Investigation of the possibility of exceeding the Greenwald density limit during ECRH in T-10

In T-10 experiments, attempts were made to significantly exceed the Greenwald limit $\bar n_{Gr} $ during high-power (P _ab=750 kW) electron-cyclotron resonance heating (ECRH) and gas puffing. Formally, the density limit $(\bar n_e )_{\lim } $ exceeding the Greenwald limit $({{(\bar n_e )_{\lim } } \mathord{\left/ {\vphantom {{(\bar n_e )_{\lim } } {\bar n_{Gr} }}} \right. \kern-0em} {\bar n_{Gr} }} = 1.8)$ was achieved for q _L=8.2. However, as q _L decreased, the ratio ${{(\bar n_e )_{\lim } } \mathord{\left/ {\vphantom {{(\bar n_e )_{\lim } } {\bar n_{Gr} }}} \right. \kern-0em} {\bar n_{Gr} }}$ also decreased, approaching unity at q _L≈3. It was suggested that the “current radius” (i.e., the radius of the magnetic surface enclosing the bulk of the plasma current I _p), rather than the limiter radius, was the parameter governing the value of $(\bar n_e )_{\lim } $ . In the ECRH experiments, no substantial degradation of plasma confinement was observed up to $\bar n_e \sim 0.9(\bar n_e )_{\lim } $ regardless of the ratio ${{(\bar n_e )_{\lim } } \mathord{\left/ {\vphantom {{(\bar n_e )_{\lim } } {\bar n_{Gr} }}} \right. \kern-0em} {\bar n_{Gr} }}$ . In different scenarios of the density growth up to $(\bar n_e )_{\lim } $ , two types of disruptions related to the density limit were observed.     

Kinetic models of coupling between H+ and Na+-translocation and ATP synthesis/hydrolysis by F0F1-ATPases: Can a cell utilize both\Delta \bar \mu _{H^ + } and\Delta \bar \mu _{Na^ + } for ATP synthesis underin vivo conditions using the same enzyme?

Kinetic models of the F₀F₁-ATPase able to transport H⁺ or/and Na⁺ ions are proposed. It is assumed that (i) H⁺ and Na⁺ compete for the same binding sites, (ii) ion translocation through F₀ is coupled to the rate-limiting step of the F₁-catalyzed reaction. The main characteristics of the dependences of ATP synthesis and hydrolysis rates on Δφ, ΔpH, and ΔpNa are predicted for various versions of the coupling model. The mechanism of the switchover from \(\Delta \bar \mu _{H^ + } \) -dependent synthesis to the \(\Delta \bar \mu _{Na^ + } \) -dependent one is demonstrated. It is shown that even with a drastic drop in \(\Delta \bar \mu _{H^ + } \) , ATP hydrolysis by the proton mode of catalysis can be effectively inhibited by Δφ and ΔpNa. The results obtained strongly support the possibility that the same F₀F₁-ATPase in bacterial cells can utilize both \(\Delta \bar \mu _{H^ + } \) and \(\Delta \bar \mu _{Na^ + } \) for ATP synthesis underin vivo conditions.     

Effect of the magnetic field structure on the intensity of electron cyclotron emission from the plasma of the L-2M stellarator

Results are presented from experimental studies of the influence of the stellarator magnetic field structure on the plasma behavior in electron-cyclotron resonance regimes with a high heating power per electron. The magnetic field structure was changed by varying the induction current I _p from ?14 to +14 kA. The plasma electrons were heated at the second harmonic of the electron gyrofrequency by an X-mode microwave beam with a power of P ～ 200 kW, the average plasma density being in the range n _e = (0.5–2) × 10¹³ cm^?3. At I _p = 0, the rotational transform varies from $\rlap{--} \iota $ (0) = 0.2 on the magnetic axis to 0.8 at the plasma boundary. At a positive current of I _p = 13.5 kA, the rotational transform was $\rlap{--} \iota $ (0) = 0.8 on the axis and $\rlap{--} \iota $ (a _p) = 0.9 at the plasma boundary. Experiments with a positive current have shown that the radiative temperature first increases with current. When the current increases to I _p = 11–14 kA, strong modulation appears in the electron cyclotron emission signals received from all the plasma radii, the emission spectrum changes, and the emission intensity decreases. At a negative current of I _p = ?(6.5–13.5) kA, the rotational transform vanishes at r/a _p = 0.4–0.6. In this regime, the number of suprathermal electrons is reduced substantially and the emission intensity decreases at both low and high plasma densities.     

Water relations of the epidermal bladder cells ofOxalis carnosa Molina

All of the cells of the upper (adaxial) epidermis of the leaves ofOxalis carnosa are transformed into large bladders, while in the lower epidermis the bladder cells are interrupted by “normal” cells with stomata. The epidermal bladders contain a high concentration of free oxalic acid (pH approx. 1). Water-relations parameters of these epidermal bladder cells have been determined using the pressure probe. Original cell turgor (P₀) of the closely packed bladders of theupper epidermis was P₀=0.7 to 2.9 bar ( \(\overline {P_0 } = 1.7 \pm 0.5 bar\) ; mean±SD;N=25 cells) and lower than that in the club-shaped bladders of thelower epidermis (P₀=1.3 to 3.7 bar; \(\overline {P_0 } = 2.5 \pm 0.7 bar\) ;N=25 cells). Large differences in the elastic modulus (ε) and the hydraulic conductivity (Lp) of the two different types of cells were observed. For the lower epidermal bladders, ε=18 to 166 bar and was similar to that of other higher plant cells. Also, for these cells it was found that ε was increasing with both, cell turgor and cell volume. By contrast, ε of the cells of the upper epidermis was by one order of magnitude smaller (ε=1.9 to 17.0 bar) and no dependence of ε on cell volume could be detected. The Lp values of the cell membranes were also different (lower epidermis: \(\overline {Lp} = (2.3 \pm 1.6) \cdot 10^{ - 5} cm s^{ - 1} bar^{ - 1}\) ; upper epidermis: \(\overline {Lp} = (3.8 \pm 2.4) \cdot 10^{ - 6} cm s^{ - 1} bar^{ - 1}\) ). These differences seem to be too large to be caused by errors in determining the exchange area for water (A) between cells and adjacent tissue. The half-times of water exchange between bladders and leaf (T_1/2) were, on average, somewhat longer for the upper than for the lower epidermis (lower epidermis: T_1/2=7 to 38 s; upper epidermis: T_1/2=22 to 213 s), but the differences in the T_1/2 values were not as distinct as for ε and Lp. This is because of the compensatory effects of ε, Lp and the different ratios of volume to exchange area. Since the bladders make up about 75% of the entire volume of the leaf, it is assumed that the rate of response of the leaf to changes in the water potential should be similar to that of the bladder cells. The results are discussed in terms of a possible function of the bladders in the leaf.     

Sur l'espace topologique lié à une nouvelle théorie de l'apprentissage

There have been two contrasting doctrines concerning learning, more generally about acquisition of knowledge: empiricism and rationalism. The theory of learning in such a field as artificial intelligence seems to fall within the empiricist framework. On the hand, N. Chomsky and his followers have discussed, during the last decade, concerning learning, especially about language learning, from the rationalist point of view (Chomsky, 1965). The main feature in the rationalist approach toward a theory of learning lies in the speculation that in order to acquire knowledge it is indispensable for a learner to be endowed with “innate ideas”, and that “experience” in the external world are merely subsidiary types of information for the learner. If this is acceptable, we can inquire: Under what kind of innate ideas can the learner understand the structure of the external world? In our previous paper (Uesaka, Aizawa, Ebara, and Ozeki, 1973), we formalized this by introducing the mathematical notion of “learnability”, and gave a partial answer to the above inquiry. In this formalization we assumed that the set F of objects to be learned consists of mappings of N to itself, where N is the set of positive integers. Then, constructing a topological space (F, \(\mathcal{O}\) ) by an appropriate family \(\mathcal{O}\) of open sets, we observed that the notion of learnability can be well described in terms of topological properties of the learning space (F, \(\mathcal{O}\) ). Many problems must be solved, however, before we raise the theory to a complete model of the rationalist theory of learning. The topological study of the space (F, \(\mathcal{O}\) ) is, we believe, the first step toward this approach. In this context, we discuss the topological aspects of this space. Now we define \(\mathcal{O}\) as follows: By N ² we mean the direct product of two N's. Let s be a subset of N ². If, for any (x, y), (x′, y′) in s, x=x′ implies y=y′, then we say that s is single-valued. Let f ∈ F, If, for any (x, y) in s, y=f(x), then f is said to be on s, denoted as \(f\underline \supseteq s\) . Let \(\pi \left( s \right) = \left\{ {g;g \in F,g\underline \supseteq s} \right\}\) . A single-valued finite subset of N ² is called datum. Let D denote the family of all data. Let \(\mathcal{O}* = \left\{ \phi \right\} \cup \left\{ {\pi \left( d \right);d \in D} \right\}\) , and \(\mathcal{O}\) denote the family of all subsets of F, each of which is written as \(\mathop \cup \limits_\alpha W_{\alpha }\) , where W _α is in \(\mathcal{O}*\) . Then, it is easily seen that \(\mathcal{O}\) satisfies the axiom of the open system of a topological space. It is shown that the learning space (F, \(\mathcal{O}\) ) has the following properties:

1. It satisfies the first and the second countability axioms.
2. It is separable and is totally disconnected.
3. It is a Hausdorff space and, further, is regular and normal.
4. It is neither compact nor locally compact.
5. It is metrizable, or more precisely there exists a complete but not totally bounded metric space which is homeomorphic to learning space.
6. Any of its subspace can be embedded into its special subspace.

     

\sqrt 2 Rule for Controlling the Tree Pattern in Forest Cut

A forest’s productivity can be optimized by the application of rules derived from monopolized circles. A monopolized circle is defined as a circle whose center is a tree and whose radius is half of the distance between the tree itself and its nearest neighbor. Three characteristics of monopolized circle are proved. (1) Monopolized circles do not overlay each other, the nearest relationship being tangent. (2) “Full uniform pattern” means that the grid of trees (a×b=N) covers the whole plot, so that the distance between each tree in a row is the same as the row spacing. The total monopolized circle area with a full uniform pattern is independent on the number of trees and $\frac{\pi }{4}$ times the plot area. (3) If a tree is removed, the area of some trees’ monopolized circle will increase without decreasing the monopolized circles of the other trees. According to the above three characteristics, “uniform index” is defined as the total area of monopolized circles in a plot divided by the total area of the monopolized circles, arranged in a uniform pattern in the same shaped plot. According to the definition of monopolized circle, the distribution of uniform index $(L) = \frac{{\chi ^2 (2n)}}{{2\pi n}}$ for a random pattern and $E(L) = \frac{1}{\pi }$ the variance of L is $D(L) = \frac{1}{{n\pi ^2 }}$ . It is evident that E(L) is independent on N and the plot area; hence, L is a relative index. L can be used to compare the uniformity among plots with different areas and the numbers of trees. In a random pattern, where L is equivalent to the tree density of the plot in which the number of trees is 1 and the area is π, the influence of tree number and plot area to L is eliminated. When n→∞, D(L)→0 and $L \to \frac{1}{\pi } = 0.318$ it indicates that the greater the number of tree is in the plots, the smaller the difference between the uniform indices will be. There are three types of patterns for describing tree distribution (aggregated, random, and uniform patterns). Since the distribution of L in the random pattern is accurately derived, L can be used to test the pattern types. The research on Moarshan showed that the whole plot has an aggregated pattern; the first, third, and sixth parts have an aggregated pattern; and the second, fourth, and fifth parts have a random pattern. None of the uniform indices is more than 0.318 (1/∏), which indicates that uniform patterns are rare in natural forests. The rules of uniform index can be applied to forest thinning. If you want to increase the value of uniform index, you must increase the total area of monopolized circles, which can be done by removing select trees. “Increasing area trees” are the removed trees and can increase the value of the uniform index. A tree is an increasing area tree if the distance between the tree and its second nearest neighbor is $\sqrt 2 $ times longer than that between the tree itself and its first nearest neighbor, which is called the $\sqrt 2 $ rule. It was very interesting to find that when six plots were randomly separated from the original plot, the proportion of increasing area trees in each plot was always about 0.5 without exception. In random pattern, the expected proportion of increasing area trees is about 0.35–0.44, which is different from the sampling value of 0.5. The reason is very difficult to explain, and further study is needed. Two criteria can be used to identify which trees should be removed to increase the uniform index during forest thinning. Those trees should be (1) trees whose monopolized circle areas are on the small side and (2) increasing area trees, which are found via the $\sqrt 2 $ rule.     

Endor characterization and D2O exchange in the \begin{array}{*{20}c} {\mathop Z\nolimits^{\begin{array}{*{20}c} + \\ . \\ \end{array} } } \\ \end{array} /{\kern 1pt} \begin{array}{*{20}c} {\mathop D\nolimits^{\begin{array}{*{20}c} + \\ . \\ \end{array} } } \\ \end{array} radical in photosystem II

The early suggestion by Lozier and Butler (Photochem. Photobiol. 17, 133–137 (1973)) that EPR Signal II arises from radicals associated with the water-splitting process in PSII has been confirmed and extended over the intervening years. Recent work has identified the Signal II radicals, \(\begin{array}{*{20}c} {\mathop D\nolimits^{\begin{array}{*{20}c} + \\ . \\ \end{array} } } \\ \end{array}\) and \(\begin{array}{*{20}c} {\mathop Z\nolimits^{\begin{array}{*{20}c} + \\ . \\ \end{array} } } \\ \end{array}\) , with plastosemiquinone cation species. In the experiments presented here we have used ENDOR spectroscopy and D₂O/H₂O exchange to characterize these paramagnets in more detail. The ENDOR matrix region, which arises from protons which interact weakly with the unpaired electron spin, is well-resolved at 4 K and at least seven resonances are apparent. A number of hyperfine couplings in the 3–8 MHz range are observed and are suggested to arise from methyl or hydroxyl protons which occur as substituents on the plastosemiquinone cation ring or from amino acid protons hydrogen-bonded to the 1,4-hydroxyl groups. Orientation selection experiments are consistent with these possibilities. D₂O/H₂O exchange shows that the D⁺/Z⁺ site is accessible to solvent. However, the exchange occurs slowly and is not complete even after 72 hours which suggests that the free radicals are functionally isolated from solvent water.     

Control of mitochondrial and cellular respiration by oxygen

Control and regulation of mitochondrial and cellular respiration by oxygen is discussed with three aims: (1) A review of intracellular oxygen levels and gradients, particularly in heart, emphasizes the dominance of extracellular oxygen gradients. Intracellular oxygen pressure, $p_{O_2 } $ , is low, typically one to two orders of magnitude below incubation conditions used routinely for the study of respiratory control in isolated mitochondria. The $p_{O_2 } $ range of respiratory control by oxygen overlaps with cellular oxygen profiles, indicating the significance of $p_{O_2 } $ in actual metabolic regulation. (2) A methodologically detailed discussion of high-resolution respirometry is necessary for the controversial topic of respiratory control by oxygen, since the risk of methodological artefact is closely connected with far-reaching theoretical implications. Instrumental and analytical errors may mask effects of energetic state and partially explain the divergent views on the regulatory role of intracellular $p_{O_2 } $ . Oxygen pressure for half-maximum respiration,p ₅₀, in isolated mitochondria at state 4 was 0.025 kPa (0.2 Torr; 0.3 ΜM O₂), whereasp ₅₀ in endothelial cells was 0.06–0.08 kPa (0.5 Torr). (3) A model derived from the thermodynamics of irreversible processes was developed which quantitatively accounts for near-hyperbolic flux/ $p_{O_2 } $ relations in isolated mitochondria. The apparentp ₅₀ is a function of redox potential and protonmotive force. The protonmotive force collapses after uncoupling and consequently causes a decrease inp ₅₀. Whereas it is becoming accepted that flux control is shared by several enzymes, insufficient attention is paid to the notion of complementary kinetic and thermodynamic flux control mechanisms.     

Dynamics of a nonquasineutral plasma in a strong magnetic field

The motion of a nonquasineutral plasma in a strong magnetic field such that is analyzed. It is shown in simple examples that, when the plasma pressure and dissipation are neglected, the only dynamic process in a magnetized plasma is the evolution of the charge-separation electric field and the related magnetic field flux. The equations derived to describe this evolution are essentially the wave Grad-Shafranov equations. The solution to these equations implies that, in a turbulent Z-pinch, a steady state can exist in which the current at a supercritical level is concentrated near the pinch axis.     

Dynamical systems under constant organization I. Topological analysis of a family of non-linear differential equations —A model for catalytic hypercycles

The paper presents a qualitative analysis of the following systems ofn differential equations: \(\dot x_i = x_i x_j - x_i \sum\nolimits_r^n { = 1} x_r x_s {\mathbf{ }}(j = i - 1 + n\delta _{i1} {\mathbf{ }}and{\mathbf{ }}s = r - 1 + n\delta _{r1} )\) , which show cyclic symmetry. These dynamical systems are of particular interest in the theory of selforganization and biological evolution as well as for application to other fields.     

Longitudinal distribution of nitrate δ15N and δ18O in two contrasting tropical rivers: implications for instream nitrogen cycling

The longitudinal variations in the nitrogen (δ¹⁵N) and oxygen (δ¹⁸O) isotopic compositions of nitrate (NO₃ ^?), the carbon isotopic composition (δ¹³C) of dissolved inorganic carbon (DIC) and the δ¹³C and δ¹⁵N of particulate organic matter were determined in two Southeast Asian rivers contrasting in the watershed geology and land use to understand internal nitrogen cycling processes. The $ \delta^{15} {\text{N}}_{{{\text{NO}}_{3} }} $ became higher longitudinally in the freshwater reach of both rivers. The $ \delta^{18} {\text{O}}_{{{\text{NO}}_{3} }} $ also increased longitudinally in the river with a relatively steeper longitudinal gradient and a less cultivated watershed, while the $ \delta^{18} {\text{O}}_{{{\text{NO}}_{3} }} $ gradually decreased in the other river. A simple model for the $ \delta^{15} {\text{N}}_{{{\text{NO}}_{3} }} $ and the $ \delta^{18} {\text{O}}_{{{\text{NO}}_{3} }} $ that accounts for simultaneous input and removal of NO₃ ^? suggested that the dynamics of NO₃ ^? in the former river were controlled by the internal production by nitrification and the removal by denitrification, whereas that in the latter river was significantly affected by the anthropogenic NO₃ ^? loading in addition to the denitrification and/or assimilation. In the freshwater-brackish transition zone, heterotrophic activities in the river water were apparently elevated as indicated by minimal dissolved oxygen, minimal δ¹³C_DIC and maximal pCO₂. The δ¹⁵N of suspended particulate nitrogen (PN) varied in parallel to the $ \delta^{15} {\text{N}}_{{{\text{NO}}_{3} }} $ there, suggesting that the biochemical recycling processes (remineralization of PN coupled to nitrification, and assimilation of NO₃ ^?-N back to PN) played dominant roles in the instream nitrogen transformation. In the brackish zone of both rivers, the $ \delta^{15} {\text{N}}_{{{\text{NO}}_{3} }} $ displayed a declining trend while the $ \delta^{18} {\text{O}}_{{{\text{NO}}_{3} }} $ increased sharply. The redox cycling of NO₃ ^?/NO₂ ^? and/or deposition of atmospheric nitrogen oxides may have been the major controlling factor for the estuarine $ \delta^{15} {\text{N}}_{{{\text{NO}}_{3} }} $ and $ \delta^{18} {\text{O}}_{{{\text{NO}}_{3} }} $ , however, the exact mechanism behind the observed trends is currently unresolved.     

The secant condition for instability in biochemical feedback control—II. Models with upper hessenberg jacobian matrices

We consider ann-component biochemical system whose Jacobian matrixJ is of upper Hessenberg form, with principal subdiagonal elementsb ₁,b ₂, ...,b _n?1 and upper right-hand corner element ?f. The open-loop Jacobian matrixJ ₀ is formed fromJ by settingf=0. It is shown that if the characteristic roots of ?J ₀ are real and non-negative then a necessary condition for instability at a critical point (steady state) is $$\frac{{b_1 b_2 ...b_{n - 1} f}}{{\left| { - J_0 } \right|}} \geqslant (\sec \pi /n)^n $$ This condition is analyzed in terms of reaction orders. For a metabolic sequence with some reversible steps, no loss of intermediate metabolites, and competitive inhibition of the first enzyme by the last metabolite, the above necessary condition becomes $$\frac{{\beta _{N - 1} X_{n + 1} }}{{\xi _{N - 1} E_{0T} }} \geqslant (\sec \pi /N)^N $$ whereN is the number of components (metabolites, enzyme-substrate complexes, and enzyme-inhibitor complex),β _N-1 the order of the enzyme-inhibitor reaction (with respect to the inhibitor),ξ _N-1 the order of reaction for the removal of the last metabolite, andX _n+1 /E _0T the fraction of first enzyme blocked by inhibitor. It is shown that, under certain assumptions, a critical point is always stable in a single two-step enzymatic process (formation of enzyme-substrate complex, followed by conversion to product, then loss of product) with slow negative feedback by competitive product inhibition. A model is constructed showing that stable oscillations can occur in a feedback system with only two metabolic steps and negative feedback by competitive inhibition with no cooperativity. The instability is due to a slow feedback reaction and saturable removal of the second metabolite.     

Proton cycles through membranes in bacteria: Relationship between proton passive and active fluxes and their dependence on some external physico-chemical factors under fermentation

This paper represents H⁺ circles through the bacterial membranes, their peculiarities and relationship with ATP synthesis or hydrolysis, utilization or accumulation of energy are considered. Data on passive and active proton (H⁺) fluxes through the bacterial membranes are analyzed and their relationship with membrane H⁺ conductance $\left( {G_m^{H^ + } } \right)$ and permeability for H⁺ $\left( {P_{H^ + } } \right)$ is discussed. Methods for determination of bacterial membrane $G_m^{H^ + }$ are presented and some difficulties in obtaining and interpreting data are pointed out. Different ways and mechanisms of passive and active H⁺ fluxes, including a role of membrane lipids in H⁺ transfer, importance of phase transitions in lipid bilayers, operation of protonophores as well as H⁺ translocation via the F₀ factor of the F₀F₁-ATPase, are discussed. Dependence of $G_m^{H^ + }$ for Escherichia coli, Enterococcus hirae, Streptococcus lactis and other bacteria on some external physico-chemical growth factors, particularly, on pH and oxidation reduction potential as well as influence of osmotic stress on $G_m^{H^ + }$ and H⁺ active fluxes through the bacterial membrane under fermentation have been shown. The relationship between $G_m^{H^ + }$ , $P_{H^ + }$ and active H⁺ fluxes through a membrane is proposed, possible mechanisms of relationship between their alterations depending on pH and oxidation reduction potential are discussed. The results are important for understanding the structural and functional properties of bacterial membranes determining H⁺ cycles operation and mechanisms of H⁺ fluxes essential in adaptation of bacteria to altered environment conditions.     

Stochastic population growth in spatially heterogeneous environments

Classical ecological theory predicts that environmental stochasticity increases extinction risk by reducing the average per-capita growth rate of populations. For sedentary populations in a spatially homogeneous yet temporally variable environment, a simple model of population growth is a stochastic differential equation dZ _t = μ Z _t dt + σ Z _t dW _t , t ≥ 0, where the conditional law of Z _t+Δt ? Z _t given Z _t = z has mean and variance approximately z μΔt and z ² σ ²Δt when the time increment Δt is small. The long-term stochastic growth rate ${\lim_{t \to \infty} t^{-1}\log Z_t}$ for such a population equals ${\mu -\frac{\sigma^2}{2}}$ . Most populations, however, experience spatial as well as temporal variability. To understand the interactive effects of environmental stochasticity, spatial heterogeneity, and dispersal on population growth, we study an analogous model ${{\bf X}_t = (X_t^1, \ldots, X_t^n)}$ , t ≥ 0, for the population abundances in n patches: the conditional law of X _t+Δt given X _t = x is such that the conditional mean of ${X_{t+\Delta t}^i - X_t^i}$ is approximately ${[x^i \mu_i + \sum_j (x^j D_{ji} - x^i D_{ij})] \Delta t}$ where μ _i is the per capita growth rate in the ith patch and D _ij is the dispersal rate from the ith patch to the jth patch, and the conditional covariance of ${X_{t+\Delta t}^i - X_t^i}$ and ${X_{t + \Delta t}^j - X_t^j}$ is approximately x ⁱ x ^j σ _ij Δt for some covariance matrix Σ = (σ _ij ). We show for such a spatially extended population that if ${S_t = X_t^1 + \cdots + X_t^n}$ denotes the total population abundance, then Y _t = X _t /S _t , the vector of patch proportions, converges in law to a random vector Y _∞ as ${t \to \infty}$ , and the stochastic growth rate ${\lim_{t \to \infty} t^{-1}\log S_t}$ equals the space-time average per-capita growth rate ${\sum_i \mu_i \mathbb{E}[Y_\infty^i]}$ experienced by the population minus half of the space-time average temporal variation ${\mathbb{E}[\sum_{i,j}\sigma_{ij}Y_\infty^i Y_\infty^j]}$ experienced by the population. Using this characterization of the stochastic growth rate, we derive an explicit expression for the stochastic growth rate for populations living in two patches, determine which choices of the dispersal matrix D produce the maximal stochastic growth rate for a freely dispersing population, derive an analytic approximation of the stochastic growth rate for dispersal limited populations, and use group theoretic techniques to approximate the stochastic growth rate for populations living in multi-scale landscapes (e.g. insects on plants in meadows on islands). Our results provide fundamental insights into “ideal free” movement in the face of uncertainty, the persistence of coupled sink populations, the evolution of dispersal rates, and the single large or several small (SLOSS) debate in conservation biology. For example, our analysis implies that even in the absence of density-dependent feedbacks, ideal-free dispersers occupy multiple patches in spatially heterogeneous environments provided environmental fluctuations are sufficiently strong and sufficiently weakly correlated across space. In contrast, for diffusively dispersing populations living in similar environments, intermediate dispersal rates maximize their stochastic growth rate.     

Mathematische Auswertung von Daten für das Wachstum von Heringen der Nordsee

The parameters of the reciprocal function (“Reziprokfunktion”) for the growth of herrings(Clupea harengus) are calculated from older and more recent measurements. The logarithmic expression of the proposed reciprocal function is as follows: \(\log y_x = \log y_{max} - \frac{1}{{\chi + \xi }}\log N\) . Values less than 1 are found for the additive age (ξ). In further calculations 0.4 is used as the estimated mean value. Measurements made before the second world war yield ca. 30 cm for the maximum value (L_max). After this period the maximum values increase to ca. 34 cm. The Scandinavian and Atlantic herrings differ from North Sea herrings by higher maximum values. The values for the constant of velocity (log N) may be different for identical ξ and Y_max values. The velocity constant determines the position of the inflection point of the growth curve. The dimension, which is only dependent on the maximum value, is at the inflection point: \(\frac{{Y_{max} }}{{7,389}}\) . From the results ofSchumacher (1967) on the growth of 3 herring populations from the North Sea it was calculated that the values for the constant of velocity rise from northern to southern areas. A low value for the constant of velocity marks an early inflection point and a high velocity of growth before this point and vice versa. The growth of the 3 populations tends to almost the same maximum value; consequently, a high velocity before the inflection point is compensated by a lower velocity after this point and vice versa. The maximum velocity of linear growth at the point of inflection is given by the expression \(\frac{{Y_{max} }}{{4,25 \cdot \log N}}\) . This expression may possibly be a useful device for quantitative comparisons of growth processes.