On the applicability of the poincaré–Birkhoff twist theorem to a class of planar periodic predator-prey models

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 Poincare–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 function-coefficients 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 \begin{document}$ T $\end{document} -periodic solutions of a non-degenerate model is ascertained as a perturbation parameter makes it degenerate.


1.
Introduction. This paper studies the non-autonomous Volterra predator-prey model where λ > 0 and α(t) 0, β(t) 0 are T -periodic continuous functions. This model was introduced in [21], [19] in the special case when αβ = 0 in R and it was later analyzed in [20], where the existence of mT -periodic coexistence states was established for all integers m ≥ 2. Figure 1 shows the graphs of α(t) and β(t) in a case when α(t)β(t) = 0 for all t ∈ [0, T ], as considered in the above quoted references and also in Section 3 below. An ecological justification for assuming α(t)β(t) = 0 in a subinterval of [0, T ] is discussed in the Appendix. A general Volterra predator-prey system with periodic coefficients takes the form with a, b, c, d : R → R T -periodic functions and b(t) 0, d(t) 0. If we are able to find a T -periodic coexistence statez(t) := (x(t),ỹ(t)) of (2), namely, a positive componentwise T -periodic solution of the system, then the change of variables x(t) = u(t)x(t), y(t) = v(t)ỹ(t), transforms (2) into the equivalent system which is of the same form as (1). In the Appendix, it will be shown that (2) and now we just look for nontrivial mT -periodic solutions of (4). Due to the Hamiltonian structure of the new system, a powerful tool to prove the existence of nontrivial periodic solutions is the Poincaré-Birkhoff twist fixed point theorem. Applications of this method to Volterra predator-prey systems can be found in [18], [10], [11], [26], [2], as well as in the recent ones [16] and [13]. A typical strategy of proof consists in showing that there is a sufficiently large gap in the rotation numbers between the small solutions and the large solutions of (4), which, in turns, guarantees a suitable twist property for the associated Poincaré map. In this context, increasing the value of the parameter λ > 0 in (1) may play the role of enlarging the gap in the rotation numbers. If we try to apply the above quoted results [10,11,13,16,18,26] to (1), we find that at least one of the two coefficients α(t), β(t) should be assumed to be always strictly positive and this prevents the applications to models where the kinetics may vanish in some time interval. On the other hand, for the special choice of α(t) and β(t) shown in Figure 1, the results of [19] establish that (1) does not admit a nontrivial T -periodic coexistence state, independently of the value of λ, and therefore it becomes apparent that the Poincaré-Birkhoff theorem cannot be applied to (1) for some specific coefficients like those in Figure 1. Otherwise, (1) should admit a pair of nontrivial T -periodic solutions for sufficiently large λ. Astonishingly, when the support of the product function α(t)β(t) is non-empty, as illustrated in Figure 2, then, according to our results in Section 2, the Poincaré-Birkhoff theorem establishes that, for every m ≥ 1, (1) possesses at least two mTperiodic coexistence states for sufficiently large λ. Moreover, the number of these periodic solutions increases with λ. And this regardless the length of the support of αβ! Thus, a rather natural question arises. What's going on with the (nontrivial) Tperiodic coexistence states of (1) when α(t)β(t) does approximate zero? In Section 4 we analyze what happens by considering α(t) and β(t) like in Figure 1 and the perturbed problem where α ε (t) := α(t) t ∈ [0, T 2 ] εϕ(t) t ∈ ( T 2 , T ] for some function ϕ, as plotted in Figure 3. According to our results of Section 2, based on the Poincaré-Birkhoff theorem, for every ε > 0 and m ≥ 1, there exists λ m (ε) > 0 such that (5) admits at least two mT -coexistence states for each λ > λ m (ε). Our main result in Section 4 establishes that, setting λ * 1 (ε) := inf{λ > 0 : (5) has at least one T -periodic coexistence state}, either lim or the v-component of any T -periodic coexistence state of (5), (u(t, ε), v(t, ε)), blows-up in [ T 2 , T ] as ε ↓ 0, which explains why the degenerate problem when αβ = 0 cannot admit any nontrivial T -periodic coexistence state.
The high multiplicity of subharmonics in these T -periodic predator-prey prototype models contrasts, very strongly, with the uniqueness of coexistence states established for their diffusive one-dimensional counterparts in [22], [6], [9], [23]. This confirms that the non-cooperative structure of all these models does not provoke the same dynamical effects on the dynamics of the underlying systems in the presence of temporal heterogeneities than in the presence of spatial heterogeneities.
2. An application of the Poincaré-Birkhoff theorem. In this section we consider the non-autonomous planar Hamiltonian system where λ > 0 and α 0 and β 0 are T -periodic continuous functions such that In model (6), λ is regarded as a real parameter, and f, g ∈ C(R) are locally Lipschitz functions such that f, g ∈ C 1 on a neighborhood of the origin and Moreover, either f , or g, satisfies, at least, one of the following conditions: Under these conditions, (x, y) = (0, 0) provides us with a steady-state solution of (6). Throughout this paper, for any given R > 0 and z = (x, y) ∈ R 2 , we will denote by D R (z) the disc of radius R centered at z. Although the next result is a direct consequence of the Lipschitz dependence of the solutions of (6) with respect to the initial conditions, we will provide with a short self-contained proof of it.
Since δ(m, ε, λ) ↓ 0 as m ↑ ∞ cannot be excluded, (0, 0) is not necessarily stable in the sense of Lyapunov towards the right.
Proof. Let us fix K > max{f (0), g (0)} and take δ 1 > 0 such that Without loss of generality, we suppose that (x(0), y(0)) = (0, 0) and hence (x(t), y(t)) is a nontrivial solution. Introducing polar coordinates, we have that Thus, differentiating with respect to t, it follows from (6) that for all t ≥ 0 for which the solution is defined, say t ∈ I := [0, T max ). Hence, Then, considering the T -periodic function it is apparent that, for every t ∈ I such that (x(s), y(s)) ∈ D δ1 for all s ∈ [0, t], the following holds: This, in turn, implies Therefore, as long as (x(s), y(s)) ∈ D δ1 for all s ∈ [0, t] with t ∈ I, we find that and choose ρ 0 := ρ(0) such that Then, ρ(t) < min{ε, δ 1 } ≤ ε for all t ∈ [0, mT ], which ends the proof. Remark 1. We observe that condition (8), together with α, β 0, imply that if (x(t), y(t)) is a solution of (6) which is not globally defined in the future, then for its maximal interval of existence [0, T max ), it happens that x 2 (t) + y 2 (t) → +∞ as t → T − max and both x(t) and y(t) have infinitely many zeros accumulating at T max . Indeed, assume, for instance that x(t) > 0 and y(t) > 0 for all t ∈ [t 0 , T max ), for some t 0 < T max . Then, from the first equation in (6) we have that x(t) is non-increasing and therefore 0 < x(t) ≤ x(t 0 ). Hence x(t) is bounded and then from the second equation in (6) y is bounded and hence y(t) is bounded as well.
Since we have find that both x(t) and y(t) are bounded on [t 0 , T max ), this gives a contradiction. From the above argument, we can infer that any solution entering the first quadrant must leave it and enter the second one. With a similar proof we can see that it is impossible for a blow-up solution to stay in the second quadrant and hence, it must enter the third one after some time. Repeating this process it becomes apparent that any blowing-up solution of (6) must perform infinitely many turns around the origin as t → T − max . As we will see in the second part of the proof of Theorem 2.2, such a situation about solutions presenting blow-up with infinitely many rotations is not possible by condition (9). Therefore, all the solutions of (6) are globally defined in the future (and also in the past).
Our main existence result in this section, that is Theorem 2.2, is a direct consequence of the Poincaré-Birkhoff theorem as presented in [8], [15] and [26] (see also [3,Remark 1] for a short discussion about this topic). It establishes the existence of, at least, two mT -periodic solutions of (6) for every m ≥ 1 provided λ is sufficiently large.
To be more specific, to any nontrivial solution (x(t), y(t)) of (6) with (x(0), y(0)) = z 0 = (0, 0) we associate an angular polar coordinate θ(t) and, for a given interval [0, mT ], we define a rotation number The rotation number is an algebraic counter of the counterclockwise turns of the solution (x(t), y(t)) around the origin during the time-interval [0, mT ]. The version of the Poincaré-Birkhoff fixed point theorem that we use in this paper is the following one. This is essentially an application of W.Y. Ding version of the twist theorem for planar annuli (see [12]), as presented also in [24, Theorem A] (see also [3] for an application of the same version of the theorem). As observed in [11, §3] once that for some m ≥ 2 we have a mT -periodic solution (x, y), then, for each j = 1, 2, . . . , m−1, also (x j , y j ) is a mT -periodic solution of the same system, for We consider all these solutions as equivalent and we say they belong to the same periodicity class.
The information about the rotation number provided by Theorem 2.1 is crucial for different reasons. First we notice that solutions with different associated rotation numbers are essentially different since as paths in R 2 \{(0, 0)} are not homotopically equivalent. Moreover, the number k is useful in order to prove the minimality of the period of the solutions. More precisely, if k is relatively prime with m, then, as consequence, the mT -periodic solutions (x(t), y(t)) that we find cannot be Tperiodic for some integer < m. Indeed, if, by contradiction, we suppose that (x(t), y(t)) is T -periodic for some = 1, . . . , m − 1, then the rotation number associated to the solution in the interval [0, T ] must be an integer, say k 1 . Then, by the obvious additivity property of the rotation numbers, we obtain, for z 0 = (x(0), y(0)), which contradicts the fact that gcd(k, m) = 1. This also shows that, in particular, for k = 1 we have a periodic solution of minimal period mT.
Theorem 2.2. For every integers k ≥ 1 and m ≥ 1, there exists a constant λ k m > 0 such that (6) admits, at least, two mT -periodic solutions of rotation number k, for each λ > λ k m .
Proof. In order to apply the version of the Poincaré-Birkhoff theorem given in Theorem 2.1 we should show that, near the origin, the solutions have a rotation number greater than k, while, far from the origin, the solutions cannot complete one turn. First, we will focus our attention on small solutions. By (8), there exists a constant η > 0 such that min{f (0), g (0)} > η.
By the limit definition, we claim that, for sufficiently small ζ ∼ 0, say for |ζ| ≤ ε, Let m ≥ 1 be an integer that we fix from now to the end of the proof. Given ε as above, for any λ > 0 we can apply Proposition 1 and find a constant δ λ = δ(m, ε, λ) > 0 such that if (x(t), y(t)) is any solution of (6) with initial point in D δ λ , then (x(t), y(t)) ∈ D ε for all t ∈ [0, mT ]. For these solutions with initial value in D δ λ \ {(0, 0)} we introduce the angular polar coordinate θ(t). Since, locally and up to an additive constant, differentiating with respect to time and using (6) yields for all t ∈ [0, mT ]. Hence, owing to (11), it is apparent that On the other hand, by the continuity of α, β in t 0 , it follows from (7) that there exist ω > 0 and τ > 0 such that Thus, for every t ∈ J + iT , with i = 0, 1, . . . , m − 1, (12) implies that θ (t) ≥ ληω.
Hence, we have that Therefore, At this point, we take an arbitrary (but fixed) λ > λ k m . Then there exists r 0 > 0 with r 0 < δ(m, ε, λ) such that, for every Now, besides the integers m and k, also the constant λ > 0 is fixed and we proceed by showing that the rotation number for large solutions is less than one.
In order to prove that the solutions of (6) with sufficiently large initial data, (x 0 , y 0 ), cannot complete a rotation in the interval [0, mT ], we argue as follows. Suppose that (9) holds with g satisfying (g − ). We shall consider only this situation as the other ones are completely symmetric.
Clearly, a solution making at least one turn around the origin must cross completely the second or the third quadrant. Indeed, if the initial point z 0 is in the first or in the fourth quadrant and rot(z 0 ; [0, mT ]) ≥ 1, then (x(t), y(t)) must cross completely both the second and the third quadrants; if z 0 is in the second (respectively third) quadrant, then the trajectory must cross completely the third (respectively the second) one. We shall show that these cases are prevented for large solutions, by finding some general bounds for a generic solution (x(t), y(t)) on the third or the second quadrant.
As a first case, we suppose that (x(t), y(t)) is a nontrivial solution which crosses entirely the third quadrant. Then, there exists an interval [t 1 , . Recalling also the sign condition for g(x) given in (9), there exists a constant, M > 0, such that Then it is clear that, for every t ∈ [t 1 , t 2 ], we have that

Thus, setting
we also find that

Hence, denoting
and we conclude that if (x(t), y(t)) is any solution of (6) such that in the third quadrant, then such a solution cannot cross entirely the third quadrant during a time interval containingt. Analogous estimates allow to find exactly the same constants such that the same non-crossing property holds with respect to the second quadrant where x ≤ 0 as well.
To complete our analysis, we now look at what happens in the other quadrants. So, let s 0 be such that x(s 0 ) = 0 and 0 < y for all t in the same interval. Integrating the first equation on [t, s 0 ] ⊆ [s 1 , s 0 ] yields to an upper bound for x(t), namely |x(t)| ≤ λN 1 mA. Now, taking R 1 as is any solution of (6) such that x 2 (t) + y 2 (t) > R 2 1 for somet ∈ [0, mT ], with (x(t), y(t)) in the first quadrant, then such a solution cannot cross entirely the second or the third quadrant.
As a final case, let us suppose now there exists s 2 such that y(s 2 ) = 0 and 0 < x(s 2 ) ≤ R 1 . Let also [s 3 , s 2 ] ⊆ [0, s 2 ] be a maximal interval such that x(t) ≥ 0 and y(t) ≤ 0 for all t ∈ [s 3 , s 2 ]. Arguing as above (and using the fact that now both x and y are non-decreasing on [s 3 , s 2 ]), we define In this manner, |g(x(t))| ≤ M 1 for all t ∈ [s 3 , s 2 ] and thus we obtain a constant , y(t)) in the fourth quadrant, then such a solution cannot cross entirely the second or the third quadrant.
Therefore, the solutions of (6) with Notice that the above proof also ensures that all the solutions are globally defined, as anticipated in Remark 1. By (14) and (16) the twist condition holds and hence, according to Theorem 2.1, the system (6) has, at least, two mT -periodic solutions with rotation number k, for each λ > λ k m . This ends the proof.
Remark 2. Theorem 2.2 when applied for m = 1, guarantees the existence of 2k geometrically distinct T -periodic solutions when λ > λ k 1 . Indeed it provides at least two (nontrivial) T -periodic solutions with rotation number j for each integer j = 1, 2, . . . , k. In view of the analysis performed in [19], [20], the condition (7) is optimal (see the next section for a detailed discussion). System (6) can be obtained from a Volterra predator-prey equation via a change of variables. In this case g(x) = e x − 1 and f (y) = e y − 1 and therefore both (f − ) and (g − ) are satisfied. This simplifies the second part of the above proof.
We stress that Theorem 2.2 considers a situation which is not covered in the previous works dealing with periodic solutions of non-autonomous Volterra equations [18], [11], [26], [16]. For instance, we do not assume monotonicity of α(t)f (·) or β(t)g(·) for all t; indeed such functions can identically vanish on some time intervals.
Remark 3. As a final observation, we should mention the fact, without any significant change in the proof, a slightly more general version of Theorem 2.2 could be proved, by assuming f, g only continuous (and not locally Lipschitz) and replacing the condition on the derivatives in (8) with the following one To this aim, instead of Theorem 2.1, one can apply a generalized version of the Poincaré-Birkhoff theorem due to Fonda and Ureña [17] for Hamiltonian systems where the uniqueness of the solutions of the initial value problems is not required (see also [14,Theorem 10.6.1] for the precise statement). Since in our application to the Volterra system, f and g are smooth functions, we preferred to state our theorem in this case, instead of considering the most general situation.
3. A critical model outside the Poincaré-Birkhoff setting. In this section we will show how condition (7) is optimal for the validity of Theorem 2.2. The non-autonomous Lotka-Volterra predator-prey model where λ > 0 and α(t) 0, β(t) 0 are T -periodic continuous functions such that αβ = 0 in R, was introduced in [21] and later analyzed in [19] and [20], where the existence of mT -periodic coexistence states was analyzed for all integers m ≥ 1. More precisely, in the rest of this paper α(t) and β(t) are T -periodic non-negative functions such that Naturally, in searching for coexistence states of (17), i.e., componentwise positive solutions (u, v), as explained in the Introduction, we can perform the change of variables x = log u, y = log v, which transforms (17) into the next Hamiltonian system which fits within the abstract setting of Section 2 with f (y) = e y − 1, g(x) = e x − 1, except for the crucial fact that, since αβ = 0, the condition (7) fails to be true in the present context. As a consequence of the next result, we have that that the Poincaré-Birkhoff cannot be applied to the problem (18) to infer the existence of, at least, two T -periodic nontrivial solutions. Therefore, the condition (7) is condition sine qua non for the validity of Theorem 2.2.
As a byproduct, since (17) cannot admit a coexistence state with minimal period T , the thesis of Theorem 2.2 fails to be true for the problem (18). This entails that the small solutions of (18) cannot rotate around the origin, for as, otherwise, the Poincaré-Birkhoff theorem could be applied to guarantee the existence of, at least, two T -periodic coexistence states. As the remaining assumptions of Theorem 2.2 hold true for model (18), it becomes apparent that condition (7) is imperative for the validity of Theorem 2.2. However, by Theorem 2.1 of [20], the problem (17) possesses some 2T -periodic coexistence state if, and only if, and, actually, in such case it has exactly two. More generally, it has been established in [20] that, in the special case when for every integer m ≥ 2, (17) possesses, at least, 2ν m mT -periodic coexistence states of (17) for each λ > 2 √ AB , where By Lemma 3.1 and a direct analysis which can be performed for m = 2, it seems that not all these coexistence states can be obtained as a direct application of the version of the Poincaré-Birkhoff theorem given in Section 2, as it occurs when αβ 0. Figure 4 provides us with a sketch of the global bifurcation diagram of subharmonics of order m of (17) for all 2 ≤ m ≤ 13; it has been borrowed from [20]. It should be noted that Figure 4 does not represents the true nature of the local bifurcations from (1, 1) of the subharmonics of order m ≥ 2. Indeed, by [20, Theorem 6.1], the local bifurcations of the 2T , 3T and 4T -periodic coexistence states is where λ > 0 and α(t), β(t) satisfy the same general assumptions as in Section 3, and, for sufficiently small ε > 0, where ϕ is a T -periodic function such that By technical reasons, β and ϕ are assumed to be differentiable functions in [ T 2 , T ] such that β(t) > 0, ϕ(t) > 0, for all t ∈ ( T 2 , T ) and Under these assumptions, (22) fits within the abstract setting of Section 2 by performing the change of variables x = log u, y = log v.
Similarly, if u(t 0 ) = 1, then This ends the proof of Part (a).
To prove Part (c), suppose that v (t 0 ) = 0 for some t 0 ∈ ( T 2 , T ). Then, by Part (b), u(t 0 ) = 1. Thus, by Part (a), u (t 0 ) = 0. Hence, differentiating the v-equation Similarly, u (t 0 ) = 0 implies that Therefore, any critical point of v, or u, is a quadratic maximum, or minimum, which ends the proof of Part (c). Part (d) is a byproduct of this property.
Finally, since β ≡ 0 on the interval [0, T 2 ], by (26) ), which shows Part (e) and ends the proof of the lemma.
The next result provides us with the non-degeneration of the T -periodic solutions with respect to the equilibrium (1, 1).
The last assertion follows readily from the second identities of (26) and (28). This ends the proof.
We are ready to prove the main theorem of this section.
Proof of Theorem 4.1: Suppose that Alternative (a) does not occur. In this case, there exists λ > 0 such that there are coexistence states of (22), for sufficiently small ε > 0. Accordingly, let (u(t, ε), v(t, ε)) denote a family of coexistence states of (22) (for a fixed λ > 0), for ε > 0 small (0 < ε < ε 0 ). We must prove that v(t, ε) blows-up in a finite time as ε ↓ 0. To accomplish the proof we will distinguish between several different cases.

Figure 5.
Behavior of u(t, ε n ) and v(t, ε n ) in Case 1.A for small n.
Nevertheless, without lost of generality, choosing an appropriate subsequence if necessary, we can assume that because, thanks to Lemma 4.3, r max ≤ T 2 cannot occur and, by periodicity, we cannot have r max ≥ T . Thus, there exists an integer n 0 such that r(ε n ) > T 2 for every n ≥ n 0 . This is the situation illustrated in Figure 6. Consequently, u( T 2 , ε n ) > 1 for sufficiently large n, which implies, by Lemma 4.2 (e), that v ( T 2 ) > 0 as illustrated by Figure 6. So, according to the properties established by Lemma 4.2, in Case 1.A the graphs of u(t, ε n ) and v(t, ε n ) for sufficiently large n should be like sketched in Figure 6, where the number of nodes of 1 − u and 1 − v have been represented in a situation of minimal complexity. On the other hand, due to (24), the l'Hôpital rule yields Thus, the next estimate holds Moreover, according to (22), the next identity holds for all t ∈ [ T 2 , T ]. Using the estimate (31) and integrating (32) in [ T 2 , r(ε n )], we find that m ε n r(εn) We are choosing the lower estimate in (31) because u (t, ε n )(−1 + u(t, ε n )) u(t, ε n ) < 0 for all t ∈ ( T 2 , r(ε n )) and n ≥ n 0 . Thus, developing the above integrals, shows that m ε n − log u(r(ε n ), Consequently, for sufficiently large n, the next estimate holds Now, we claim that there exists an integer n 1 such that for all n ≥ n 1 . Indeed, by Lemma 4.3, we have that, for sufficiently large n,
Therefore, the components v(·, ε n ) blow up at r max as n → ∞.
According to Lemma 4.2 (e), since u( T 2 , ε n ) ≤ 1 − ρ < 1, it is apparent that v ( T 2 , ε n ) < 0, as illustrated in Figure 7. By the properties established by Lemma 4.2, the graphs of u(t, ε n ) and v(t, ε n ) for sufficiently large n look like shows Figure 7. As in the previous cases, the number of nodes of 1 − u and 1 − v have been represented in a situation of minimal complexity to be T -periodic. Let s(ε n ) be the very last node of u(·, ε n ) − 1 in [0, T ]. Since r(ε n ) < s(ε n ), it is clear that s(ε n ) ∈ ( T 2 , T ) for sufficiently large n. Finally, let z(ε n ) denote the very last node of v(t, ε n ) − 1 (see Figure 7).
Indeed, by Lemma 4.3, there exists an integer n 0 such that, for every n ≥ n 0 , Thus, the first estimate of (37) holds. Similarly, since for every n ≥ n 0 the second estimate of (37) also holds. Consequently, arguing as in the previous case, the estimate (36) provides us with the next one In other words, the components v(·, ε n ) blow up at s max .
When, in addition, u 0 (ε n ) > 1 for all n ≥ 1, arguing as in the Subcase 1.A, it is easily seen that v(t, ε n ) blows-up as n → ∞. Figure 8 shows the minimal complexity of T -periodic components in this case; a(ε n ) stands for the first node of v(·, ε n ) − 1. If instead of u n (ε n ) < 1, we assume that u 0 (ε n ) < 1 for all n ≥ 1, then adapting the proof of the Subcase 1.B, it is apparent that v(t, ε n ) blows-up as n → ∞. Figure  9 shows the minimal complexity of T -periodic components in this particular case. Figure 9. Admissible u(t, ε n ) and v(t, ε n ) in Case 2 for large n.

Subcase 4.A:
Suppose, in addition, that u 0 (ε n ) > 1 for each n ≥ 1, like in Figure  10. According to Lemma 4.2, 1 − v(·, ε n ) has finitely many zeroes in the interval ( T 2 , T ). Figure 10 shows a case with exactly one zero. Necessarily, the number of zeroes of 1 − v(·, ε n ) is odd. Indeed, if it would be even, 1 − v(·, ε n ) would have an odd number of critical points, which entails the existence of an odd number of zeroes of the function 1 − u(·, ε n ) in the interval ( T 2 , T ). But this would contradict the periodicity of u. Therefore, 1 − v(·, ε n ) has an odd number of zeroes. Now, adapting the argument of the Subcase 1.A, it readily follows that v blows-up as n → ∞. Subcase 4.B: Now, suppose that, in addition, that u 0 (ε n ) < 1 for all n ≥ 1. then, the same argument of the previous case shows that 1 − v(·, ε n ) has an odd number of zeroes in the interval ( T 2 , T ), like shown in Figure 11. The blow-up of v(·, ε n ) as n → ∞ can be shown by adapting the argument of the Subcase 1.B. This ends the proof. Appendix: On the Volterra model with periodic coefficients. The classical predator-prey model is a planar system of the form proposed by Vito Volterra in 1926 to explain some statistics (provided by the biologist Umberto D'Ancona) about the percentage of selachians in the north Adriatic sea (see [4], [25]). The main assumptions of the model involve the fact that all the coefficients a, b, c, d are positive. In this manner, in the absence of predators, the prey population x(t) is supposed to grow in a Malthusian way, exponentially, while, in absence of the prey, the predator population decays exponentially. According to [25], "the term xy can be thought of as representing the conversion of energy from one source to another: bxy is taken from the prey and dxy accrues to the predators".
This model is also named after Alfred J. Lotka, who in 1920 proposed it for mimicking an hypothetical chemical reaction mechanism. A natural, more realistic, extension of (A1) considers the fact that all the dynamics occurs in the frame of a periodic (seasonally varying) environment. Incorporating seasonal effects to the prototype model (A1), one is naturally lead to where a, b, c, d : R → R are periodic coefficients of a common period T > 0. Although the intrinsic growth rates and the decay coefficients may change during the time interval [0, T ], to be consistent with the original hypotheses in the Volterra model, one has to assume the positivity of the averages of these coefficients. Similarly, to be consistent with the fact that x(t) and y(t) represent, respectively, the prey and the predators, requires to impose that The fact that b(t) and d(t) may vanish on a subinterval of [0, T ] mimics the (real) possibility that, seasonally, the predation/hunting is absent. The existence of periodic solutions for predator-prey systems with periodic coefficients was established by Cushing in [7] and Butler and Freedman [5] using bifurcation techniques. The next result provides a general existence principle for coexistence states of (A2) under a minimal set of assumptions. In this result, we restrict ourselves to consider the case of continuous coefficients, in the same vein as in the previous sections, though the same result holds for general T -periodic coefficients in L 1 (0, T ). Proof. Suppose that (A2) has a componentwise positive T -periodic solution, (x(t), y(t)). Then, by the uniqueness of the associated Cauchy problem,x(t) > 0 and y(t) > 0 for all t ∈ [0, T ] and hence, for every t ∈ [0, T ], Therefore, integrating on [0, T ] establishes the necessity of (A4) for the existence of a coexistence state. Conversely, let us assume (A4) and re-write the system (A2) in the form x = xP (t, y), y = yQ(t, x), Using the fact that P (t, y) is non-increasing in y and Q(t, x) is non-decreasing in x, it becomes apparent that are satisfied. Although the second part of [11,Th. 3], the one discussing the existence of subharmonics, requires the strict monotonicity of either P (t, ·) or Q(t, ·) for all t ∈ [0, T ], its first part, not requiring any strict monotonicity, can be applied here.
Clearly, Theorem A.1 justifies the passage from the system (2) to the system (3), an hence our motivation in studying (1).
We conclude this paper with a brief additional discussion on our model, which is a prototype that allows α(t)β(t) not only vanishing on a subinterval of [0, T ] but also the possibility that the supports of α(t) and β(t) do not coincide. Although from the point of view of its applications in Population Dynamics, there is a strong debate concerning the validity of the classical predator-prey model of Lotka-Volterra type, which is (1) with α(t) and β(t) positive constants, the basic features of this model still conform the basis upon which the mathematical theory of predation is built. According to ecologists, predation is essentially consumption of one organism (the prey) by another one (the predator), in which the prey is alive when the predator first attacks it, [1]. Incorporating to the simplest prototype models saturation effects for the predators does not really change the most basic features of these models, though might make them more realistic, of course.
In nature, there are many examples in which a predator, or a prey, maintains a fairly constant density in spite of the fluctuations of its prey. In general, the huntingcollecting processes adapt to these dynamics. While within a certain period of the year the predators damage the prey population by hunting their individuals and collecting them on their stocks, in another period they stop hunting the preys and simply consume the ones collected, the prey being unaffected by the action of the predators during these periods. Naturally, these periods can overlap or not, which is reflected by the length of the support of the product function α(t)β(t) in model (1).
A very concrete example which fits very well within the setting of the model (1) takes place in the Mediterranean pine woods, where the Thaumetopoea pityocampa, commonly refereed to as the pine processionary, defoliates the pines before being ready to pupate. Indeed, while attacking the pines and becoming pupates, the pine processionary density stays constant, whereas the pine density fluctuates. Moreover, depending on the impact of the defoliation, the new generation of caterpillars can increase or decrease before the beginning of a new cycle. Naturally, a very severe defoliation of one generation might seriously damage the individuals of the next one. The model (1) is the simplest prototype model mimicking those real situations which are far from understood yet.