This paper ascertains the global topological structure of the set of subharmonics of arbitrary order of the periodic predator-prey model introduced in [14]. By constructing the iterates of the monodromy operator of the system, it is shown that the system admits subharmonics of all orders for the appropriate ranges of values of the parameters. Then, some sharp results of topological nature in the context of global bifurcation theory provide us with the fine topological structure of the components of subharmonics emanating from the T -periodic coexistence state. 2 α(t) > 0 if t ∈ (0, T 2 ), and β(t) > 0 if t ∈ ( T 2 , T ), which entail αβ = 0. This model was introduced by J. López-Gómez, R. Ortega and A. Tineo [14] as a simple example of a predator-prey model with an unstable coexistence state. Later, it was shown in [9] that it actually admits three T -periodic (non-degenerate) coexistence states: one T -periodic and two additional 2T -periodic solutions. The non-degeneration of these solutions facilitated the construction of some examples of T -periodic Lotka-Volterra modelswith at least three coexistence states (see [9]). In (1.3), λ, µ, a, b, c, d are smooth positive T -periodic functions. Such multiplicity results contrast very strongly with the main theorem of J. López-Gómez and R. Pardo [15], where it was established the uniqueness of the coexistence state for the boundary value problem 4) where λ, µ, a, b, c, d are positive (arbitrary) continuous functions in [0, L]. Inheriting the same non-cooperative structure, at first glance causes some perplexity that (1.3) and (1.4) behave so differently. The original theorem of [15] was later refined in a series of papers by A. Casal et al. [2], E. N. Dancer et al. [5] and J. López-Gómez and R. Pardo [16].An important feature of model (1.1) is that it does not fit within the general setting of T. Ding and F. Zanolin [6], where the existence of higher order subharmonics for a general class of predator-prey models was established. Precisely, [6, Th.3] gives some general conditions on the nonlinearities f (t, v) and g(t, u) so that the Lotka-Volterra predator-prey system(1.5) can admit higher order subharmonics. In (1.5), f (t, v) and g(t, u) are continuous functions T -periodic in time, t, satisfying certain bounds for the existence of T-periodic solutions and such that, for every t ∈ [0, T ], either v → f (t, v) is (strictly) decreasing, or u → g(t, u) is (strictly) increasing. Under these assumptions, [6, Th. 3] establishes the existence of an integer m * ≥ 2 such that (1.5) admits, at least, one mT -periodic solution for all m ≥ m * . Although setting(1.1) can be also written down in the form of (1.5), by (1.2), neither α(t)(1 − v) can be decreasing for all t ∈ [0, T ], nor β(t)(−1 + u) can be increasing for all t ∈ [0, T ]. Thus, (1.1) remains outside the class of models considered in [6]. In particular, [6, Th. 3] cannot be applied to establish the existence of higher order subharmonics for (1.1).
This paper introduces a spatially heterogeneous diffusive predatorprey model unifying the classical Lotka-Volterra and Holling-Tanner ones through a prey saturation coefficient, m(x), which is spatially heterogenous and it is allowed to 'degenerate'. Thus, in some patches of the territory the species can interact according to a Lotka-Volterra kinetics, while in others the prey saturation effects play a significant role on the dynamics of the species. As we are working under general mixed boundary conditions of non-classical type, we must invoke to some very recent technical devices to get some of the main results of this paper.
This paper studies the existence of subharmonics of arbitrary order in a generalized class of non-autonomous predator-prey systems of Volterra type with periodic coefficients. When the model is non-degenerate it is shown that the Poincaré-Birkhoff twist theorem can be applied to get the existence of subharmonics of arbitrary order. However, in the degenerate models, whether or not the twist theorem can be applied to get subharmonics of a given order might depend on the particular nodal behavior of the several weight functioncoefficients involved in the setting of the model. Finally, in order to analyze how the subharmonics might be lost as the model degenerates, the exact point-wise behavior of the T-periodic solutions of a non-degenerate model is ascertained as a perturbation parameter makes it degenerate.
In this paper, we investigate the problem of the existence and multiplicity of periodic solutions to the planar Hamiltonian system x ′ = - λ α ( t ) f ( y ) x^{\prime}=-\lambda\alpha(t)f(y) , y ′ = λ β ( t ) g ( x ) y^{\prime}=\lambda\beta(t)g(x) , where α , β \alpha,\beta are non-negative 𝑇-periodic coefficients and λ > 0 \lambda>0 . We focus our study to the so-called “degenerate” situation, namely when the set Z := supp α ∩ supp β Z:=\operatorname{supp}\alpha\cap\operatorname{supp}\beta has Lebesgue measure zero. It is known that, in this case, for some choices of 𝛼 and 𝛽, no nontrivial 𝑇-periodic solution exists. On the opposite, we show that, depending of some geometric configurations of 𝛼 and 𝛽, the existence of a large number of 𝑇-periodic solutions (as well as subharmonic solutions) is guaranteed (for λ > 0 \lambda>0 and large). Our proof is based on the Poincaré–Birkhoff twist theorem. Applications are given to Volterra’s predator-prey model with seasonal effects.
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