Periodic solutions of delayed predator–prey model with the Beddington–DeAngelis functional response

Chaos, Solitons and Fractals 33 (2007) 505–512 www.elsevier.com/locate/chaos

Periodic solutions of delayed predator–prey model with the Beddington–DeAngelis functional response Hai-Feng Huo

a,*,1

, Wan-Tong Li

b,2

, Juan J. Nieto

c,3

a

c

Institute of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, PR China b Department of Mathematics, Lanzhou University Lanzhou, Gansu 730000, PR China Departamento de Ana´lisis Matema´tico, Facultad de Matema´ticas, Universidad de Santiago de Compostela 15782, Spain Accepted 16 December 2005

Communicated by Prof. A. Helal

Abstract By using the continuation theorem based on Gaines and Mawhin’s coincidence degree, sufficient and realistic conditions are obtained for the global existence of positive periodic solutions for a delayed predator–prey model with the Beddington–DeAngelis functional response. Our results are applicable to state dependent and distributed delays. Ó 2006 Elsevier Ltd. All rights reserved.

1. Introduction Cantrell and Cosner [2] consider the following predator–prey model with the Beddington–DeAngelis functional response 8 dx cxy > > < dt ¼ xð1  xÞ  1 þ nx þ my ; ð1:1Þ > dy fxy > : ¼  dy; dt 1 þ nx þ my

*

Corresponding author. E-mail addresses: [email protected] (H.-F. Huo), [email protected] (W.-T. Li), [email protected] (J.J. Nieto). 1 Supported by the NSF of Gansu Province of China (3ZS042-B25-013), the NSF of Bureau of Education of Gansu Province of China (0416B-08), the Key Research and Development Program for Outstanding Groups of Lanzhou University of Technology and the Development Program for Outstanding Young Teachers in Lanzhou University of Technology. 2 Supported by the NNSF of China (10571078) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China. 3 Supported by Ministerio de Educacio´ y Ciencia and FEDER, project MTM2004-06652-C03-01, and by Xunta de Galicia and FEDER, project PGIDIT05PXIC20702PN. 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.12.045

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where x and y represent prey and predator densities, respectively. The functional response in (1.1) was introduced by Beddington [1] and DeAngelis et al. [4]. It is similar to the well known Holling type 2 functional response but has an extra term my in the denominator which models mutual interference between predators. It can be derived mechanistically via considerations of time utilization [1,16] or spatial limits on predation [3]. There has been considerable interest in ratio-dependent predator–prey model; see [3] and the references therein. A ratio-dependent version of (1.1) would have the form 8 dx cxy > > < dt ¼ xð1  xÞ  nx þ my ; ð1:2Þ > dy fxy > : ¼  dy. dt nx þ my The ratio-dependent form (1.2) also incorporates mutual interference by predators, but it has somewhat singular behavior at low densities and has been criticized on other grounds. See [12] for a mathematical analysis and the references in [3] for some aspects of the debate among biologists about ratio dependence. The Beddington–DeAngelis form of functional response has some of the same qualitative features as the ratio-dependent models form but avoids some of the same behaviors of ratio-dependent models at low densities which have been the source of controversy. Hence it seems worthy of further study. On the other hand, we often observe that populations in the real world tend to fluctuate. There are three typical approaches for modeling such behavior: (i) introduce more species into the model, and consider higher dimensional systems (like predator–prey interactions, May [15]); (ii) assume that the per capita growth function is time dependent and periodic in time; (iii) take into account the time delay effect in the population dynamics [17,20]. Although they are good mechanism of generating periodic solutions (and therefore offer some explanations to the often observed oscillatory behavior in population densities), it does not give us any insight as which is the real generating or dominating force behind the oscillatory behavior if only one of such mechanism is considered. Naturally, more realistic and interesting models of population interactions should take into account both the seasonality of the changing environment and the effects of time delay. Therefore, it is interesting and important to study the following nonautonomous delayed predator–prey model with the Beddington–DeAngelis functional response 8 dxðtÞ cðtÞxðtÞyðtÞ > > < dt ¼ xðtÞ½aðtÞ  bðtÞxðt  sðt; xðtÞ; yðtÞÞ  1 þ nxðtÞ þ myðtÞ ;   ð1:3Þ > dyðtÞ f ðtÞxðt  rðt; xðtÞ; yðtÞÞÞ > : ¼ yðtÞ  dðtÞ ; dt 1 þ nxðt  rðt; xðtÞ; yðtÞÞÞ þ myðt  rðt; xðtÞ; yðtÞÞ where x(t) and y(t) stand for the prey’s and the predator’s density at time t, respectively; a(t), b(t), c(t), d(t) 2 C(R, R+), R+ = (0, +1) are x-periodic function; s and r 2 C(R, R) are nonnegative x-periodic function with respect to their first argument; m and n are positive constants. In system (1.3), we are assuming that the growth of the predator is influenced by the amount of prey in the past and the prey growth rate response to resources limitations involves a delay. 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 solution which plays the role played by the equilibrium of the autonomous models. But existing results on the existence of periodic solutions in periodic systems (population models, in particular) often fall into one of these three categories: (1) the results of the applications of the contraction principle or the fluctuation, which establish both the existence and attractivity of the periodic solutions in periodic equations with time delay [11, p. 181]; (2) the existence simply follows from the observation that the periodic solution exists when there is no time delay and this periodic solution remains so when the time delay is a multiple of the period of the periodic equation [8,19]; (3) the results of the application of Horn’s asymptotic fixed-point theorem [6,18]. While these methods often allow the investigator to address the stability issues of the periodic solutions, the conditions for existence are often unnecessary, numerous, tedious, stringent and difficult to satisfy. Specifically, all the above methods are ill suited to problems with state-dependently delay equation. By employing the powerful and effective coincidence degree method, we found that the existence of periodic solutions in periodic equations with or without state-dependent delay require only a set of natural and easily verifiable conditions. These conditions are readily satisfied in many realistic population models. Such approach was adopted in [5,9,13,14]. To our knowledge, no such work has been done on the global existence of positive periodic solutions of systems (1.3). The main purpose of this paper is to derive easily verifiable sufficient conditions for the global existence of positive periodic solutions of systems (1.3). The method used here will be the coincidence degree theory developed by Gaines and Mawhin [7].

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2. Existence of a positive periodic solution To obtain the existence of a positive periodic solution of the system (1.3), we first make the following preparations. Let X and Z be two Banach spaces. Consider the operator equation Lx ¼ kNx;

k 2 ð0; 1Þ;

where L : Dom L \ X ! Z is a linear operator and k is a parameter. Let P and Q denote two projectors such that P : X \ Dom L ! Ker L and

Q : Z ! Z=Im L.

In the sequel, we will use the following result of Gaines and Mawhin [7, p. 40]. Recall that a linear mapping L : Dom L \ X ! Z with Ker L = L1(0) and Im L = L(Dom L), will be called a Fredholm mapping if the following two conditions hold: (i) Ker L has a finite dimension; (ii) Im L is closed and has a finite 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–codimIm L. We shall say that a mapping N is L-compact on X if the mapping QN : X ! Z is continuous, QN ðXÞ is bounded, and K p ðI  QÞN : X ! X is compact, i.e., it is continuous and K p ðI  QÞN ðXÞ 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 KpL = I  P. Lemma 2.1. Let X and Z be two Banach spaces and L a Fredholm mapping of index zero. Assume that N : X ! Z is Lcompact on X with X open bounded in X. Furthermore assume: (a) for each k 2 (0, 1), x 2 oX \ Dom L, Lx 6¼ kNx; (b) for each x 2 oX \ Ker L, QNx 6¼ 0 and degfJQNx; X \ Ker L; 0g 6¼ 0. Then the equation Lx = Nx has at least one solution in X. For convenience, we shall introduce the notation: Z 1 x uðtÞ dt; u ¼ x 0 where u is a periodic continuous function with period x. Now we state our first theorem for the existence of a positive x-periodic solution of system (1.3). Theorem 2.1. If ma  c > 0, expf2axg

 ðma  cÞðf  ndÞ  1 > 0; bmd

and the system of algebraic equations cv2 ¼ 0; 1 þ nv1 þ mv2 f v1 ¼ 0; d  1 þ nv1 þ mv2

a  bv1 

has a unique solution ðv1 ; v2 Þ with vi > 0, i = 1, 2. Then Eq. (1.3) has at least one positive x-periodic solution.

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Proof. Since Z t    cðtÞyðtÞ ds ; aðtÞ  bðtÞxðt  sðt; xðtÞ; yðtÞÞÞ  1 þ nxðtÞ þ myðtÞ 0 Z t    f ðtÞxðt  rðt; xðtÞ; yðtÞÞÞ yðtÞ ¼ yð0Þ exp  dðtÞ ds 1 þ nxðt  rðt; xðtÞ; yðtÞÞÞ þ myðt  rðt; xðtÞ; yðtÞÞ 0 xðtÞ ¼ xð0Þ exp

the solution of system (1.3) remains positive for t P 0, we can let xðtÞ ¼ expfx1 ðtÞg;

yðtÞ ¼ expfx2 ðtÞg

ð2:1Þ

and derive that 8 dx1 ðtÞ cðtÞ expfx2 ðtÞg > > ¼ aðtÞ  bðtÞ expfx1 ðt  sðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg  ; > < dt 1 þ n expfx1 ðtÞg þ m expfx2 ðtÞg > dx2 ðtÞ f ðtÞ expfx1 ðt  rðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg > > ¼  dðtÞ; : dt 1 þ n expfx1 ðt  rðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg þ m expfx2 ðt  rðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg

ð2:2Þ

In order to use Lemma 2.1 to system (1.3), we take X ¼ Z ¼ fx ðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞT 2 CðR; R2 Þ : x ðt þ xÞ ¼ x ðtÞg and denote kx k ¼ kðx1 ðtÞ; x2 ðtÞÞT k ¼ max jx1 ðtÞj þ max jx2 ðtÞj. t2½0;x

t2½0;x

Then X and Z are Banach spaces when they are endowed with the norms kÆk. Set  cðtÞ expfx2 ðtÞg Nx ¼ aðtÞ  bðtÞ expfx1 ðt  sðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg  1 þ n expfx1 ðtÞg þ m expfx2 ðtÞg 

f ðtÞ expfx1 ðt  rðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg  dðtÞ 1 þ n expfx1 ðt  rðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg þ m expfx2 ðt  rðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg

and 0

Lx ¼ x ; Px ¼

1 x

Z 0

x

x ðtÞ dt;

x 2 X ;

Qz ¼

1 x

Z



x

zðtÞ dt;

z 2 Z.

0

Rx Evidently, Ker L = {x*jx* 2 X, x = R2}, Im L ¼ fzjz 2 Z; 0 zðtÞ dt ¼ 0g is closed in Z and dimKer L = codimIm L = 2. 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 Z Z Z t 1 x t zðsÞ ds  zðsÞ ds dt. K p ðzÞ ¼ x 0 0 0 Thus



  cðtÞ expfx2 ðtÞg x1 ðtÞ x2 ðtÞ dt aðtÞ  bðtÞ expfx1 ðt  sðt; e ; e Þg  1 þ n expfx1 ðtÞg þ m expfx2 ðtÞg 0   Z x 1 f ðsÞ expfx1 ðs  rðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg  dðtÞ; dt  x 0 1 þ n expfx1 ðt  rðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg þ m expfx2 ðt  rðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg

1 QNx ¼ x 

Z

and K p ðI  QÞN ¼

x

Z t  aðsÞ  bðsÞ expfx1 ðs  sðs; ex1 ðsÞ ; ex2 ðsÞ ÞÞg 

 cðsÞ expfx2 ðsÞg ds 1 þ n expfx1 ðsÞg þ m expfx2 ðsÞg 0   Z t f ðsÞ expfx1 ðs  rðs; ex1 ðsÞ ; ex2 ðsÞ ÞÞg   dðsÞ ds 1 þ n expfx1 ðs  rðs; ex1 ðsÞ ; ex2 ðsÞ ÞÞg þ m expfx2 ðs  rðs; ex1 ðsÞ ; ex2 ðsÞ ÞÞg 0  Z x Z t  1 cðsÞ expfx2 ðsÞg ds dt  aðsÞ  bðsÞ expfx1 ðs  sðs; ex1 ðsÞ ; ex2 ðsÞ ÞÞg  x 0 1 þ n expfx1 ðsÞg þ m expfx2 ðsÞg 0

H.-F. Huo et al. / Chaos, Solitons and Fractals 33 (2007) 505–512

509

  f ðsÞ expfx1 ðs  rðs; ex1 ðsÞ ; ex2 ðsÞ ÞÞg  dðsÞ ds dt 1 þ n expfx1 ðs  rðs; ex1 ðsÞ ; ex2 ðsÞ ÞÞg þ m expfx2 ðs  rðs; ex1 ðsÞ ; ex2 ðsÞ ÞÞg 0 0 Z x    t 1 cðtÞ expfx2 ðtÞg  aðtÞ  bðtÞ expfx1 ðt  sðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg  dt  x 2 0 1 þ n expfx1 ðtÞg þ m expfx2 ðtÞg  Z x    t 1 f ðtÞ expfx1 ðt  rðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg   dðtÞ dt .  x 2 0 1 þ n expfx1 ðt  sðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg þ m expfx2 ðt  sðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg 

1 x

Z

x

Z t

Clearly, QN and Kp(I  Q)N are continuous and, moreover, QN ðXÞ, K p ðI  QÞN ðXÞ are relatively compact for any open bounded set X  X. Hence, N is L-compact on X, here X is any open bounded set in X. Now we reach the position to search for an appropriate open bounded subset X for the application of Lemma 2.1. Corresponding to equation Lx* = kNx*, k 2 (0, 1), we have 8   cðtÞ expfx2 ðtÞg > 0 x1 ðtÞ x2 ðtÞ > ; ðtÞ ¼ k aðtÞ  bðtÞ expfx ðt  sðt; e ; e ÞÞg  x > 1 < 1 1 þ n expfx1 ðtÞg þ m expfx2 ðtÞg ð2:3Þ   > f ðtÞ expfx1 ðt  rðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg > 0 >  dðtÞ . : x2 ðtÞ ¼ k 1 þ n expfx1 ðt  sðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg þ m expfx2 ðt  sðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg Suppose that x*(t) = (x1, x2) 2 X is a solution of system (2.3) for a certain k 2 (0, 1). By integrating (2.3) over the interval [0, x], we obtain 8   Rx cðtÞ expfx2 ðtÞg > x1 ðtÞ x2 ðtÞ > dt ¼ 0; aðtÞ  bðtÞ expfx ðt  sðt; e ; e ÞÞg  > 1 < 0 1 þ n expfx1 ðtÞg þ m expfx2 ðtÞg   Rx > f ðtÞ expfx1 ðt  rðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg > >  dðtÞ dt ¼ 0. : 0 1 þ n expfx1 ðt  sðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg þ m expfx2 ðt  sðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg Hence, Z

x

 bðtÞ expfx1 ðt  sðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg þ

 cðtÞ expfx2 ðtÞg dt ¼  ax 1 þ n expfx1 ðtÞg þ m expfx2 ðtÞg

ð2:4Þ

 f ðtÞ expfx1 ðt  rðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg  dt ¼ dx. 1 þ n expfx1 ðt  sðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg þ m expfx2 ðt  sðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg

ð2:5Þ

0

and Z

x

0



From (2.3), (2.4) and (2.5), we obtain  Z x Z x Z x cðtÞ expfx2 ðtÞg jx01 ðtÞj dt < ½bðtÞ expfx1 ðt  sðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg dt þ dt 1 þ n expfx1 ðtÞg þ m expfx2 ðtÞg 0 0 0 Z x þ jaðtÞj dt 0

¼ 2ax;

ð2:6Þ

and Z

x

jx02 ðtÞj dt <

0

Z

x

0



 f ðtÞ expfx1 ðt  rðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg  ¼ 2dx.  dt þ dx 1 þ n expfx1 ðt  sðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg þ m expfx2 ðt  sðt; ex1 ðtÞ ; ex2 ðtÞ ÞÞg

ð2:7Þ

Note that (x1(t), x2(t))T 2 X, then there exists ni, gi 2 [0, x], i = 1, 2 such that xi ðni Þ ¼ min xi ðtÞ; t2½0;x

xi ðgi Þ ¼ max xi ðtÞ;

By (2.4) and (2.8), we have  expfx1 ðn1 Þg. ax P bx That is   a x1 ðn1 Þ 6 ln  . b

t2½0;x

i ¼ 1; 2.

ð2:8Þ

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Then x1 ðtÞ 6 x1 ðn1 Þ þ

Z

x

0

  a jx01 ðtÞj dt < ln  þ 2ax. b

ð2:9Þ

By virtue of (2.4) and (2.8), we also have ax P bx expfx1 ðg1 Þg þ

c x. m

So x1 ðg1 Þ P ln

  ma  c . mb

Then x1 ðtÞ P x1 ðg1 Þ 

Z

x

jx01 ðtÞj dt 0

  ma  c P ln  2 ax. mb

It follows from (2.9) and (2.10) that             a ma  c  :¼ B1 .  2 a x max jx1 ðtÞj 6 max ln  þ 2ax; ln   t2½0;x b mb

ð2:10Þ

ð2:11Þ

Similarly, by (2.5) and (2.8), we obtain  P f x dx

expfx1 ðn1 Þg ; 1 þ n expfx1 ðg1 Þg þ m expfx2 ðg2 Þg

which together with (2.10) implies that expfx2 ðg2 Þg P

   f  nd 1 1 ðm a  cÞðf  ndÞ ¼ expf2 a xg ðn Þg  expfx  1 :¼ H 1 1 1 m m md b md

and consequently x2 ðtÞ P x2 ðg2 Þ 

Z

x

 jx02 ðtÞj dt P H 1  2dx.

ð2:12Þ

0

In addition, By (2.5), (2.8) and (2.9), we also obtain  6 f x expfx1 ðg1 Þg . dx m expfx2 ðn2 Þg Thus x2 ðn2 Þ 6 ln

    f expfx1 ðg1 Þg f a expf2axg :¼ H 2 . 6 ln mdb md

Then x2 ðtÞ 6 x2 ðn2 Þ þ

Z

x

 jx02 ðtÞj dt 6 H 2 þ 2dx.

ð2:13Þ

0

Eqs. (2.12) and (2.13) imply that  jH 2 þ 2dxjg  :¼ B2 . max jx2 ðtÞj 6 maxfjH 1  2dxj; t2½0;x

ð2:14Þ

Clearly, Hi and Bi, i = 1, 2 are independent of k. Under the assumption in Theorem 2.1, it is easy to show that the system of algebraic equations cv2 ¼ 0; 1 þ nv1 þ mv2 f v1 ¼ 0; d  1 þ nv1 þ mv2

a  bv1 

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511

has a unique solution ðv1 ; v2 ÞT with vi > 0, i = 1, 2. Denote B = B1 + B2 + B3, where B3 > 0 is taken sufficiently large such that kðlnfv1 g; lnfv2 gÞk ¼ j lnfv1 gj þ j lnfv2 gj < B3 , and define X ¼ fx ðtÞ 2 X : kx k < Bg. It is clear that X satisfies the condition (a) of Lemma 2.1. When x* = (x1, x2)T 2 oX \ Ker L = oX \ R2, x* is a constant vector in R2 with kx*k = M. Then   c expfx2 g f expfx1 g    QNx ¼ a  b expfx1 g  ¼ 0  d þ ¼ 0 6¼ 0. 1 þ n expfx1 g þ m expfx2 g 1 þ n expfx1 g þ m expfx2 g Furthermore, in view of assumption in Theorem 2.1, it can easily be seen that degfJQNx ; X \ Ker L; 0g 6¼ 0. By now we know that X verifies all the requirements of Lemma 2.1 and then system (2.2) has at least one x-periodic solution. By the medium of (2.1), we derive that (1.3) has at least one positive x-periodic solution. The proof is complete. h Next, we consider the following predator–prey systems with distributed delays 8 Z 0 dxðtÞ cðtÞxðtÞyðtÞ > > > xðt þ hÞ dlðhÞ  ¼ xðtÞ½aðtÞ  bðtÞ ; > < dt 1 þ nxðtÞ þ myðtÞ s " # R0 > f ðtÞ r xðt þ hÞ dgðhÞ dyðtÞ > > ¼ yðtÞ  dðtÞ ; > R0 R0 : dt 1 þ n r xðt þ hÞ dgðhÞ þ m r yðt þ hÞ dgðhÞ

ð2:15Þ

where s, r are positive constants and l, g are nondecreasing functions such that gð0þ Þ  gðr Þ ¼ 1;

lð0þ Þ  lðs Þ ¼ 1.

Theorem 2.2. If the conditions of Theorem 2.1 are satisfied, then (2.15) has at least one positive x-periodic solution. Proof. The proof is similar to the proof of Theorem 2.1 and hence is omitted here. h Remark 1. From the proofs of Theorems 2.1 and 2.2, one can see that in (2.15), even if some of the s’s and r’s or all of them are 1, the conclusion of Theorem 2.2 remains true.

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