The Shigesada–Kawasaki–Teramoto cross-diffusion system beyond detailed balance
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Abstract:
The existence of global weak solutions to the cross-diffusion model of Shigesada, Kawasaki, and Teramoto for an arbitrary number of species is proved. The model consists of strongly coupled parabolic equations for the population densities in a bounded domain with no-flux boundary conditions, and it describes the dynamics of the segregation of the population species. The diffusion matrix is neither symmetric nor positive semidefinite. A new logarithmic entropy allows for an improved condition on the coefficients of heavily nonsymmetric diffusion matrices, without imposing the detailed-balance condition that is often assumed in the literature. Furthermore, the large-time convergence of the solutions to the constant steady state is proved by using the relative entropy associated to the logarithmic entropy.Keywords:
Population model
Constant (computer programming)
Detailed balance
In this note, we prove that the displacement of a linearly damped string under bounded distributed forces is bounded. The proof is achieved by showing that an energy-like (Lyapunov) function of the string is bounded.
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Abstract In this paper, we prove that the displacement of a damped nonlinear string under bounded distributed forces is bounded. The proof is achieved by showing that an energy-like (Lyapunov) function corresponding to the string is bounded.
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By a bounded backward sequence of the operator $T$ we mean a bounded sequence $\{x_n\}$ satisfying $Tx_{n+1}=x_n$. In \cite{Pa} we have characterized contractions with strongly stable nonunitary part in terms of bounded backward sequences. The main purpose of this work is to extend that result to power-bounded operators. Aditionally, we show that a power-bounded operator is strongly stable ($C_{0 \cdot} $) if and only if its adjoint does not have any nonzero bounded backward sequence. Similarly, a power-bounded operator is non-vanishing ($C_{1 \cdot} $) if and only if its adjoint has a lot of bounded backward sequences.
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The central theme of this article can be given a very simple operational description, namely, that both from the point of view of developing deterministic models or simply from the point of view of looking at data, there is a considerable advantage in thinking in terms of the logarithm of population size rather than population size itself. This advantage has been frequently observed in the case of single-population models. The advantage is at least as great for more than one population. To demonstrate this, a logarithmic model is compared with classical multipopulation linear and totally linear models from two points of view. From the point of view of mathematical tractability, that is, the ease with which a model can answer questions asked of it, the logarithmic and totally linear models are equivalent and considerably more tractable than the popular linear model. The tractability of the logarithmic model is demonstrated by looking in detail at its solution for a two-population predator-prey system and by presenting a general solution for the case of any number of populations. From the point of view of ecological relevance, that is, how faithfully a model represents natural populations, the logarithmic and linear models are generally superior to the totally linear model. The logarithmic and linear models are nearly equivalent over limited ranges of population size. The model which is most relevant over less restricted ranges depends, of course, on the populations they are to represent. For a two-population predator-prey system, a comparison of the two models suggests that the logarithmic model is probably as good as, if not better than, the linear model. Some natural population data suggest the same conclusion in the case of single-population systems. In addition to comparison with other models, a number of qualitative implications of the logarithmic model are summarized. The model predicts a scalloped shape, with rounded minima and sharp peaks, for population oscillations. Under the model a single-population system cannot oscillate. Two-population systems can oscillate at one frequency, four-population systems at two frequencies, six-population systems at three frequencies, etc. The model indicates, however, that self-maintained oscillations are unlikely. It suggests that some driving force, such as added periodic variation in the relative growth rate or stochastic variation in the model parameters, is necessary for sustained oscillation.
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A Logarithm as generally understood and used by the ordinary student is the expression of a number as a power of 10. In order to multiply two numbers together he adds the logarithms of these numbers and takes as the result the anti-logarithm of this sum. Similarly for division he takes the anti-logarithm of the difference of the logarithms of the two numbers.
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