Periodic solution of a delayed predator–prey system with Michaelis–Menten type functional response

Journal of Computational and Applied Mathematics 166 (2004) 453 – 463 www.elsevier.com/locate/cam

Periodic solution of a delayed predator–prey system with Michaelis–Menten type functional response Hai-Feng Huoa;∗ , Wan-Tong Lib a

Department of Applied Mathematics, Gansu University of Technology, Lanzhou, Gansu 730050, PR China b Department of Mathematic, Lanzhou University, Lanzhou, Gansu 730000, PR China Received 12 September 2002; received in revised form 4 August 2003

Abstract By using the continuation theorem base on Gaines and Mawhin’s coincidence degree, su4cient and realistic conditions are obtained for the global existence of positive periodic solution for a delayed predator–prey system with Michaelis–Menten type functional response. Indeed, our results are applicable to state-dependent delays. c 2003 Elsevier B.V. All rights reserved.
MSC: 34K15; 92D25; 34C25 Keywords: Predator–prey system; Positive periodic solution; Coincidence degree

1. Introduction Recently, Xu and Chaplain [11] considered the following delayed predator–prey model with Michaelis–Menton type functional response: 
d x1 a12 x2 (t) = x1 (t) a1 − a11 x1 (t − 11 ) − ; dt m1 + x1 (t) 
d x2 a21 x1 (t − 21 ) a23 x3 (t) − a22 x2 (t − 22 ) − ; = x2 (t) −a2 + dt m1 + x1 (t − 21 ) m2 + x2 (t)
 d x3 a32 x2 (t − 32 ) = x3 (t) −a3 + − a33 x3 (t − 33 ) ; (1) dt m2 + x2 (t − 32 ) where ai and aij (i; j = 1; 2; 3) are positive constants, and ij (i; j = 1; 2; 3) are nonnegative constants. ∗

Corresponding author. E-mail addresses: [email protected] (H.-F. Huo), [email protected] (W.-T. Li).

c 2003 Elsevier B.V. All rights reserved. 0377-0427/$ - see front matter  doi:10.1016/j.cam.2003.08.042

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In [11], the authors proved that system (1) is uniformly persistent under appropriate conditions and obtained su4cient conditions for global asymptotic stability of the positive equilibrium of system (1). Since the variation of the environment plays an important role in many biological and ecological systems. In particular, the eIects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a Juctuating environment diIer from those in a stable environment. Thus, the assumption of periodicity of the parameters in the way (in a way) incorporates the periodicity of the environment (e.g., seasonal aIects of weather, food supplies, mating habits, etc.). In fact, it has been suggested in [8] that any periodic change of climate tends to impose its period upon oscillations of internal origin or to cause such oscillations to have a harmonic relation to periodic climatic changes. In view of this it is realistic to assume that the parameters in the models are periodic functions of period !. Thus, the modiKcation of (1) according to the environmental variation is the nonautonomous delay diIerential system 
d x1 a12 (t) x2 (t) = x1 (t) a1 (t) − a11 (t) x1 (t − 11 ) − ; dt m1 + x1 (t) 
d x2 a21 (t) x1 (t − 21 ) a23 (t) x3 (t) = x2 (t) −a2 (t) + − a22 (t) x2 (t − 22 ) − ; dt m1 + x1 (t − 21 ) m2 + x2 (t)
 d x3 a32 (t) x2 (t − 32 ) = x3 (t) −a3 (t) + − a33 (t) x3 (t − 33 ) ; (2) dt m2 + x2 (t − 32 ) where ai (t) and aij (t) (i; j=1; 2; 3) are continuous, bounded, and strictly positive functions on [0; ∞), and ij (i; j = 1; 2; 3) are nonnegative constants. A very basic and important ecological problem associated with the study of multispecies population interaction in a periodic environment is the global existence of positive periodic solution which plays the role played by the equilibrium of the autonomous models. The main purpose of this paper is to derive easily veriKable su4cient conditions for the global existence of a positive periodic solution of systems (2). The method used here will be the coincidence degree theory developed by Gaines and Mawhin [2]. Such approach was adopted in [1,4,6,7]. Finally, we remark that in recent years periodic population dynamics has become a very popular subject. In fact, several diIerent periodic models have been studied in [5,9,10,12]. 2. Main results In order to obtain the existence of a positive periodic solution of system (2), we Krst make the following preparations. Let Y and Z be two Banach spaces. Consider an operator equation Ly = Ny;

 ∈ (0; 1);

where L : Dom L ∩ Y → Z is a linear operator and  is a parameter. Let P and Q denote two projectors such that P : Y ∩ Dom L → Ker L

and Q : Z → Z=Im L:

In the sequel, we will use the following result of Mawhin [2, p. 40].

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455

Lemma 2.1. Let Y and Z be two Banach spaces and L a Fredholm mapping of index zero. Assume that N : M → Z is L-compact on M with  open bounded in Y . Furthermore, assume: (a) for each  ∈ (0; 1), y ∈ 9 ∩ Dom L, Ly = Ny; (b) for each y ∈ 9 ∩ Ker L, QNy = 0 and deg{JQNy;  ∩ Ker L; 0} = 0: Then the equation Ly = Ny has at least one solution in M ∩ L. Recall that a linear mapping L : Dom L ∩ y → Z with Ker L = L−1 (0) and Im L = L(Dom L), will be called a Fredholm mapping if the following two conditions hold: (i) Ker L has a Knite dimension; (ii) Im L is closed and has a Knite codimension. Recall also that the codimension of Im L is the dimension of Z=Im L, i.e., the dimension of the cokernel coker L of L. When L is a Fredholm mapping, its index is the integer Ind L = dim ker L − codim Im L. We shall say that a mapping N is L-compact on  if the mapping QN : M → Z is continuous, M is bounded, and Kp (I − Q)N : M → Y is compact, i.e., it is continuous and Kp (I − Q)N () M QN () is relatively compact, where Kp : Im L → Dom L ∩ Ker P is a inverse of the restriction Lp of L to Dom L ∩ Ker P, so that LKp = I and Kp L = I − P. For convenience, we shall introduce the notation:  ! 1 uM = u(t) dt; ! 0 where u is a periodic continuous function with period !. In system (2), we always assume the following: (H1 ) ai (t) and aij (t) (i; j = 1; 2; 3) are continuous, bounded, and strictly positive periodic functions with ! ¿ 0. Now we state our Krst theorem for the existence of a positive !-periodic solution of system (2). Theorem 2.1. In addition to (H1 ), assume further that system (2) satis
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(H4 ) The system of the equations aM1 − aM11 v1 −

aM12 v2 = 0; m1 + v 1

−aM2 +

aM21 v1 aM23 v3 − aM22 v2 − = 0; m1 + v 1 m2 + v 2

−aM3 +

aM32 v2 − aM33 v3 = 0 m2 + v 2

(3)

has a unique positive solution (v1 ; v2 ; v3 ) ∈ R3 . Then system (2) has at least one positive !-periodic solution. Proof. Since

   t  a12 (s) x2 (s) ds ; x1 (t) = x1 (0) exp a1 (s) − a11 (s) x1 (s − 11 ) − m1 + x1 (s) 0   t
a21 (s) x1 (s − 21 ) a23 (s) x3 (s) ds; x2 (t) = x2 (0) exp −a2 (s) + − a22 (s) x2 (s − 22 ) − m1 + x1 (s − 21 ) m2 + x2 (s) 0  t    a32 (s) x2 (s − 32 ) − a33 (s) x3 (s − 33 ) ds x3 (t) = x3 (0) exp −a3 (s) + m2 + x2 (s − 32 ) 0

the solution of system (2) remains positive for t ¿ 0, we can let x1 (t) = exp{y1 (t)};

x2 (t) = exp{y2 (t)};

x3 (t) = exp{y3 (t)}

(4)

and derive that dy1 a12 (t) exp{y2 (t)} = a1 (t) − a11 (t) exp{y1 (t − 11 )} − ; dt m1 + exp{y1 (t)} a21 (t) exp{y1 (t − 21 )} a23 (t) exp{y3 (t)} dy2 − a22 (t) exp{y2 (t − 22 )} − ; = −a2 (t) + dt m1 + exp{y1 (t − 21 )} m2 + exp{y2 (t)} dy3 a32 (t) exp{y2 (t − 32 )} = −a3 (t) + − a33 (t) exp{y3 (t − 33 )}: dt m2 + exp{y2 (t − 32 )} In order to use Lemma 2.1 to system (2), we take Y = Z = {y(t) = (y1 (t); y2 (t); y3 (t))T ∈ C(R; R3 ) : y(t + !) = y(t)} and denote

y = (y1 (t); y2 (t); y3 (t))T = max |y1 (t)| + max |y2 (t)| + max |y3 (t)|: t ∈[0;!]

t ∈[0;!]

t ∈[0;!]

Then Y and Z are Banach spaces when they are endowed with the norms · .

(5)

H.-F. Huo, W.-T. Li / Journal of Computational and Applied Mathematics 166 (2004) 453 – 463

Set



a1 (t) − a11 (t) exp{y1 (t − 11 )} −

a12 (t) exp{y2 (t)} m1 + exp{y1 (t)}

457



     a21 (t) exp{y1 (t − 21 )} a23 (t) exp{y3 (t)}    − a22 (t) exp{y2 (t − 22 )} − Ny =  −a2 (t) +   m1 + exp{y1 (t − 21 )} m2 + exp{y2 (t)}      a32 (t) exp{y2 (t − 32 )} − a33 (t) exp{y3 (t − 33 )} −a3 (t) + m2 + exp{y2 (t − 32 )} and

 ! 1 y(t) dt; y ∈ Y; Qz = z(t) dt; z ∈ Z: Ly = y ; ! 0 0 ! Evidently, Ker L = {y|y ∈ Y; y = R3 }; Im L = {z|z ∈ Z; 0 z(t) dt = 0} is closed in Z and dim Ker L = codim Im L = 3. Hence, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) Kp : Im L → Ker P ∩ dom L has the form  ! t  t 1 Kp (z) = z(s) ds − z(s) ds dt: ! 0 0 0 

Thus,

1 Py = !



!

  !  1 a12 (t) exp{y2 (t)} dt a1 (t) − a11 (t) exp{y1 (t − 11 )} −  ! 0 m1 + exp{y1 (t)}       1 ! a21 (t) exp{y1 (t − 21 )}   −a2 (t) + − a22 (t) exp{y2 (t − 22 )}    ! 0 m1 + exp{y1 (t − 21 )}  QNy =     a23 (t) exp{y3 (t)}   dt −     m2 + exp{y2 (t)}      !   1 a32 (t) exp{y2 (t − 32 )} − a33 (t) exp{y3 (t − 33 )} dt −a3 (t) + ! 0 m2 + exp{y2 (t − 32 )} 

and Kp (I − Q)Ny  t    a12 (s) exp{y2 (s)} ds a1 (s) − a11 (s) exp{y1 (s − 11 )} −   m1 + exp{y1 (s)}  0   t     a21 (s) exp{y1 (s − 21 )}   − a22 (s) exp{y2 (s − 22 )} −a2 (s) +   m + exp{y (s −  )} 1 1 21  0  =     (s) exp{y (s)} a 23 3  −  ds   m2 + exp{y2 (s)}      t    a32 (s) exp{y2 (s − 32 )} − a33 (s) exp{y3 (s − 33 )} ds −a3 (s) + m2 + exp{y2 (s − 32 )} 0

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   !  t 1 a12 (s) exp{y2 (s)} ds dt a1 (s) − a11 (s) exp{y1 (s − 11 )} −  ! m1 + exp{y1 (s)}   0 0    !  t   1 a21 (s) exp{y1 (s − 21 )}   −a2 (s) + − a22 (s) exp{y2 (s − 22 )}  ! m + exp{y (s −  )} 1 1 21 0 0   −     (s) exp{y (s)} a 3   − 23 ds dt   m + exp{y (s)} 2 2       !  t   1 a32 (s) exp{y2 (s − 32 )} − a33 (s) exp{y3 (s − 33 )} ds dt −a3 (s) + ! 0 m2 + exp{y2 (s − 32 )} 0
      ! t 1 a12 (t) exp{y2 (t)} − dt a1 (t) − a11 (t) exp{y1 (t − 11 )} −  ! 2  m1 + exp{y1 (t)} 0 
  !  t  1 (t) exp{y (t −  )} a   21 1 21 − − a22 (t) exp{y2 (t − 22 )} −a2 (t) +    ! 2  m + exp{y (t −  )} 1 1 21 0  : −   (t) exp{y (t)} a 23 3   dt  −    m2 + exp{y2 (t)} 
 !    t  1 a32 (t) exp{y2 (t − 32 )} − − a33 (t) exp{y3 (t − 33 )} dt −a3 (t) + ! 2 m2 + exp{y2 (t − 32 )} 0 

M Kp (I − Q)N () M are relatively Clearly, QN and Kp (I − Q)N are continuous and, moreover, QN (); M compact for any open bounded set  ⊂ X . Hence, N is L-compact on , here  is any open bounded set in X . Now, we reach the position to search for an appropriate open bounded subset  for the application of Lemma 2.1. Corresponding to equation Ly = Ny,  ∈ (0; 1), we have
 dy1 a12 (t) exp{y2 (t)} =  a1 (t) − a11 (t) exp{y1 (t − 11 )} − ; dt m1 + exp{y1 (t)}
 dy2 a21 (t) exp{y1 (t − 21 )} a23 (t) exp{y3 (t)} =  −a2 (t) + ; − a22 (t) exp{y2 (t − 22 )} − dt m1 + exp{y1 (t − 21 )} m2 + exp{y2 (t)}
 dy3 a32 (t) exp{y2 (t − 32 )} =  −a3 (t) + − a33 (t) exp{y3 (t − 33 )} : (6) dt m2 + exp{y2 (t − 32 )} Suppose that y(t) = (y1 ; y2 ; y3 ) ∈ Y is a solution of system (6) for a certain  ∈ (0; 1). By integrating (6) over the interval [0; !], we obtain   ! a12 (t) exp{y2 (t)} dt = 0; a1 (t) − a11 (t) exp{y1 (t − 11 )} − m1 + exp{y1 (t)} 0   ! a21 (t) exp{y1 (t − 21 )} a23 (t) exp{y3 (t)} dt = 0; −a2 (t) + − a22 (t) exp{y2 (t − 22 )} − m1 + exp{y1 (t − 21 )} m2 + exp{y2 (t)} 0   ! a32 (t) exp{y2 (t − 32 )} − a33 (t) exp{y3 (t − 33 )} dt = 0: −a3 (t) + m2 + exp{y2 (t − 32 )} 0

H.-F. Huo, W.-T. Li / Journal of Computational and Applied Mathematics 166 (2004) 453 – 463

Hence, 

  a12 (t) exp{y2 (t)} dt = aM1 !; a11 (t) exp{y1 (t − 11 )} + m1 + exp{y1 (t)} 0   ! a21 (t) exp{y1 (t − 21 )} a23 (t) exp{y3 (t)} − a22 (t) exp{y2 (t − 22 )} − dt = aM2 ! m1 + exp{y1 (t − 21 )} m2 + exp{y2 (t)} 0

and



!

!

0



 a32 (t) exp{y2 (t − 32 )} − a33 (t) exp{y3 (t − 33 )} dt = aM3 !: m2 + exp{y2 (t − 32 )}

From (6)–(9), we obtain   !  ! a12 (t) exp{y2 (t)}  dt a11 (t) exp{y1 (t − 11 )} + |y1 (t)| dt ¡ m1 + exp{y1 (t)} 0 0  ! + |a1 (t)| dt = 2aM1 !;  ! 0 |y2 (t)| dt ¡ 2aM2 ! 0

and

 0

!

|y3 (t)| dt ¡ 2aM3 !:

459

(7) (8)

(9)

(10) (11)

(12)

Note that (y1 (t); y2 (t); y3 (t)) ∈ Y , then there exists "i ; #i ∈ [0; !]; i = 1; 2; 3 such that yi ("i ) = min yi (t); yi (#i ) = max yi (t); t ∈[0;!]

t ∈[0;!]

i = 1; 2; 3:

(13)

By (7) and (13), we have aM1 ! ¿ aM11 ! exp {y1 ("1 )} : That is



y1 ("1 ) 6 ln

aM1 aM11

Then

 : 

y1 (t) 6 y1 ("1 ) +

0

!

|y1 (t)| dt ¡ ln



aM1 aM11

 + 2aM1 !:

By virtue of (7), (13) and (14), we also have exp{y2 ("2 )} exp{y2 ("2 )} ¿ aM12 ! : aM1 ! ¿ aM12 ! m1 + exp{y1 (t)} m1 + (aM1 = aM11 ) exp{2aM1 !} So

 y2 ("2 ) 6 ln

aM1 (m1 + (aM1 = aM11 ) exp{2aM1 !}) aM12

 :

(14)

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Then

 y2 (t) 6 y2 ("2 ) +

!

0





|y2 (t)| dt ¡ ln

aM1 (m1 + (aM1 = aM11 ) exp{2aM1 !}) aM12

 + 2aM2 !:

(15)

By (8) and (13), we have aM21 ! exp{y1 (#1 )} ¿ aM2 !: m1 + exp{y1 (#1 )} So

 y1 (#1 ) ¿ ln

aM2 m1 aM21 − aM2

Then

 y1 (t) ¿ y1 (#1 ) −

!

0

(16)

 :





|y1 (t)| dt ¿ ln

aM2 m1 aM21 − aM2

 − 2aM1 !:

(17)

It follows from (9) and (13) that aM32 ! exp{y2 (#2 )} ¿ aM3 !: m2 + exp{y2 (#2 )} So

 y2 (#2 ) ¿ ln

aM3 m2 aM32 − aM3

Then

 y2 (t) ¿ y2 (#2 ) −

0

!

(18)

 :



|y2 (t)| dt ¿ ln



aM3 m2 aM32 − aM3

 − 2aM2 !:

Hence, (14), (15), (17) and (19) imply that             aM1 aM2 m1    max |y1 (t)| 6 max ln + 2aM1 ! ; ln − 2aM1 ! := B1 t ∈[0;!] aM11 aM21 − aM2 and

(19)

(20)

  

    aM1  max |y2 (t)| 6 max ln aM1 m1 + exp{2aM1 !} aM12 + 2aM2 ! ; t ∈[0;!] aM11       aM3 m2 ln  − 2 a M ! 2   aM32 − aM3 := B2 :

By (9), (13) and (15), we also obtain aM32 (aM1 = aM12 ) exp{2aM2 !} − aM33 exp{y3 ("3 )} ¿ aM3 : m2 + (aM1 = aM12 ) exp{2aM2 !}

(21)

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Thus


y3 ("3 ) 6 ln

aM32 (aM1 = aM12 ) exp{2aM2 !} − aM3 m2 + (aM1 = aM12 ) exp{2aM2 !}

Then

 y3 (t) 6 y3 ("3 ) +

!

0






|y3 (t)| dt 6 ln



461

 aM33 :

aM32 (aM1 = aM12 ) exp{2aM2 !} − aM3 m2 + (aM1 = aM12 ) exp{2aM2 !}



 aM33

+ 2aM3 !: (22)

Furthermore, in view of (9) and (13), we obtain aM32 − aM33 exp{y3 (#3 )} 6 aM3 : m2 + 1 That is   [aM32 =(m2 + 1)] − aM3 y3 (#3 ) ¿ ln : aM33 Then

 y3 (t) ¿ y3 (#3 ) −

0

!



|y3 (t)| dt ¿ ln



[aM32 =(m2 + 1)] − aM3 aM33

 − 2aM3 !:

(23)

(22) and (23) imply that

    
  aM32 (aM1 = aM12 ) exp{2aM2 !}  − aM3 max |y3 (t)| 6 max ln aM33 + 2aM3 ! ; t ∈[0;!] m2 + (aM1 = aM12 ) exp{2aM2 !}       ln [aM32 =(m2 + 1)] − aM3 − 2aM3 !   aM33 := B3 :

(24)

Clearly, Bi , i = 1; 2; 3 are independent of , Under the assumption in Theorem 2.1, it is easy to show that the system of algebraic equations (3) has a unique solution (v1∗ ; v2∗ ; v3∗ )T ∈ int R3+ with vi∗ ¿ 0; i = 1; 2; 3. Denote B = B1 + B2 + B3 + B4 , where B4 ¿ 0 is taken su4ciently large such that

(ln{v1∗ }; ln{v2∗ }; ln{v3∗ }) = |ln{v1∗ }| + |ln{v2∗ } + ln{v3∗ }| ¡ B4 , and deKne  = {y(t) ∈ Y : y ¡ B}: It is clear that  satisKes condition (a) of the Lemma 2.1. When y = (y1 ; y2 ; y3 )T ∈ 9 ∩ Ker L = 9 ∩ R3 , y is a constant vector in R3 with y = M . Then   aM12 exp{y2 } rM1 − aM11 exp{y1 } −   m1 + exp{y1 }     exp{y } exp{y } a M a M   21 1 23 3 − aM22 exp{y2 } − QNx =  −rM2 +  = 0:  m1 + exp{y1 } m2 + exp{y2 }      aM32 exp{y2 } −rM3 + − aM33 exp{y3 } m2 + exp{y2 } Furthermore, in view of assumption in Theorem 2.1, it can easily be seen that deg{JQNx;  ∩ Ker L; 0} = 0:

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By now we know that  veriKes all the requirements of Lemma 2.1 and then system (5) has at least one !-periodic solution. By the medium of (4), we derive that (2) has at least one positive !-periodic solution. The proof is complete. Remark 2.1. From [11] we know that, under certain conditions, system (2) is uniformly persistent, then system (2) must have a positive equilibrium [3], which also implies that system (3) has a positive solution. Next, we consider the following predator–prey systems with state dependent delays: 
d x1 a12 (t) exp{y2 (t)} = x1 (t) a1 (t) − a11 (t)x1 (t − 11 (t; x1 (t); x2 (t); x3 (t))); ; dt m1 + exp{y1 (t)}
d x2 a21 (t)x1 (t − 21 (t; x1 (t); x2 (t); x3 (t))) = x2 (t) −a2 (t) + dt m1 + x1 (t − 21 (t; x1 (t); x2 (t); x3 (t)))  a23 (t)x3 (t) ; − a22 (t)x2 (t − 22 (t; x1 (t); x2 (t); x3 (t))) − m2 + x2 (t) 
d x3 a32 (t)x2 (t − 32 (t; x1 (t); x2 (t); x3 (t))) = x3 (t) −a3 (t) + dt m2 + x2 (t − 32 (t; x1 (t); x2 (t); x3 (t))) − a33 (t)x3 (t − 33 (t; x1 (t); x2 (t); x3 (t))):

(25)

Theorem 2.2. If the assumptions of Theorem 2.1 hold, then the system (25) has at least one positive !-periodic solution. Proof. The proof is similar to that of Theorem 2.1 and hence is omitted here. Acknowledgements This work is supported by the NNSF of China (10171040), the NSF of Gansu Province of China (ZS011-A25-007-Z), the Foundation for University Key Teacher by the Ministry of Education of China, the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China, the Key Research and Development Program for Outstanding Groups of Gansu University of Technology and the Development Program for Outstanding Young Teachers in Gansu University of Technology. References [1] M. Fan, K. Wang, Periodicity in a delayed ratio-dependent predator–prey system, J. Math. Anal. Appl. 262 (2001) 179–190. [2] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear DiIerential Equations, Springer, Berlin, 1977. [3] V. Hotson, The existence of an equilibrium for permanent systems, Rocky Mountain J. Math. 20 (1990) 1033–1040. [4] H.F. Huo, W.T. Li, Periodic solution of a periodic two-species competition model with delays, Internat. J. Appl. Math. 12 (2003) 13–21.

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