In this paper, a system consisting of two harmful phytoplanktons and one zooplankton with two time delays is investigated. Firstly, the global existence, nonnegativity and boundedness of the solutions of the system are discussed. Secondly, using time delays as bifurcating parameters, the existence of local Hopf bifurcations at the positive equilibrium of the system is investigated in details. The phenomenon of stability switches is confirmed under some certain conditions. It is shown by numerical simulations under some suitable parameters that the system can exhibit complicated dynamic properties, and undergoes changes from stable periodic solution or equilibrium to chaos or from chaos to stable periodic solution or equilibrium. At last, some conclusions are given. The direction of the Hopf bifurcations and the stability of the bifurcation periodic solutions are given in Appendix by using the center manifold and normal form theory.

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In this paper, we will study the irreducibility of the discrete Painlevé equation of type D₇⁽¹⁾ in the sense of decomposable extensions. The irreducibility here particularly implies that the transcendental function solution cannot be built from rational functions by reiterating algebraic operations, the taking of a solution of a linear difference equation and the taking of a solution of a first-order algebraic difference equation. We also study non-existence of algebraic function solution. A modification to the definition of the decomposable extension is mentioned.

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We study the Cauchy problem for a general inhomogeneous linear moment partial differential equation of two complex variables with constant coefficients, where the inhomogeneity is given by the formal power series. We state sufficient conditions for the convergence, analytic continuation and summability of formal power series solutions in terms of properties of the inhomogeneity. We consider both the summability in one variable t (with coefficients belonging to some Banach space of Gevrey series with respect to the second variable z) and the summability in two variables (t,z).

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We prove the existence of ground states for the semi-relativistic Schrödinger-Poisson-Slater energy
$$I^{\alpha,\beta}(\rho)=\inf_{\substack{u\in H^\frac 12(\mathbb R^3)\\\int_{\mathbb R^3}|u|^2 dx=\rho}} \frac{1}{2}\|u\|^2_{H^\frac 12(\mathbb R^3)}+\alpha\int\int_{\mathbb R^{3}\times\mathbb R^{3}} \frac{|u(x)|^{2}|u(y)|^2}{|x-y|}dxdy-\beta\int_{\mathbb R^{3}}|u|^{\frac{8}{3}}dx$$
α, β > 0 and ρ > 0 is small enough. The minimization problem is L² critical and in order to characterize the values α, β > 0 such that I^α,β(ρ) > -∞ for every ρ > 0, we prove a new lower bound on the Coulomb energy involving the kinetic energy and the exchange energy. We prove the existence of a constant S > 0 such that
$$\frac{1}{S}\frac{\|\varphi\|_{L^\frac 83(\mathbb R^3)}}{\|\varphi\|_{\dot H^\frac 12(\mathbb R^3)}^\frac 12}\leq \left (\int\int_{\mathbb R^3\times \mathbb R^3} \frac{|\varphi(x)|^2|\varphi(y)|^2}{|x-y|}dxdy\right )^\frac 18$$
for all φ ∈ C₀^∞(R³). Besides, we show that similar compactness property fails if we replace the inhomogeneous Sobolev norm ||u||²_H^1/2(R³) by the homogeneous one ||u||_{$\dot{H}$^1/2(R³)} in the energy above.

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In this paper we consider formal solutions of the following form for some linear partial differential equations: $\hat{u}$(t,x) = Σ_i≥1 Σ_k≤miu_i,k(x)tⁱ(log t)^k. Under the same conditions as those in Ōuchi [8], we show multisummability of the formal solutions $\hat{u}$(t,x). It is our key of proofs to find a solution of exponential growth for convolution equations with polynomial coefficients with respect to y via the change of variable log t = y.

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