IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation37686210.1155/2011/376862376862Research ArticlePositive Periodic Solutions for Neutral Delay Ratio-Dependent Predator-Prey Model with Holling-Tanner Functional ResponseLiuGuirong1WangSanhu2YanJurang1RedfieldRobert1School of Mathematical SciencesShanxi UniversityTaiyuanShanxi 030006Chinasxu.edu.cn2Department of ComputerLuliang UniversityLuliangShanxi 033000China2011287201120113112201025042011290520112011Copyright © 2011 Guirong Liu et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By using a continuation theorem based on coincidence degree theory, we establish some easily verifiable criteria for the existence of positive periodic solutions for neutral delay ratio-dependent predator-prey model with Holling-Tanner functional response x'(t)=x(t)[r(t)-a(t)x(t-σ(t))-b(t)x'(t-σ(t))]-c(t)x(t)y(t)/h(t)y(t)+x(t), y'(t)=y(t)d(t)-f(t)y(t-τ(t))/x(t-τ(t)).

1. Introduction

The dynamic relationship between the predator and the prey has long been and will continue to be one of the dominant themes in population dynamics due to its universal existence and importance in nature . In order to precisely describe the real ecological interactions between species such as mite and spider mite, lynx and hare, and sparrow and sparrow hawk, described by Tanner  and Wollkind et al. , May  developed the Holling-Tanner prey-predator model x(t)=rx(t)(1-x(t)K)-mx(t)y(t)x(t)+q,y(t)=y(t)[s(1-hy(t)x(t))]. In system (1.1),  x(t) and y(t) stand for prey and predator density at time t. r, K, m, q, s, h are positive constants that stand for prey intrinsic growth rate, carrying capacity, capturing rate, half-capturing saturation constant, predator intrinsic growth rate, and conversion rate of prey into predators biomass, respectively.

Nowadays attention have been paid by many authors to Holling-Tanner predator-prey model (see ).

Recently, there is a growing explicit biological and physiological evidence  that in many situations, especially when predators have to search for food (and, therefore, have to share or compete for food), a more suitable general predator-prey theory should be based on the so-called ratio-dependent theory, which can be roughly stated as that the per capita predator growth rate should be a function of the ratio of prey to predator abundance and so should be the so-called ratio-dependent functional response. This is strongly supported by numerous field and laboratory experiments and observations [11, 12]. Generally, a ratio-dependent Holling-Tanner predator-prey model takes the form of x(t)=rx(t)(1-x(t)K)-mx(t)y(t)qy(t)+x(t),y(t)=y(t)[s(1-hy(t)x(t))]. Liang and Pan  obtained results for the global stability of the positive equilibrium of (1.2).

However, time delays of one type or another have been incorporated into biological models by many researchers; we refer to the monographs of Cushing , Gopalsamy , Kuang , and MacDonald  for general delayed biological systems. Time delay due to gestation is a common example, because generally the consumption of prey by the predator throughout its past history governs the present birth rate of the predator. Therefore, more realistic models of population interactions should take into account the effect of time delays.

Recently, Saha and Chakrabarti  considered the following delayed ratio-dependent Holling-Tanner predator-prey model: x(t)=rx(t)(1-x(t-τ)K)-mx(t)y(t)qy(t)+x(t),y(t)=y(t)[s(1-hy(t)x(t))].

In addition, based on the investigation on laboratory populations of Daphnia magna, Smith  argued that the neutral term should be added in population models, since a growing population is likely to consume more or less food than a matured one, depending on individual species (for details, see Pielou ). In addition, as one may already be aware, many real systems are quite sensitive to sudden changes. This fact may suggest that proper mathematical models of the systems should consist of some neutral delay equations. In 1991, Kuang  studied the local stability and oscillation of the following neutral delay Gause-type predator-prey system: x(t)=rx(t)[1-x(t-τ)+ρx(t-τ)K]-y(t)p(x(t)),y(t)=y(t)[-α+βp(x(t-σ))].

In this paper, motivated by the above work, we will consider the following neutral delay ratio-dependent predator-prey model with Holling-Tanner functional response: x(t)=x(t)[r(t)-a(t)x(t-σ(t))-b(t)x(t-σ(t))]-c(t)x(t)y(t)h(t)y(t)+x(t),y(t)=y(t)[d(t)-f(t)y(t-τ(t))x(t-τ(t))].

As pointed out by Freedman and Wu  and Kuang , it would be of interest to study the existence of periodic solutions for periodic systems with time delay. The periodic solutions play the same role played by the equilibria of autonomous systems. In addition, in view of the fact that many predator-prey systems display sustained fluctuations, it is thus desirable to construct predator-prey models capable of producing periodic solutions. To our knowledge, no such work has been done on the global existence of positive periodic solutions of (1.5).

For convenience, we will use the following notations: |p|0=maxt[0,ω]{|p(t)|},  p̅=1ω0ωp(t)dt,  p̂=1ω0ω|p(t)|dt, where p(t) is a continuous ω-periodic function.

In this paper, we always make the following assumptions for system (1.5).

r(t), a(t), b(t), c(t), d(t), f(t), h(t), τ(t), and σ(t) are continuous ω-periodic functions. In addition, r̅>0, d̅>0, and  a(t)>0, c(t)>0, f(t)>0, h(t)>0 for any  t[0,ω];

bC1(R,[0,)), σC2(R,R), σ(t)<1, and g(t)>0, where g(t)=a(t)-q(t),q(t)=b(t)1-σ(t),tR.

eBmax{|b|0,|q|0}<1, where B=ln[2r̅maxt[0,ω]{1-σ(t)g(t)}]+|q|0maxt[0,ω]{2r̅g(t)}+(r̂+r̅)ω.

k̅<r̅, where k(t)=c(t)/h(t).

Our aim in this paper is, by using the coincidence degree theory developed by Gaines and Mawhin , to derive a set of easily verifiable sufficient conditions for the existence of positive periodic solutions of system (1.5).

2. Existence of Positive Periodic Solution

In this section, we will study the existence of at least one positive periodic solution of system (1.5). The method to be used in this paper involves the applications of the continuation theorem of coincidence degree. For the readers' convenience, we introduce a few concepts and results about the coincidence degree as follows.

Let X,Z be real Banach spaces, L:DomLXZ a linear mapping, and N:XZ a continuous mapping.

The mapping L is said to be a Fredholm mapping of index zero if dimKerL=codimImL<+ and ImL is closed in Z.

If L is a Fredholm mapping of index zero, then there exist continuous projectors P:XX and Q:ZZ, such that ImP=KerL, KerQ=ImL=Im(I-Q). It follows that the restriction LP of L to DomLKerP:(I-P)XImL is invertible. Denote the inverse of LP by KP.

The mapping N is said to be L-compact on Ω¯ if Ω is an open bounded subset of X, QN(Ω¯) is bounded, and KP(I-Q)N:Ω¯X is compact.

Since ImQ is isomorphic to KerL, there exists an isomorphism J:ImQKerL.

Lemma 2.1 (Continuation Theorem [<xref ref-type="bibr" rid="B23">23</xref>, page 40]).

Let ΩX be an open bounded set, L be a Fredholm mapping of index zero, and NL-compact on Ω¯. Suppose

for each λ(0,1), xΩDomL,  LxλNx;

for each xΩKerL, QNx0;

deg{JQN,ΩKerL,0}0.

Then Lx=Nx has at least one solution in Ω¯DomL.

We are now in a position to state and prove our main result.

Theorem 2.2.

Assume that (H1)–(H4) hold. Then system (1.5) has at least one ω-periodic solution with strictly positive components.

Proof.

Consider the following system: u1(t)=r(t)-a(t)eu1(t-σ(t))-b(t)eu1(t-σ(t))u1(t-σ(t))-c(t)eu2(t)h(t)eu2(t)+eu1(t),u2(t)=d(t)-f(t)eu2(t-τ(t))eu1(t-τ(t)), where all functions are defined as ones in system (1.5). It is easy to see that if system (2.1) has one ω-periodic solution (u1*(t),u2*(t))T, then (x*(t),y*(t))T=(eu1*(t),eu2*(t))T is a positive ω-periodic solution of system (1.5). Therefore, to complete the proof it suffices to show that system (2.1) has one ω-periodic solution.

Take X={u=(u1(t),u2(t))TC1(R,R2):ui(t+ω)=ui(t),  tR,  i=1,2},Z={u=(u1(t),u2(t))TC(R,R2):ui(t+ω)=ui(t),  tR,  i=1,2} and denote |u|=maxt[0,ω]{|u1(t)|+|u2(t)|},u=|u|+|u|. Then X and Z are Banach spaces when they are endowed with the norms · and |·|, respectively. Let L:XZ and N:XZ be L(u1(t),u2(t))T=(u1(t),u2(t))T,N[u1(t)u2(t)]=[r(t)-a(t)eu1(t-σ(t))-b(t)eu1(t-σ(t))u1(t-σ(t))-c(t)eu2(t)h(t)eu2(t)+eu1(t)d(t)-f(t)eu2(t-τ(t))eu1(t-τ(t))]. With these notations, system (2.1) can be written in the form Lu=Nu,uX. Obviously, KerL=R2, ImL={(u1(t),u2(t))TZ:0ωui(t)dt=0,  i=1,2} is closed in Z, and dimKerL=codimImL=2. Therefore, L is a Fredholm mapping of index zero. Now define two projectors P:XX and Q:ZZ as P(u1(t),u2(t))T=(u̅1,u̅2)T,(u1(t),u2(t))TX,Q(u1(t),u2(t))T=(u̅1,u̅2)T,(u1(t),u2(t))TZ. Then P and Q are continuous projectors such that ImP=KerL, KerQ=ImL=Im(I-Q). Furthermore, the generalized inverse (to L) KP:ImLKerPDomL has the form KP(u)=0tu(s)ds-1ω0ω0tu(s)dsdt. Note that 0ωb(t)eu1(t-σ(t))u1(t-σ(t))dt=0ωb(t)1-σ(t)(eu1(t-σ(t)))dt=0ωq(t)(eu1(t-σ(t)))dt=[q(t)eu1(t-σ(t))]0ω-0ωq(t)eu1(t-σ(t))dt=-0ωq(t)eu1(t-σ(t))dt.   Then QN:XZ and KP(I-Q)N:XX read (QN)u=[1ω0ω[r(t)-g(t)eu1(t-σ(t))-c(t)eu2(t)h(t)eu2(t)+eu1(t)]dt1ω0ω[d(t)-f(t)eu2(t-τ(t))eu1(t-τ(t))]dt],(KP(I-Q)N)u=[0t[r(s)-g(s)eu1(s-σ(s))-c(s)eu2(s)h(s)eu2(s)+eu1(s)]ds-q(t)eu1(t-σ(t))+q(0)eu1(-σ(0))0t[d(s)-f(s)eu2(s-τ(s))eu1(s-τ(s))]ds]-[1ω0ω0t[r(s)-g(s)eu1(s-σ(s))-c(s)eu2(s)h(s)eu2(s)+eu1(s)]dsdt-1ω0ω[q(t)eu1(t-σ(t))-q(0)eu1(-σ(0))]dt1ω0ω0t[d(s)-f(s)eu2(s-τ(s))eu1(s-τ(s))]dsdt]-[(tω-12)0ω[r(s)-g(s)eu1(s-σ(s))-c(s)eu2(s)h(s)eu2(s)+eu1(s)]ds(tω-12)0ω[d(s)-f(s)eu2(s-τ(s))eu1(s-τ(s))]ds]. It is obvious that QN and KP(I-Q)N are continuous by the Lebesgue theorem, and using the Arzela-Ascoli theorem it is not difficult to show that QN(Ω¯) is bounded and KP(I-Q)N(Ω¯)¯ is compact for any open bounded set ΩX. Hence N is L-compact on Ω¯ for any open bounded set ΩX.

In order to apply Lemma 2.1, we need to search for an appropriate open, bounded subset ΩX.

Corresponding to the operator equation Lu=λNu, λ(0,1), we have u1(t)=λ[r(t)-a(t)eu1(t-σ(t))-b(t)eu1(t-σ(t))u1(t-σ(t))-c(t)eu2(t)h(t)eu2(t)+eu1(t)],u2(t)=λ[d(t)-f(t)eu2(t-τ(t))eu1(t-τ(t))]. Suppose that (u1(t),u2(t))TX is a solution of (2.10) for a certain λ(0,1). Integrating (2.10) over the interval [0,ω] leads to 0ω[r(t)-a(t)eu1(t-σ(t))-b(t)eu1(t-σ(t))u1(t-σ(t))-c(t)eu2(t)h(t)eu2(t)+eu1(t)]dt=0,0ω[d(t)-f(t)eu2(t-τ(t))eu1(t-τ(t))]dt=0. It follows from (2.8) and (2.11) that 0ω[g(t)eu1(t-σ(t))+c(t)eu2(t)h(t)eu2(t)+eu1(t)]dt=r̅ω. From (2.12), we have 0ωf(t)eu2(t-τ(t))eu1(t-τ(t))dt=d̅ω. From (H2), (2.10), and (2.13), one can find 0ω|ddt[u1(t)+λq(t)eu1(t-σ(t))]|dt=λ0ω|r(t)-g(t)eu1(t-σ(t))-c(t)eu2(t)h(t)eu2(t)+eu1(t)|dt0ω|r(t)|dt+0ω[g(t)eu1(t-σ(t))+c(t)eu2(t)h(t)eu2(t)+eu1(t)]dt=(r̂+r̅)ω. Let t=ψ(v) be the inverse function of v=t-σ(t). It is easy to see that g(ψ(v)) and σ(ψ(v)) are all ω-periodic functions. Further, it follows from (2.13), (H1), and (H2) that r̅ω0ωg(t)eu1(t-σ(t))dt=-σ(0)ω-σ(ω)g(ψ(v))eu1(v)11-σ(ψ(v))dv=0ωg(ψ(v))1-σ(ψ(v))eu1(v)dv=0ωg(ψ(t))1-σ(ψ(t))eu1(t)dt, which yields that 2r̅ω0ω[g(ψ(t))1-σ(ψ(t))eu1(t)+g(t)eu1(t-σ(t))]dt. According to the mean value theorem of differential calculus, we see that there exists ξ[0,ω] such that g(ψ(ξ))1-σ(ψ(ξ))eu1(ξ)+g(ξ)eu1(ξ-σ(ξ))2r̅. This, together with (H2), yields u1(ξ)ln[2r̅maxt[0,ω]{1-σ(t)g(t)}],eu1(ξ-σ(ξ))maxt[0,ω]{2r̅g(t)}, which, together with (2.15) and (H3), imply that, for any t[0,ω], u1(t)+λq(t)eu1(t-σ(t))u1(ξ)+λq(ξ)eu1(ξ-σ(ξ))+0ω|ddt[u1(t)+λq(t)eu1(t-σ(t))]|dtln[2r̅maxt[0,ω]{1-σ(t)g(t)}]+|q|0maxt[0,ω]{2r̅g(t)}+(r̂+r̅)ω=B. As λq(t)eu1(t-σ(t))0, one can find that u1(t)B,t[0,ω]. Since (u1(t),u2(t))TX, there exist ξi,ηi[0,ω]  (i=1,2) such that ui(ξi)=mint[0,ω]{ui(t)},ui(ηi)=maxt[0,ω]{ui(t)}. From (2.13) and (H4), we obtain r̅ω0ω[g(t)eu1(t-σ(t))+c(t)h(t)]dtk̅ω+eu1(η1)g̅ω, which, together with (H4), implies that u1(η1)lnr̅-k̅g̅. In view of (2.10), (2.13), and (2.21), we obtain 0ω|u1(t)|dt=λ0ω|r(t)-a(t)eu1(t-σ(t))-b(t)eu1(t-σ(t))u1(t-σ(t))-c(t)eu2(t)h(t)eu2(t)+eu1(t)|dt0ω|r(t)|dt+0ω[c(t)eu2(t)h(t)eu2(t)+eu1(t)]dt+0ω|a(t)eu1(t-σ(t))|dt+0ω|b(t)eu1(t-σ(t))u1(t-σ(t))|dt(r̂+r̅)ω+|a|0eBω+eB0ω|b(t)u1(t-σ(t))|dt. In addition, 0ω|b(t)u1(t-σ(t))|dt=-σ(0)ω-σ(ω)|b(ψ(v))u1(v)11-σ(ψ(v))|dv,|q|0-σ(0)ω-σ(ω)|u1(v)|dv=|q|00ω|u1(v)|dv=|q|00ω|u1(t)|dt, which, together with (H3), implies that 0ω|u1(t)|dtω1-|q|0eB(r̂+r̅+|a|0eB). It follows from (2.24) and (2.27) that, for any t[0,ω], u1(t)u1(η1)-0ω|u1(t)|dtlnr̅-k̅g̅-ω1-|q|0eB(r̂+r̅+|a|0eB)=:β1, which, together with (2.21), implies that |u1|0max{|B|,|β1|}=:β2. From (2.10), we obtain |u1|0|r|0+|a|0eB+|b|0eB|u1|0+|k|0. It follows from (H3) that |u1|011-|b|0eB[|r|0+|a|0eB+|k|0]=:β3. In view of (2.14), we obtain d̅ω0ωf(t)eu2(ξ2)eBdt=f̅ωeu2(ξ2)eB,d̅ω0ωf(t)eu2(η2)eβ1dt=f̅ωeu2(η2)eβ1. Further, u2(ξ2)B+lnd̅f̅,u2(η2)β1+lnd̅f̅. It follows that from (2.10) and (2.14), we obtain 0ω|u2(t)|dt0ω|d(t)|dt+0ωf(t)eu2(t-τ(t))eu1(t-τ(t))dt=(d̂+d̅)ω. From (2.33) and (2.34), one can find that, for any t[0,ω], u2(t)u2(η2)-0ω|u2(t)|dtβ1+lnd̅f̅-(d̂+d̅)ω=:β4,u2(t)u2(ξ2)+0ω|u2(t)|dtB+lnd̅f̅+(d̂+d̅)ω=:β5, which imply that |u2|0max{|β4|,|β5|}=:β6. In view of (2.10), we have |u2|0|d|0+|f|0eβ5eβ1=:β7. From (2.29), (2.31), (2.36), and (2.37), we obtain u=|u|+|u|β2+β3+β6+β7. From (H4), the algebraic equations r̅-g̅eu1-1ω0ωc(t)eu2h(t)eu2+eu1dt=0,d̅-f̅eu2-u1=0 have a unique solution (u1*,u2*)TR2, where u1*=ln[1g̅(r̅-1ω0ωd̅c(t)d̅h(t)+f̅dt)],u2*=u1*+lnd̅f̅. Set β=β2+β3+β6+β7+β0, where β0 is taken sufficiently large such that the unique solution of (2.39) satisfies (u1*,u2*)T=|u1*|+|u2*|<β0. Clearly, β is independent of λ.

We now take Ω={(u1(t),u2(t))TX:(u1(t),u2(t))T<β}. This satisfies condition (i) in Lemma 2.1. When (u1(t),u2(t))TΩKerL=ΩR2, (u1(t),u2(t))T is a constant vector in R2 with |u1|+|u2|=β. Thus, we have QN[u1u2]=[r̅-g̅eu1-1ω0ωc(t)eu2h(t)eu2+eu1dtd̅-f̅eu2-u1]. This proves that condition (ii) in Lemma 2.1 is satisfied.

Taking J=I:ImQKerL,(u1,u2)T(u1,u2)T, a direct calculation shows that deg{JQN,ΩKerL,0}=sgndet[-g̅eu1*+1ωeu1*eu2*0ωc(t)(h(t)eu2*+eu1*)2dt-1ωeu1*eu2*0ωc(t)(h(t)eu2*+eu1*)2dtf̅eu2*-u1*-f̅eu2*-u1*]=sgn{f̅g̅eu2*}0. By now we have proved that Ω satisfies all the requirements in Lemma 2.1. Hence, (2.1) has at least one ω-periodic solution. Accordingly, system (1.5) has at least one ω-periodic solution with strictly positive components. The proof of Theorem 2.2 is complete.

Remark 2.3.

From the proof of Theorem 2.2, we see that Theorem 2.2 is also valid if b(t)0 for tR. Consequently, we can obtain the following corollary.

Corollary 2.4.

Assume that (H1), (H4) hold, and σC2(R,R), σ(t)<1. Then the following delay ratio-dependent predator-prey model with Holling-Tanner functional response x(t)=x(t)[r(t)-a(t)x(t-σ(t))]-c(t)x(t)y(t)h(t)y(t)+x(t),y(t)=y(t)[d(t)-f(t)y(t-τ(t))x(t-τ(t))] has at least one ω-periodic solution with strictly positive components.

Next consider the following neutral ratio-dependent predator-prey system with state-dependent delays: x(t)=x(t)[r(t)-a(t)x(t-σ(t))-b(t)x(t-σ(t))]-c(t)x(t)y(t)h(t)y(t)+x(t),y(t)=y(t)[d(t)-f(t)y(t-τ1(t,x(t),y(t)))x(t-τ1(t,x(t),y(t)))], where τ1(t,x,y) is a continuous function and ω-periodic function with respect to t.

Theorem 2.5.

Assume that (H1)–(H4) hold. Then system (2.45) has at least one ω-periodic solution with strictly positive components.

Proof.

The proof is similar to that of Theorem 2.2 and hence is omitted here.

3. Discussion

In this paper, we have discussed the combined effects of periodicity of the ecological and environmental parameters and time delays due to the negative feedback of the predator density and gestations on the dynamics of a neutral delay ratio-dependent predator-prey model. By using Gaines and Mawhin's continuation theorem of coincidence degree theory, we have established sufficient conditions for the existence of positive periodic solutions to a neutral delay ratio-dependent predator-prey model with Holling-Tanner functional response. By Theorem 2.2, we see that system (1.5) will have at least one ω-periodic solution with strictly positive components if a (the density-dependent coefficient of the prey) is sufficiently large, the neutral coefficient b is sufficiently small, and c/h<r, where c, h, r stand for capturing rate, half-capturing saturation coefficient, and prey intrinsic growth rate, respectively.

We note that τ (the time delay due to the negative feedback of the predator density) and f (the conversion rate of prey into predators biomass) have no influence on the existence of positive periodic solutions to system (1.5). However, σ (the time delay due to gestation) plays the important role in determining the existence of positive periodic solutions of (1.5).

From the results in this paper, we can find that the neutral term effects are quite significant.

Acknowledgments

This work was supported by the Natural Science Foundation of China (no. 11001157), Tianyuan Mathematics Fund of China (no. 10826080) and the Youth Science Foundation of Shanxi Province (no. 2009021001-1, no. 2010021001-1).

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