We study persistence in general seasonally varying predator-prey models. Using the notion of basic reproduction number R0 and the theoretical results proved in [23] in the framework of epidemiological models, we show that uniform persistence is obtained as long as R0 > 1. In this way, we extend previous results obtained in the autonomous case for models including competition among predators, prey-mesopredator-superpredator models and Leslie-Gower systems.
We study the existence of monotone traveling waves u(t, x) = u(x + ct), connecting two equilibria, for the reaction-diffusion PDE u t = (for the reaction term f (u) (among which we have the so-called types A, B, and C), we show that, concerning the admissible speeds, the situation presents both similarities and differences with respect to the classical case. We use a first order model obtained after a suitable change of variables. The model contains a singularity and therefore has some features which are not present in the case of linear diffusion. The technique used involves essentially shooting arguments and lower and upper solutions. Some numerical simulations are provided in order to better understand the features of the model.
We prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincaré–Birkhoff fixed point theorem. The first part of the paper deals with periodic perturbations of a completely integrable system, while in the second part we focus on some suitable global conditions, so to deal with weakly coupled systems
We prove the existence of periodic solutions for first order planar systems at resonance. The nonlinearity is indeed allowed to interact with two positively homogeneous Hamiltonians, both at resonance, and some kind of Landesman-Lazer conditions are assumed at both sides. We are thus able to obtain, as particular cases, the existence results proposed in the pioneering papers by Lazer and Leach (1969) [27], and by Frederickson and Lazer (1969) [18]. Our theorem also applies in the case of asymptotically piecewise linear systems, and in particular generalizes Fabry's results in Fabry (1995) [10], for scalar equations with double resonance with respect to the Dancer-Fučik spectrum.
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