Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments

1 Introduction

The analysis of subharmonic solutions to differential systems with periodic coefficients is a classical research topic which has been widely investigated also with respect to its relevant significance in several applications, including the study of differential equations models arising from celestial mechanics and engineering. Generally speaking, given a first-order differential system

for z = ( z 1 , … , z d ) ∈ Ω , where Ω is an open domain of R d and F : R × Ω → R d is a sufficiently regular vector field which is T -periodic in the t -variable, by a subharmonic solution of order n ≥ 2 to system (1.1) we mean a n T -periodic solution of the system which is not k T -periodic for all integers k ∈ { 1 , … , n − 1 } . As pointed out by Rabinowitz in [1]:

This latter quest is complicated by the fact that any T -periodic solution is a fortiori k T -periodic. Thus an additional argument is required to show that any subharmonics are indeed distinct.

In particular, it will also be important to check whether a subharmonic solution of order n has indeed n T as its minimal period. This is a difficult task that nevertheless can be overcome in some circumstances thanks to the special structure of the vector field F . Some sufficient conditions have been already proposed in the literature (see, e.g., Michalek and Tarantello [2]).

The general aim of this work is to investigate, as deeply as possible, the subharmonics to a class of Lotka Volterra systems under seasonal effects. Precisely, the model is a planar Hamiltonian system of the form

where f : R → R is a locally Lipschitz continuous function with f ( 0 ) = 0 and f ( s ) s > 0 for s ≠ 0 such that f is bounded on ( − ∞ , 0 ] and has a superlinear growth at + ∞ . The assumptions on f are motivated by the paradigmatic case

coming from the original Volterra’s equations.

In 1926, Volterra in [3] proposed a mathematical model for the predator-prey interactions as an answer to the statistics on the fishing data in Northern Adriatic sea, provided by the biologist Umberto D’Ancona. His model, extraordinarily famous today, since it is discussed in most of textbooks on Ordinary Differential Equations (ODEs) and Ecology, has shown to be a milestone for the development of more realistic predator-prey models in environmental sciences and population dynamics (see, e.g., Begon et al. [4]). It can be formulated as follows

where N 1 ( t ) > 0 and N 2 ( t ) > 0 represent, respectively, the density of the prey and the predator populations at time t . In (1.4), the coefficients, a , b , c , and d , are assumed to be positive constants (see Braun [5]). The same system had been already introduced few years before by Alfred J. Lotka from a hypothetical chemical reaction exhibiting periodic behavior in the chemical concentrations (see Murray [6, §3.1]), which reveals how the same models can mimic a variety of phenomenologies of a different nature. After these pioneering contributions, differential equations involving the interaction of two or more species are usually named as Lotka-Volterra systems and are represented in the general form:

However, in this article we will focus attention on those satisfying b i i = 0 . The choice of the sign of the coefficients a i and b i j allows describing different kinds of interactions, as competition, cooperation, or parasitism.

Although Volterra [7] and Lotka [8] considered the possibility of some varying in time coefficients, an extensive study of the Lotka-Volterra systems with periodic coefficients has not been carried out until more recently (see, e.g., Butler and Freedman [9], Cushing [10,11], and Rosenblat [12], for some early works in this direction). However, in the past four decades, the researches in this area have originated a great number of contributions, also in connection with the study of periodically perturbed Hamiltonian systems and reaction-diffusion systems arising in genetics and population dynamics, beginning with the influential monograph of Hess [13] and the refinements of López-Gómez [14], where a general class of spatially heterogeneous diffusive Lotka-Volterra systems with periodic coefficients was studied. To understand the role played by the spatial heterogeneities in these models, the reader is sent to the early works of López-Gómez [15] and Hutson et al. [16], as well as to the monographs [17,18]. Further models covering more general spatial interactions, outside the Lotka-Volterra World, were introduced by López-Gómez and Molina-Meyer [19,20].

Although the assumption that the coefficients of the model are periodic with a common period might seem restrictive, it is based on the natural assumption that the species are affected by identical seasonal effects, which is rather natural as they interact in the same environment. Precisely, the article focuses into the following periodic counterpart of the autonomous system (1.4):

where, typically, a , b , c , d : R → R are T -periodic functions such that

In this case, by [21, Th. A.1], the system (1.6) has a component-wise positive T -periodic solution, ( N ˜ 1 ( t ) , N ˜ 2 ( t ) ) , and the change of variables

leads to the study of the equivalent system

where ( N ˜ 1 ( t ) , N ˜ 2 ( t ) ) becomes ( 1 , 1 ) (see the Appendix of [21] for any further details). In (1.7), λ > 0 is regarded as a parameter, while α ( t ) ⪈ 0 and β ( t ) ⪈ 0 are T -periodic function coefficients, where T > 0 is their minimal period. The component-wise positive periodic solutions of (1.7) are called (periodic) coexistence states and, obviously, are relevant in population dynamics, as they represent states where none of the interacting species is driven to extinction by the other.

Among the periodic coexistence states, the subharmonics are of particular interest. Focussing attention into the simplest prototype model with constant coefficients (1.4), it is folklore since Volterra [7] that all its positive solutions are periodic and oscillate around the equilibrium P 0 ≡ ( c / d , a / b ) in the counterclockwise sense, lying on the “energy levels”:

for k ∈ ( k 0 , + ∞ ) , where

Therefore, P 0 is a global center in the open first quadrant. The fact that the period of the orbit at the level k , τ ( k ) , is an increasing function of k is a more recent finding of Rothe [22], Shaaf [23], and Waldvogel [24], where it was also shown that

Thus, for every T ∈ ( 0 , τ 0 ) , the equilibrium point P 0 is the unique T -periodic coexistence state (harmonic solution), though there are infinitely many nontrivial subharmonic coexistence states corresponding to the energy levels k for which τ ( k ) = m T for sufficiently large m ≥ 2 . This classical example also illustrates how subharmonics can be packaged in equivalence classes (two subharmonics are equivalent when they are a time-shift of the other). This also holds for nonautonomous systems. Indeed, if z ( t ) is a n T -periodic solution of (1.1), then z ( t + j T ) is also a n T -periodic solution for every j = 0 , 1 , … , n − 1 . Michalek and Tarantello [2] referred to the set

as the Z n -orbit of z. This set is also usually denoted as the periodicity class of the n T -periodic solution z ( t ) . Hence, searching subharmonics in nonautonomous systems is a task fraught with a number of difficulties. The main goals of this article are as follows:

• finding out n T -periodic solutions having n T as minimal period, namely, subharmonics of order n;

• among the subharmonics having the same order, ascertaining whether, they belong to the same periodicity class, namely, if they are, or not, a shift in time by an integer multiple of T of another subharmonic of the same order.

López-Gómez et al. [34] and López-Gómez [35, §5] carried out a thorough investigation of the positive coexistence states for the generalized Lotka-Volterra system with periodic coefficients

which includes the presence of logistic terms incorporating to the model setting some interspecific competition effects. This model has, in addition, semi-trivial coexistence states of the form ( N ˆ 1 ( t ) , 0 ) and ( 0 , N ˆ 2 ( t ) ) , where N ˆ i stands for the (unique) positive T -periodic solution of the logistic equation:

As in the periodic-parabolic counterpart of (1.8), already analyzed by Hess [13] and López-Gómez [14], the local character of the semitrivial positive solutions plays a crucial role in determining the dynamics of (1.8), being a challenging task to ascertain the stability, or instability, of the coexistence states.

López-Gómez et al. [34] made the crucial observation that, for the special choice

(1.7) has a (unique) linearly unstable coexistence state, though the system can admit two coexistence states [34, Rem. 7.5]. This is in strong contrast with some previous one-dimensional uniqueness results available for the diffusive counterparts of these models (see the detailed discussion of [35]). Actually, this prototype model has shown to be rather paradigmatic for analyzing the local character and the multiplicity of harmonic and subharmonic coexistence states. Indeed, setting

it follows from [34, Pr. 7.1] that P 0 ≡ ( 1 , 1 ) is linearly unstable if

Moreover, by [35, Th. 5.3], the instability of P 0 guarantees the existence of three coexistence states for (1.8) within the appropriate ranges of values of its function coefficients. It turns out that, besides the unique (harmonic) coexistence state P 0 , there are, at least, two additional 2 T -periodic coexistence states if (1.11) holds. After two decades, López-Gómez and Muñoz-Hernández [36] were able to construct n T -periodic solutions for every n ≥ 2 , providing simultaneously with a sharp estimate of their minimal cardinals. Figure 1 shows the (minimal) global bifurcation diagram of subharmonics of (1.7) found in [36] for an arbitrary choice of α ( t ) and β ( t ) satisfying the orthogonality condition (1.9).

Figure 1

Admissible global bifurcation diagram under (1.9), after [36].

These findings should be also true, at least, when α ( t ) and β ( t ) are nearly orthogonal, in the sense that the product α β is sufficiently small in [ 0 , T ] , and actually they might be also true for very general classes of weight functions α ( t ) and β ( t ) far from satisfying (1.9). Thus, it is rather natural to investigate the structure of the set of subharmonics of (1.7) for more general configurations of α ( t ) and β ( t ) than those satisfying (1.9). Besides its intrinsic interest, this analysis might reveal some new important features, based, e.g., on the shapes of α ( t ) and β ( t ) , which might be significative in population dynamics. Indeed, as already discussed by the authors in [21], when β ≡ 0 , one is considering a seasonal time-interval where predator is still aggressive with respect to the prey but does not take advantage of the harvest, which may simulate the difference between the hunting and the gathering period. Analog interpretations may be given for the time-intervals when α ≡ 0 .

From the mathematical analysis of (1.7), it becomes apparent that its dynamics might vary according to the measure of the intersection of the supports of α and β . To differentiate the two extreme cases, we will name as “degenerate” the case when ∣ supp α ∩ supp β ∣ = 0 , while the case when

will be referred to as the “nondegenerate” case. In [21,37], the authors have already shown the applicability of the Poincaré-Birkhoff theorem in searching subharmonics in both cases (see also the recent refinements of Boscaggin and Muñoz-Hernández [33]).

As a sharp analysis of (1.7) seems imperative to classify all the possible dynamics associated to a general predator-prey system with periodic coefficients, throughout this article we will focus our attention into the simplest prototype model (1.7), with special emphasis towards the problem of analyzing its subharmonic solutions depending on whether, the chosen configuration of α ( t ) and β ( t ) is degenerate.

Among the various existing approaches in finding out subharmonics in Hamiltonian systems, the most successful ones are the following:

• variational methods (critical point theory),

• Kolmogorov-Arnold-Moser (KAM) theory, Moser twist theorem, and Poincaré-Birkhoff theorem,

• symbolic dynamics associated to Smale’s horseshoe-type structures, and

• reduction via symmetry and bifurcation theory.

Variational methods (critical point theory): In this framework, a number of different arguments have been given to show that the periodic solutions that are critical points of the associated functional are indeed subharmonics of large period. For instance, Rabinowitz [1] achieves it by analyzing the critical levels of the functional (see also Fonda and Lazer [38] and Serra et al. [39]), whereas Michalek and Tarantello [2] apply a combination of estimates on critical levels and Z p -index theory. Assuming an additional nondegeneracy condition on the solutions, Conley and Zhender [40] can get subharmonics through their Morse indices (see also Abbondandolo [41] and the references therein).

KAM theory, Moser twist theorem, and Poincaré-Birkhoff theorem: In this setting, a typical approach consists in constructing an annular region enclosed by two curves, Γ int and Γ out , which are invariant by the Poincaré map associated to the given planar system. Recall that the Poincaré map is the homeomorphism Φ = Φ 0 T transforming an initial point z 0 to ζ ( T ; 0 , z 0 ) , where ζ ( ⋅ ; 0 , z 0 ) stands for the solution of (1.1) with z ( 0 ) = z 0 . The associated flow is area-preserving when div z F ( t , z ) ≡ 0 , a condition that is always satisfied by Hamiltonian systems. Then, introducing a suitable rotation number, rot ( [ 0 , n T ] , z 0 ) , the Poincaré-Birkhoff fixed point theorem guarantees the existence of at least two fixed points for Φ n in the interior of the annulus as soon as the following twist condition

holds for some integer j . In such situation, the fixed points of Φ n have j as an associated rotation number. This actually entails the existence of subharmonics of order n if n and j are co-prime integers. Theorems 2.1 and 3.1 of Neumann [42] give some sufficient conditions so that these subharmonic solutions belong to different periodicity classes.

A typical definition of rotation number can be given by passing to polar coordinates and counting the number of turns (counterclockwise, or clockwise) that the solution ζ ( t ; 0 , z 0 ) makes around the origin in a given time interval. The choice of the origin as a “pivotal point” is merely conventional, but it fits well for systems like (1.2), where F ( t , 0 ) ≡ 0 . By adopting this methodology, for any given interval [ t 0 , t 1 ] , we will set

where δ = ± 1 and ∧ denotes the wedge product, i.e., ( F 1 , F 2 ) ∧ ( ζ 1 , ζ 2 ) = F 2 ζ 1 − F 1 ζ 2 . Boscaggin [30] and Boscaggin and Muñoz-Hernández [33] give some other equivalent notions.

Ding [43], Fonda and Ureña [32], Franks [44], Qian and Torres [45], and Rebelo [46] have found some refinements of the Poincaré-Birkhoff fixed-point theorem adapted to more general settings where the boundary invariance of the underlying annular region is lost. Dalbono and Rebelo [47] and more recently Fonda et al. [48] have reviewed a series of results on the applicability of the Poincaré-Birkhoff theorem in noninvariant annuli.

According to some recent findings of Boscaggin and Muñoz-Hernández [33], there is a link between the variational approach discussed in the previous paragraph and the rotation number estimates necessary to apply the Poincaré-Birkhoff theorem based on the Conley-Zehnder-Maslov index (see Abbondandolo [41] and Long [49]).

Nevertheless, establishing the existence of the invariant curves is a hard task that typically requires to invoke the KAM theory or some of its variants (see Laederich and Levi [50]), like the Moser twist theorem [51], as in Dieckerhoff and Zehnder [52] and Levi [53] for the scalar second-order differential equation:

or as in Hausrath [25] and Liu [26] for the most sophisticated predator-prey system.

Symbolic dynamics associated with horseshoe-type structures: In connection with the previous discussion, subharmonics can be also detected by applying to the Poincaré map and its iterates various methods coming from the theory of dynamical systems, the most paradigmatic being the celebrated Smale’s horseshoe (see Smale [54,55], Moser [56, Ch. III], and Wiggins [57]). Essentially, it consists of a toy-diffeomorphism stretching and bending, recursively, a square onto a horseshoe-type domain.

In this article, we will work in a slightly weaker setting entering into the theory of topological horseshoes as developed, adapting different perspectives, by Carbinatto et al. [58], Mischaikow and Mrozek [59], Srzednicki [60], Srzednicki and Wójcik [61], Wójcik and Zgliczyński [62], Zgliczyński [63], and Zgliczyński and Gidea [64], just to quote some of the most illustrative contributions among a vast literature on this topic. The theory of topological horseshoes, as discussed by Burns and Weiss [65] and Kennedy and Yorke [66], consists of a series of methods introduced to extend the classical geometry of the Smale horseshoe to more general dynamical situations involving topological crossings and, crucially, avoiding any hyperbolicity assumptions on the diffeomorphism, as it might be a challenge to verify them in applications.

In particular, in this article, we will benefit of a topological approach developed by Papini and Zanolin [67,68] and Pascoletti et al. [69], leading to the following notion of chaos in the coin-tossing sense.

Definition 1.1 is inspired in the concept of chaotic dynamics as a situation where a deterministic map can reproduce, along its iterates, all the possible outcomes of a coin-flipping experiment, as discussed by Smale [70] (see also Kirchgraber and Stoffer [71]). According to Medio et al. [72], any map Φ inducing chaotic dynamics on ℓ symbols in Q is semi-conjugate to the Bernoulli shift automorphism

in the sense that there exist a compact subset, Λ ⊂ ⋃ i = 0 , … ℓ − 1 ℋ i ⊂ Q , invariant for Φ , whose set of periodic points, Per Φ , is dense in Λ , and a continuous and surjective map g : Λ → Σ ℓ such that:

1. g ∘ Φ = σ ∘ g , and

2. for every periodic sequence s ∈ Σ ℓ , the set g − 1 ( s ) contains a periodic point of Φ with the same period.

Clearly, for any map Φ satisfying the requirements of Definition 1.1, one can infer the existence of periodic points of arbitrary minimal period, and hence subharmonics if Φ is the Poincaré map of an ODE with periodic coefficients. For instance, if ℓ = 2 , given any periodic sequence in { 0 , 1 } of minimal period n , there is also a periodic point of Φ in Q with minimal period n . Similar notions of “chaos” can be given by means of a number of topological methods, based on the Conley index, fixed point theories, or topological degree (see, e.g., Carbinatto et al. [58], Mischaikow and Mrozek [59], Srzednicki [60], Srzednicki and Wójcik [61], Wójcik and Zgliczyński [62], Zgliczyński [63], and Zgliczyński and Gidea [64]). Typically, in any setting entailing Definition 1.1, it is possible to detect a larger number of subharmonics than merely applying the Poincaré-Birkhoff theorem (see Feltrin [79, Rem. 4.1]).

Among the main advantages of using the theory of topological horseshoes, instead of the methods discussed in the two previous items, it is worth mentioning that, since the system is not required to inherit any Hamiltonian structure, there are no constraints on the dimension and, hence, they could be studied with this theory systems with an odd dimension.

Reduction via symmetry and bifurcation theory: Another successful approach for solving a huge variety of nonlinear differential equations, both ODEs and partial differential equations (PDEs), relies on the topological degree through local and global bifurcation theory. Essentially, in the context of bifurcation theory, the continuation methods in parametric models do substitute the implicit function theorem in the presence of degenerate solutions, provided that a change of degree occurs as the parameter, λ , crosses some critical value, λ 0 . The relevance of bifurcation theory in studying nonlinear differential equations was first understood by Krasnosel’skii [80], who was able to show that any eigenvalue of the linearization at a given state with an odd (classical) algebraic multiplicity is a nonlinear eigenvalue. By a nonlinear eigenvalue, we mean a bifurcation value from the given state, regardless the nature of the nonlinear terms of the differential equation. In other words, nonlinear eigenvalues are those for which the fact that bifurcation occurs is based on the linear part, as discussed by Chow and Hale [81]. Some years later, Rabinowitz [82,83] established his celebrated global alternative within the setting of the Krasnosel’skii’s theorem founding global bifurcation theory. According to the Rabinowitz’s global alternative, the global connected component, C , bifurcating from the given state at an eigenvalue with an odd (classical) algebraic multiplicity must be unbounded, or it bifurcates from the given state at, at least, two different values of λ . The relevant fact that if C is bounded, then the number of bifurcation points from the given state with an odd algebraic multiplicity must be even as observed by Nirenberg [84].

However, the precise role played by the classical spectral theory in the context of the emerging bifurcation theory remained a real mystery for two decades. The mystery was confirmed by the astonishing circumstance that the extremely popular transversality condition studied by Crandall and Rabinowitz [85,86] for bifurcation from simple eigenvalues was not known to entail a change of the Leray-Schauder degree until Theorem 5.6.2 of López-Gómez [17] could be derived through the generalized algebraic multiplicity, χ , of Esquinas and López-Gómez [17,87,88]. The multiplicity χ is far more general that the one introduced in [85,86] for algebraically simple eigenvalues, and it was used, e.g., by López-Gómez and Mora Corral [89], to characterize the existence of the Smith canonical form. According to [17, Ch.4], the oddity of χ characterizes whether, λ 0 is a nonlinear eigenvalue of the problem, and this occurs if, and only if, the local degree changes as λ crosses λ 0 . And this regardless if we are dealing with the Leray-Schauder degree, or with the degree for Fredholm operators of Fitzpatrick and Pejsachowicz [90], Fitzpatrick et al. [91], or Benevieri and Furi [92,93], which are almost equivalent. Therefore, by Corollary 2.5 of López-Gómez and Mora-Corral [94], the local theorem of Crandall and Rabinowitz [85] is actually global. This important feature was later rediscovered by Shi and Wang [95] in a much less general context.

Some more specific important information in the context of dynamical bifurcation theory and singularity theory can be found in the textbooks of Guckenheimer and Holmes [96] and Golubitsky and Shaeffer [97]. Essentially, the singularity theory tries to classify canonically all the possible local structures at the bifurcation values, while dynamical bifurcation theory focuses attention in bifurcation phenomena not involving only equilibria.

These abstract developments have tremendously facilitated the mathematical analysis of a huge variety of nonlinear bvps related to a huge variety of nonlinear differential equations and systems (see, e.g., the monographs of López-Gómez [17,18], Cantrell and Cosner [98], and Ni [99], as well as their abundant lists of references). However, the underlying mathematical analysis is more involved when dealing with periodic conditions, instead of mixed boundary conditions, especially when searching for branches of subharmonic solutions bifurcating from a given state. Indeed, although the pioneering strategy for constructing coexistence states as bifurcating from the semitrivial solution branches in reaction-diffusion systems of Lotka-Volterra type was developed by Cushing [10,11] for their classical nondiffusive periodic counterparts, rather astonishingly, except for certain technicalities inherent to nonlinear PDEs, the level of difficulty in establishing the existence of the coexistence states in the diffusive prototype models inherits the same order of magnitude as getting them in their nondiffusive periodic counterparts. Not to talk about finding out infinitely many subharmonics of large order. Although some additional contributions in this direction were done by Táboas [100], the analysis of the periodic-parabolic counterparts of these classical models, extraordinarily facilitated by the pioneering results of Cushing [10,11] for the nonspatial models, was already ready to be developed by Hess [13] and López-Gómez [14].

Nevertheless, in spite of the huge amount of literature on bifurcation for reaction-diffusion systems in population dynamics, almost no reference is available about harmonic and subharmonic solutions for predator-prey systems of Volterra type, except for those already discussed in this section, beginning with [34] and [35], and continuing, after two decades, with [36], where the weight functions α ( t ) and β ( t ) were assumed to have nonoverlapping supports so that the underlying Poincaré map associated with (1.7) could take a special form to allow solving the periodic problem via a one-dimensional reduction.

As mentioned earlier, the main goal of this article is to analyze the existence, multiplicity, and structure of subharmonic solutions to planar systems, including the periodic Volterra’s predator-prey model (1.7). Naturally, as the underlying Poincaré maps play a crucial role in this analysis, we will benefit of a number of methods and tools among those already described in the previous paragraphs. As a consequence of our analysis, the richness of the dynamics of (1.7) will become apparent even for the simplest configurations of α ( t ) and β ( t ) . Some recent applications of the Poincaré-Birkhoff fixed-point theorem to equations directly related to (1.7) have been given by Boscaggin [30], Ding and Zanolin [28,29], Fonda and Toader [31], Hausrath and Manásevich [27], and Rebelo [46]. In the more recent papers [21,36,37], the authors have studied in detail some simple prototype models, nondegenerate and degenerate. Essentially, this article continues the research program initiated in [21,36,37] trying to understand how the relative position of the supports of the weight functions α ( t ) and β ( t ) might influence the dynamics of (1.7) and the global structure of the set of its subharmonics.

These goals will be achieved in Section 2 for the degenerate case by means of the bifurcation approach introduced in [36]. Precisely, we will consider the general case of weight functions having multiple nonoverlapping humps as in Figure 2. Then, depending on the mutual distributions of the supports of α ( t ) and β ( t ) , some sharp estimates on the parameter λ ensuring the existence of nontrivial subharmonics will be given. These objectives will be accomplished in Theorems 2.1–2.5, to deal with the most general configuration admissible for the validity of these results.

Figure 2

Some admissible examples of weight functions α and β .

Further, in Section 3, we will analyze some simple prototype models, not previous considered in the literature, for which the associated Poincaré consists of a superposition of a stretching and a twist producing a horseshoe-type geometry. The new findings have been collected in Theorems 3.1 and 3.2. As this topic is more sophisticated technically and not well understood by most of experts in reaction-diffusion systems, we will begin the proof of Theorem 3.1 by giving a rather direct proof for stepwise-constant functions α ( t ) and β ( t ) before completing the proof in the general case. At a further step, we will discuss the problem of the semi-conjugation/conjugation of the Poincaré map to the Bernoulli shift, which, essentially, depends on the shape of the weight functions. Finally, we will end Section 3 describing in full detail the geometric horseshoe nature of the Poincaré map associated with the periodic Volterra predator-prey system. This provides us with a (new) simple mechanism to mimic the Smale’s horseshoe from one of the most paradigmatic models in population dynamics.

As a byproduct of our mathematical analysis, it becomes apparent how the evolution in seasonal environments where predator-prey interaction plays a role might be random. To catch the attention of experts in reaction-diffusion systems and population dynamics, note that, actually, the harmonics and subharmonics of (1.7) are the nonspatial periodic harmonic and subharmonic solutions of the following reaction-diffusion periodic-parabolic problem:

where Ω stands for a C 2 bounded domain of R N , N ≥ 1 , n x stands for the outward normal vector-field to Ω on its boundary, d 1 and d 2 are two positive constants, and Δ stands for the Laplace’s operator in R N . Therefore, the findings of this article seem extremely relevant from the point of view of population dynamics. As a byproduct of our analysis, noncooperative systems in periodic environments can provoke chaotic dynamics. This cannot occur in cooperative and quasi-cooperative dynamics, as a consequence of the ordering imposed by the maximum principle. Therefore, it is just the lack of a maximum principle for the predator-prey models the main mechanism provoking chaos in these models, though this sharper analysis will be accomplished in a forthcoming paper.

To avoid unnecessary repetitions, throughout this article, we will assume that α , β : R → R + ≔ [ 0 , + ∞ ) are continuous and T -periodic functions, though our results are easily extended to the Carathéodory setting with coefficients measurable and integrable in L 1 ( [ 0 , T ] , R + ) . In particular, the case of bounded and piecewise continuous α , β falls within our functional setting. Hence, piecewise constant coefficients are admissible in Section 3.

2 A bifurcation approach: minimal complexity of subharmonics for a class of degenerate predator-prey models

We consider the nonautonomous Volterra predator-prey model

where λ > 0 is regarded as a real parameter, and the intersection of the supports of the nonnegative weight functions α and β , denoted by

is assumed to have Lebesgue measure zero, ∣ Z ∣ = 0 . This is the reason why the model (2.1) is said to be degenerate. In (2.1), given a real number T > 0 , α , and β are T -periodic continuous functions such that

These kind of degenerate Volterra predator-prey models were introduced in [34] and [35] and, then, deeply analyzed in [36]. Actually, these references dealt with the very special, but interesting, case when

where we could benefit of the degenerate character of the model to ascertain the global structure of the set of n T -periodic coexistence states of (2.1). In the nondegenerate case when ∣ Z ∣ ≠ 0 , the techniques developed in [36] do not work. In such case, the existence of high-order subharmonics can be established through the celebrated Poincaré-Birkhoff twist theorem, or appropriate variants of it (see, e.g., [32,101–104], as well as [21] for a specific application to (2.1)). However, the twist theorem cannot provide us with the global bifurcation diagram of subharmonics constructed in [36]. The main goal of this section is sharpening and generalizing as much as possible the main findings of [36].

Precisely, we analyze the existence of n T -periodic coexistence states of (2.1) for any integer n ≥ 1 under the following structural constraints on the weight functions α and β . For some integers k , ℓ ≥ 1 , with ∣ k − ℓ ∣ ≤ 1 , it is assumed the existence of k + ℓ continuous functions in the interval [ 0 , T ] , α i ⪈ 0 , 1 ≤ i ≤ k , and β j ⪈ 0 , 1 ≤ j ≤ ℓ , such that

with

for some partition of [ 0 , T ]

if k = ℓ , or

if k = ℓ + 1 . Similarly, we also consider the case when, instead of (2.2),

for some partition of [ 0 , T ]

if ℓ = k , or

if ℓ = k + 1 .

Moreover, we refer to an α -interval (resp. β -interval) as a maximal interval, where β ≡ 0 (resp. α ≡ 0 ), and we set

Figure 2 shows a series of examples satisfying the previous requirements. Note that the support of the α i ’s and the β j ’s on each of the intervals [ t r i , t r + 1 i ] , 1 ≤ i ≤ k , and [ t r j , t r + 1 j ] , 1 ≤ j ≤ ℓ , might not be connected.

There are two fundamental aims in this section. The first one is to show that the complexity of the global bifurcation diagram of subharmonics of (2.1) when k = ℓ depends on the size of k , rather than on the particular structure of the α i ’s and the β j ’s on each of the intervals of the partition of [ 0 , T ] . In fact, all admissible global bifurcation diagrams when k = ℓ can be constructed systematically, through a certain algorithm, from the one already found in [36], regardless the particular locations of each of the points of the partitions, t r s ’s. The second aim of this section is to determine a lower bound for the number of n T -periodic coexistence states of model (2.1).

It is elementary to show that, for any initial point z 0 ≔ ( u 0 , v 0 ) (with u 0 , v 0 > 0 ) and each initial time τ 0 , there exists a unique solution ( u ( t ; τ 0 , z 0 ) , v ( t ; τ 0 , z 0 ) ) to system (2.1), which is globally defined in time. In the sequel, by convention, when studying n T -periodic solutions (for any n ≥ 1 ), we will be looking for the fixed and periodic points of the Poincaré map with τ 0 = 0 , i.e.,

This does not exclude the possibility of the existence of other fixed points for the Poincaré maps defined with a different initial point τ 0 . Typically, the corresponding solutions will be equivalent through an appropriate time translation, and hence, they will be not considered in counting the multiplicity of the solutions.

2.1 The case when k = ℓ = 1 and supp α 1 ⊆ [ t 0 1 , t 1 1 ]

Then,

(2.5) supp α 1 ⊆ [ t 0 1 , t 1 1 ] and supp β 1 ⊆ [ t 2 1 , t 3 1 ] ,

where

0 ≤ t 0 1 < t 1 1 ≤ t 2 1 < t 3 1 ≤ T .

Figure 3 illustrates this case.

Figure 3

α and β satisfying (2.5).

Under condition (2.5), the next result holds.

Theorem 2.1

Assume (2.5). Then, equilibrium ( 1 , 1 ) is the unique T -periodic coexistence state of (2.1). Thus, (2.1) cannot admit nontrivial T -periodic coexistence states. Moreover, (2.1) possesses exactly two nontrivial 2 T -periodic coexistence states for every

λ > 2 A 1 B 1 ,

where A 1 and B 1 are those defined in (2.4). Furthermore, in the special case when A 1 = B 1 and u ( 0 ) = v ( 0 ) , for every λ > 2 A 1 and n ≥ 2 , (2.1) has, at least, n nontrivial n T -periodic coexistence states if n is even, and n − 1 if n is odd.

Proof

Thanks to (2.5), (2.1) can be integrated. Indeed, for any given ( u 0 , v 0 ) ∈ R 2 , the (unique) solution of (2.1) such that ( u ( 0 ) , v ( 0 ) ) = ( u 0 , v 0 ) is given by

u ( t ) = u 0 e ( 1 − v 0 ) λ ∫ 0 t α ( s ) d s , v ( t ) = v 0 e ( − 1 + u ( T ) ) λ ∫ 0 t β ( s ) d s , t ∈ [ 0 , T ] .

Thus, the associated T -time and 2 T -time Poincaré maps are defined through

( u 1 , v 1 ) ≔ P 1 ( u 0 , v 0 ) ≔ ( u ( T ) , v ( T ) ) = ( u 0 e ( 1 − v 0 ) λ A 1 , v 0 e ( − 1 + u 1 ) λ B 1 )

and

( u 2 , v 2 ) ≔ P 2 ( u 0 , v 0 ) = P 1 ( u 1 , v 1 ) = ( u 1 e ( 1 − v 1 ) λ A 1 , v 1 e ( − 1 + u 2 ) λ B 1 ) = ( u 0 e ( 2 − v 0 − v 1 ) λ A 1 , v 0 e ( − 2 + u 1 + u 2 ) λ B 1 ) .

It is apparent that the unique T -periodic coexistence state, i.e., the unique solution of (2.1) such that u 0 , v 0 > 0 and P 1 ( u 0 , v 0 ) = ( u 0 , v 0 ) , is equilibrium (1,1). Similarly, the 2 T -periodic coexistence states are the solutions such that u 0 , v 0 > 0 and P 2 ( u 0 , v 0 ) = ( u 0 , v 0 ) , i.e., those solutions satisfying

(2.6) u 0 , v 0 > 0 , 2 − u 0 = u 1 = u 0 e ( 1 − v 0 ) λ A 1 , 2 − v 0 = v 1 = v 0 e ( 1 − u 0 ) λ B 1 .

Thus, expressing x ≡ v 0 in terms of u 0 , setting ( A , B ) ≡ ( A 1 , B 1 ) , and adapting the corresponding argument on [36, p. 41], it is easily seen that the 2 T -periodic coexistence states are given by the zeroes of the function

φ ( x ) = x e e ( 1 − x ) λ A − 1 e ( 1 − x ) λ A + 1 λ B + 1 − 2 .

This function has been already analyzed in [36, Th. 2.1], where it was established that it possesses exactly two zeros different from the equilibrium x = 1 for every λ > 2 A B . This proves the first part of the theorem.

By iterating n times, it follows that the n T -time map is defined through

(2.7) ( u n , v n ) ≔ P n ( u 0 , v 0 ) = P 1 n ( u 0 , v 0 ) = ( u 0 e ( n − v 0 − v 1 − ⋯ − v n − 1 ) λ A , v 0 e ( u 1 + u 2 + ⋯ + u n − n ) λ B ) .

Hence, due to (2.7), a solution ( u ( t ) , v ( t ) ) of (2.1) provides us with a n T -periodic coexistence state if, and only if, u 0 > 0 , v 0 > 0 and

(2.8) n = v 0 + v 1 + ⋯ + v n − 1 , n = u 0 + u 1 + ⋯ + u n − 1 .

Thus, assuming that A = B and u 0 = v 0 = x , system (2.8) reduces to one equation (cf. [36, Le. 3.1]). Hence, setting

(2.9) E n ( λ , x ) ≔ exp n + 1 2 − x ∑ i = 0 i ∈ 2 N n − 1 E i ( λ , x ) λ A , n ∈ 2 N + 1 , exp x ∑ i = 1 i ∈ 2 N + 1 n − 1 E i ( λ , x ) − n 2 λ A , n ∈ 2 N ,

where E 0 ( λ , x ) = 1 , it follows from [36, Th. 3.3] that, for every n ≥ 1 ,

φ n ( x ) = φ n − 1 ( x ) − 1 + x E n − 1 ( λ , x )

and φ 0 ≡ 0 , where these φ n ’s are the functions whose zeroes provide us with the n T -periodic coexistence states of (2.1) that were constructed in [36]. Therefore, we are within the setting of [36], where it was inferred from these features (Sections 2–6 of [36]) the existence of, at least, n nontrivial n T -periodic coexistence states if n is even and n − 1 if n is odd. This concludes the proof.□

Note that in [36], we dealt with continuous nonnegative weight functions α and β such that

supp α ≡ 0 , T 2 and supp β ≡ T 2 , T ,

whereas in Theorem 2.1, the weight functions α and β are two arbitrary nonnegative continuous functions with disjoint supports. As the set of subharmonics obeys identical equations as in [36], it is apparent that, much like in [36], also in this more general case, the global bifurcation diagram of the positive subharmonics of (2.1) follows the general patterns sketched in Figure 1, which has been reproduced from [36]. Similarly, at the light of the analysis of [36], Theorem 2.1 establishes that the global topological structure of the bifurcation diagram sketched in Figure 1 remains invariant regardless the concrete values of t 0 1 , t 1 1 , t 2 1 , t 3 1 and the number and distribution of the components of the supports of the weight functions α ( t ) and β ( t ) on each of the intervals of the partition of [ 0 , T ] induced by these values. Naturally, much like in [36], Figure 1 shows an ideal global bifurcation diagram, for as the local behavior of most of the bifurcations from ( 1 , 1 ) is unknown, except for n ∈ { 2 , 3 , 4 } .

2.2 The case when k = ℓ = 1 and supp β 1 ⊆ [ t 0 1 , t 1 1 ]

Then,

(2.10) supp β 1 ⊆ [ t 0 1 , t 1 1 ] and supp α 1 ⊆ [ t 2 1 , t 3 1 ] ,

where

0 ≤ t 0 1 < t 1 1 ≤ t 2 1 < t 3 1 ≤ T .

Figure 4 shows a simple example within this case.

Figure 4

α and β satisfying (2.10).

Remark 2.1

On the basis of Theorem 2.1, one can obtain, very easily, solutions of (2.1) satisfying (2.10) in the interval [ 0 , T ] . Indeed, if ( u 0 , v 0 ) is the initial value to an n T -periodic solution of (2.1) for the weight distribution (2.5), then a solution with initial values ( u 0 e ( 1 − v 0 ) A 1 , v 0 ) provides us with an n T -periodic solution of (2.1) for the configuration (2.10). The next section goes further by establishing that, for the distribution (2.10), there are periodic solutions of (2.5) with initial data on the line u = v by means of similar symmetry reductions and techniques as in the proof of Theorem 2.1. Equivalently, fixing a time τ 0 ∈ ( t 1 1 , t 2 1 ) and setting

α ˜ ( t ) ≔ α ( t + τ 0 ) , β ˜ ( t ) ≔ β ( t + τ 0 ) ,

the pair ( α ˜ , β ˜ ) lies within the configuration of Figure 3. Thus, Theorem 2.1 applies. Note that this is equivalent to consider the Poincaré map with τ 0 as initial time.

The next result focuses attention into the case when condition (2.10) holds. A genuine situation where this occurs is represented in Figure 4. As this case was left outside the general scope of [36], it is a novelty here.

Theorem 2.2

Under condition (2.10), the equilibrium ( 1 , 1 ) is the unique T -periodic coexistence state of (2.1). Moreover, (2.1) possesses exactly two nontrivial 2 T -periodic coexistence states for every

(2.11) λ > 2 A 1 B 1 .

If, in addition, A 1 = B 1 , then problem (2.1) has, for every λ > 2 A 1 and n ≥ 3 , at least n − 1 nontrivial n T -periodic coexistence states with u 0 = v 0 if n is odd, whereas if n is even, then (2.1) possesses, at least, n − 2 nontrivial n T -periodic coexistence states with u 0 = v 0 , and exactly two with u 0 + v 0 = 2 .

The main difference between Theorems 2.1 and 2.2 relies on the fact that all the solutions of (2.1) when A 1 = B 1 and n is even have been constructed to satisfy u 0 = v 0 under condition (2.5), while (2.1) only admits n − 2 solutions with u 0 = v 0 and 2 solutions with u 0 + v 0 = 2 when (2.10) holds.

Proof

Since, for every ( u 0 , v 0 ) ∈ R 2 , the unique solution of (2.1) with ( u ( 0 ) , v ( 0 ) ) = ( u 0 , v 0 ) is given through

u ( t ) = u 0 e ( 1 − v ( T ) ) λ ∫ 0 t α ( s ) d s , v ( t ) = v 0 e ( − 1 + u 0 ) λ ∫ 0 t β ( s ) d s , t ∈ [ 0 , T ] ,

the T -time and 2 T -time Poincaré maps of (2.1) are given by

( u 1 , v 1 ) ≔ P 1 ( u 0 , v 0 ) ≔ ( u ( T ) , v ( T ) ) = ( u 0 e ( 1 − v 1 ) λ A 1 , v 0 e ( − 1 + u 0 ) λ B 1 )

and

( u 2 , v 2 ) ≔ P 2 ( u 0 , v 0 ) = P 1 ( u 1 , v 1 ) = ( u 1 e ( 1 − v 2 ) λ A 1 , v 1 e ( − 1 + u 1 ) λ B 1 ) = ( u 0 e ( 2 − v 1 − v 2 ) λ A 1 , v 0 e ( − 2 + u 0 + u 1 ) λ B 1 ) .

It is easily seen that ( u 0 , v 0 ) = ( 1 , 1 ) is the unique fixed point of P 1 . Moreover, a solution of (2.1), ( u ( t ) , v ( t ) ) , is a 2 T -periodic coexistence state if, and only if, u 0 > 0 , v 0 > 0 and

( u 2 , v 2 ) = P 2 ( u 0 , v 0 ) = ( u 0 , v 0 ) .

In other words,

(2.12) u 0 , v 0 > 0 , 2 − u 0 = u 1 = u 0 e ( v 0 − 1 ) λ A 1 , 2 − v 0 = v 1 = v 0 e ( u 0 − 1 ) λ B 1 .

Setting ( A , B ) ≡ ( A 1 , B 1 ) and arguing as in the proof of Theorem 2.1, it becomes apparent that the nontrivial 2 T -periodic coexistence states of (2.1) are given by the zeroes of the map

(2.13) ψ ( x ) ≔ x e 1 − e ( x − 1 ) λ A 1 + e ( x − 1 ) λ A λ B + 1 − 2

with x = v 0 ≠ 1 . By definition, it is obvious that

ψ ( x ) < 0 for all x ≤ 0 , ψ ( 1 ) = 0 , ψ ( x ) > 0 for all x ≥ 2 .

Moreover, by differentiating with respect to x , after rearranging terms, yields

ψ ′ ( x ) = e 1 − e ( x − 1 ) λ A 1 + e ( x − 1 ) λ A λ B 1 − 2 λ 2 A B x e ( x − 1 ) λ A ( 1 + e ( x − 1 ) λ A ) 2 + 1 , ψ ″ ( x ) = e 1 − e ( x − 1 ) λ A 1 + e ( x − 1 ) λ A λ B x 2 λ 2 A B e ( x − 1 ) λ A ( 1 + e ( x − 1 ) λ A ) 2 2 − 4 λ 2 A B e ( x − 1 ) λ A ( 1 + e ( x − 1 ) λ A ) 2 + 2 λ 3 A 2 B x e ( x − 1 ) λ A ( e ( x − 1 ) 2 λ A − 1 ) ( 1 + e ( x − 1 ) λ A ) 4 ,

for all x ≥ 0 . In particular,

ψ ′ ( 1 ) = 2 − λ 2 A B 2 .

Thus, owing to (2.11), ψ ′ ( 1 ) < 0 . Summarizing, (2.11) implies that

ψ ( 0 ) = − 2 < 0 , ψ ( 1 ) = 0 , ψ ′ ( 1 ) < 0 , ψ ( 2 ) > 0 .

Hence, the function ψ possesses, at least, one zeroes in each of the intervals ( 0 , 1 ) and ( 1 , 2 ) . Therefore, (2.1) has, at least, two 2 T -periodic coexistence states. Moreover, adapting the analysis carried out in [36, Sec. 2], from the previous value of ψ ″ ( x ) , it is easily seen that any critical point, x c , of ψ satisfies ψ ″ ( x c ) < 0 if x c ∈ ( 0 , 1 ) and ψ ″ ( x c ) > 0 if x c ∈ ( 1 , 2 ) . Consequently, (2.1) possesses exactly two 2 T -periodic coexistence states under condition (2.11). This ends the proof of the first assertion.

Subsequently, we assume that

(2.14) A = B , u 0 = v 0 = x .

In this case, the n T -time map is defined as follows:

( u n , v n ) ≔ P n ( u 0 , v 0 ) = P 1 n ( u 0 , v 0 ) = ( u 0 e ( n − v 1 − v 2 − ⋯ − v n ) λ A , v 0 e ( u 0 + u 2 + ⋯ + u n − 1 − n ) λ A ) .

Thus, ( u n , v n ) = ( u 0 , v 0 ) , i.e., ( u 0 , v 0 ) provides us with a subharmonic of order n ≥ 1 of (2.1), if and only if

(2.15) n = v 0 + v 1 + ⋯ + v n − 1 , n = u 0 + u 1 + ⋯ + u n − 1 .

By adapting the argument of the proof of [36, Lem. 3.1], it is easily seen that the two equations of (2.15) coincide under condition (2.14). Thus, the solutions of the system (2.15) are the zeroes of the function

ψ n ( x ) ≔ x + u 1 ( x ) + … + u n − 1 ( x ) − n , x > 0 .

Consequently, the n T -periodic coexistence states of (2.1) are given through ψ − 1 ( 0 ) .

Arguing as in the proof of [36, Prop. 3.2], it becomes apparent that

( u n , v n ) ≔ P n ( x , x ) = ( x E 2 n ( − λ , x ) , x E 2 n − 1 ( − λ , x ) )

for all n ≥ 1 . Hence,

(2.16) ( 1 , 1 ) = P n ( 1 , 1 ) = ( E 2 n ( − λ , 1 ) , E 2 n − 1 ( − λ , 1 ) )

for all n ≥ 1 . Further, by the proof of [36, Th. 3.3], we find that

ψ n ( λ , x ) = x ∑ j = 0 n − 1 E j ( − λ , x ) − n = ψ n − 1 ( x ) − 1 + x E n − 1 ( − λ , x )

for all x > 0 . Note that ψ n also depends on the parameter λ > 0 . To make explicit this dependence, we will subsequently write ψ n ( λ , x ) , instead of ψ n ( x ) , for all n ≥ 1 .

By (2.16), differentiating with respect to x and particularizing at x = 1 yields

(2.17) q n ( λ ) ≔ ∂ ψ n ∂ x ( λ , 1 ) = n + ∑ j = 1 n − 1 E ′ ( − λ , 1 ) .

Adapting the induction argument of the proof of [36, Lemma 4.1], it follows from the definition of the E j ’s that the function q n ( λ ) is a polynomial for all n ≥ 1 .

The next result relates the sequence { q n } n ≥ 1 with the corresponding sequence { p n } n ≥ 1 constructed in [36] under condition (2.5).

Proposition 2.1

For every n ≥ 1 ,

(2.18) q 2 n − 1 ( λ ) = p 2 n − 1 ( λ ) , q 2 n ( λ ) 2 + A λ = p 2 n ( λ ) 2 − A λ

and

(2.19) q n ( λ ) = [ 2 + ( − 1 ) n A ł ] q n − 1 ( λ ) − q n − 2 ( λ ) .

Proof

According to [36, (4.6)], p n is defined as follows:

p n ( λ ) ≔ ∂ φ n ∂ x ( λ , 1 ) = n + ∑ j = 1 n − 1 E ′ ( λ , 1 ) .

Thus, by (2.17), q n ( λ ) = p n ( − λ ) . By [36, Cor. 4.7], p 2 n − 1 ( λ ) and p 2 n ( λ ) 2 − A λ are even functions in λ . Hence,

q 2 n − 1 ( λ ) = p 2 n − 1 ( λ ) and q 2 n ( λ ) = p 2 n ( − λ ) = p 2 n ( − λ ) 2 − A ( − λ ) ( 2 + A λ ) = p 2 n ( λ ) 2 − A λ ( 2 + A λ ) .

This concludes the proof of (2.18).

On the other hand, by [36, Th. 4.6], we already know that

(2.20) p 2 n − 1 ( λ ) = ( 2 + A λ ) p 2 n − 2 ( λ ) − p 2 n − 3 ( λ )

and

(2.21) p 2 n ( λ ) 2 − A λ = p 2 n − 1 ( λ ) − p 2 n − 2 ( λ ) 2 − A λ .

Therefore, owing to (2.18), (2.20), and (2.21), we find that, for every n ≥ 1 ,

q 2 n − 1 ( λ ) 2 − A λ = p 2 n − 1 ( λ ) 2 − A λ = ( 2 + A λ ) p 2 n − 2 ( λ ) 2 − A λ − p 2 n − 3 ( λ ) 2 − A λ = q 2 n − 2 ( λ ) − q 2 n − 3 ( λ ) 2 − A λ

and

q 2 n ( λ ) 2 + A λ = p 2 n ( λ ) 2 − A λ = p 2 n − 1 ( λ ) − p 2 n − 2 ( λ ) 2 − A λ = q 2 n − 1 ( λ ) − q 2 n − 2 ( λ ) 2 + A λ .

So, (2.19) holds, and the proof is complete.□

As a direct consequence of (2.18), the corresponding sets of bifurcation points from the curve ( λ , x ) = ( λ , 1 ) coincide under conditions (2.5) and (2.10) as soon as u 0 = v 0 ( = x ) and A = B , except for the bifurcation point ( λ , x ) = ( 2 / A , 1 ) , because

2 / A ∈ p 2 n − 1 ( 0 ) \ q 2 n − 1 ( 0 ) for all n ≥ 1 .

Moreover, also by (2.18), the mathematical analysis carried out in Sections 5 and 6 of [36] applies mutatis mutandis to cover the case when (2.10) holds, instead of (2.5). As a byproduct, also in the case when (2.10) holds, all the zeroes of the polynomials q n ( λ ) are simple. Thus, the Crandall-Rabinowitz theorem [85] provides us with a local analytic curve of n T -periodic solutions. Moreover, since the generalized algebraic multiplicity of Esquinas and López-Gómez [87] equals one, according to [17, Th. 6.2.1] and the unilateral theorem [17, Th. 6.4.3], these local curves of subharmonics can be extended to maximal connected components of the set of n T -subharmonics of (2.1). This proves the theorem when u 0 = v 0 = x and A = B , regardless the oddity of n ≥ 1 .

Finally, assume that 2 = u 0 + v 0 and A = B . As we already know that a solution ( u ( t ) , v ( t ) ) is 2 T -periodic if, and only if,

2 = u 0 + v 0 2 = v 0 + v 1 = v 0 + v 0 e ( u 0 − 1 ) λ A ,

it becomes apparent that the nontrivial 2 T -periodic coexistence states of (2.1) are the zeroes of the function

φ 2 ( x ) = x [ e ( 1 − x ) λ A + 1 ] − 2 , x ∈ ( 0 , 2 ) \ { 1 } .

As, according to the proof of Theorem 2.1, φ 2 possesses exactly two zeroes, the proof of Theorem 2.2 is completed.□

Since, according to (2.18), we already know that

{ r ∈ q n − 1 ( 0 ) : r > 0 , n ≥ 1 } = { r ∈ p n − 1 ( 0 ) : r > 0 , n ≥ 1 } \ { 2 / A } ,

the global bifurcation diagram of subharmonics of (2.1) when (2.10), instead of (2.5), holds true, can be obtained from the global bifurcation diagram plotted in Figure 1 by removing the component of subharmonics of order two. However, even the local behavior of the corresponding components of subharmonics of order n in each of the cases (2.5) and (2.10) might be different, because, in general, φ n ≠ ψ n for all n ≥ 2 and, hence, the identity φ n − 1 ( 0 ) = ψ n − 1 ( 0 ) cannot be guaranteed.

By (2.6), 0 < v 0 < 1 if 0 < u 0 < 1 . Similarly, 1 < v 0 < 2 if 1 < u 0 < 2 . Thus, the 2 T -periodic coexistence states of (2.1) under condition (2.5) are localized in the shadowed region of the left plot in Figure 5. Moreover, by (2.12), 1 < v 0 < 2 if 0 < u 0 < 1 , and 0 < v 0 < 1 if 1 < u 0 < 2 . Thus, under condition (2.10), the 2 T -periodic coexistence states of (2.1) lye in the shadowed area of the right plot of Figure 5. This explains why (2.1) cannot admit a subharmonic of order two if u 0 = v 0 .

Figure 5

The shadowed regions on each of these figures represent the quadrants of the phase-plane containing the initial data to 2 T -periodic coexistence states of (2.1) under the conditions (2.5) (left) and (2.10) (right).