<abstract><p>In this paper, a homogeneous diffusive system of plant-herbivore interactions with toxin-determined functional responses is considered. We are mainly interested in studying the existence of global steady state bifurcations of the diffusive system. In particular, we also consider the case when the bifurcation parameter, one of the diffusion rates, tends to infinity. The corresponding system is called shadow system. By using time-mapping methods, we can show the existence of the positive non-constant steady state solutions. The results tend to describe the mechanism of the spatial pattern formations for this particular system of plant-herbivore interactions.</p></abstract>

In this paper, we study a class of the Kirchhoff-Schrödinger-Poisson system. By using the quantitative deformation lemma and degree theory, the existence result of the least energy sign-changing solution u0 is obtained. Meanwhile, the energy doubling property is proved, that is, we prove that the energy of any sign-changing solution is strictly larger than twice that of the least energy. Moreover, we also get the convergence properties of u0 as the parameters b↘0 and λ↘0.

In this paper, a homogeneous diffusive system of plant-herbivore interactions with toxin-determined functional responses is considered. We are mainly interested in studying the existence of global steady state bifurcations of the diffusive system. In particular, we also consider the case when the bifurcation parameter, one of the diffusion rates, tends to infinity. The corresponding system is called shadow system. By using time-mapping methods, we can show the existence of the positive non-constant steady state solutions. The results tend to describe the mechanism of the spatial pattern formations for this particular system of plant-herbivore interactions.