This paper is devoted to the study of a predator-prey model in a patchy environment. The model represents the interactions between phytoplankton and zooplankton in the water column. Two patches are considered with respect to light availability: one patch is associated to the surface layer, the second patch describes the bottom layer. We show that this spatial heterogeneity may destabilize the predator-prey system, even in oligotrophic system where the nutrient is low enough to avoid ”paradox-enrichment”...

A bifurcation problem for variational inequalities $$U\left(t\right)\in K,(\dot{U}\left(t\right)-B_{\lambda }U\left(t\right)-G(\lambda ,U\left(t\right)),Z-U\left(t\right))\ge 0\text{for}\text{all}Z\in K,\text{a.a.}t\ge 0$$ is studied, where $K$ is a closed convex cone in ${ℝ}^{\kappa }$, $\kappa \ge 3$, $B_{\lambda }$ is a $\kappa \times \kappa$ matrix, $G$ is a small perturbation, $\lambda$ a real parameter. The main goal of the paper is to simplify the assumptions of the abstract results concerning the existence of a bifurcation of periodic solutions developed in the previous paper and to give examples in more than three dimensional case.

We consider the nonlinear Dirichlet problem $$-u^{\text{'}\text{'}}-r\left(x\right){\left|u\right|}^{\sigma }u=\lambda u\text{in}(0,\infty ),u\left(0\right)=0\text{and}\underset{x\to \infty }{lim}u\left(x\right)=0,$$ and develop conditions for the function $r$ such that the considered problem has a positive classical solution. Moreover, we present some results showing that $\lambda =0$ is a bifurcation point in $W^{1,2}(0,\infty )$ and in $L^p(0,\infty )(2\le p\le \infty )$.

In this paper, we are concerned with the existence of one-signed solutions of four-point boundary value problems $$-u^{\text{'}\text{'}}+Mu=rg\left(t\right)f\left(u\right),u\left(0\right)=u\left(\epsilon \right),u\left(1\right)=u(1-\epsilon )$$ and $$u^{\text{'}\text{'}}+Mu=rg\left(t\right)f\left(u\right),u\left(0\right)=u\left(\epsilon \right),u\left(1\right)=u(1-\epsilon ),$$ where $\epsilon \in (0,1/2)$, $M\in (0,\infty )$ is a constant and $r>0$ is a parameter, $g\in C\left(\right[0,1],(0,+\infty \left)\right)$, $f\in C(ℝ,ℝ)$ with $sf\left(s\right)>0$ for $s\ne 0$. The proof of the main results is based upon bifurcation techniques.