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.

It is shown that the dormancy of predators induces mixed-mode oscillations and chaos in the population dynamics of a prey-predator system under certain conditions. The mixed-mode oscillations and chaos are shown to bifurcate from a coexisting equilibrium by means of the theory of fast-slow systems. These results may help to find experimental conditions under which one can demonstrate chaotic population dynamics in a simple phytoplankton-zooplankton (-resting eggs) community in a microcosm with a short duration.

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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.