We consider parameter-dependent cocycles generated by nonautonomous difference equations. One of them is a discrete-time cardiac conduction model. For this system with a control variable a cocycle formulation is presented. We state a theorem about upper Hausdorff dimension estimates for cocycle attractors which includes some regulating function. We also consider the existence of invariant measures for cocycle systems using some elements of Perron-Frobenius theory and discuss the bifurcation of parameter-dependent measures.
Ontogenetic changes of rates of photon-saturated photosynthesis (Psat) and dark respiration (RD) of individual leaves were examined in relation to nitrogen content (Nc) in rice, winter wheat, maize, soybean, field bean, tomato, potato, and beet. Psat was positively correlated with Nc as follows: Psat = CfNc + Psat0, where Cf and Psat0 are coefficients. The value of Cf was high in maize, medium in rice and soybean, and low in field bean, potato, tomato, and beet, of which difference was not explained by ribulose-1,5-bisphoshate carboxylase/oxygenase (RuBPCO) content. RD was explained by Psat and/or Nc, however, two models must be applied according to plant species. RD related linearly with Psat and Nc in maize, field bean, and potato as follows: RD = a Psat + b, or RD = a'Nc + b', where a, a', b and b' are coefficients. In other species, the RD/Psat ratio increased exponentially with the decrease of Nc as follows: RD/Psat = a exp(b Nc), where a and b are coefficients. Therefore, RD in these crops was expressed as follows: In(RD) = ln(a Psat) + b Nc, indicating that RD in these crops was regulated by both Psat and Nc. and M. Osaki ... [et al.].
We study solutions tending to nonzero constants for the third order differential equation with the damping term (a1(t)(a2(t)x ′ (t))′ ) ′ + q(t)x ′ (t) + r(t)f(x(ϕ(t))) = 0 in the case when the corresponding second order differential equation is oscillatory.