Positive periodic solutions of functional differential equations and population models

Transcription

1 Elecronic Journal of Differenial Equaions, Vol. 22(22), No. 71, pp ISSN: URL: hp://ejde.mah.sw.edu or hp://ejde.mah.un.edu fp ejde.mah.sw.edu (login: fp) Posiive periodic soluions of funcional differenial equaions and populaion models Daqing Jiang, Junjie Wei, & Bo Zhang Absrac In his paper, we employ Krasnosel skii s fixed poin heorem for cones o sudy he exisence of posiive periodic soluions o a sysem of infinie delay equaions, x () = A()x() + f(, x ). We prove wo general heorems and esablish new periodiciy condiions for several populaion growh models. 1 Inroducion In his paper we sudy he exisence of posiive periodic soluions of he sysem of funcional differenial equaions ẋ() = A()x() + f(, x ) (1.1) in which A() = diag[a 1 (), a 2 (),..., a n ()], a j C(R, R) is ω-periodic, f(, x ) is a funcion defined on R BC, and f(, x ) is ω-periodic whenever x is ω- periodic, where BC denoes he Banach space of bounded coninuous funcions φ : R R n wih he norm φ = sup θ R n φ j(θ) where φ = (φ 1, φ 2,..., φ n ) T, ω > is a consan. If x BC, hen x BC for any R is defined by x (θ) = x( + θ) for θ R. One of he mos used models, a prooype of (1.1), is he sysem of Volerra inegrodifferenial equaions [ ẋ i () = x i () a i () b ij ()x j () ] C ij (, s)g ij (x j (s))ds (1.2) which governs he populaion growh of ineracing species x j (), j = 1, 2,..., n. The inegral erm here specifies how much weigh o aach o he populaion a varies pas imes, in order o arrive a heir presen effec on he resources Mahemaics Subjec Classificaions: 34K13, 92B5. Key words: Funcional differenial equaions, posiive periodic soluion, populaion models. c 22 Souhwes Texas Sae Universiy. Submied July 5, 22. Published July 3, 22. Jiang & Wei were parially suppored by he Naional Naural Science Fundaion of China. 1

2 2 Posiive periodic soluions EJDE 22/71 availabiliy. For an auonomous sysem (1.2), here may be a sable equilibrium poin of he populaion. When he poenial equilibrium poin becomes unsable, here may exis a nonrivial periodic soluion. Then he oscillaion of soluions occurs. The exisence of such sable periodic soluion is of quie fundamenal imporance biologically since i concerns he long ime survival of species. The sudy of such phenomena has become an essenial par of qualiaive heory of differenial equaions. For hisorical background, basic heory of periodiciy, and discussions of applicaions of (1.1) o a variey of dynamical models, we refer o he reader o, for example, he work of Buron [1], Buron and Zhang [2], Cavani, Lizana, and Smih [3], Cushing [5], Gopalsamy [7], Hadeler [8], Havani and Kriszin [9], Hino, Murakami, Taio [1], Kuang [12], Makay [13], May [14], May and Leonard [15], Sech [17], and he references herein. I is our view ha one of he mos imporan problems in he sudy of dynamical models and heir applicaions is ha of describing he naure of he soluions for a large range of parameers involved. From a numerical poin of view, he exisence of a periodic soluion of he approximaion scheme mus also be sudied. The usual approach o fulfill such requiremens is o have a se of es equaions which are as general as possible and for which explici analyic condiions can be given. The nex sep is o characerize he numerical mehods which show he same exisence resul under he same condiions when applied o he es equaions. In his par of our invesigaion, we provide a unified approach o he sudy of exisence of posiive periodic soluions of sysem (1.1) under general condiions and apply he resuls o some well-known models in populaion dynamics. In Secion 2, we prove he main resuls concerning equaion (1.1). Applicaions of hese resuls o populaion models will be given in Secion 3. Le R = (, + ), R + = [, + ), and R = (, ] respecively. For each x = (x 1, x 2,..., x n ) T R n, he norm of x is defined as x = n x j. R n + = {(x 1, x 2,..., x n ) T R n : x j, j = 1, 2,..., n}. We say ha x is posiive whenever x R n +. BC(X, Y ) denoes he se of bounded coninuous funcions φ : X Y. 2 Exisence of Posiive Periodic Soluions We esablish he exisence of posiive periodic soluions of equaion (1.1) by applying Krasnosel skii s fixed poin heorem (see [6],[11]) on cones. A compac operaor will be consruced. I will be shown ha he operaor has a fixed poin, which corresponds o a periodic soluion of (1.1). We denoe f = (f 1, f 2,..., f n ) T and assume (H1) a j(s)ds for j = 1, 2,..., n. (H2) f j (, φ ) a j(s)ds for all (, φ) R BC(R, R n +), j = 1, 2,..., n. (H3) f(, x ) is a coninuous funcion of for each x BC(R, R n +).

3 EJDE 22/71 D. Jiang, J. Wei, & B. Zhang 3 (H4) For any L > and ε >, here exiss δ > such ha [φ, ψ BC, φ L, ψ L, φ ψ < δ, s ω] imply f(s, φ s ) f(s, ψ s ) < ε. (2.1) Definiion. Le X be a Banach space and K be a closed, nonempy subse of X. K is a cone if (i) αu + βv K for all u, v K and all α, β (ii) u, u K imply u =. Theorem 2.1 (Krasnosel skii [11]) Le X be a Banach space, and le K X be a cone in X. Assume ha Ω 1, Ω 2 are open subses of X wih Ω 1, Ω 1 Ω 2, and le Φ : K ( Ω 2 \ Ω 1 ) K be a compleely coninuous operaor such ha eiher (i) Φy y y K Ω 1 and Φy y y K Ω 2 ; or (ii) Φy y y K Ω 1 and Φy y y K Ω 2. Then Φ has a fixed poin in K ( Ω 2 \ Ω 1 ). We now for (, s) R 2, j = 1, 2,..., n, we define { } σ := min exp( 2 a j (s) ds), j = 1, 2,..., n We also define G j (, s) = (2.2) exp( s a j(ν)dν) exp( a j(ν)dν) 1. (2.3) G(, s) = diag[g 1 (, s), G 2 (, s),..., G n (, s)]. I is clear ha G(, s) = G( + ω, s + ω) for all (, s) R 2 and by (H2), G j (, s)f j (u, φ u ) for (, s) R 2 and (u, φ) R BC(R, R n +). Nex, we inroduce wo ses wih ω and σ as given above. C ω = {x C(R, R n ) : x( + ω) = x(), ; R}, K = {x C ω : x j () σ x j, [, ω], x = (x 1, x 2,..., x n ) T }. (2.4) One may readily verify ha K is a cone. Finally, we define an operaor Φ : K K as (Φx)() = +ω G(, s)f(s, x s )ds for x K, R, where G(, s) is defined following (2.3). We denoe ( T (Φx) = Φ 1 x, Φ 2 x,..., Φ n x). Lemma 2.2 Φ : K K is well-defined.

4 4 Posiive periodic soluions EJDE 22/71 Proof. For each x K, since f(, x ) is a coninuous funcion of, we have (Φx)() is coninuous in and (Φx)( + ω) = = = +2ω +ω +ω +ω G( + ω, s)f(s, x s )ds G( + ω, v + ω)f(v + ω, x v+ω )dv G(, v)f(v, x v )dv = (Φx)(). Thus, (Φx) C ω. Observe ha p j := exp( a j(ν) dν) exp( a j(ν)dν) 1 G j (, s) for all s [, + ω]. Hence, for x K, we have exp( a j(ν) dν) exp( a j(ν)dν) 1 =: q j (2.5) Φ j x q j f j (s, x s ) ds (2.6) and (Φ j x)() p j Therefore, (Φx) K. This complees he proof. f j (s, x s ) ds p j q j Φ j x σ Φ j x. Lemma 2.3 Φ : K K is compleely coninuous, and x = x() is an ω- periodic soluion of (1.1) whenever x is a fixed poin of Φ. Proof. We firs show ha Φ is coninuous. By (H4), for any L > and ε >, here exiss a δ > such ha [φ, ψ C ω, φ L, ψ L, φ ψ < δ] imply sup f(s, φ s ) f(s, ψ s ) < ε sω qω where q = max 1jn q j. If x, y K wih x L, y L, and x y < δ, hen (Φx)() (Φy)() +ω q G(, s) f(s, x s ) f(s, y s ) ds f(s, x s ) f(s, y s ) ds < ε for all [, ω], where G(, s) = max 1jn G j (, s). (Φy) < ε. Thus, Φ is coninuous. This yields (Φx)

5 EJDE 22/71 D. Jiang, J. Wei, & B. Zhang 5 Nex, we show ha f maps bounded ses ino bounded ses. Indeed, le ε = 1. By (H4), for any µ > here exiss δ > such ha [x, y BC, x µ, y µ, x y < δ] imply f(s, x s ) f(s, y s ) < 1. Choose a posiive ineger N such ha µ N < δ. Le x BC and define xk () = x() k N for k =, 1, 2,..., N. If x µ, hen Thus, x k x k 1 = sup x() k R N x()k 1 N x 1 N µ N < δ. for all s [, ω]. This yields f(s, x k s) f(s, x k 1 s ) < 1 f(s, x s ) N k=1 f(s, x k s) f(s, x k 1 s ) + f(s, ) < N + sup f(s, ) =: M µ. (2.7) s [,ω] I follows from (2.6) ha for [, ω], Φx = sup θ R Finally, for R we have (Φ j x)(θ) q j f j (s, x s ) ds qωm µ. d d (Φx)() = G(, + ω)f( + ω, x +ω) G(, )f(, x ) + A()(Φx)() = A()(Φx)() + [G(, + ω) G(, )]f(, x ) = A()(Φx)() + f(, x ). (2.8) Combine (2.6), (2.7), and (2.8) o obain d d (Φx)() A q f(s, x s ) ds + f(, x ) A qωm µ + M µ where A = max 1jn a j. Hence, {(Φx) : x K, x µ} is a family of uniformly bounded and equiconinuous funcions on [, ω]. By a heorem of Ascoli-Arzela (Royden [16, p.169]), he funcion Φ is compleely coninuous. I is clear from (2.8) ha x = x() is an ω-periodic soluion of (1.1) whenever x is a fixed poin of Φ. This proves he lemma.

6 6 Posiive periodic soluions EJDE 22/71 Theorem 2.4 Assume (H1)-(H4) and ha here are posiive consans R 1 and R 2 wih R 1 < R 2 such ha sup φ =R 1, φ K inf φ =R 2, φ K f(s, φ s ) ds R 1 /q (2.9) f(s, φ s ) ds R 2 /p (2.1) for all [, ω] wih p = min 1jn p j and q = max 1jn q j, where p j, q j are defined in (2.5). Then equaion (1.1) has an ω-periodic soluion x wih R 1 x R 2. Proof. Le x K and x = R 1. By (2.9), we have (Φx)() q +ω f(s, x s ) ds qr 1 /q = R 1. This implies ha (Φx) x for x K Ω 1, Ω 1 = {ψ C ω : ψ < R 1 }. If x K and x = R 2, hen Use (2.1) o obain (Φx)() p +ω (Φ j x)() p j f j (s, x s ) ds. +ω f(s, x s ) ds p(r 2 /p) = R 2 Thus, (Φx) x for x K Ω 2, Ω 2 = {ψ C ω : ψ < R 2 }. By Theorem 2.1, Φ has a fixed poin in K ( Ω 2 \ Ω 1 ). I follows from Lemma 2.3 ha (1.1) has an ω-periodic soluion x wih R 1 x R 2. This complees he proof. Corollary 2.5 Assume (H1)-(H4) and lim φ K, φ lim φ K, φ Then (1.1) has an ω-periodic soluion. f(s, φ s) ds φ f(s, φ s) ds φ =, (2.11) =. (2.12) Theorem 2.6 Assume (H1)-(H4) and ha here are posiive consans R 1 and R 2 wih R 1 < R 2 such ha inf φ =R 1, φ K sup φ =R 2, φ K f(s, φ s ) ds R 1 /p, (2.13) f(s, φ s ) ds R 2 /q. (2.14) Then equaion (1.1) has an ω-periodic soluion x wih R 1 x R 2.

7 EJDE 22/71 D. Jiang, J. Wei, & B. Zhang 7 The proof will be omied here since i is similar o ha of Theorem 2.4. We now sae a corollary o he above heorem. Corollary 2.7 Assume (H1)-(H4) and ha lim φ K, φ lim φ K, φ Then (1.1) has an ω-periodic soluion. 3 Applicaions f(s, φ s) ds φ f(s, φ s) ds φ =, (2.15) =. (2.16) We now apply he main resul obained in he previous secion o some wellknown models in populaion dynamics. Firs, we consider he scalar inegrodifferenial equaion [ Ṅ() = α()n() 1 1 ] B(s)N( + s)ds N () (3.1) which governs he growh of he populaion N() of a single species whose members compee among hemselves for a limied amoun of food and living spaces. Equaion (3.1) is a modificaion of he simple logisic equaion [ Ṅ() = αn() 1 N() ] N (3.2) where α is he inrinsic per capia growh rae and N is he oal carrying capaciy. We refer he reader o May [14] for a deailed model consrucion from (3.2) o (3.1). Suppose ha (P1) α(), N () are real-valued and ω-periodic funcions on R wih N () posiive and α()d >. (P2) B() is nonnegaive and piecewise coninuous on R wih B()d = 1. Theorem 3.1 Under assumpions (P1) and (P2), equaion (3.1) has a leas one posiive ω-periodic soluion. Le a() = α() and f(, x ) = x()α() N () B(s)x( + s)ds. I is clear ha f(, x ) is ω-periodic whenever x is ω-periodic. We need o show ha (H1)-(H4) hold. In fac, a()d > and f(, φ ) for all

8 8 Posiive periodic soluions EJDE 22/71 (, φ) R BC(R, R + ). Thus, (H1) and (H2) are saisfied. To verify (H3), le x BC(R, R + ) wih x B 1. Noice ha x()α()/n () is coninuous. By Lebesgue Dominan Convergence Theorem, one can easily see ha B(s)x( + s)ds is coninuous for all R. Thus, f(, x ) is coninuous in. Now le x, y BC(R, R + ) wih x L, y L for some L >. Then f(, x ) f(, y ) = x()α() N () x()α() N () + (x() y())α() N () L α N B(s)x( + s)ds y()α() N () B(s) x( + s) y( + s) ds sup x( + s) y( + s) + s R B(s) y( + s) ds x() y() α L N, B(s)y( + s)ds where N = inf{n(s) : s ω}. For any ε >, choose δ = εn /(2L a ). If x y < δ, hen f(, x ) f(, y ) < L α δ/n + δ α L/N = 2L α δ/n = ε. This implies ha (H4) holds. We now verify (2.11) and (2.12) for equaion (3.1). For φ K, we have φ() σ φ for all [, ω]. This yields as φ and f(, φ) φ f(, φ) φ inf τ [,ω] sup τ [,ω] α(τ) B(s)ds φ N (τ) α(τ) B(s)ds(σ 2 φ ) + N (τ) as φ. Thus, (2.11) and (2.12) are saisfied. By Corollary 2.5, equaion (3.1) has a posiive ω-periodic soluion. Nex, consider a hemaopoiesis model (Weng and Liang [19]) Ṅ() = γ()n() + α() B(s)e β()n( s) ds (3.3) where N() is he number of red blood cells a ime, α, β, γ C(R, R) are ω-periodic, and B L 1 (R + ) is nonnegaive and piecewise coninuous. This is a generalized model of he red cell sysem inroduced by Wazewska-Czyzewska and Lasoa [18] ṅ() = γn() + αe βn( r) (3.4)

9 EJDE 22/71 D. Jiang, J. Wei, & B. Zhang 9 where α, β, γ, r are consans wih r >. Periodiciy in equaion (3.4) has been invesigaed exensively in he lieraure (see Chow [4]). Using Corollary 2.7 wih a() = γ() and f(, φ ) = α() B(s)e β()φ( s) ds, we obain he following heorem whose proof is essenially he same as ha of Theorem 3.1, and herefore will be omied. Theorem 3.2 Suppose ha α() >, β() for all R +, γ()d >. + B(s)ds >. Then equaion (3.3) has a posiive ω-periodic soluion. We now consider he Volerra equaion menioned in Secion 1.1. [ ] ẋ i () = x i () a i () b ij ()x j () C ij (, s)g ij (x j (s))ds (3.5) where x i () is he populaion of he ih species, a i, b ij C(R, R) are ω-periodic and C ij (, s) is piecewise coninuous on R 2. Theorem 3.3 Suppose ha he following condiions hold for i, j = 1, 2,..., n. (i) (ii) b ij () a i (s)ds, a i (s)ds, C ij (, s) (iii) g ij C(R +, R + ) is increasing wih g ij () =, (iv) b ii (s)ds, (v) C ij ( + ω, s + ω) = C ij (, s) for all (, s) R 2 wih sup R C ij(, s) ds < +. Then equaion (3.5) has an ω-periodic soluion. Proof. For x = (x 1, x 2,..., x n ) T, define f i (, x ) = x i () b ij ()x j () x i () a i (ν)dν for all (, s) R 2, C ij (, s)g ij (x j (s))ds for i = 1, 2,..., n and se f = (f 1, f 2,..., f n ) T. Then (3.5) can be wrien in he form of (1.1) wih (H 1 ) (H4) saisfied. Define b = max{ b ij : i, j = 1, 2,,,, n} { } C = max sup C ij (, s) ds : i = 1, 2,,,, n R g (u) = max{g ij (u) : i, j = 1, 2,,,, n}

10 1 Posiive periodic soluions EJDE 22/71 Le x K, where K is defined in (2.4). Then and This implies [ f i (, x ) x i () b x + x i () [b x + C g ( x )] ] C ij (, s) g ij ( x j )ds f(, x s ) ds ω x [b x + C g ( x )]. f(, x s) ds ω[b x + C g ( x )] x as x. Since x i () σ x i for all R whenever x K and b ij (), C ij (, s) have he same sign, we obain and = f i (, x s ) ds x i () b ij () x j ()d + b ii () x i () 2 d σ 2 x i 2 b ii () d f(, x s ) ds σ 2 n i=1 x i 2 min 1in Here we have applied he inequaliy x i () C ij (, s) g ij (x j (s))dsd b ii () d σ2 n x 2 min 1in b ii () d. ( n ) 2 i=1 x n i n i=1 x i 2. Thus, f(, x s) ds x + as x +. By Corollary 2.5, equaion (3.5) has an ω-periodic soluion. Remark. I is clear from he proof of Theorem 3.3 ha condiion (iv) can be replaced by (iv ) C ii (, s) dsd and g ii (u) + as u +. We conclude his paper by invesigaing he following scalar equaion of advanced and delay ype which is highly nonlinear and akes a quie general form. ẋ() = a()x() + C(s)g(, x( τ ()), x( τ 1 ()), x( + s))ds (3.6)

11 EJDE 22/71 D. Jiang, J. Wei, & B. Zhang 11 where a, τ, τ 1 C(R, R) are ω-periodic, g C(R 4, R + ), and C L(R, R) is piecewise coninuous. Le u = (u, u 1, u 2 ) T R 3 and define g(, u) = g(, u, u 1, u 2 ). We assume ha g( + ω, u) = g(, u) for all (, y) R 4 and (P 1) (P 2) C() a(s)ds, a(s)ds for all R and C(u) du = 1. Theorem 3.4 Under assumpions (P 1) and (P 2), equaion (3.6) has a leas one posiive ω-periodic soluion, provided one of he following condiions hold (P 3) (P 4) lim u max [,ω] g(, u) u lim u, u j σ u, j2 = and lim u, u j σ u, j2 g(, u) min = and lim [,ω] u u + where σ is given in (2.2) and u = (u, u 1, u 2 ) T. g(, u) min =, [,ω] u max [,ω] g(, u) u Proof. We firs show ha under (P 3), condiions (2.11) and (2.12) of Corollary 2.5 hold. Le f(, x ) = C(s)g(, x( τ ()), x( τ 1 ()), x( + s))ds. By (P 3), for any ε >, here exiss R 1 > such ha u R 1 implies g(, u) ε u for all [, ω]. Now le x C ω wih x < R 1 /3. Then This yields x( τ ()) + x( τ 1 ()) + x( + s) 3 x < R 1. = f(, x ) C(s) g(, x( τ ()), x( τ 1 ()), x( + s))ds C(s) [ε( x( τ ()) + x( τ 1 ()) + x( + s) )]ds C(s) ds(3ε x ) = 3ε x and f(, x ) d 3ωε. x

12 12 Posiive periodic soluions EJDE 22/71 Thus, (2.11) holds. Nex, i follows from he second par of (P 3) ha for any M > here exiss R 2 > such ha [ u R 2, u j σ u, j 2] implies g(, u) M u. (3.7) Le K be given in (2.4) and observe ha if x K, hen x( τ j ()) σ x, x( + s) σ x for all R and j =, 1, and x( τ ()) + x( τ 1 ()) + x( + s) 3σ x 3σ R 2 /(3σ) = R 2 whenever x R 2 /(3σ). By (3.7), we have g(, x( τ ()), x( τ 1 ()), x( + s)) M( x( τ ()) + x( τ 1 ()) + x( + s) ) M(3σ x ). Using he above inequaliy, we ge Thus, f(, x ) d = C(s) g(, x( τ ()), x( τ 1 ()), x( + s))ds d C(s) (3σM x )ds d = 3σωM x. f(, x ) d 3σωM. x This proves (2.12) holds. By Corollary 2.5, equaion (3.6) has an ω-periodic soluion. We omi he case when (P 4) holds. This complees he proof. References [1] T. A. Buron, Volerra Inegral and Differenial equaions, Academic Press, New York, [2] T. A. Buron and B. Zhang, Uniform ulimae boundedness and periodiciy in funcional differenial equaions, Tohoku Mah J. 42(199), [3] M. Cavani, M. Lizana, and H. L. Smih, Sable periodic orbis for a predaor- prey model wih delay, J. Mah. Anal. Appl. 249(2), [4] S. -N. Chow, Exisence of periodic soluions of auonomous funcional differenial equaions, 15(1974), [5] J. M. Cushing, Inegro-Differenial Equaions wih Delay Models in Populaion Dynamics, Lecure Noes in Biomah. 2, Springer-Verlag, Berlin, [6] K. Deimling, Nonlinear Funcional Analysis, Springer-Verlag, New York, 1985.

13 EJDE 22/71 D. Jiang, J. Wei, & B. Zhang 13 [7] K. Gopalsamy, Sabiliy and Oscillaions in Delay Differenial Equaions of Populaion Dynamics, Kluwer Academic Publishers, London, [8] K. Hadeler, On he sabiliy of he saionary sae of a populaion growh equaion wih ime-lag, J. Mah. Biology 3(1976), [9] L. Havani and T. Kriszin, On he exisence of periodic soluions for linear inhomogeneous and quasilinear funcional differenial equaions, J. Differenial equaions 97(1992), [1] Y. Hino, S. Murakami, and T. Naio, Funcional Differenial Equaions wih Infinie Delay, Springer-Verlag, New York, [11] M. A. Krasnoselskii, Posiive Soluions of Operaor Equaions, Gorninggen, Noordhoff, [12] Y. Kuang, Delay Differenial Equaions wih Applicaion in Populaion Dynamics, Academic Press, New York, [13] G. Makay, Periodic soluions of dissipaive funcional differenial equaions, Tohoku Mah. J. 46(1994), [14] R. M. May, Sabiliy and Complexiy in Model Ecosysems, Princeon Universiy Press, [15] R. M. May and W. J. Leonard, Nonlinear aspec of compeiion beween hree species, SIAM J. Appl. Mah. 29(1975), [16] H. L. Royden, Real Analysis, Macmillan Publishing Company, New York, [17] W. Sech, The effec of ime lags on he sabiliy of he equilibrium sae of a populaion growh equaion, J. Mah. Biology 5(1978), [18] M. Wazewska-Czyzewska and A. Lasoa, Mahemaical models of he red cell sysem, Maemaya Sosowana 6(1976), [19] P. Weng and M. Liang, The exisence and behavior of periodic soluions of a Hemaopoiesis model, Mahemaica Applicaa 8(1995), Daqing Jiang ( daqingjiang@vip.163.com) Junjie Wei ( weijj@nenu.edu.cn) Deparmen of Mahemaics, Norheas Normal Universiy Changchun, Jilin 1324, China Bo Zhang Deparmen of Mahemaics and Compuer Science Fayeeville Sae Universiy Fayeeville, NC 2831, USA bzhang@uncfsu.edu