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Coexistence states for periodic planar Kolmogorov systems
Nonlinear Analysis 37 (1999) 735 – 749
Coexistence states for periodic planar Kolmogorov systems1 Anna Battauz ∗ Dipartimento di Matematica e Informatica, UniversitÃa di Udine, 33100 Udine, Italy Received 21 August 1997; accepted 2 September 1997
Keywords: Positive periodic solutions; Kolmogorov systems; Continuation theorems; Preypredator; Competing species
1. Introduction and statement of the main Lemma The object of this paper is to study the existence of coexistence states (i.e. positive and T -periodic solutions) for the two-dimensional Kolmogorov system ( 0 x = x F(t; x; y) (1) y0 = y G(t; x; y) where, once for all, we assume (H) F; G : R × (R+ )2 →R are continuous functions which are T-periodic (T ¿ 0) in the t-variable. (Here and in what follows, we denote by R+ := [0; + ∞), the set of non-negative real numbers and by R+ := ]0; +∞), the set of positive reals.) Systems of the form (1) have been widely analyzed in the literature for their signi cance in various mathematical models, generalizing the classical Lotka–Volterra planar systems. There are many articles for some speci c forms of F and G (see, for instance, [2, 4, 5, 10, 11, 15, 17, 19, 22–25]). Results for general F and G have been also obtained in [3] in the autonomous case, in [22] for competitive systems and in [13] for the predator–prey model. In some of such cases (see, e.g., [1, 2, 4, 19, 21, 22]), interesting information on the behaviour of the solutions is obtained. On the contrary, in 1 Work performed under the auspices of GNAFA and supported by a fellowship of the Faculty of Sciences of the University of Udine, 1997. ∗
E-mail: [email protected]
0362-546X/99/$ – see front matter ? 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 0 6 9 - 8
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this paper we limit ourselves only to the study of the existence of positive T -periodic solutions, but, on the other hand, we are able to consider a rather general class of systems, which generalize, under natural monotonicity hypotheses and appropriate conditions on the averages of the functions F and G, some aspects of the characteristic behaviour of the predator-prey or the competing species models. To this aim, we use a continuation result, which is a slightly modi ed formulation of Lemma 1 in [7], where N -dimensional Kolmogorov systems, with N ≥ 1, are considered. First of all, following a common approach within the degree theory, we embed system (1) into a one-parameter family of systems of the form 0 x = xF ∗ (t; x; y; ) ∈ [0; 1]; (2) 0 y = y G ∗ (t; x; y; ) where, F ∗ ; G ∗ : R × (R+ )2 × [0; 1]→R are continuous functions which are T -periodic (T ¿ 0) in the t-variable and such that F ∗ (t; x; y; 1) = F(t; x; y);
G ∗ (t; x; y; 1) = G(t; x; y)
∀t ∈ [0; T ]; ∀(x; y) ∈ (R+ )2 :
As in [9], we will often assume that for = 0 system (2) is autonomous, that is F ∗ (t; x; y; 0) = F 0 (x; y);
G ∗ (t; x; y; 0) = G 0 (x; y);
where F 0 (x; y), G 0 (x; y) : (R+ )2 → R are continuous functions. In this paper we make use of the following notation: C is the Banach space of the continuous functions f : [0; T ] → R2 , with the “sup” norm, | · |∞ . If g : R2 ⊃ D → R2 is a continuous function such that, for some open set ⊂ D, −1 g (0) ∩ is compact, then we can de ne the integer deg(g; ) := d(g; O; 0); where O is any open and bounded set with g−1 (0) ∩ ⊂ O ⊂ O ⊂ and “d” is the Brouwer degree. Note that the de nition of deg(g; ) is well-posed, as it is independent of the set O, by virtue of the excision property of the degree (see [12]). We give now a lemma about coexistence states, which is our main tool in the proof of the subsequent existence results. Lemma 1. Consider system (2) and assume RT RT 6 0; for all ∈ [0; 1]. (K0 ) | 0 F ∗ (t; 0; 0; ) dt| + | 0 G ∗ (t; 0; 0; ) dt| = (K1 ) There is M ¿ 0 such that any positive T-periodic solution (x(·); y(·)) of (2) satisÿes x(t) ≤ M; y(t) ≤ M for all t ∈ [0; T ]: (K2 ) For each positive T-periodic solution u(·) of the equation x0 = x F ∗ (t; x; 0; ); with RT ∈ [0; 1]; 0 G ∗ (t; u(t); 0; ) dt 6= 0; and; similarly; for each positive T-periodic RT solution v(·) of the equation y0 = y G ∗ (t; 0; y; ); with ∈ [0; 1]; 0 F ∗ (t; 0; v(t); ) dt 6= 0. Then; there is a compact set K ⊂ (R+ )2 that contains all the positive T-periodic solutions of system (2); that is; if (x(·); y(·)) is a positive and T-periodic solution
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of (2), for some ∈ [0; 1]; then (x(t); y(t)) ∈ K; ∀t ∈ R. Moreover; the topological degree of a suitable functional operator in C associated to system (2) does not change when the parameter runs in [0; 1]. In particular; if for = 0; system (2) is autonomous and (K3 ) deg((F 0 ; G 0 ); (R+ )2 ) 6= 0; then there exists a coexistence state for system (1). We notice that the condition in (K2 ) has to be considered as vacuously satis ed when system (2) has no semitrivial positive T -periodic solutions of the form (u; 0) or (0; v). Proof. The rst part of this lemma is a simple application of the homotopic invariance of the Leray–Schauder degree for a suitable operator equation in the function space C, which is associated to the problem (2) (see [18, 20]). For instance, we could take for a two-dimensional system of the form z 0 = Z(t; x; );
∈ [0; 1]
Rt
(3)
the operator : (z(·); ) 7→ w(·), where w(t) = z(T )+ 0 Z(s; z(s); ) ds. Note that : C × [0; 1] ⊃ dom → C is a completely continuous operator, having as xed points the T periodic solutions of Eq. (3). So we need to verify the admissibility of the homotopy that we have de ned by introducing the parameter into system (1). In other words, we prove now the existence of a compact set K ⊂ (R+ )2 where all the positive T periodic solutions of (2) are contained. Assume, by contradiction, that for any n ∈ N there is a positive T -periodic solution (xn (·); yn (·)) of (2), for some = n ∈ [0; 1], 6 [1=n; n]2 , for some t ∈ [0; T ]. Since (xn ·); yn (·)) satis es consuch that (xn (t); yn (t)) ⊂ dition (K1 ), we have that, for n ¿ M , mint∈[0; T ] xn (t)¡1=n or mint∈[0; T ] yn (t)¡1=n and that the derivatives x0 (·) and yn0 (·) are uniformly bounded by a constant, say LM , where L ≥ max{sup{|F ∗ (t; x; y; )| : t ∈ [0; T ]; (x; y) ∈ [0; M ]2 ; ∈ [0; 1]}; sup{|G ∗ (t; x; y; )| : t ∈ [0; T ]; (x; y) ∈ [0; M ]2 ; ∈ [0; 1]}}. So it is possible, owing to the AscoliArzela theorem, to extract a subsequence that converges to a non-negative function (u(·); v(·)) with mint∈R u(t) = 0 or mint∈R v(t) = 0. Passing to the limit as n→∞ into the relations satis ed by (xn (·); yn (·)), we obtain that (u(t); v(t)) is a T -periodic solution of system (2), for some = , such that 0 ≤ u(t) ≤ M and 0 ≤ v(t) ≤ M for all t ∈ [0; T ]. MoreRT RT over, as by the T -periodicity, 0 = 0 xn0 (t)=xn (t) dt = 0 F ∗ (t; xn (t); yn (t); n ) dt as well RT 0 RT ∗ as 0 = 0 yn (t)=yn (t) dt = 0 G (t; xn (t); yn (t); n ) dt; passing to the limit, we have Z T Z T F ∗ (t; u(t); v(t); ) dt = G ∗ (t; u(t); v(t); ) dt = 0: (4) 0
0
Now, we observe that, if mint∈[0; T ] u(t) = u(t∗ ) = 0, then u(t) = 0 for all t ∈ [0; T ], because, if t ∗ : t∗ ≤ t ∗ ¡t∗ + T is a maximum point for the function u(·), then R t∗ R t∗ xn (t ∗ ) = xn (t∗ )exp( t∗ xn0 (t)=xn (t) dt) = xn (t∗ ) t∗ F ∗ (t; xn (t); yn (t); n ) dt ≤ xn (t∗ )LT , and so, passing to the limit as n→∞, we have that maxt∈[0; T ] u(t) = u(t ∗ ) = 0. Likewise, if mint∈R v(t) = 0, then v(t) = 0 for all t ∈ [0; T ]. So, if u = v = 0, then (4) contradicts (K0 ). On the other hand, if u = 0 and v ¿ 0 or u ¿ 0 and v = 0, we obtain a
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contradiction between (4) and assumption (K2 ). Thus, we have proved that there is a compact set K ⊂ (R+ )2 containing all the possible positive and T -periodic solutions of (2), for each ∈ [0; 1]. Then the homotopic invariance of the Leray–Schauder degree (denoted here by D) implies that D(I − (·; ); A ; 0) is constant with respect to , where A = {(x; y) ∈ C: (x(t); y(t)) ∈]; −1 [2 ; ∀t ∈ [0; T ]}, with ¿ 0 such that K ⊂ ]; −1 [2 ⊂ (R+ )2 . Concerning the nal part of this result, a continuation theorem in [9] reduces the estimate of the Leray–Schauder degree of the operator associated to problem (1) to the estimate of the Brouwer degree for the function (F 0 ; G 0 ). Indeed, if for = 0 system (2) is autonomous, then D(I − (·; ); A ; 0) = d((xF 0 (x; y); yG 0 (x; y)); ]; −1 [2 ; 0) = deg((F 0 ; G 0 ); (R+ )2 ). So, if condition (K3 ) holds, we have that D(I − (·; 1); A ; 0) 6= 0 and this ensures the existence of positive and T -periodic . solutions for system (1). We conclude this section with a list of some further notations to be used in the next results. If H : R × (R+ )2 → R is a continuous function, T -periodic in the t-variable, we set Z 1 T H (t; z) dt; H (z) = T 0 the mean value in a period T of the function H (·; z), for a given z ∈ R2 . The same notation h is used for the mean value of a continuous and T -periodic function h : R → R. As we said above, we are going to require often in the applications some monotonicity properties for system (1). With this respect, note that in this paper a increasing (resp. decreasing) function is just a weakly increasing (resp. weakly decreasing) function. In the sequel, the following property of the scalar equation, u0 = uh(t; u);
(5)
where h : R × R → R is a continuous function, which is T -periodic in the t-variable, will be used: Assume that there are a constant L ¿ 0 and a continuous and T -periodic function
: R → R such that h(t; s) ≤ (t) and
Z 0
T
for all t ∈ [0; T ] and s ≥ L Z
h(t; 0) dt ¿ 0 ¿
T
0
(t) dt:
Under this assumption, that we call of Landesman–Lazer type (cf. [20, Ch. VI]) it is possible to prove that (5) has a minimal and a maximal positive T -periodic solutions uL and uM such that any positive T -periodic solution u˜ of (5) satis es ˜ ≤ uM (t) uL (t) ≤ u(t)
for all t = [0; T ]:
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It is also easy to see that if h(t; ·) is decreasing for all t ∈ R and strictly decreasing for at least a tˆ ∈ [0; T ], then uL = uM , that is the positive T -periodic solution of (5) is unique. 2. Applications of Lemma 1 We use now Lemma 1 in order to prove the existence of positive and T -periodic solutions of system (1), under suitable assumptions. The idea is that of applying the rst part of Lemma 1 several times, thereby constructing appropriate homotopies, which verify conditions (K0 ), (K1 ) and (K2 ). So we reduce every time our problem to an easier one until we come, by the last homotopy, to an autonomous system satisfying also condition (K3 ). At this point we can apply Lemma 1 entirely and ensure the existence of at least one coexistence state for (1). As a rst application of Lemma 1 we study system (1) under hypotheses which describe some typical aspects of the model of two competing species. Theorem 1. Consider system (1) and suppose that the functions F(t; x; ·); G(t; ·; y) are decreasing and that F(t; ·; 0) and G(t; 0; ·) satisfy the Landesman–Lazer type assumptions. Let uM and uL be the maximal and the minimal positive T -periodic solutions of the equation x0 = x F(t; x; 0); and; respectively; vM and vL the maximal and the minimal positive T -periodic solutions of the equation y0 = y G(t; 0; y). Assume also Rthat RT T (I) 0 F(t; 0; vM (t)) dt ¿ 0 and 0 G(t; uM (t); 0) dt ¿ 0 RT RT (II) 0 F(t; 0; vL (t)) dt ¡ 0 and 0 G(t; uL (t); 0) dt ¡ 0: Then there exists at least one T -periodic positive solution of Eq. (1). RT RT Proof. By the assumptions, we have that 0 F(t; 0; 0) dt ¿ 0; 0 G(t; 0; 0) dt ¿ 0 and there exists L ¿ 0 such that, for all t ∈ R and for all x; y ≥ L; F(t; x; 0) ≤ (t); G(t; 0; y) RT ≤ (t), where ; : R → R are continuous and T -periodic functions with 0 (t) dt RT ¡ 0; 0 (t) dt ¡ 0. To apply Lemma 1, assume condition (I) and let F ∗ (t; x; y; ) = F(t; x; y), G ∗ (t; x; y; ) = G(t; x; y); F ∗ ; G ∗ : R × (R+ )2 × [0; 1] → R are continuous and T -periodic functions in the t-variable, which preserve the monotonicity properties of F and G. We can verify immediately (K0 ) of Lemma 1; as to (K1 ), if (x(·); y(·)) is a positive T periodic solution of (2) for some ∈ [0; 1], then we obtain, supposing by contradiction RT RT that x(t) ≥ L for all t ∈ [0; T ], that 0 = 0 F ∗ (t; x(t); y(t); ) dt ≤ 0 F(t; x(t); 0) dt ¡ 0, which is absurd. So there is t1 ∈ [0; T ] such that x(t1 ) ¡ L; if T = {t ∈ R: x(t) ¡ L} and t ∗ ∈ [t1 ; t1 + T ] is a maximum point for the function x(·), we have that L is a bound for R t∗ R x(·) if t1 = t ∗ , otherwise log x(t ∗ )=x(t1 )= t1 F ∗ (t; x(t); y(t); ) dt ≤ [t1 ; t ∗ ]∩T F ∗ (t; x(t); 0; R ) dt + [t1 ; t ∗ ]∩Tc F ∗ (t; x(t); 0; ) dt ≤ |F ∗ |L; 0 T + | |∞ T , where |F ∗ |L; 0 = sup{|F ∗ (t; x; 0; )|: t ∈ [0; T ]; x ∈ [0; L]; ∈ [0; 1]} and | |∞ = sup{| (t)|: t ∈ [0; T ]}. This implies |x|∞ ¡ L exp((|F ∗ |L; 0 + | |∞ ))T ). Likewise |y|∞ ¡ L exp((|G ∗ |0; L + | |∞ )T ). We check now that condition (K2 ) is satis ed: if u(·) is a positive T -periodic solution of the
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equation x0 = xF ∗ (t; x; 0; ) for some ∈ [0; 1], we have that u0 = uF(t; u; 0); therefore RT RT uL (t)≤ u(t) ≤ uM (t) for all t ∈ [0; T ] and 0 G ∗ (t; u(t); 0; ) dt = 0 G(t; u(t); 0) dt RT ≥ 0 G(t; uM (t); 0) dt ¿ 0, owing to condition (I). Likewise we obtain that all the positive and T -periodic solutions v(·) of the equation y0 = yG ∗ (t; 0; y; ); ∈ [0; 1], are RT such that 0 F ∗ (t; 0; v(t); ) dt ¿ 0. So there is a compact set K ⊂ (R+ )2 that contains all the positive T -periodic solutions of system (2) and the topological degree of the functional operator associated to system (2) is the same for = 1 and = 0; for this last value of the parameter , we obtain the system ( 0 x = xF(t; x; 0) y0 = yG(t; 0; y) that we reduce to an autonomous system, by the homotopy F1∗ (t; x; y; ) = F1∗ (t; x; ) = y), with 0); G1∗ (t; x; y; ) = G1∗ (t; y; ) = G(t; 0; y)+(1−)G(0; F(t; x; 0)+(1−)F(x; ∈ [0; 1]. We see immediately that system (2), obtained using the last homotopy, veri es the hypotheses of Lemma 1, as the conditions in (K0 ); (K1 ) and (K2 ) are obviously 0) satis ed. Consistent with the notation of Lemma 1, we have that F 0 (x; y) = F(x; 0 and G (x; y) = G(0; y). To complete this proof we only need to check that d((F(·; 0); ·)); ]; −1 [2 ; 0) 6= 0, for ¿ 0 small enough, that follows from d(F(·; 0); ]; −1 [; 0) G(0; −1 = d(G(0; ·); ]; [; 0) = − 1. Assume now condition (II) and consider the homotopy F1∗ (t; x; y; ) = F(t; x; y) − (1−)y; G1∗ (t; x; y; ) = G(t; x; y)−(1−)x; this homotopy determines a system (2), that satis es obviously (K0 ); (K1 ) and, owing to condition (II), (K2 ). So, for = 0 we are led to consider the system x0 = x(F(t; x; y)−y); y0 = y(G(t; x; y)−x). We construct now a second homotopy, putting F2∗ (t; x; y; ) = F(t; x; y)−y; G2∗ (t; x; y; ) = G(t; x; y)−x and apply again Lemma 1: it’s easy to check that condition (K0 ) is satis ed and that x(·); y(·), with ∈ [0; 1] are bounded at least in a t ∈ [0; T ], when (x(·); y(·)) is a positive T -periodic solution of (2). From the di erential inequalities x0 ≤ xF(t; x; 0) and y0 ≤ yG(t; 0; y), with F(t; s; 0) and G(t; 0; s) upper bounded, the existence of a M ¿ 0 such that |x|∞ ; |y|∞ ¡ M (with M independent of ) follows. So we have that F(t; x; y)−y ≤ F(t; x; 0)−y ≤ R1 −y, where R1 = sup{|F(t; x; 0)|: t ∈ [0; RT RT T ]; x ∈ [0; M ]}, and therefore 0 = 0 x0 (t)=x(t) dt ≤ 0 (R1 − y(t)) dt. This implies that there exists t ∗ ∈ [0; T ] such that y(t ∗ ) ¡ R1 and from this fact we obtain easily a bound for |y|∞ . The proof of the boundedness of x(·) is similar. In order to prove condition (K2 ), observe that, if u(·) is a positive T -periodic solution of x0 = xF2∗ (t; x; 0; ) = xF(t; RT RT x; 0) for some ¿ 0, then u ≥ uL and 0 G2∗ (t; u(t); 0) dt ≤ 0 G(t; (1=)uL (t); 0) dt ≤ RT G(t; uL (t); 0) dt ¡ 0; likewise, if v(·) is a positive T -periodic solution of the equa0 RT tion y0 = yG2∗ (t; 0; y; ) for some ¿ 0, we have that 0 F2∗ (t; 0; v(t)) dt ¡ 0 (note that for = 0 there are no positive T -periodic solutions of these equations). We can apply Lemma 1 and reduce system (2), obtained for = 0, to an autonomous system y), with the nal homotopy F3∗ (t; x; y; ) = F3∗ (t; y; ) = (F(t; 0; y) − y) + (1 − )F(0; ∗ ∗ G3 (t; x; y; ) = G3 (t; x; ) = (G(t; x; 0) − x) + (1 − )G(x; 0), (with ∈ [0; 1]), that satis es the hypotheses of Lemma 1. In fact, (K0 ) is immediately veri ed and it is easy to check that, if (x(·); y(·)) is a positive and T -periodic solution of system (2) for some
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∈ [0; 1], then |uL |∞ and |vL |∞ provide a bound to, resp., x(·) and y(·) in at least a point of [0; T ]. So we can nd a bound for both |x|∞ ; |y|∞ , since these functions satisfy the di erential inequalities x0 ≤ xF3∗ (t; 0; 0; ) and y0 ≤ yG3∗ (t; 0; 0; ). Condition (K2 ) is vacuously satis ed, because there are no positive T -periodic solutions of the equations x0 = xF3∗ (t; 0; 0; ) and y0 = yG3∗ (t; 0; 0; ), with ∈ [0; 1]. Consistent with the notation 0). In order to esti y) and G 0 (x; y) = G(x; of Lemma 1, we have that F 0 (x; y) = F(0; mate the Brouwer degree for (F(0; ·); G(·; 0)), we can choose an ¿ 0 so small that 0); ]; −1 [; 0) = −1. Therefore, since d((F(0; 0)); ]; ·); G(·; ·); ]; −1 [; 0) = d(G(·; d(F(0; −1 2 [ ; 0) = − 1 6= 0, condition (K3 ) of Lemma 1 is veri ed and we can ensure that system (1) under our assumptions has at least one T -periodic and positive solution. Remark 1. Theorem 1 in case (I) was previously obtained in [26]: here we provide a new proof of that result. On the other hand; for the case (II) there is no such a general formulation in the known literature. As observed in the introduction we note that in the special case when F(t; ·; 0) is decreasing for every t and strictly decreasing for at least a tˆ ∈ [0; T ] and the same holds with respect to G(t; 0; ·); then the maximal and minimal positive T -periodic solutions coincide. In other words; there is a unique positive T -periodic solution u0 (resp. v0 ) such that u0 = u F(t; u; 0) (resp. v0 = v G(t; 0; v)). Thus; we obtain that; if RT RT RT RT F(t; 0; 0) dt ¿ 0 ¿ 0 F(t; ∞; 0) dt and 0 G(t; 0; 0) dt ¿ 0 ¿ 0 G(t; 0; ∞) dt; then sys0 tem (1) has at least one coexistence state when Z T Z T G(t; u0 (t); 0) dt F(t; 0; v0 (t)) dt ¿ 0: 0
0
Under stronger regularity or monotonicity hypotheses on the functions F and G; this result could be obtained from previous theorems in [16; 21; 22]; where the authors reduce the problem of the existence of positive and T -periodic solutions for a twodimensional competitive system to the study of the Poincare map on a one-dimensional manifold. As in [26] we have an immediate application of Theorem 1, studying a system with periodic time-dependent coecients, which describes more familiarly the typical interaction of two competing species with logistic growth e ects. Corollary 1. Consider the system ( 0 x = x (a(t) − b(t)x − c(t)y) y0 = y (d(t) − e(t)x − f(t)y)
(6)
where a; b; c; d; e; f : R → R are continuous and T -periodic functions such that b; c; e; d; f ¿ 0 and assume that f≥ 0 and a; b; Z T Z T (a(t) − c(t)v0 (t)) dt (d(t) − e(t)u0 (t)) dt ¿ 0; (7) 0
0
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where u0 (·) and v0 (·) are the unique positive T-periodic solutions of the equations x0 = x (a(t) − b(t)x) and y0 = y (d(t) − f(t)y); respectively: Then system (6) has at least one positive T-periodic solution. Remark 2. Our assumptions leave us to apply Theorem 1 to system (6) and obtain a generalization (under weaker conditions on the coecients) of an existence result of de Mottoni and Schiano [21] (see also [23]): In fact; system (6) was studied by several authors since the work of [21] under similar hypotheses; generally stonger; on the positiveness of the coecients and their mean values (see; for example; [17; 26] and the references therein): As for Theorem 1, we apply now Lemma 1 to system (1) under hypotheses which describe some typical aspects of a predator–prey model. Theorem 2. Consider system (1) and suppose that the functions F(t; x; ·); G(t; x; ·) are decreasing; G(t; ·; y) is increasing and F(t; ·; 0) satisÿes the Landesman–Lazer type condition: Assume that Z T Z T G(t; 0; 0) dt ¡0¡ G(t; uL (t); 0) dt; 0
0
where uL (·) is the minimal positive and T-periodic solution of x0 = xF(t; x; 0): Moreover; suppose that for each M ¿ 0 there exists N ¿ 0 such that (pp)I
RT 0
F(t; x; N ) dt¡0 for all x ∈ [0; M ];
or; alternatively; (pp)II
RT 0
G(t; M; N ) dt¡0:
Then there exists at least one T-periodic positive solution of Eq. (1): RT Proof. By the Landesman–Lazer type condition, we have that 0 F(t; 0; 0) dt ¿ 0 and RT that there are L ¿ 0; : R → R a continuous and T -periodic function, with 0 (t) dt ¡ 0, such that F(t; x; 0) ≤ (t), for all x ≥ L: Assume (pp)I and consider the homotopy F1∗ (t; x; y; ) = F(t; x; y); G1∗ (t; x; y; ) = G(t; x; y); with ∈ [0; 1]: This homotopy is such that the corresponding system (2) satis es condition (K0 ) of Lemma 1 and (K1 ), since if (x(·); y(·)) is a positive and T -periodic solution of system (2) for some ∈ [0; 1]; we have that |x|∞ ¡ M = RT L exp{T (| |∞ +sup{|F(t; x; 0)|: t ∈ [0; T ]; x ∈ [0; L]})}. From 0 F(t; x(t); y(t)) dt = 0; we nd that there is a point where the function y is bounded by N , and therefore |y|∞ ¡ N exp{T sup{|G(t; M; 0)|: t ∈ [0; T ]}}. Condition (K2 ) is also veri ed, because the equation y0 = yG1∗ (t; 0; y; ) for ∈ [0; 1] has no positive T -periodic solutions and, if x is a positive T -periodic solution of x0 = xF1∗ (t; x; 0; ) for some ∈ [0; 1]; RT RT then x(t) ≥ uL (t) for all t ∈ [0; T ] and so 0 G1∗ (t; x(t); 0; ) dt ≥ 0 G(t; uL (t); 0) dt ¿ 0.
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Therefore, we can apply the rst part of Lemma 1, reducing our system to x0 = xF(t; x; y); y0 = yG(t; x; 0). Consider now a second homotopy F2∗ (t; x; y; ) = F(t; x; y), G2∗ (t; x; y; ) = G(t; x; 0) and observe that condition (K0 ) is obviously satis ed. In order to prove condition (K1 ), we note that, if (x(·); y(·)) is a positive T -periodic solution of (2) for some ∈ [0; 1], then |x|∞ ¡M and there exists RT RT t ∗ ∈ [0; T ] such that x(t ∗ )¡uL (t ∗ ) ≤ |uL |∞ ; otherwise 0 = 0 G(t; x(t); 0) dt ≥ 0 G(t; uL (t); 0) dt ¿ 0. Therefore |x|∞ ¡ M1 := |uL |∞ exp{T (| |∞ + sup{|F(t; z; 0)|: t ∈[0; T ]; z ∈ [0; M ]})}. Moreover, y(·) is bounded in at least a point of [0; T ] by the constant RT N , otherwise 0 = 0 F(t; x(t); y(t)) dt ¡ 0, since |x|∞ ¡M . Now it is easy to check that |y|∞ ¡N exp{T sup{|G(t; M1 ; 0)|: t ∈ [0; T ]}}. In order to apply the rst part of Lemma 1, we need only to verify condition (K2 ); observe that, if u(·) is a positive T -periodic solution of the equation x0 = xF2∗ (t; x; 0; ); then ∈ ]0; 1], otherwise, if RT RT = 0, 0 = 0 u0 (t)=u(t) dt = 0 F(t; 0; 0) dt, against our assumptions on the integrals of F. This means that u(·) is a positive T -periodic solution of the equation x0 = xF(t; RT x; 0), so that u(t) ≥ uL (t); for all t ∈ [0; T ]. Therefore we have that 0 G(t; u(t); 0) dt RT RT ≥ 0 G(t; uL (t)=; 0) dt ≥ 0 G(t; uL (t); 0) dt ¿ 0. Since the equation y0 = yG2∗ (t; 0; y; ) = yG(t; 0; 0) has no positive T -periodic solutions, condition (K2 ) is completely satis ed and we can apply the rst part of Lemma 1 again, reaching, for = 0, the equation x0 = xF(t; 0; y); y0 = yG(t; x; 0). Finally, in order to come to an autonomous y); system, consider the homotopy F3∗ (t; x; y; ) = F3∗ (t; y; ) = F(t; 0; y) + (1 − )F(0; 0): it is easy to check that condiG3∗ (t; x; y; ) = G3∗ (t; y; ) = G(t; x; 0) + (1 − )G(x; tions (K0 ) and (K1 ) of Lemma 1 are veri ed, since if (x; y) is a positive and T -periodic solution of system (2) then x(t1 ) ¡ |uL |∞ and, thanks to condition (pp)I , there exists a suitable N ¿ 0 such that y(t2 ) ¡ N for some (t1 ; t2 ) ∈ [0; T ]2 . From this, one immediately nds the a priori bounds. Condition (K2 ) is vacuously satis ed. Consistently with 0), so y) and G 0 (x; y) = G(x; the notation of Lemma 1, we have that F 0 (x; y) = F(0; ·); ]; −1 [; 0) = −1 that we can apply Lemma 1 and choose ¿ 0 so small that d(F(0; 0); ]; −1 [; 0) = 1. We can therefore assert that condition (K3 ) is veri ed and d(G(·; and so there exists a positive T -periodic solution for system (1). Assume (pp)II and consider the homotopy F ∗ (t; x; y; ) = F(t; x; y); G ∗ (t; x; y; ) = G(t; x; y). System (2) so obtained satis es condition (K0 ) and (K1 ), since, if (x(·); y(·)) is a positive T -periodic solution, |x|∞ ¡ M and therefore, by (pp)II and the second equation in (2), |y|∞ is bounded by a suitable constant too. Finally, condition (K2 ) is veri ed thanks to our hypotheses on the integrals of G. We can then apply the rst part of Lemma 1 in order to arrive to the system x0 = xF(t; x; 0); y0 = yG(t; x; y): Consider now another homotopy: F1∗ (t; x; y; ) = F(t; x; 0); G1∗ (t; x; y; ) = G(t; x + (1 − )uL (t); y), which satis es (K0 ), (K1 ) and (K2 ) of Lemma 1. We check here only the last condition, since the others are immediate. Assume that there are positive and T -periodic solutions of system (1) for some ∈ [0; 1] of the form (u(·); 0) or (0; v(·)); RT in the rst case we have that u(t) ≥ uL (t) for all t ∈ [0; T ] and 0 G1∗ (t; u(t); 0; ) dt ≥ RT RT G(t; uL (t); 0) dt ¿ 0; in the second case we observe that 0 F1∗ (t; 0; v(t); ) dt = 0 RT F(t; 0; 0) dt ¿ 0. We can so apply Lemma 1, reaching, for = 0, the equation 0 x0 = xF(t; x; 0); y0 = yG(t; uL (t); y). Lastly, consider the nal homotopy: F2∗ (t; x; y; ) =
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ˆ ˆ ˆ F(t; x; 0) + (1−)F(x); G ∗ (t; x; y; ) = G(t; uL (t); y) + (1−)G(y), where F(x) = F(x; RT 2 ˆ 0) and G(y) = (1=T ) 0 G(t; uL (t); y) dt. System (2) so obtained satis es (K0 ), (K1 ), thanks to assumption (pp)II , and (K2 ). We can so apply again Lemma 1 and observe ˆ ]; −1 [; 0) = −1, such that conˆ ]; −1 [; 0) = d(G; that, for ¿ 0 suciently small, d(F; dition (K3 ) is veri ed and the existence of a positive T -periodic solution of Eq. (1) is proved. Remark 3. We can modify slightly some hypotheses of Theorem 2; obtaining the same existence result: RT (i) Under assumption (pp)II we can weaken the hypothesis 0 G(t; 0; 0) dt ¡0; reRT quiring that 0 G(t; 0; 0) dt ≤ 0 and supposing that there exists tˆ ∈ [0; T ] such that G(tˆ; 0; ·) is strictly decreasing: (ii) Under a stronger formulation of condition (pp)II : for each M ¿ 0 there exists an RT N ¿ 0 and a function : R → R; continuous and T -periodic; with 0 (t) dt¡0; such that; for all y ¿ N; G(t; M; y) ≤ (t)
∀t ∈ [0; T ]
we can give up the hypothesis G(t; x; ·) to be decreasing: Conditions (pp)I and (pp)II could seem unnatural at the rst look. However we point out that it is possible to construct examples of no existence of coexistence states (even in the autonomous case), where (pp)I and (pp)II both fail. Using Theorem 2 we study, in the following corollary, a predator–prey system with periodic coecients, where some logistic e ects are considered. As one can see in [13] and in the references therein, there are many articles about the existence of positive and T -periodic solutions for this model. Moreover, some authors, assuming stronger constraints on the postiveness of the coecients (or their mean values), obtain results even about uniqueness and asymptotic stability of the solutions [5, 10, 19]. Corollary 2. Consider the system x0 = x (a(t) − b(t)x − c(t)y) y0 = y (−d(t) + e(t)x − f(t)y)
(8)
where a; b; c; d; e; f : R → R are continuous and T -periodic functions such that b; c; e; f d ¿ 0. Assume ≥ 0 and a; b; Z T (−d(t) + e(t)u0 (t)) dt ¿ 0; 0
where u0 (·) is the unique positive T -periodic solution of the equation x0 = x(a(t) − b(t)x); and; moreover; c ¿ 0 or; alternatively; f ¿ 0: Then system (8) has at least one positive T-periodic solution:
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Remark 4. Following the previous remark; under the hypothesis f ¿ 0; we can weaken the assumption d ¿ 0; asking d ≥ 0: This enable us to improve slightly a corresponding result in [13]: We can modify the hypotheses of Theorem 2 in order to obtain a result, which is applicable to system (8) when − d ¿ 0. Indeed we have the following: Theorem 3. Consider system (1) and assume that the functions F(t; x; ·); G(t; x; ·) are decreasing and G(t; ·; y) is increasing: Suppose that F(t; ·; 0) and G(t; 0; ·) satisfy the Landesman–Lazer type assumption and that Z T F(t; 0; vM (t)) dt ¿ 0; 0
where vM (·) is the maximal positive and T-periodic solution of y0 = y G(t; 0; y): Finally; assume that for each M ¿ 0 there exists N ¿ 0 such that (pp)I
RT 0
F(t; x; N ) dt¡0 for all x ∈ [0; M ];
or; alternatively; (pp)II
RT 0
G(t; M; N ) dt¡0:
Then there exists at least one T-periodic positive solution of Eq. (1): Proof. First of all, thanks to the Landesman–Lazer-type condition, we have that RT RT F(t; 0; 0) dt ¿ 0 and 0 G(t; 0; 0) dt ¿ 0; moreover, there exists L ¿ 0 such that, 0 for all x; y ≥ L; F(t; x; 0) ≤ (t) and G(t; 0; y) ≤ (t), where ; : R → R are continuous RT RT and T -periodic functions such that 0 (t) dt ¡0; 0 (t) dt ¡0: In order to prove our theorem, we consider at rst the homotopy F1∗ (t; x; y; ) = F(t; x; y); G1∗ (t; x; y; ) = G(t; x; y); which satis es conditions (K0 ), (K1 ) and (K2 ). We omit the details of the veri cation of these conditions, since the arguments follow closely the computation described in the proof of Theorem 2. We apply the rst part of Lemma 1 and arrive to the system x0 = xF(t; x; y); y0 = yG(t; 0; y). Then construct a second homotopy: F2∗ (t; x; y; ) = F(t; x; y + (1−)vM (t)), G2∗ (t; x; y; ) = G(t; 0; y) and use Lemma 1 again, since system (2), so obtained, veri es (K0 ), (K1 ) and (K2 ). Thus, we reach, for = 0, the equation x0 = xF(t; x; vM (t)); y0 = yG(t; 0; y). Finally, putting ˆ ˆ G3∗ (t; x; y; ) = G(t; 0; y) + (1 − )G(y); F3∗ (t; x; y; ) = F(t; x; vM (t)) + (1 − )F(x); RT ˆ ˆ where F(x) = (1=T ) 0 F(t; x; vM (t)) dt and G(y) = G(0; y); we obtain system (2), that ˆ ]; −1 [; 0) = −1 for = 0 is autonomous. Moreover, chosen a suitably small ¿ 0, d(F; −1 ˆ ]; [; 0) = −1. So, conditions (K0 ), (K1 ), (K2 ) and (K3 ) are veri ed and and d(G; the existence of at least one positive T -periodic solution for (1) is proved. Applying Theorem 3, we come to the following existence result for a system with continuous and periodic time-dependent coecients, which was achieved by a di erent proof in [19].
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Corollary 3. Consider system ( 0 x = x (a(t) − b(t)x − c(t)y) y0 = y (d(t) + e(t)x − f(t)y)
(9)
where a; b; c; d; e; f : R → R are continuous and T -periodic functions such that b; c; e; d; f ¿ 0. Moreover; assume that f ≥ 0 and a; b; Z
T
0
(a(t) − c(t)v0 (t)) dt ¿ 0;
where v0 (·) is the unique positive T -periodic solution of the equation y0 = y (d(t) − f(t)y). Then system (9) has at least one positive T -periodic solution. We can still modify the assumptions of Theorem 2 on the functions F and G, preserving some typical aspects of a predator–prey model, which characterize even the classical Lotka–Volterra system and obtaining the following Theorem 4. Consider system (1) and assume that the function F(t; x; y) is decreasing in x and y and the function G(t; x; y) is increasing in x and decreasing in y. Assume that there exists L ¿ 0 such that, for all x ≥ 0; Z
T 0
Z F(t; x; 0) dt ¿ 0 ¿
0
T
F(t; 0; L) dt
and, for all y ≥ 0; Z 0
T
Z G(t; 0; 0) dt¡0¡
0
T
G(t; L; y) dt:
Then there exists at least one T -periodic positive solution of Eq. (1). Proof. Putting F1∗ (t; x; y; ) = F(t; x; y); G1∗ (t; x; y; ) = G(t; x; y), we obtain system (2) that satis es (K0 ), (K1 ) and (K2 ), since the equations v0 = vG(t; 0; v) and u0 = uF(t; u; 0) have no positive T -periodic solutions. We can use the rst part of Lemma 1 in order to pass to the system x0 = xF(t; 0; y); y0 = yG(t; x; 0). Then reduce the problem y); to an autonomous system by the homotopy F2∗ (t; x; y; ) = F(t; 0; y) + (1 − ) F(0; ∗ G2 (t; x; y; ) = G(t; x; 0) + (1 − )G(x; 0). Since (K0 ), (K1 ) and (K2 ) are satis ed and 0); ]; −1 [; 0) ·), ]; −1 [; 0) = − 1 and d(G(·; we can choose ¿ 0 so small that d(F(0; = 1; then condition (K3 ) is also veri ed and, using Lemma 1, we complete the proof. Using Theorem 4 we can prove the existence of positive and T -periodic solutions for another system with periodic time dependent coecients, that was studied in [14] and generalizes the classical Lotka–Volterra model with constant coecients.
A. Battauz / Nonlinear Analysis 37 (1999) 735 – 749
Corollary 4. Consider system ( x0 = x (a(t) − c(t)y) y0 = y (−d(t) + e(t)x)
747
(10)
where a; c; d; e : R → R are continuous and T -periodic functions such that c; e ≥ 0 and a; c ; d; e ¿ 0. Then system (10) has at least one positive T -periodic solution. As a nal application of Lemma 1 we study here a system, which generalizes a result obtained recently in [25]. Theorem 5. Consider system (1) and assume that the functions F(t; x; y); G(t; x; y) are decreasing in x and y. Moreover, suppose that, for all x; y ≥ 0; Z T Z T F(t; x; 0) dt ¿ 0 and G(t; 0; y) dt ¿ 0 0
and there is L ¿ 0 such that Z T F(t; 0; L) dt¡0 and 0
0
Z 0
T
G(t; L; 0) dt¡0:
Then there exists at least one T -periodic positive solution of Eq. (1). Proof. In order to apply Lemma 1 we consider a rst homotopy F1∗ (t; x; y; ) = F(t; x; y); G1∗ (t; x; y; ) = G(t; x; y), which satis es conditions (K0 ) and (K1 ). Indeed, if (x(·), y(·)) is a positive T -periodic solution of (2), then x and y are bounded in some point of the interval [0; T ] by L and, as the functions F and G are decreasing in both their space variables, also x0 =x and y0 =y are upper bounded. Hence (K1 ) easily follows. Condition (K2 ) is vacuously satis ed. We can so apply the rst part of Lemma 1 and move to the system x0 = xF(t; 0; y); y0 = yG(t; x; 0). Then, construct a second homotopy, depending on the parameter ∈ [0; 1], which reduces our problem to an au y); G2∗ (t; x; y; ) = tonomous system for = 0: F2∗ (t; x; y; ) = F(t; 0; y) + (1 − )F(0; 0). This homotopy veri es condition (K0 ), (K1 ) and, vacuG(t; x; 0) + (1 − )G(x; ·); ]; −1 [; 0) = − 1 and ously, (K2 ). Moreover, chosen a suitably small ¿ 0, d(F(0; −1 0); ]; [; 0) = − 1, so that condition (K3 ) is satis ed and the existence of a d(G(·; positive T -periodic solution of (1) is proved. Using Theorem 5, we come immediately to the following corollary, obtaining the existence of at least one coexistence state for the considered system under assumptions which are weaker than those supposed in [25]. Corollary 5. Consider the system ( x0 = x (a(t) − c(t)y) y0 = y (d(t) − e(t)x)
(11)
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where a; c; d; e : R → R are continuous and T -periodic functions such that c; e ≥ 0 and a; c ; d; e ¿ 0. Then system (11) has at least one positive T -periodic solution. Acknowledgements I would like to thank Professor F. Zanolin for helpful discussions on topics related to this paper. References [1] S. Ahmad, On the nonautonomous Volterra–Lotka competition equations, Proc. Amer. Math. Soc. 117 (1993) 199 –204. [2] S. Ahmad, On almost periodic solutions of the competing species problem, Proc. Amer. Math. Soc. 102 (1988) 855– 861. [3] F. Albrecht, H. Gatzke, A. Haddad, N. Wax, The dynamics of two interacting populations, J. Math. Anal. Appl. 46 (1974) 658 – 670. [4] C. Alvarez, A.C. Lazer, An application of topological degree to the periodic competing species problem, J. Austral. Math. Soc. Ser. B 28 (1986) 202–219. [5] Z. Amine, R. Ortega, A periodic prey-predator system, J. Math. Anal. Appl. 185 (1994) 477 – 489. [6] A. Battauz, “Grado topologico e applicazioni a sistemi di erenziali nella dinamica della popolazioni”, Tesi di Laurea, Universita di Udine, 1996. [7] A. Battauz, F. Zanolin, Coexistence states for periodic competitive Kolmogorov systems, J. Math. Anal. Appl. 219 (1998) 179–199. [8] T.A. Burton, V. Hutson, Permanence for nonautonomous predator-prey systems, Di . Int. Eq. 4 (1991) 1269 –1280. [9] A. Capietto, J. Mawhin, F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc. 329 (1992) 41–72. [10] J.M. Cushing, Periodic time-dependent predator prey-systems, SIAM J. Appl. Math. 32 (1977) 82–95. [11] J.M. Cushing, Two species competition in a periodic environment, J. Math. Biol. 10 (1980) 385– 400. [12] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. [13] T. Ding, H. Huang, F. Zanolin, A priori bounds and periodic solutions for a class of planar systems with application to Lotka–Volterra equations, Discrete Continuous Dyn. Systems 1 (1995) 103 –118. [14] T. Ding, F. Zanolin, Periodic solutions and subharmonic solutions for a class of planar systems of Lotka–Volterra type, in: Proc. 1st WCNA, Tampa, FL, 1992, vol. I, de Gruyter, New York, 1996, pp. 395– 406. [15] K. Gopalsamy, Exchange of equilibria in two species Lotka–Volterra competition models, J. Austral. Math. Soc. Ser. B 24 (1982) 160 –170. [16] J.K. Hale, A.S. Somolinos, Competition for uctuating nutrient, J. Math. Biol. 18 (1983) 255–280. [17] P. Korman, Some new results on the periodic competition model, J. Math. Anal. Appl. 171 (1992) 131–138. [18] M.A. Kranosel’skii, P.P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin, 1984. [19] J. Lopez-Gomez, R. Ortega, A. Tineo, The periodic predator-prey Lotka–Volterra model, Adv. Di erential Equations 1 (1996) 403 – 423. [20] J. Mawhin, Topological degree methods in nonlinear boundary value problems, CBMS Regional Conf. Ser. Math., vol. 40, Amer. Math. Soc., Providence, RI, 1979. [21] P. de Mottoni, A. Schiano, Competition systems with periodic coecients: A geometric approach, J. Math. Biol. 11 (1981) 319 –335. [22] H.L. Smith, Periodic competitive di erential equations and the discrete dynamics of competitive maps, J. Di erential Equations 64 (1986) 165–194.
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[23] A. Tineo, Sobre las equaciones de Lotka–Volterra para dos especies en competencia, Notas de Matematica, Universidad de Los Andes, Merida-Venezuela 96 (1988) 1–20. [24] W. Wang, Z. Ma, Permanence of populations in a polluted environment, Math. Biosci. 122 (1994) 235–248. [25] K. Wojcik, On existence of periodic positive solutions, to appear. [26] F. Zanolin, Permanence and positive periodic solutions for Kolmogorov competing species systems, Results Math. 21 (1992) 224 –250. [27] F. Zanolin, Continuation theorems for the periodic problem via the translation operator, Rend. Sem. Mat. Univ. Pol. Torino 54 (1996) 1–23.
Coexistence states for periodic planar Kolmogorov systems1 Anna Battauz ∗ Dipartimento di Matematica e Informatica, UniversitÃa di Udine, 33100 Udine, Italy Received 21 August 1997; accepted 2 September 1997
Keywords: Positive periodic solutions; Kolmogorov systems; Continuation theorems; Preypredator; Competing species
1. Introduction and statement of the main Lemma The object of this paper is to study the existence of coexistence states (i.e. positive and T -periodic solutions) for the two-dimensional Kolmogorov system ( 0 x = x F(t; x; y) (1) y0 = y G(t; x; y) where, once for all, we assume (H) F; G : R × (R+ )2 →R are continuous functions which are T-periodic (T ¿ 0) in the t-variable. (Here and in what follows, we denote by R+ := [0; + ∞), the set of non-negative real numbers and by R+ := ]0; +∞), the set of positive reals.) Systems of the form (1) have been widely analyzed in the literature for their signi cance in various mathematical models, generalizing the classical Lotka–Volterra planar systems. There are many articles for some speci c forms of F and G (see, for instance, [2, 4, 5, 10, 11, 15, 17, 19, 22–25]). Results for general F and G have been also obtained in [3] in the autonomous case, in [22] for competitive systems and in [13] for the predator–prey model. In some of such cases (see, e.g., [1, 2, 4, 19, 21, 22]), interesting information on the behaviour of the solutions is obtained. On the contrary, in 1 Work performed under the auspices of GNAFA and supported by a fellowship of the Faculty of Sciences of the University of Udine, 1997. ∗
E-mail: [email protected]
0362-546X/99/$ – see front matter ? 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 0 6 9 - 8
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this paper we limit ourselves only to the study of the existence of positive T -periodic solutions, but, on the other hand, we are able to consider a rather general class of systems, which generalize, under natural monotonicity hypotheses and appropriate conditions on the averages of the functions F and G, some aspects of the characteristic behaviour of the predator-prey or the competing species models. To this aim, we use a continuation result, which is a slightly modi ed formulation of Lemma 1 in [7], where N -dimensional Kolmogorov systems, with N ≥ 1, are considered. First of all, following a common approach within the degree theory, we embed system (1) into a one-parameter family of systems of the form 0 x = xF ∗ (t; x; y; ) ∈ [0; 1]; (2) 0 y = y G ∗ (t; x; y; ) where, F ∗ ; G ∗ : R × (R+ )2 × [0; 1]→R are continuous functions which are T -periodic (T ¿ 0) in the t-variable and such that F ∗ (t; x; y; 1) = F(t; x; y);
G ∗ (t; x; y; 1) = G(t; x; y)
∀t ∈ [0; T ]; ∀(x; y) ∈ (R+ )2 :
As in [9], we will often assume that for = 0 system (2) is autonomous, that is F ∗ (t; x; y; 0) = F 0 (x; y);
G ∗ (t; x; y; 0) = G 0 (x; y);
where F 0 (x; y), G 0 (x; y) : (R+ )2 → R are continuous functions. In this paper we make use of the following notation: C is the Banach space of the continuous functions f : [0; T ] → R2 , with the “sup” norm, | · |∞ . If g : R2 ⊃ D → R2 is a continuous function such that, for some open set ⊂ D, −1 g (0) ∩ is compact, then we can de ne the integer deg(g; ) := d(g; O; 0); where O is any open and bounded set with g−1 (0) ∩ ⊂ O ⊂ O ⊂ and “d” is the Brouwer degree. Note that the de nition of deg(g; ) is well-posed, as it is independent of the set O, by virtue of the excision property of the degree (see [12]). We give now a lemma about coexistence states, which is our main tool in the proof of the subsequent existence results. Lemma 1. Consider system (2) and assume RT RT 6 0; for all ∈ [0; 1]. (K0 ) | 0 F ∗ (t; 0; 0; ) dt| + | 0 G ∗ (t; 0; 0; ) dt| = (K1 ) There is M ¿ 0 such that any positive T-periodic solution (x(·); y(·)) of (2) satisÿes x(t) ≤ M; y(t) ≤ M for all t ∈ [0; T ]: (K2 ) For each positive T-periodic solution u(·) of the equation x0 = x F ∗ (t; x; 0; ); with RT ∈ [0; 1]; 0 G ∗ (t; u(t); 0; ) dt 6= 0; and; similarly; for each positive T-periodic RT solution v(·) of the equation y0 = y G ∗ (t; 0; y; ); with ∈ [0; 1]; 0 F ∗ (t; 0; v(t); ) dt 6= 0. Then; there is a compact set K ⊂ (R+ )2 that contains all the positive T-periodic solutions of system (2); that is; if (x(·); y(·)) is a positive and T-periodic solution
A. Battauz / Nonlinear Analysis 37 (1999) 735 – 749
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of (2), for some ∈ [0; 1]; then (x(t); y(t)) ∈ K; ∀t ∈ R. Moreover; the topological degree of a suitable functional operator in C associated to system (2) does not change when the parameter runs in [0; 1]. In particular; if for = 0; system (2) is autonomous and (K3 ) deg((F 0 ; G 0 ); (R+ )2 ) 6= 0; then there exists a coexistence state for system (1). We notice that the condition in (K2 ) has to be considered as vacuously satis ed when system (2) has no semitrivial positive T -periodic solutions of the form (u; 0) or (0; v). Proof. The rst part of this lemma is a simple application of the homotopic invariance of the Leray–Schauder degree for a suitable operator equation in the function space C, which is associated to the problem (2) (see [18, 20]). For instance, we could take for a two-dimensional system of the form z 0 = Z(t; x; );
∈ [0; 1]
Rt
(3)
the operator : (z(·); ) 7→ w(·), where w(t) = z(T )+ 0 Z(s; z(s); ) ds. Note that : C × [0; 1] ⊃ dom → C is a completely continuous operator, having as xed points the T periodic solutions of Eq. (3). So we need to verify the admissibility of the homotopy that we have de ned by introducing the parameter into system (1). In other words, we prove now the existence of a compact set K ⊂ (R+ )2 where all the positive T periodic solutions of (2) are contained. Assume, by contradiction, that for any n ∈ N there is a positive T -periodic solution (xn (·); yn (·)) of (2), for some = n ∈ [0; 1], 6 [1=n; n]2 , for some t ∈ [0; T ]. Since (xn ·); yn (·)) satis es consuch that (xn (t); yn (t)) ⊂ dition (K1 ), we have that, for n ¿ M , mint∈[0; T ] xn (t)¡1=n or mint∈[0; T ] yn (t)¡1=n and that the derivatives x0 (·) and yn0 (·) are uniformly bounded by a constant, say LM , where L ≥ max{sup{|F ∗ (t; x; y; )| : t ∈ [0; T ]; (x; y) ∈ [0; M ]2 ; ∈ [0; 1]}; sup{|G ∗ (t; x; y; )| : t ∈ [0; T ]; (x; y) ∈ [0; M ]2 ; ∈ [0; 1]}}. So it is possible, owing to the AscoliArzela theorem, to extract a subsequence that converges to a non-negative function (u(·); v(·)) with mint∈R u(t) = 0 or mint∈R v(t) = 0. Passing to the limit as n→∞ into the relations satis ed by (xn (·); yn (·)), we obtain that (u(t); v(t)) is a T -periodic solution of system (2), for some = , such that 0 ≤ u(t) ≤ M and 0 ≤ v(t) ≤ M for all t ∈ [0; T ]. MoreRT RT over, as by the T -periodicity, 0 = 0 xn0 (t)=xn (t) dt = 0 F ∗ (t; xn (t); yn (t); n ) dt as well RT 0 RT ∗ as 0 = 0 yn (t)=yn (t) dt = 0 G (t; xn (t); yn (t); n ) dt; passing to the limit, we have Z T Z T F ∗ (t; u(t); v(t); ) dt = G ∗ (t; u(t); v(t); ) dt = 0: (4) 0
0
Now, we observe that, if mint∈[0; T ] u(t) = u(t∗ ) = 0, then u(t) = 0 for all t ∈ [0; T ], because, if t ∗ : t∗ ≤ t ∗ ¡t∗ + T is a maximum point for the function u(·), then R t∗ R t∗ xn (t ∗ ) = xn (t∗ )exp( t∗ xn0 (t)=xn (t) dt) = xn (t∗ ) t∗ F ∗ (t; xn (t); yn (t); n ) dt ≤ xn (t∗ )LT , and so, passing to the limit as n→∞, we have that maxt∈[0; T ] u(t) = u(t ∗ ) = 0. Likewise, if mint∈R v(t) = 0, then v(t) = 0 for all t ∈ [0; T ]. So, if u = v = 0, then (4) contradicts (K0 ). On the other hand, if u = 0 and v ¿ 0 or u ¿ 0 and v = 0, we obtain a
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contradiction between (4) and assumption (K2 ). Thus, we have proved that there is a compact set K ⊂ (R+ )2 containing all the possible positive and T -periodic solutions of (2), for each ∈ [0; 1]. Then the homotopic invariance of the Leray–Schauder degree (denoted here by D) implies that D(I − (·; ); A ; 0) is constant with respect to , where A = {(x; y) ∈ C: (x(t); y(t)) ∈]; −1 [2 ; ∀t ∈ [0; T ]}, with ¿ 0 such that K ⊂ ]; −1 [2 ⊂ (R+ )2 . Concerning the nal part of this result, a continuation theorem in [9] reduces the estimate of the Leray–Schauder degree of the operator associated to problem (1) to the estimate of the Brouwer degree for the function (F 0 ; G 0 ). Indeed, if for = 0 system (2) is autonomous, then D(I − (·; ); A ; 0) = d((xF 0 (x; y); yG 0 (x; y)); ]; −1 [2 ; 0) = deg((F 0 ; G 0 ); (R+ )2 ). So, if condition (K3 ) holds, we have that D(I − (·; 1); A ; 0) 6= 0 and this ensures the existence of positive and T -periodic . solutions for system (1). We conclude this section with a list of some further notations to be used in the next results. If H : R × (R+ )2 → R is a continuous function, T -periodic in the t-variable, we set Z 1 T H (t; z) dt; H (z) = T 0 the mean value in a period T of the function H (·; z), for a given z ∈ R2 . The same notation h is used for the mean value of a continuous and T -periodic function h : R → R. As we said above, we are going to require often in the applications some monotonicity properties for system (1). With this respect, note that in this paper a increasing (resp. decreasing) function is just a weakly increasing (resp. weakly decreasing) function. In the sequel, the following property of the scalar equation, u0 = uh(t; u);
(5)
where h : R × R → R is a continuous function, which is T -periodic in the t-variable, will be used: Assume that there are a constant L ¿ 0 and a continuous and T -periodic function
: R → R such that h(t; s) ≤ (t) and
Z 0
T
for all t ∈ [0; T ] and s ≥ L Z
h(t; 0) dt ¿ 0 ¿
T
0
(t) dt:
Under this assumption, that we call of Landesman–Lazer type (cf. [20, Ch. VI]) it is possible to prove that (5) has a minimal and a maximal positive T -periodic solutions uL and uM such that any positive T -periodic solution u˜ of (5) satis es ˜ ≤ uM (t) uL (t) ≤ u(t)
for all t = [0; T ]:
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It is also easy to see that if h(t; ·) is decreasing for all t ∈ R and strictly decreasing for at least a tˆ ∈ [0; T ], then uL = uM , that is the positive T -periodic solution of (5) is unique. 2. Applications of Lemma 1 We use now Lemma 1 in order to prove the existence of positive and T -periodic solutions of system (1), under suitable assumptions. The idea is that of applying the rst part of Lemma 1 several times, thereby constructing appropriate homotopies, which verify conditions (K0 ), (K1 ) and (K2 ). So we reduce every time our problem to an easier one until we come, by the last homotopy, to an autonomous system satisfying also condition (K3 ). At this point we can apply Lemma 1 entirely and ensure the existence of at least one coexistence state for (1). As a rst application of Lemma 1 we study system (1) under hypotheses which describe some typical aspects of the model of two competing species. Theorem 1. Consider system (1) and suppose that the functions F(t; x; ·); G(t; ·; y) are decreasing and that F(t; ·; 0) and G(t; 0; ·) satisfy the Landesman–Lazer type assumptions. Let uM and uL be the maximal and the minimal positive T -periodic solutions of the equation x0 = x F(t; x; 0); and; respectively; vM and vL the maximal and the minimal positive T -periodic solutions of the equation y0 = y G(t; 0; y). Assume also Rthat RT T (I) 0 F(t; 0; vM (t)) dt ¿ 0 and 0 G(t; uM (t); 0) dt ¿ 0 RT RT (II) 0 F(t; 0; vL (t)) dt ¡ 0 and 0 G(t; uL (t); 0) dt ¡ 0: Then there exists at least one T -periodic positive solution of Eq. (1). RT RT Proof. By the assumptions, we have that 0 F(t; 0; 0) dt ¿ 0; 0 G(t; 0; 0) dt ¿ 0 and there exists L ¿ 0 such that, for all t ∈ R and for all x; y ≥ L; F(t; x; 0) ≤ (t); G(t; 0; y) RT ≤ (t), where ; : R → R are continuous and T -periodic functions with 0 (t) dt RT ¡ 0; 0 (t) dt ¡ 0. To apply Lemma 1, assume condition (I) and let F ∗ (t; x; y; ) = F(t; x; y), G ∗ (t; x; y; ) = G(t; x; y); F ∗ ; G ∗ : R × (R+ )2 × [0; 1] → R are continuous and T -periodic functions in the t-variable, which preserve the monotonicity properties of F and G. We can verify immediately (K0 ) of Lemma 1; as to (K1 ), if (x(·); y(·)) is a positive T periodic solution of (2) for some ∈ [0; 1], then we obtain, supposing by contradiction RT RT that x(t) ≥ L for all t ∈ [0; T ], that 0 = 0 F ∗ (t; x(t); y(t); ) dt ≤ 0 F(t; x(t); 0) dt ¡ 0, which is absurd. So there is t1 ∈ [0; T ] such that x(t1 ) ¡ L; if T = {t ∈ R: x(t) ¡ L} and t ∗ ∈ [t1 ; t1 + T ] is a maximum point for the function x(·), we have that L is a bound for R t∗ R x(·) if t1 = t ∗ , otherwise log x(t ∗ )=x(t1 )= t1 F ∗ (t; x(t); y(t); ) dt ≤ [t1 ; t ∗ ]∩T F ∗ (t; x(t); 0; R ) dt + [t1 ; t ∗ ]∩Tc F ∗ (t; x(t); 0; ) dt ≤ |F ∗ |L; 0 T + | |∞ T , where |F ∗ |L; 0 = sup{|F ∗ (t; x; 0; )|: t ∈ [0; T ]; x ∈ [0; L]; ∈ [0; 1]} and | |∞ = sup{| (t)|: t ∈ [0; T ]}. This implies |x|∞ ¡ L exp((|F ∗ |L; 0 + | |∞ ))T ). Likewise |y|∞ ¡ L exp((|G ∗ |0; L + | |∞ )T ). We check now that condition (K2 ) is satis ed: if u(·) is a positive T -periodic solution of the
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equation x0 = xF ∗ (t; x; 0; ) for some ∈ [0; 1], we have that u0 = uF(t; u; 0); therefore RT RT uL (t)≤ u(t) ≤ uM (t) for all t ∈ [0; T ] and 0 G ∗ (t; u(t); 0; ) dt = 0 G(t; u(t); 0) dt RT ≥ 0 G(t; uM (t); 0) dt ¿ 0, owing to condition (I). Likewise we obtain that all the positive and T -periodic solutions v(·) of the equation y0 = yG ∗ (t; 0; y; ); ∈ [0; 1], are RT such that 0 F ∗ (t; 0; v(t); ) dt ¿ 0. So there is a compact set K ⊂ (R+ )2 that contains all the positive T -periodic solutions of system (2) and the topological degree of the functional operator associated to system (2) is the same for = 1 and = 0; for this last value of the parameter , we obtain the system ( 0 x = xF(t; x; 0) y0 = yG(t; 0; y) that we reduce to an autonomous system, by the homotopy F1∗ (t; x; y; ) = F1∗ (t; x; ) = y), with 0); G1∗ (t; x; y; ) = G1∗ (t; y; ) = G(t; 0; y)+(1−)G(0; F(t; x; 0)+(1−)F(x; ∈ [0; 1]. We see immediately that system (2), obtained using the last homotopy, veri es the hypotheses of Lemma 1, as the conditions in (K0 ); (K1 ) and (K2 ) are obviously 0) satis ed. Consistent with the notation of Lemma 1, we have that F 0 (x; y) = F(x; 0 and G (x; y) = G(0; y). To complete this proof we only need to check that d((F(·; 0); ·)); ]; −1 [2 ; 0) 6= 0, for ¿ 0 small enough, that follows from d(F(·; 0); ]; −1 [; 0) G(0; −1 = d(G(0; ·); ]; [; 0) = − 1. Assume now condition (II) and consider the homotopy F1∗ (t; x; y; ) = F(t; x; y) − (1−)y; G1∗ (t; x; y; ) = G(t; x; y)−(1−)x; this homotopy determines a system (2), that satis es obviously (K0 ); (K1 ) and, owing to condition (II), (K2 ). So, for = 0 we are led to consider the system x0 = x(F(t; x; y)−y); y0 = y(G(t; x; y)−x). We construct now a second homotopy, putting F2∗ (t; x; y; ) = F(t; x; y)−y; G2∗ (t; x; y; ) = G(t; x; y)−x and apply again Lemma 1: it’s easy to check that condition (K0 ) is satis ed and that x(·); y(·), with ∈ [0; 1] are bounded at least in a t ∈ [0; T ], when (x(·); y(·)) is a positive T -periodic solution of (2). From the di erential inequalities x0 ≤ xF(t; x; 0) and y0 ≤ yG(t; 0; y), with F(t; s; 0) and G(t; 0; s) upper bounded, the existence of a M ¿ 0 such that |x|∞ ; |y|∞ ¡ M (with M independent of ) follows. So we have that F(t; x; y)−y ≤ F(t; x; 0)−y ≤ R1 −y, where R1 = sup{|F(t; x; 0)|: t ∈ [0; RT RT T ]; x ∈ [0; M ]}, and therefore 0 = 0 x0 (t)=x(t) dt ≤ 0 (R1 − y(t)) dt. This implies that there exists t ∗ ∈ [0; T ] such that y(t ∗ ) ¡ R1 and from this fact we obtain easily a bound for |y|∞ . The proof of the boundedness of x(·) is similar. In order to prove condition (K2 ), observe that, if u(·) is a positive T -periodic solution of x0 = xF2∗ (t; x; 0; ) = xF(t; RT RT x; 0) for some ¿ 0, then u ≥ uL and 0 G2∗ (t; u(t); 0) dt ≤ 0 G(t; (1=)uL (t); 0) dt ≤ RT G(t; uL (t); 0) dt ¡ 0; likewise, if v(·) is a positive T -periodic solution of the equa0 RT tion y0 = yG2∗ (t; 0; y; ) for some ¿ 0, we have that 0 F2∗ (t; 0; v(t)) dt ¡ 0 (note that for = 0 there are no positive T -periodic solutions of these equations). We can apply Lemma 1 and reduce system (2), obtained for = 0, to an autonomous system y), with the nal homotopy F3∗ (t; x; y; ) = F3∗ (t; y; ) = (F(t; 0; y) − y) + (1 − )F(0; ∗ ∗ G3 (t; x; y; ) = G3 (t; x; ) = (G(t; x; 0) − x) + (1 − )G(x; 0), (with ∈ [0; 1]), that satis es the hypotheses of Lemma 1. In fact, (K0 ) is immediately veri ed and it is easy to check that, if (x(·); y(·)) is a positive and T -periodic solution of system (2) for some
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∈ [0; 1], then |uL |∞ and |vL |∞ provide a bound to, resp., x(·) and y(·) in at least a point of [0; T ]. So we can nd a bound for both |x|∞ ; |y|∞ , since these functions satisfy the di erential inequalities x0 ≤ xF3∗ (t; 0; 0; ) and y0 ≤ yG3∗ (t; 0; 0; ). Condition (K2 ) is vacuously satis ed, because there are no positive T -periodic solutions of the equations x0 = xF3∗ (t; 0; 0; ) and y0 = yG3∗ (t; 0; 0; ), with ∈ [0; 1]. Consistent with the notation 0). In order to esti y) and G 0 (x; y) = G(x; of Lemma 1, we have that F 0 (x; y) = F(0; mate the Brouwer degree for (F(0; ·); G(·; 0)), we can choose an ¿ 0 so small that 0); ]; −1 [; 0) = −1. Therefore, since d((F(0; 0)); ]; ·); G(·; ·); ]; −1 [; 0) = d(G(·; d(F(0; −1 2 [ ; 0) = − 1 6= 0, condition (K3 ) of Lemma 1 is veri ed and we can ensure that system (1) under our assumptions has at least one T -periodic and positive solution. Remark 1. Theorem 1 in case (I) was previously obtained in [26]: here we provide a new proof of that result. On the other hand; for the case (II) there is no such a general formulation in the known literature. As observed in the introduction we note that in the special case when F(t; ·; 0) is decreasing for every t and strictly decreasing for at least a tˆ ∈ [0; T ] and the same holds with respect to G(t; 0; ·); then the maximal and minimal positive T -periodic solutions coincide. In other words; there is a unique positive T -periodic solution u0 (resp. v0 ) such that u0 = u F(t; u; 0) (resp. v0 = v G(t; 0; v)). Thus; we obtain that; if RT RT RT RT F(t; 0; 0) dt ¿ 0 ¿ 0 F(t; ∞; 0) dt and 0 G(t; 0; 0) dt ¿ 0 ¿ 0 G(t; 0; ∞) dt; then sys0 tem (1) has at least one coexistence state when Z T Z T G(t; u0 (t); 0) dt F(t; 0; v0 (t)) dt ¿ 0: 0
0
Under stronger regularity or monotonicity hypotheses on the functions F and G; this result could be obtained from previous theorems in [16; 21; 22]; where the authors reduce the problem of the existence of positive and T -periodic solutions for a twodimensional competitive system to the study of the Poincare map on a one-dimensional manifold. As in [26] we have an immediate application of Theorem 1, studying a system with periodic time-dependent coecients, which describes more familiarly the typical interaction of two competing species with logistic growth e ects. Corollary 1. Consider the system ( 0 x = x (a(t) − b(t)x − c(t)y) y0 = y (d(t) − e(t)x − f(t)y)
(6)
where a; b; c; d; e; f : R → R are continuous and T -periodic functions such that b; c; e; d; f ¿ 0 and assume that f≥ 0 and a; b; Z T Z T (a(t) − c(t)v0 (t)) dt (d(t) − e(t)u0 (t)) dt ¿ 0; (7) 0
0
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where u0 (·) and v0 (·) are the unique positive T-periodic solutions of the equations x0 = x (a(t) − b(t)x) and y0 = y (d(t) − f(t)y); respectively: Then system (6) has at least one positive T-periodic solution. Remark 2. Our assumptions leave us to apply Theorem 1 to system (6) and obtain a generalization (under weaker conditions on the coecients) of an existence result of de Mottoni and Schiano [21] (see also [23]): In fact; system (6) was studied by several authors since the work of [21] under similar hypotheses; generally stonger; on the positiveness of the coecients and their mean values (see; for example; [17; 26] and the references therein): As for Theorem 1, we apply now Lemma 1 to system (1) under hypotheses which describe some typical aspects of a predator–prey model. Theorem 2. Consider system (1) and suppose that the functions F(t; x; ·); G(t; x; ·) are decreasing; G(t; ·; y) is increasing and F(t; ·; 0) satisÿes the Landesman–Lazer type condition: Assume that Z T Z T G(t; 0; 0) dt ¡0¡ G(t; uL (t); 0) dt; 0
0
where uL (·) is the minimal positive and T-periodic solution of x0 = xF(t; x; 0): Moreover; suppose that for each M ¿ 0 there exists N ¿ 0 such that (pp)I
RT 0
F(t; x; N ) dt¡0 for all x ∈ [0; M ];
or; alternatively; (pp)II
RT 0
G(t; M; N ) dt¡0:
Then there exists at least one T-periodic positive solution of Eq. (1): RT Proof. By the Landesman–Lazer type condition, we have that 0 F(t; 0; 0) dt ¿ 0 and RT that there are L ¿ 0; : R → R a continuous and T -periodic function, with 0 (t) dt ¡ 0, such that F(t; x; 0) ≤ (t), for all x ≥ L: Assume (pp)I and consider the homotopy F1∗ (t; x; y; ) = F(t; x; y); G1∗ (t; x; y; ) = G(t; x; y); with ∈ [0; 1]: This homotopy is such that the corresponding system (2) satis es condition (K0 ) of Lemma 1 and (K1 ), since if (x(·); y(·)) is a positive and T -periodic solution of system (2) for some ∈ [0; 1]; we have that |x|∞ ¡ M = RT L exp{T (| |∞ +sup{|F(t; x; 0)|: t ∈ [0; T ]; x ∈ [0; L]})}. From 0 F(t; x(t); y(t)) dt = 0; we nd that there is a point where the function y is bounded by N , and therefore |y|∞ ¡ N exp{T sup{|G(t; M; 0)|: t ∈ [0; T ]}}. Condition (K2 ) is also veri ed, because the equation y0 = yG1∗ (t; 0; y; ) for ∈ [0; 1] has no positive T -periodic solutions and, if x is a positive T -periodic solution of x0 = xF1∗ (t; x; 0; ) for some ∈ [0; 1]; RT RT then x(t) ≥ uL (t) for all t ∈ [0; T ] and so 0 G1∗ (t; x(t); 0; ) dt ≥ 0 G(t; uL (t); 0) dt ¿ 0.
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Therefore, we can apply the rst part of Lemma 1, reducing our system to x0 = xF(t; x; y); y0 = yG(t; x; 0). Consider now a second homotopy F2∗ (t; x; y; ) = F(t; x; y), G2∗ (t; x; y; ) = G(t; x; 0) and observe that condition (K0 ) is obviously satis ed. In order to prove condition (K1 ), we note that, if (x(·); y(·)) is a positive T -periodic solution of (2) for some ∈ [0; 1], then |x|∞ ¡M and there exists RT RT t ∗ ∈ [0; T ] such that x(t ∗ )¡uL (t ∗ ) ≤ |uL |∞ ; otherwise 0 = 0 G(t; x(t); 0) dt ≥ 0 G(t; uL (t); 0) dt ¿ 0. Therefore |x|∞ ¡ M1 := |uL |∞ exp{T (| |∞ + sup{|F(t; z; 0)|: t ∈[0; T ]; z ∈ [0; M ]})}. Moreover, y(·) is bounded in at least a point of [0; T ] by the constant RT N , otherwise 0 = 0 F(t; x(t); y(t)) dt ¡ 0, since |x|∞ ¡M . Now it is easy to check that |y|∞ ¡N exp{T sup{|G(t; M1 ; 0)|: t ∈ [0; T ]}}. In order to apply the rst part of Lemma 1, we need only to verify condition (K2 ); observe that, if u(·) is a positive T -periodic solution of the equation x0 = xF2∗ (t; x; 0; ); then ∈ ]0; 1], otherwise, if RT RT = 0, 0 = 0 u0 (t)=u(t) dt = 0 F(t; 0; 0) dt, against our assumptions on the integrals of F. This means that u(·) is a positive T -periodic solution of the equation x0 = xF(t; RT x; 0), so that u(t) ≥ uL (t); for all t ∈ [0; T ]. Therefore we have that 0 G(t; u(t); 0) dt RT RT ≥ 0 G(t; uL (t)=; 0) dt ≥ 0 G(t; uL (t); 0) dt ¿ 0. Since the equation y0 = yG2∗ (t; 0; y; ) = yG(t; 0; 0) has no positive T -periodic solutions, condition (K2 ) is completely satis ed and we can apply the rst part of Lemma 1 again, reaching, for = 0, the equation x0 = xF(t; 0; y); y0 = yG(t; x; 0). Finally, in order to come to an autonomous y); system, consider the homotopy F3∗ (t; x; y; ) = F3∗ (t; y; ) = F(t; 0; y) + (1 − )F(0; 0): it is easy to check that condiG3∗ (t; x; y; ) = G3∗ (t; y; ) = G(t; x; 0) + (1 − )G(x; tions (K0 ) and (K1 ) of Lemma 1 are veri ed, since if (x; y) is a positive and T -periodic solution of system (2) then x(t1 ) ¡ |uL |∞ and, thanks to condition (pp)I , there exists a suitable N ¿ 0 such that y(t2 ) ¡ N for some (t1 ; t2 ) ∈ [0; T ]2 . From this, one immediately nds the a priori bounds. Condition (K2 ) is vacuously satis ed. Consistently with 0), so y) and G 0 (x; y) = G(x; the notation of Lemma 1, we have that F 0 (x; y) = F(0; ·); ]; −1 [; 0) = −1 that we can apply Lemma 1 and choose ¿ 0 so small that d(F(0; 0); ]; −1 [; 0) = 1. We can therefore assert that condition (K3 ) is veri ed and d(G(·; and so there exists a positive T -periodic solution for system (1). Assume (pp)II and consider the homotopy F ∗ (t; x; y; ) = F(t; x; y); G ∗ (t; x; y; ) = G(t; x; y). System (2) so obtained satis es condition (K0 ) and (K1 ), since, if (x(·); y(·)) is a positive T -periodic solution, |x|∞ ¡ M and therefore, by (pp)II and the second equation in (2), |y|∞ is bounded by a suitable constant too. Finally, condition (K2 ) is veri ed thanks to our hypotheses on the integrals of G. We can then apply the rst part of Lemma 1 in order to arrive to the system x0 = xF(t; x; 0); y0 = yG(t; x; y): Consider now another homotopy: F1∗ (t; x; y; ) = F(t; x; 0); G1∗ (t; x; y; ) = G(t; x + (1 − )uL (t); y), which satis es (K0 ), (K1 ) and (K2 ) of Lemma 1. We check here only the last condition, since the others are immediate. Assume that there are positive and T -periodic solutions of system (1) for some ∈ [0; 1] of the form (u(·); 0) or (0; v(·)); RT in the rst case we have that u(t) ≥ uL (t) for all t ∈ [0; T ] and 0 G1∗ (t; u(t); 0; ) dt ≥ RT RT G(t; uL (t); 0) dt ¿ 0; in the second case we observe that 0 F1∗ (t; 0; v(t); ) dt = 0 RT F(t; 0; 0) dt ¿ 0. We can so apply Lemma 1, reaching, for = 0, the equation 0 x0 = xF(t; x; 0); y0 = yG(t; uL (t); y). Lastly, consider the nal homotopy: F2∗ (t; x; y; ) =
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ˆ ˆ ˆ F(t; x; 0) + (1−)F(x); G ∗ (t; x; y; ) = G(t; uL (t); y) + (1−)G(y), where F(x) = F(x; RT 2 ˆ 0) and G(y) = (1=T ) 0 G(t; uL (t); y) dt. System (2) so obtained satis es (K0 ), (K1 ), thanks to assumption (pp)II , and (K2 ). We can so apply again Lemma 1 and observe ˆ ]; −1 [; 0) = −1, such that conˆ ]; −1 [; 0) = d(G; that, for ¿ 0 suciently small, d(F; dition (K3 ) is veri ed and the existence of a positive T -periodic solution of Eq. (1) is proved. Remark 3. We can modify slightly some hypotheses of Theorem 2; obtaining the same existence result: RT (i) Under assumption (pp)II we can weaken the hypothesis 0 G(t; 0; 0) dt ¡0; reRT quiring that 0 G(t; 0; 0) dt ≤ 0 and supposing that there exists tˆ ∈ [0; T ] such that G(tˆ; 0; ·) is strictly decreasing: (ii) Under a stronger formulation of condition (pp)II : for each M ¿ 0 there exists an RT N ¿ 0 and a function : R → R; continuous and T -periodic; with 0 (t) dt¡0; such that; for all y ¿ N; G(t; M; y) ≤ (t)
∀t ∈ [0; T ]
we can give up the hypothesis G(t; x; ·) to be decreasing: Conditions (pp)I and (pp)II could seem unnatural at the rst look. However we point out that it is possible to construct examples of no existence of coexistence states (even in the autonomous case), where (pp)I and (pp)II both fail. Using Theorem 2 we study, in the following corollary, a predator–prey system with periodic coecients, where some logistic e ects are considered. As one can see in [13] and in the references therein, there are many articles about the existence of positive and T -periodic solutions for this model. Moreover, some authors, assuming stronger constraints on the postiveness of the coecients (or their mean values), obtain results even about uniqueness and asymptotic stability of the solutions [5, 10, 19]. Corollary 2. Consider the system x0 = x (a(t) − b(t)x − c(t)y) y0 = y (−d(t) + e(t)x − f(t)y)
(8)
where a; b; c; d; e; f : R → R are continuous and T -periodic functions such that b; c; e; f d ¿ 0. Assume ≥ 0 and a; b; Z T (−d(t) + e(t)u0 (t)) dt ¿ 0; 0
where u0 (·) is the unique positive T -periodic solution of the equation x0 = x(a(t) − b(t)x); and; moreover; c ¿ 0 or; alternatively; f ¿ 0: Then system (8) has at least one positive T-periodic solution:
A. Battauz / Nonlinear Analysis 37 (1999) 735 – 749
745
Remark 4. Following the previous remark; under the hypothesis f ¿ 0; we can weaken the assumption d ¿ 0; asking d ≥ 0: This enable us to improve slightly a corresponding result in [13]: We can modify the hypotheses of Theorem 2 in order to obtain a result, which is applicable to system (8) when − d ¿ 0. Indeed we have the following: Theorem 3. Consider system (1) and assume that the functions F(t; x; ·); G(t; x; ·) are decreasing and G(t; ·; y) is increasing: Suppose that F(t; ·; 0) and G(t; 0; ·) satisfy the Landesman–Lazer type assumption and that Z T F(t; 0; vM (t)) dt ¿ 0; 0
where vM (·) is the maximal positive and T-periodic solution of y0 = y G(t; 0; y): Finally; assume that for each M ¿ 0 there exists N ¿ 0 such that (pp)I
RT 0
F(t; x; N ) dt¡0 for all x ∈ [0; M ];
or; alternatively; (pp)II
RT 0
G(t; M; N ) dt¡0:
Then there exists at least one T-periodic positive solution of Eq. (1): Proof. First of all, thanks to the Landesman–Lazer-type condition, we have that RT RT F(t; 0; 0) dt ¿ 0 and 0 G(t; 0; 0) dt ¿ 0; moreover, there exists L ¿ 0 such that, 0 for all x; y ≥ L; F(t; x; 0) ≤ (t) and G(t; 0; y) ≤ (t), where ; : R → R are continuous RT RT and T -periodic functions such that 0 (t) dt ¡0; 0 (t) dt ¡0: In order to prove our theorem, we consider at rst the homotopy F1∗ (t; x; y; ) = F(t; x; y); G1∗ (t; x; y; ) = G(t; x; y); which satis es conditions (K0 ), (K1 ) and (K2 ). We omit the details of the veri cation of these conditions, since the arguments follow closely the computation described in the proof of Theorem 2. We apply the rst part of Lemma 1 and arrive to the system x0 = xF(t; x; y); y0 = yG(t; 0; y). Then construct a second homotopy: F2∗ (t; x; y; ) = F(t; x; y + (1−)vM (t)), G2∗ (t; x; y; ) = G(t; 0; y) and use Lemma 1 again, since system (2), so obtained, veri es (K0 ), (K1 ) and (K2 ). Thus, we reach, for = 0, the equation x0 = xF(t; x; vM (t)); y0 = yG(t; 0; y). Finally, putting ˆ ˆ G3∗ (t; x; y; ) = G(t; 0; y) + (1 − )G(y); F3∗ (t; x; y; ) = F(t; x; vM (t)) + (1 − )F(x); RT ˆ ˆ where F(x) = (1=T ) 0 F(t; x; vM (t)) dt and G(y) = G(0; y); we obtain system (2), that ˆ ]; −1 [; 0) = −1 for = 0 is autonomous. Moreover, chosen a suitably small ¿ 0, d(F; −1 ˆ ]; [; 0) = −1. So, conditions (K0 ), (K1 ), (K2 ) and (K3 ) are veri ed and and d(G; the existence of at least one positive T -periodic solution for (1) is proved. Applying Theorem 3, we come to the following existence result for a system with continuous and periodic time-dependent coecients, which was achieved by a di erent proof in [19].
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A. Battauz / Nonlinear Analysis 37 (1999) 735 – 749
Corollary 3. Consider system ( 0 x = x (a(t) − b(t)x − c(t)y) y0 = y (d(t) + e(t)x − f(t)y)
(9)
where a; b; c; d; e; f : R → R are continuous and T -periodic functions such that b; c; e; d; f ¿ 0. Moreover; assume that f ≥ 0 and a; b; Z
T
0
(a(t) − c(t)v0 (t)) dt ¿ 0;
where v0 (·) is the unique positive T -periodic solution of the equation y0 = y (d(t) − f(t)y). Then system (9) has at least one positive T -periodic solution. We can still modify the assumptions of Theorem 2 on the functions F and G, preserving some typical aspects of a predator–prey model, which characterize even the classical Lotka–Volterra system and obtaining the following Theorem 4. Consider system (1) and assume that the function F(t; x; y) is decreasing in x and y and the function G(t; x; y) is increasing in x and decreasing in y. Assume that there exists L ¿ 0 such that, for all x ≥ 0; Z
T 0
Z F(t; x; 0) dt ¿ 0 ¿
0
T
F(t; 0; L) dt
and, for all y ≥ 0; Z 0
T
Z G(t; 0; 0) dt¡0¡
0
T
G(t; L; y) dt:
Then there exists at least one T -periodic positive solution of Eq. (1). Proof. Putting F1∗ (t; x; y; ) = F(t; x; y); G1∗ (t; x; y; ) = G(t; x; y), we obtain system (2) that satis es (K0 ), (K1 ) and (K2 ), since the equations v0 = vG(t; 0; v) and u0 = uF(t; u; 0) have no positive T -periodic solutions. We can use the rst part of Lemma 1 in order to pass to the system x0 = xF(t; 0; y); y0 = yG(t; x; 0). Then reduce the problem y); to an autonomous system by the homotopy F2∗ (t; x; y; ) = F(t; 0; y) + (1 − ) F(0; ∗ G2 (t; x; y; ) = G(t; x; 0) + (1 − )G(x; 0). Since (K0 ), (K1 ) and (K2 ) are satis ed and 0); ]; −1 [; 0) ·), ]; −1 [; 0) = − 1 and d(G(·; we can choose ¿ 0 so small that d(F(0; = 1; then condition (K3 ) is also veri ed and, using Lemma 1, we complete the proof. Using Theorem 4 we can prove the existence of positive and T -periodic solutions for another system with periodic time dependent coecients, that was studied in [14] and generalizes the classical Lotka–Volterra model with constant coecients.
A. Battauz / Nonlinear Analysis 37 (1999) 735 – 749
Corollary 4. Consider system ( x0 = x (a(t) − c(t)y) y0 = y (−d(t) + e(t)x)
747
(10)
where a; c; d; e : R → R are continuous and T -periodic functions such that c; e ≥ 0 and a; c ; d; e ¿ 0. Then system (10) has at least one positive T -periodic solution. As a nal application of Lemma 1 we study here a system, which generalizes a result obtained recently in [25]. Theorem 5. Consider system (1) and assume that the functions F(t; x; y); G(t; x; y) are decreasing in x and y. Moreover, suppose that, for all x; y ≥ 0; Z T Z T F(t; x; 0) dt ¿ 0 and G(t; 0; y) dt ¿ 0 0
and there is L ¿ 0 such that Z T F(t; 0; L) dt¡0 and 0
0
Z 0
T
G(t; L; 0) dt¡0:
Then there exists at least one T -periodic positive solution of Eq. (1). Proof. In order to apply Lemma 1 we consider a rst homotopy F1∗ (t; x; y; ) = F(t; x; y); G1∗ (t; x; y; ) = G(t; x; y), which satis es conditions (K0 ) and (K1 ). Indeed, if (x(·), y(·)) is a positive T -periodic solution of (2), then x and y are bounded in some point of the interval [0; T ] by L and, as the functions F and G are decreasing in both their space variables, also x0 =x and y0 =y are upper bounded. Hence (K1 ) easily follows. Condition (K2 ) is vacuously satis ed. We can so apply the rst part of Lemma 1 and move to the system x0 = xF(t; 0; y); y0 = yG(t; x; 0). Then, construct a second homotopy, depending on the parameter ∈ [0; 1], which reduces our problem to an au y); G2∗ (t; x; y; ) = tonomous system for = 0: F2∗ (t; x; y; ) = F(t; 0; y) + (1 − )F(0; 0). This homotopy veri es condition (K0 ), (K1 ) and, vacuG(t; x; 0) + (1 − )G(x; ·); ]; −1 [; 0) = − 1 and ously, (K2 ). Moreover, chosen a suitably small ¿ 0, d(F(0; −1 0); ]; [; 0) = − 1, so that condition (K3 ) is satis ed and the existence of a d(G(·; positive T -periodic solution of (1) is proved. Using Theorem 5, we come immediately to the following corollary, obtaining the existence of at least one coexistence state for the considered system under assumptions which are weaker than those supposed in [25]. Corollary 5. Consider the system ( x0 = x (a(t) − c(t)y) y0 = y (d(t) − e(t)x)
(11)
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A. Battauz / Nonlinear Analysis 37 (1999) 735 – 749
where a; c; d; e : R → R are continuous and T -periodic functions such that c; e ≥ 0 and a; c ; d; e ¿ 0. Then system (11) has at least one positive T -periodic solution. Acknowledgements I would like to thank Professor F. Zanolin for helpful discussions on topics related to this paper. References [1] S. Ahmad, On the nonautonomous Volterra–Lotka competition equations, Proc. Amer. Math. Soc. 117 (1993) 199 –204. [2] S. Ahmad, On almost periodic solutions of the competing species problem, Proc. Amer. Math. Soc. 102 (1988) 855– 861. [3] F. Albrecht, H. Gatzke, A. Haddad, N. Wax, The dynamics of two interacting populations, J. Math. Anal. Appl. 46 (1974) 658 – 670. [4] C. Alvarez, A.C. Lazer, An application of topological degree to the periodic competing species problem, J. Austral. Math. Soc. Ser. B 28 (1986) 202–219. [5] Z. Amine, R. Ortega, A periodic prey-predator system, J. Math. Anal. Appl. 185 (1994) 477 – 489. [6] A. Battauz, “Grado topologico e applicazioni a sistemi di erenziali nella dinamica della popolazioni”, Tesi di Laurea, Universita di Udine, 1996. [7] A. Battauz, F. Zanolin, Coexistence states for periodic competitive Kolmogorov systems, J. Math. Anal. Appl. 219 (1998) 179–199. [8] T.A. Burton, V. Hutson, Permanence for nonautonomous predator-prey systems, Di . Int. Eq. 4 (1991) 1269 –1280. [9] A. Capietto, J. Mawhin, F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc. 329 (1992) 41–72. [10] J.M. Cushing, Periodic time-dependent predator prey-systems, SIAM J. Appl. Math. 32 (1977) 82–95. [11] J.M. Cushing, Two species competition in a periodic environment, J. Math. Biol. 10 (1980) 385– 400. [12] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. [13] T. Ding, H. Huang, F. Zanolin, A priori bounds and periodic solutions for a class of planar systems with application to Lotka–Volterra equations, Discrete Continuous Dyn. Systems 1 (1995) 103 –118. [14] T. Ding, F. Zanolin, Periodic solutions and subharmonic solutions for a class of planar systems of Lotka–Volterra type, in: Proc. 1st WCNA, Tampa, FL, 1992, vol. I, de Gruyter, New York, 1996, pp. 395– 406. [15] K. Gopalsamy, Exchange of equilibria in two species Lotka–Volterra competition models, J. Austral. Math. Soc. Ser. B 24 (1982) 160 –170. [16] J.K. Hale, A.S. Somolinos, Competition for uctuating nutrient, J. Math. Biol. 18 (1983) 255–280. [17] P. Korman, Some new results on the periodic competition model, J. Math. Anal. Appl. 171 (1992) 131–138. [18] M.A. Kranosel’skii, P.P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin, 1984. [19] J. Lopez-Gomez, R. Ortega, A. Tineo, The periodic predator-prey Lotka–Volterra model, Adv. Di erential Equations 1 (1996) 403 – 423. [20] J. Mawhin, Topological degree methods in nonlinear boundary value problems, CBMS Regional Conf. Ser. Math., vol. 40, Amer. Math. Soc., Providence, RI, 1979. [21] P. de Mottoni, A. Schiano, Competition systems with periodic coecients: A geometric approach, J. Math. Biol. 11 (1981) 319 –335. [22] H.L. Smith, Periodic competitive di erential equations and the discrete dynamics of competitive maps, J. Di erential Equations 64 (1986) 165–194.
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[23] A. Tineo, Sobre las equaciones de Lotka–Volterra para dos especies en competencia, Notas de Matematica, Universidad de Los Andes, Merida-Venezuela 96 (1988) 1–20. [24] W. Wang, Z. Ma, Permanence of populations in a polluted environment, Math. Biosci. 122 (1994) 235–248. [25] K. Wojcik, On existence of periodic positive solutions, to appear. [26] F. Zanolin, Permanence and positive periodic solutions for Kolmogorov competing species systems, Results Math. 21 (1992) 224 –250. [27] F. Zanolin, Continuation theorems for the periodic problem via the translation operator, Rend. Sem. Mat. Univ. Pol. Torino 54 (1996) 1–23.