tains a stable feasible equilibrium and the system spirals in phase space to an isolated 5 (2.7) (2.8)A. B. Ryabov and B. Blasius The role of diffusion and advection fixed point (Fig. 1C), see e.g., =-=[38, 10, 39]-=-. Furthermore, for perfect nutrient recycling (ε = 1) one can easily show that S(t) follows the simple dynamics dS dt = δ(Ni − S) . (2.9) Therefore, after some transient time, determined by the exchan... |

6 |
Spatial models of competition
- Klausmeier, Tilman
- 2002
(Show Context)
Citation Context |

6 |
Advection-diffusion in the presence of surface convergence
- Okubo
- 1978
(Show Context)
Citation Context ...ritical patch size L0 of the KiSS model. Many other examples of critical patch models can be found in the books by Okubo and Levin [72] and Murray [62]. We just briefly mention some extensions. Okubo =-=[68, 70]-=- considered a model for growth and diffusion under an attractive force toward the centre of a patch. He found that the modified critical size equals Lc = L0f (v 2 /4αD), where f(0) = 1 and the functio... |

6 |
A general equation for the mesoscale distribution of phytoplankton
- Platt, L
- 1975
(Show Context)
Citation Context ...abitat centre. This approach is more realistic since it includes a gradual transition from favourable to unfavourable areas. Wroblewski et al. [116], Wroblewski and O’Brien [114] and Platt and Denman =-=[81]-=- included the effect of grazing and obtained an expression similar to (3.12) for the critical patch size, in which, however, µ0 is replaced by µ0 − g, where g characterises the grazing rate. Influence... |

6 |
Modelling phytoplankton dynamics in lakes and reservoirs: the problem of in-situ growth rates. Hydrobiologia 349
- Reynolds, Irish
- 1997
(Show Context)
Citation Context ...on by light and the nutrient. The specific form of these functions depends on many factors, for instance, strong light may photoinhibit photosynthesis and reduce the growth rate for large values of I =-=[83, 22, 86]-=-. However, usually it is suggested that f(x) → 1 as x → ∞, that is, the maximum growth rate is achieved when all resources are unlimited. For phytoplankton modelling, the most frequently used form is ... |

5 |
Diffusion and drift in a medium with randomly distributed traps
- Grassberger, Procaccia
- 1982
(Show Context)
Citation Context ...orm of the second equation of system (3.17) for the problem without flux. Thus we can conclude that the presence of advection for a population under boundary conditions simply reduces the eigenvalues =-=[24, 64, 14]-=- λ v = λ − v2 4D . With the same arguments as in Section 3.1., to provide the persistence of a population, the largest eigenvalue λ v must be positive. Therefore, a population is only able to persist ... |

5 |
Diffusion-induced instability in model ecosystems
- Okubo
- 1974
(Show Context)
Citation Context ...ditions, that can lead to new patterns and new dynamical behaviour. A coupled system of reaction-diffusion equations describing nutrient-phytoplankton cycling was probably first investigated by Okubo =-=[69, 71]-=-. Radach and Maier-Reimer [85] suggested a mathematical model of phytoplankton growth which included light-nutrient-phytoplankton dynamics. This model was extended by Jamart et al. [41] who considered... |

5 |
Mixing-induced global modes in open active
- Straube, Pikovsky
- 2007
(Show Context)
Citation Context ...or x > 0. In both plots P(x, 0) = 1 for x ≤ 0. crucial factor not only for an invasion process but also for the survival of a population in the presence of drift, sinking or other advective processes =-=[14, 99, 101]-=-. There is a variety of ways to derive relation (3.19). The more common and rigorous approach suggests to assume a travelling solution of the form P(x − vt) and then to prove that this solution is sta... |

4 |
Plankton cycles disguised by turbulent advection.
- Koszalka, Bracco, et al.
- 2007
(Show Context)
Citation Context |

4 |
Persistence and stability of single-species and prey-predator systems in spatially heterogeneous environments
- McMurtrie
- 1978
(Show Context)
Citation Context ...ch is better for the persistence of a species than two smaller patches of the same total size, since two patches would have four ends, that would double the loss rate. Diamond and May [15], McMurtrie =-=[58]-=-, Cantrell and Cosner [8] applied this concept of a critical patch size to the design of national parks and natural reserves of optimal size and form. Logistic growth As the growth of biomass in the K... |

4 |
Horizontal Turbulence and Turbulent Exchange in an Ocean
- Ozmidov
- 1968
(Show Context)
Citation Context ... Below we will provide another heuristic derivation confirming the validity of this expression. Note that in large aquatic basins the horizontal turbulent mixing increases with the scale of phenomena =-=[73, 67, 66, 74]-=-. Petrovskii [77, 78] showed that this should result in the increase of the front propagation velocity. Moreover, this velocity should grow with the size of the area occupied by a population. Pseudo w... |

4 |
Effet3 of a singular patch on population persistence in a multi-patch system
- Seno
- 1991
(Show Context)
Citation Context ...aced by µ0 − g, where g characterises the grazing rate. Influence of boundary conditions and spatial arrangement It is interesting to extend the analysis for more complicated spatial geometries. Seno =-=[93]-=- and Cantrell and Cosner [7] investigated the influence of a spatial sequence of favourable and unfavourable habitats and boundary conditions. Seno considered a population of organisms migrating betwe... |

4 |
Akira okubo and the theory of blooms
- Slobodkin
- 1999
(Show Context)
Citation Context ... sides. For its simplicity, Akiro Okubo assigned to the model the name KiSS, which on the one hand includes the authors’ initials, and from the other hand can be deciphered as “Keep it Simple Stupid” =-=[97]-=-. The KiSS model suggests a population growing on a finite patch of size L, surrounded by an absolutely hostile environment with infinite mortality (Fig. 2). Thus, per definition the population densit... |

3 |
A note on the transient stage of the random dispersal of logistic populations
- Barakat
- 1959
(Show Context)
Citation Context ...wth function ( ∂P(x, t) = µ0P 1 − ∂t P ) + D K ∂2P , for 0 ≤ x ≤ L (3.14) ∂x2 and the same (hostile) boundary conditions (3.8). For this model only approximate solutions are available (see e.g. [52], =-=[2]-=-, [61], [72]). The stationary solution can be expressed in terms of elliptic functions and was investigated by Skellam [96], Levandowsky and White [54], and Ludwig et al. [56]. However, applying the i... |

3 |
Spring blooms and stratification
- Ellertsen
- 1993
(Show Context)
Citation Context ...plankton production cannot exceed the destruction by respiration and phytoplankton blooming is not possible. The concept of the maximal diffusivity is also consistent with field experiments, see e.g. =-=[106, 18, 76]-=-. 4.2. Light and Nutrient limitation In the last section we will discuss models which take into account both light and nutrient limitation of phytoplankton growth. These models are more difficult to a... |

3 |
Population dynamics in spatially heterogeneous systems with drift: The generalized contact process
- Joo, Lebowitz
(Show Context)
Citation Context ...be easily expressed from equation (3.34). Examining bacterial growth, Lin et al. [55] confirmed the results of Dahmen et al. [14] experimentally and by means of numerical simulation. Joo and Lebowitz =-=[43]-=- carried out computer simulations in a stochastic spatially discrete population model and obtained similar results, confirming the robustness of the model. Locally elevated diffusivity Consider a popu... |

3 |
A note on population growth under random dispersal
- Landahl
- 1959
(Show Context)
Citation Context ...ic growth function ( ∂P(x, t) = µ0P 1 − ∂t P ) + D K ∂2P , for 0 ≤ x ≤ L (3.14) ∂x2 and the same (hostile) boundary conditions (3.8). For this model only approximate solutions are available (see e.g. =-=[52]-=-, [2], [61], [72]). The stationary solution can be expressed in terms of elliptic functions and was investigated by Skellam [96], Levandowsky and White [54], and Ludwig et al. [56]. However, applying ... |

3 |
time scales, and the evolution of biological communities
- Randomness
- 1977
(Show Context)
Citation Context ...proximate solutions are available (see e.g. [52], [2], [61], [72]). The stationary solution can be expressed in terms of elliptic functions and was investigated by Skellam [96], Levandowsky and White =-=[54]-=-, and Ludwig et al. [56]. However, applying the invasibility criteria, we can conclude that this system possesses the same critical values as the KiSS model. Indeed, as the population density approach... |

3 |
Review of three-dimensional ecol. model. related to the north sea shelf system part 1: models and their results. Prog
- Moll, Radach
(Show Context)
Citation Context ...latory behaviour may lead to spatio-temporal chaotic waves [79, 80]. Finally, we did not even touch such important problems as model validation and the simulation of multicomponent natural ecosystems =-=[60]-=-. The review is structured as follows. In chapter two, we shortly discuss some non-spatial models, which will be used later to build-up spatial explicit models. In chapter three we consider critical p... |

3 |
Lectures on nonlinear rate equations, especially those with quadratic nonlinearities
- Montroll
- 1968
(Show Context)
Citation Context ...unction ( ∂P(x, t) = µ0P 1 − ∂t P ) + D K ∂2P , for 0 ≤ x ≤ L (3.14) ∂x2 and the same (hostile) boundary conditions (3.8). For this model only approximate solutions are available (see e.g. [52], [2], =-=[61]-=-, [72]). The stationary solution can be expressed in terms of elliptic functions and was investigated by Skellam [96], Levandowsky and White [54], and Ludwig et al. [56]. However, applying the invasib... |

3 |
A note on small organism diffusion around an attractive center: A mathematical model
- Okubo
- 1972
(Show Context)
Citation Context ...ritical patch size L0 of the KiSS model. Many other examples of critical patch models can be found in the books by Okubo and Levin [72] and Murray [62]. We just briefly mention some extensions. Okubo =-=[68, 70]-=- considered a model for growth and diffusion under an attractive force toward the centre of a patch. He found that the modified critical size equals Lc = L0f (v 2 /4αD), where f(0) = 1 and the functio... |

3 |
Phytoplankton patches in the ocean under various modes of ocean turbulence
- Ozmidov
- 1998
(Show Context)
Citation Context ... Below we will provide another heuristic derivation confirming the validity of this expression. Note that in large aquatic basins the horizontal turbulent mixing increases with the scale of phenomena =-=[73, 67, 66, 74]-=-. Petrovskii [77, 78] showed that this should result in the increase of the front propagation velocity. Moreover, this velocity should grow with the size of the area occupied by a population. Pseudo w... |

3 |
On the diffusion of a plankton patch in a turbulent ocean
- Petrovskii
- 1999
(Show Context)
Citation Context ...r heuristic derivation confirming the validity of this expression. Note that in large aquatic basins the horizontal turbulent mixing increases with the scale of phenomena [73, 67, 66, 74]. Petrovskii =-=[77, 78]-=- showed that this should result in the increase of the front propagation velocity. Moreover, this velocity should grow with the size of the area occupied by a population. Pseudo waves We should stress... |

3 |
sinking velocity, and apparent diffusivity within marine snow and zooplankton fecal pellets: Implications for substrate turnover by attached bacteria
- Ballast
(Show Context)
Citation Context ...(4.12) ∂N(z, t) = −µ(N, I)P + rT +D ∂t ∂2N , ∂z2 where vP and vT are the sinking velocities of phytoplankton and detritus respectively. Note that usually detritus sinks much faster than phytoplankton =-=[94, 84]-=-. In this model the cycle of chemicals includes three stages: the transfer of biomass to detritus with mortality m, the remineralisation of detritus back into nutrients with remineralisation rate r, a... |

3 |
The influence of an upper mixed layer on the vertical distribution and composition of phytoplankton biomass
- Ryabov, Rudolf, et al.
(Show Context)
Citation Context ...arrier. Thus, in an incompletely mixed water column, a more light limited or buoyant species obtains a competitive advantage to another species, whose favourable patch is located in subsurface layers =-=[34, 89]-=-. Arbitrary spatial dependence of the growth rate The behaviour of model (3.6) in the case where the growth rate µ(x) is an arbitrary function of the coordinates was investigated by Cantrell and Cosne... |

1 |
What minimal models can tell: A reply to van Nes and Scheffer
- Huppert, Blasius, et al.
(Show Context)
Citation Context ...tains a stable feasible equilibrium and the system spirals in phase space to an isolated 5 (2.7) (2.8)A. B. Ryabov and B. Blasius The role of diffusion and advection fixed point (Fig. 1C), see e.g., =-=[38, 10, 39]-=-. Furthermore, for perfect nutrient recycling (ε = 1) one can easily show that S(t) follows the simple dynamics dS dt = δ(Ni − S) . (2.9) Therefore, after some transient time, determined by the exchan... |