Masataka Kuwamura

A minimum model of prey-predator system with dormancy of predators and the paradox of enrichment

In this paper, a mathematical model of a prey-predator system is proposed to resolve the paradox of enrichment in ecosystems. The model is based on the natural strategy that a predator takes, i.e, it produces resting eggs in harsh environment. Our result gives a criterion for a functional response, which ensures that entering dormancy stabilizes the population dynamics. It is also shown that the hatching of resting eggs can stabilize the population dynamics when the switching between non-resting and resting eggs is sharp. Furthermore, the bifurcation structure of our model suggests the simultaneous existence of a stable equilibrium and a large amplitude cycle in natural enriched environments.

On the Turing patterns in one-dimensional gradient/skew-gradient dissipative systems

In this article, fundamental properties concerning the Turing patterns are considered in one-dimensional dissipative systems with gradient/skew-gradient structure introduced in [M. Kuwamura and E. Yanagida, Phys. D, 175 (2003), pp. 185--195]. It is a natural extension of free energy, which covers reaction-diffusion systems of activator-inhibitor type. The theory based on this concept provides a new perspective on a fundamental problem of what unique Turing pattern is to be selected among many.

The Eckhaus and zigzag instability criteria in gradient/skew-gradient dissipative systems

Abstract The concept of gradient/skew-gradient structure—an extension of free energy—is introduced. In dissipative systems with this structure, the Eckhaus and zigzag instability criteria can be represented in terms of first integral and free energy per unit length for steady states. Moreover, an exact relation between these criteria is presented. These results are valid for steady states of arbitrary profile and amplitude, which implies that the instability criteria are valid for steady states far from threshold. Furthermore, our results support the marginal stability hypothesis for roll solutions in the wavenumber selection problem in gradient systems. The derivation for the results is simple and rigorous.

Turing instabilities in prey–predator systems with dormancy of predators

In this paper, we study the stationary and oscillatory Turing instabilities of a homogeneous equilibrium in prey–predator reaction–diffusion systems with dormant phase of predators. We propose a simple criterion which is useful in classifying these Turing instabilities. Moreover, numerical simulations reveal transient spatio-temporal complex patterns which are a mixture of spatially periodic steady states and traveling/standing waves. In this mixture, the steady part is the stable Turing pattern bifurcated primarily from the homogeneous equilibrium, while wave parts are unstable oscillatory solutions bifurcated secondarily from the same homogeneous equilibrium. Although our criterion does not exclude the occurrence of oscillatory Turing instability, we have not yet found stable traveling/standing waves due to oscillatory Turing instability in our simulations. These results suggest that dormancy of predators is not a generator but an enhancer of spatio-temporal Turing patterns in prey–predator reaction–diffusion systems.

Implications of resting eggs of zooplankton for the paradox of enrichment

In this study, we numerically investigated to what extent introducing resting-egg dynamics would stabilize simple Daphnia–algae consumer–resource models. In the models, the density of viable resting eggs was explicitly expressed, and we assumed that zooplankton produced resting eggs seasonally or in response to food deficiency and that resting eggs hatched seasonally. The models predicted that, although the paradox of enrichment was not completely resolved (i.e., the system was destabilized by eutrophication), we found the following conditions under which the stabilizing effects of resting eggs would be significantly large: (1) resting eggs are produced seasonally (rather than in response to food deficiency), (2) the annual average allocation ratio to resting eggs is large, and (3) the annual average hatching rate of resting eggs is low. The results suggest that resting-egg dynamics can significantly reduce the paradox of enrichment within the biologically meaningful parameter space and contribute to the stability of plankton community dynamics.

The stability of roll solutions of the two-dimensional Swift-Hohenberg equation and the phase-diffusion equation

A stability criterion of roll solutions of the two-dimensional Swift–Hohenberg equation is presented. It clarifies the effect of the system size on the primary instabilty of rolls. An interpretation of the phase-diffusion equation is also given from the viewpoint of spectral analysis. The key to carrying out the spectral analysis is that the infinite-dimensional system of linear equations naturally induced by the Fourier decomposition for the linearized eigenvalue problem of the roll solution can be reduced to the three-dimensional system.

Dormancy of predators dependent on the rate of variation in prey density

In this paper, a simple model is used to demonstrate that the dormancy of predators dependent on the rate of decline in the prey density can strongly stabilize the population dynamics of prey-predator systems. This result may help explain why the population dynamics of phytoplankton-zooplankton (-resting eggs) is frequently observed to be strongly stable in nature. Moreover, it is numerically shown that the model can have two stable prey-predator cycles with different amplitudes and periods, which suggests that prey-predator (-dormant predator) systems have the potential to generate multiple stable cycles without any other mechanism.

Mathematical modelling and experiments for the proliferation and differentiation of Drosophila intestinal stem cells II

We study the proliferation and differentiation of stem cells in the Drosophila posterior midgut epithelium, which mainly consists of intestinal stem cells (ISCs); semi-differentiated cells, i.e. enteroblasts (EBs); and two types of fully differentiated cells, i.e. enteroendocrine cells (EEs) and enterocytes (ECs). The cellular system of ISCs is controlled by Wnt and Notch signalling pathways. In this article, we experimentally show that EBs are not capable of efficiently differentiating into ECs in the absence of Wnt signalling. On the basis of the experimental results and known facts, we propose a scheme and a simple ordinary differential equation (ODE) model for the proliferation and differentiation of ISCs. This is a first step towards understanding the universal mechanism for the maintenance of the cellular system of tissue stem cells controlled by signalling pathways.

Diffusion-driven destabilization of spatially homogeneous limit cycles in reaction-diffusion systems

We study the diffusion-driven destabilization of a spatially homogeneous limit cycle with large amplitude in a reaction-diffusion system on an interval of finite size under the periodic boundary condition. Numerical bifurcation analysis and simulations show that the spatially homogeneous limit cycle becomes unstable and changes to a stable spatially nonhomogeneous limit cycle for appropriate diffusion coefficients. This is analogous to the diffusion-driven destabilization (Turing instability) of a spatially homogeneous equilibrium. Our approach is based on a reaction-diffusion system with mass conservation and its perturbed system considered as an infinite dimensional slow-fast system (relaxation oscillator).

A perspective of renormalization group approaches

In this note, we present an elementary example of an ODE problem which clearly illustrates how various renormalization group approaches are used in theoretical physics. We attempt to clarify relations between the perturbative and constructive renormalizations through the example.