Positive Periodic Solutions for Neutral Delay Ratio-Dependent Predator-Prey Model with Holling-Tanner Functional Response

Abstract

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 [1]. 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 [2] and Wollkind et al. [3], May [4] developed the Holling-Tanner prey-predator model 𝑥^′(𝑡)(𝑡)𝑥(𝑡)=𝑟𝑥1−𝐾−𝑚𝑥(𝑡)𝑦(𝑡)𝑥(𝑡),𝑦+𝑞_′(𝑡)(𝑡𝑠𝑦(𝑡)=𝑦)1−ℎ𝑥(𝑡). (1.1) 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 [5–7]).

Recently, there is a growing explicit biological and physiological evidence [8–10] 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)(𝑡)𝑥(𝑡)=𝑟𝑥1−𝐾−𝑚𝑥(𝑡)𝑦(𝑡)(𝑡)(𝑡),𝑦𝑞𝑦+𝑥_′(𝑡)(𝑡𝑠𝑦(𝑡)=𝑦)1−ℎ𝑥(𝑡). (1.2) Liang and Pan [13] 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 [14], Gopalsamy [15], Kuang [16], and MacDonald [17] 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 [18] considered the following delayed ratio-dependent Holling-Tanner predator-prey model: 𝑥^′(𝑡)(𝑡)𝑥()=𝑟𝑥1−𝑡−𝜏𝐾−𝑚𝑥(𝑡)𝑦(𝑡)(𝑡)(𝑡),𝑦𝑞𝑦+𝑥_′(𝑡)(𝑡𝑠𝑦(𝑡)=𝑦)1−ℎ𝑥(𝑡). (1.3)

In addition, based on the investigation on laboratory populations of Daphnia magna, Smith [19] 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 [20]). 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 [21] studied the local stability and oscillation of the following neutral delay Gause-type predator-prey system: 𝑥^′(𝑡)(𝑡𝑥()=𝑟𝑥)1−𝑡−𝜏+𝜌𝑥^′(𝑡−𝜏)𝐾(𝑡)𝑝(𝑥(𝑡,𝑦−𝑦))_′(𝑡)=𝑦(𝑡)−𝛼+𝛽𝑝(𝑥(𝑡−𝜎)). (1.4)

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−𝑐(𝑡)𝑥(𝑡)𝑦(𝑡)𝑡−𝜎))ℎ(𝑡)𝑦(𝑡)(𝑡),𝑦+𝑥_′(𝑡)(𝑡𝑑(𝑡)(𝑡)𝑦((𝑡=𝑦)−𝑓𝑡−𝜏))𝑥((𝑡.𝑡−𝜏)) (1.5)

As pointed out by Freedman and Wu [22] and Kuang [16], 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: 𝑝₀=max_{[]𝑡∈0,𝜔}𝑝(𝑡),1𝑝=𝜔^𝜔₀𝑝(𝑡)1d𝑡,𝑝=𝜔^𝜔₀𝑝(𝑡)d𝑡, (1.6) where 𝑝(𝑡) is a continuous 𝜔-periodic function.

In this paper, we always make the following assumptions for system (1.5).(H₁)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, 𝑟>0, 𝑑>0, and𝑎(𝑡)>0, 𝑐(𝑡)>0, 𝑓(𝑡)>0, ℎ(𝑡)>0 for any𝑡∈[0,𝜔];(H₂)𝑏∈𝐶¹(𝑅,[0,∞)), 𝜎∈𝐶²(𝑅,𝑅), 𝜎^′(𝑡)<1, and 𝑔(𝑡)>0, where 𝑔(𝑡)=𝑎(𝑡)−𝑞^′(𝑡)(𝑡)=𝑏(𝑡),𝑞1−𝜎^′(𝑡),𝑡∈𝑅. (1.7)(H₃)𝑒^𝐵max{|𝑏|₀,|𝑞|₀}<1, where 𝐵=ln2𝑟max_{𝑡∈[0,𝜔]}1−𝜎^′(𝑡)𝑔(𝑡)+𝑞₀max_{[]𝑡∈0,𝜔}2𝑟𝑔(𝑡)+(𝑟̂+𝑟)𝜔. (1.8)(H₄)𝑘<𝑟, where 𝑘(𝑡)=𝑐(𝑡)/ℎ(𝑡).

Our aim in this paper is, by using the coincidence degree theory developed by Gaines and Mawhin [23], 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∶Dom𝐿⊂𝑋→𝑍 a linear mapping, and 𝑁∶𝑋→𝑍 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∶𝑋→𝑋 and 𝑄∶𝑍→𝑍, such that Im𝑃=Ker𝐿, Ker𝑄=Im𝐿=Im(𝐼−𝑄). It follows that the restriction 𝐿_𝑃 of 𝐿 to Dom𝐿∩Ker𝑃∶(𝐼−𝑃)𝑋→Im𝐿 is invertible. Denote the inverse of 𝐿_𝑃 by 𝐾_𝑃.

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 𝐾_𝑃(𝐼−𝑄)𝑁∶Ω→𝑋 is compact.

Since Im𝑄 is isomorphic to Ker𝐿, there exists an isomorphism 𝐽∶Im𝑄→Ker𝐿.

Lemma 2.1 (Continuation Theorem [23, page 40]). Let Ω⊂𝑋 be an open bounded set, 𝐿 be a Fredholm mapping of index zero, and 𝑁𝐿-compact on Ω. Suppose(i)for each 𝜆∈(0,1), 𝑥∈𝜕Ω∩Dom𝐿,𝐿𝑥≠𝜆𝑁𝑥;(ii)for each 𝑥∈𝜕Ω∩Ker𝐿, 𝑄𝑁𝑥≠0;(iii)deg{𝐽𝑄𝑁,Ω∩Ker𝐿,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: 𝑢^′₁(𝑡)=𝑟(𝑡)−𝑎(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}−𝑏(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}𝑢^′₁((𝑡−𝑐(𝑡)𝑒𝑡−𝜎))^𝑢2(𝑡)ℎ(𝑡)𝑒^𝑢2(𝑡)+𝑒^𝑢1(𝑡),𝑢^′₂(𝑡)(𝑡)(𝑡)𝑒=𝑑−𝑓^{𝑢2(𝑡−𝜏(𝑡))}𝑒^{𝑢1(𝑡−𝜏(𝑡))}, (2.1) 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 (𝑢^∗₁(𝑡),𝑢^∗₂(𝑡))^𝑇, then (𝑥^∗(𝑡),𝑦^∗(𝑡))^𝑇=(𝑒^{𝑢∗1(𝑡)},𝑒^{𝑢∗2(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)^𝑇∈𝐶¹𝑅,𝑅²∶𝑢_𝑖(𝑡+𝜔)=𝑢_𝑖(𝑡),𝑢,𝑡∈𝑅,𝑖=1,2𝑍=𝑢=₁(𝑡),𝑢₂(𝑡)^𝑇∈𝐶𝑅,𝑅²∶𝑢_𝑖(𝑡+𝜔)=𝑢_𝑖(𝑡),𝑡∈𝑅,𝑖=1,2 (2.2) and denote |𝑢|_∞=max_{[]𝑡∈0,𝜔}𝑢₁(𝑡)+𝑢₂(𝑡),‖𝑢‖=|𝑢|_∞+𝑢^′_∞. (2.3) Then 𝑋 and 𝑍 are Banach spaces when they are endowed with the norms ‖⋅‖ and |⋅|_∞, respectively. Let 𝐿∶𝑋→𝑍 and 𝑁∶𝑋→𝑍 be 𝐿𝑢₁(𝑡),𝑢₂(𝑡)^𝑇=𝑢^′₁(𝑡),𝑢^′₂(𝑡)^𝑇,𝑁𝑢₁(𝑡)𝑢₂(𝑡)=𝑟(𝑡)−𝑎(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}−𝑏(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}𝑢^′₁((𝑡−𝑐(𝑡)𝑒𝑡−𝜎))^𝑢2(𝑡)ℎ(𝑡)𝑒^𝑢2(𝑡)+𝑒^𝑢1(𝑡)𝑑(𝑡)(𝑡)𝑒−𝑓^{𝑢2(𝑡−𝜏(𝑡))}𝑒^{𝑢1(𝑡−𝜏(𝑡))}. (2.4) With these notations, system (2.1) can be written in the form 𝐿𝑢=𝑁𝑢,𝑢∈𝑋. (2.5) Obviously, Ker𝐿=𝑅², Im𝐿={(𝑢₁(𝑡),𝑢₂(𝑡))^𝑇∈𝑍∶∫^𝜔₀𝑢_𝑖(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∶𝑋→𝑋 and 𝑄∶𝑍→𝑍 as 𝑃𝑢₁(𝑡),𝑢₂(𝑡)^𝑇=𝑢₁,𝑢₂^𝑇,𝑢₁(𝑡),𝑢₂(𝑡)^𝑇𝑄𝑢∈𝑋,₁(𝑡),𝑢₂(𝑡)^𝑇=𝑢₁,𝑢₂^𝑇,𝑢₁(𝑡),𝑢₂(𝑡)^𝑇∈𝑍. (2.6) Then 𝑃 and 𝑄 are continuous projectors such that Im𝑃=Ker𝐿, Ker𝑄=Im𝐿=Im(𝐼−𝑄). Furthermore, the generalized inverse (to 𝐿) 𝐾_𝑃∶Im𝐿→Ker𝑃∩Dom𝐿 has the form 𝐾_𝑃(𝑢)=^𝑡₀𝑢(𝑠)1d𝑠−𝜔^𝜔₀^𝑡₀𝑢(𝑠)d𝑠d𝑡. (2.7) Note that ^𝜔₀𝑏(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}𝑢^′₁(𝑡−𝜎(𝑡))d𝑡=^𝜔₀𝑏(𝑡)1−𝜎^′(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))′}=d𝑡_𝜔0𝑞(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))′}=𝑞(𝑡)𝑒d𝑡_{𝑢1(𝑡−𝜎(𝑡))𝜔0}−^𝜔₀𝑞^′(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}d𝑡=−^𝜔₀𝑞^′(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}d𝑡. (2.8) Then 𝑄𝑁∶𝑋→𝑍 and 𝐾_𝑃(𝐼−𝑄)𝑁∶𝑋→𝑋 read ()𝑢=1𝑄𝑁𝜔^𝜔₀𝑟(𝑡)−𝑔(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}−𝑐(𝑡)𝑒^𝑢2(𝑡)ℎ(𝑡)𝑒^𝑢2(𝑡)+𝑒^𝑢1(𝑡)1d𝑡𝜔^𝜔₀𝑑(𝑡)(𝑡)𝑒−𝑓^{𝑢2(𝑡−𝜏(𝑡))}𝑒^{𝑢1(𝑡−𝜏(𝑡))},𝐾d𝑡_𝑃()𝑁𝑢=𝐼−𝑄_𝑡0𝑟(𝑠)−𝑔(𝑠)𝑒^{𝑢1(𝑠−𝜎(𝑠))}−𝑐(𝑠)𝑒^𝑢2(𝑠)ℎ(𝑠)𝑒^𝑢2(𝑠)+𝑒^𝑢1(𝑠)d𝑠−𝑞(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}+𝑞^{𝑢1(−𝜎(0))}_𝑡0(0)𝑒𝑑(𝑠)(𝑠)𝑒−𝑓^{𝑢2(𝑠−𝜏(𝑠))}𝑒^{𝑢1(𝑠−𝜏(𝑠))}−1d𝑠𝜔^𝜔₀^𝑡₀𝑟(𝑠)−𝑔(𝑠)𝑒^{𝑢1(𝑠−𝜎(𝑠))}−𝑐(𝑠)𝑒^𝑢2(𝑠)ℎ(𝑠)𝑒^𝑢2(𝑠)+𝑒^𝑢1(𝑠)1d𝑠d𝑡−𝜔^𝜔₀𝑞(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}−𝑞(0)𝑒^{𝑢1(−𝜎(0))}1d𝑡𝜔^𝜔₀^𝑡₀𝑑(𝑠)(𝑠)𝑒−𝑓^{𝑢2(𝑠−𝜏(𝑠))}𝑒^{𝑢1(𝑠−𝜏(𝑠))}−𝑡d𝑠d𝑡𝜔−12^𝜔₀𝑟(𝑠)−𝑔(𝑠)𝑒^{𝑢1(𝑠−𝜎(𝑠))}−𝑐(𝑠)𝑒^𝑢2(𝑠)ℎ(𝑠)𝑒^𝑢2(𝑠)+𝑒^𝑢1(𝑠)𝑡d𝑠𝜔−12^𝜔₀𝑑(𝑠)(𝑠)𝑒−𝑓^{𝑢2(𝑠−𝜏(𝑠))}𝑒^{𝑢1(𝑠−𝜏(𝑠))}.d𝑠 (2.9) It is obvious that 𝑄𝑁 and 𝐾_𝑃(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 𝐾_𝑃(𝐼−𝑄)𝑁(Ω) is compact for any open bounded set Ω⊂𝑋. Hence 𝑁 is 𝐿-compact on Ω for any open bounded set Ω⊂𝑋.
In order to apply Lemma 2.1, we need to search for an appropriate open, bounded subset Ω⊂𝑋.
Corresponding to the operator equation 𝐿𝑢=𝜆𝑁𝑢, 𝜆∈(0,1), we have 𝑢^′₁(𝑡)=𝜆𝑟(𝑡)−𝑎(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}−𝑏(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}𝑢^′₁((𝑡−𝑐(𝑡)𝑒𝑡−𝜎))^𝑢2(𝑡)ℎ(𝑡)𝑒^𝑢2(𝑡)+𝑒^𝑢1(𝑡),𝑢^′₂(𝑡)𝑑(𝑡)(𝑡)𝑒=𝜆−𝑓^{𝑢2(𝑡−𝜏(𝑡))}𝑒^{𝑢1(𝑡−𝜏(𝑡))}. (2.10) Suppose that (𝑢₁(𝑡),𝑢₂(𝑡))^𝑇∈𝑋 is a solution of (2.10) for a certain 𝜆∈(0,1). Integrating (2.10) over the interval [0,𝜔] leads to ^𝜔₀𝑟(𝑡)−𝑎(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}−𝑏(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}𝑢₁′((𝑡−𝑐(𝑡)𝑒𝑡−𝜎))^𝑢2(𝑡)ℎ(𝑡)𝑒^𝑢2(𝑡)+𝑒^𝑢1(𝑡)d𝑡=0,_𝜔0𝑑(𝑡)(𝑡)𝑒−𝑓^{𝑢2(𝑡−𝜏(𝑡))}𝑒^{𝑢1(𝑡−𝜏(𝑡))}d𝑡=0. (2.11)(2.12) It follows from (2.8) and (2.11) that ^𝜔₀𝑔(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}+𝑐(𝑡)𝑒^𝑢2(𝑡)ℎ(𝑡)𝑒^𝑢2(𝑡)+𝑒^𝑢1(𝑡)d𝑡=𝑟𝜔. (2.13) From (2.12), we have ^𝜔₀𝑓(𝑡)𝑒^{𝑢2(𝑡−𝜏(𝑡))}𝑒^{𝑢1(𝑡−𝜏(𝑡))}d𝑡=𝑑𝜔. (2.14) From (H₂), (2.10), and (2.13), one can find ^𝜔₀d𝑢d𝑡¹(𝑡)+𝜆𝑞(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}d𝑡=𝜆^𝜔₀𝑟(𝑡)−𝑔(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}−𝑐(𝑡)𝑒^𝑢2(𝑡)ℎ(𝑡)𝑒^𝑢2(𝑡)+𝑒^𝑢1(𝑡)≤d𝑡_𝜔0|𝑟(𝑡)|d𝑡+^𝜔₀𝑔(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}+𝑐(𝑡)𝑒^𝑢2(𝑡)ℎ(𝑡)𝑒^𝑢2(𝑡)+𝑒^𝑢1(𝑡)=(̂d𝑡𝑟+𝑟)𝜔. (2.15) 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 𝑟𝜔≥^𝜔₀𝑔(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}=d𝑡_{(0)𝜔−𝜎(𝜔)−𝜎}𝑔𝜓(𝑣)𝑒^𝑢1(𝑣)11−𝜎^′𝜓(𝑣)=d𝑣_𝜔0𝑔𝜓(𝑣)1−𝜎^′𝜓(𝑣𝑒)^𝑢1(𝑣)d𝑣=^𝜔₀𝑔𝜓(𝑡)𝜓(𝑡𝑒1−𝜎′)^𝑢1(𝑡)d𝑡, (2.16) which yields that 2𝑟𝜔≥^𝜔₀𝑔𝜓(𝑡)1−𝜎^′𝜓(𝑡)𝑒^𝑢1(𝑡)+𝑔(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}d𝑡. (2.17) According to the mean value theorem of differential calculus, we see that there exists 𝜉∈[0,𝜔] such that 𝑔𝜓(𝜉)𝜓(𝜉)𝑒1−𝜎′^𝑢1(𝜉)+𝑔(𝜉)𝑒^{𝑢1(𝜉−𝜎(𝜉))}≤2𝑟. (2.18) This, together with (H₂), yields 𝑢₁(𝜉)≤ln2𝑟max_{[]𝑡∈0,𝜔}1−𝜎^′(𝑡)𝑔(𝑡),𝑒^{𝑢1(𝜉−𝜎(𝜉))}≤max_{[]𝑡∈0,𝜔}2𝑟𝑔(𝑡), (2.19) which, together with (2.15) and (H₃), imply that, for any 𝑡∈[0,𝜔], 𝑢₁(𝑡)+𝜆𝑞(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}≤𝑢₁(𝜉)+𝜆𝑞(𝜉)𝑒^{𝑢1(𝜉−𝜎(𝜉))}+^𝜔₀d𝑢d𝑡¹(𝑡)+𝜆𝑞(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}2d𝑡≤ln𝑟max_{[]𝑡∈0,𝜔}1−𝜎^′(𝑡)𝑔(𝑡)+𝑞₀max_{[]𝑡∈0,𝜔}2𝑟𝑔(𝑡)+(𝑟̂+𝑟)𝜔=𝐵. (2.20) As 𝜆𝑞(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}≥0, one can find that 𝑢₁(𝑡)≤𝐵,𝑡∈[0,𝜔]. (2.21) Since (𝑢₁(𝑡),𝑢₂(𝑡))^𝑇∈𝑋, there exist 𝜉_𝑖,𝜂_𝑖∈[0,𝜔](𝑖=1,2) such that 𝑢_𝑖𝜉_𝑖=min_{[]𝑡∈0,𝜔}𝑢_𝑖(𝑡),𝑢_𝑖𝜂_𝑖=max_{[]𝑡∈0,𝜔}𝑢_𝑖(𝑡). (2.22) From (2.13) and (H₄), we obtain 𝑟𝜔≤^𝜔₀𝑔(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}+𝑐(𝑡)ℎ(𝑡)d𝑡≤𝑘𝜔+𝑒^𝑢1(𝜂1)𝑔𝜔, (2.23) which, together with (H₄), implies that 𝑢₁𝜂₁≥ln𝑟−𝑘𝑔. (2.24) In view of (2.10), (2.13), and (2.21), we obtain ^𝜔₀𝑢^′₁(𝑡)d𝑡=𝜆^𝜔₀𝑟(𝑡)−𝑎(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}−𝑏(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}𝑢^′₁((𝑡−𝑐(𝑡)𝑒𝑡−𝜎))^𝑢2(𝑡)ℎ(𝑡)𝑒^𝑢2(𝑡)+𝑒^𝑢1(𝑡)≤d𝑡_𝜔0|𝑟(𝑡)|d𝑡+^𝜔₀𝑐(𝑡)𝑒^𝑢2(𝑡)ℎ(𝑡)𝑒^𝑢2(𝑡)+𝑒^𝑢1(𝑡)+d𝑡_𝜔0𝑎(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}d𝑡+^𝜔₀𝑏(𝑡)𝑒^{𝑢1(𝑡−𝜎(𝑡))}𝑢^′₁((𝑡≤(̂𝑡−𝜎))d𝑡𝑟+𝑟)𝜔+|𝑎|₀𝑒^𝐵𝜔+𝑒^𝐵𝜔₀𝑏(𝑡)𝑢^′₁(𝑡−𝜎(𝑡))d𝑡. (2.25) In addition, ^𝜔₀𝑏(𝑡)𝑢^′₁(𝑡−𝜎(𝑡))d𝑡=₍₀₎^{𝜔−𝜎(𝜔)}_−𝜎𝑏𝜓(𝑣)𝑢^′₁(𝑣)11−𝜎^′𝜓(𝑣)≤𝑞d𝑣,_{0(0)𝜔−𝜎(𝜔)−𝜎}𝑢^′₁(𝑣)d𝑣=𝑞₀^𝜔₀𝑢^′₁(𝑣)d𝑣=𝑞₀^𝜔₀𝑢^′₁(𝑡)d𝑡, (2.26) which, together with (H₃), implies that ^𝜔₀𝑢^′₁(𝑡𝜔)d𝑡≤1−𝑞₀𝑒^𝐵𝑟̂+𝑟+|𝑎|₀𝑒^𝐵. (2.27) It follows from (2.24) and (2.27) that, for any 𝑡∈[0,𝜔], 𝑢₁(𝑡)≥𝑢₁𝜂₁−^𝜔₀𝑢^′₁(𝑡)d𝑡≥ln𝑟−𝑘𝑔−𝜔1−𝑞₀𝑒^𝐵𝑟̂+𝑟+|𝑎|₀𝑒^𝐵=∶𝛽₁, (2.28) which, together with (2.21), implies that 𝑢₁₀≤max|𝐵|,𝛽₁=∶𝛽₂. (2.29) From (2.10), we obtain 𝑢₁′₀≤|𝑟|₀+|𝑎|₀𝑒^𝐵+|𝑏|₀𝑒^𝐵𝑢₁′₀+|𝑘|₀. (2.30) It follows from (H₃) that 𝑢₁′₀≤11−|𝑏|₀𝑒^𝐵|𝑟|₀+|𝑎|₀𝑒^𝐵+|𝑘|₀=∶𝛽₃. (2.31) In view of (2.14), we obtain 𝑑𝜔≥^𝜔₀𝑓(𝑡)𝑒^𝑢2(𝜉2)𝑒^𝐵d𝑡=𝑒𝑓𝜔^𝑢2(𝜉2)𝑒^𝐵,𝑑𝜔≤^𝜔₀𝑓(𝑡)𝑒^𝑢2(𝜂2)𝑒^𝛽1d𝑡=𝑒𝑓𝜔^𝑢2(𝜂2)𝑒^𝛽1. (2.32) Further, 𝑢₂𝜉₂≤𝐵+ln𝑑𝑓,𝑢₂𝜂₂≥𝛽₁+ln𝑑𝑓. (2.33) It follows that from (2.10) and (2.14), we obtain ^𝜔₀𝑢^′₂(𝑡)d𝑡≤^𝜔₀|𝑑(𝑡)|d𝑡+^𝜔₀𝑓(𝑡)𝑒^{𝑢2(𝑡−𝜏(𝑡))}𝑒^{𝑢1(𝑡−𝜏(𝑡))}d𝑡=𝑑+𝑑𝜔. (2.34) From (2.33) and (2.34), one can find that, for any 𝑡∈[0,𝜔], 𝑢₂(𝑡)≥𝑢₂𝜂₂−^𝜔₀𝑢^′₂(𝑡)d𝑡≥𝛽₁+ln𝑑𝑓−𝑑+𝑑𝜔=∶𝛽₄,𝑢₂(𝑡)≤𝑢₂𝜉₂+^𝜔₀𝑢^′₂(𝑡)d𝑡≤𝐵+ln𝑑𝑓+𝑑+𝑑𝜔=∶𝛽₅, (2.35) which imply that 𝑢₂₀≤max𝛽₄,𝛽₅=∶𝛽₆. (2.36) In view of (2.10), we have 𝑢₂′₀≤|𝑑|₀+𝑓₀𝑒^𝛽5𝑒^𝛽1=∶𝛽₇. (2.37) From (2.29), (2.31), (2.36), and (2.37), we obtain ‖𝑢‖=|𝑢|_∞+𝑢^′_∞≤𝛽₂+𝛽₃+𝛽₆+𝛽₇. (2.38) From (H₄), the algebraic equations 𝑟−𝑔𝑒^𝑢1−1𝜔^𝜔₀𝑐(𝑡)𝑒^𝑢2ℎ(𝑡)𝑒^𝑢2+𝑒^𝑢1d𝑡=0,𝑑−𝑓𝑒^{𝑢2−𝑢1}=0 (2.39) have a unique solution (𝑢^∗₁,𝑢^∗₂)^𝑇∈𝑅², where 𝑢^∗₁1=ln𝑔1𝑟−𝜔^𝜔₀𝑑𝑐(𝑡)𝑑ℎ(𝑡)+𝑓d𝑡,𝑢^∗₂=𝑢^∗₁+ln𝑑𝑓. (2.40) Set 𝛽=𝛽₂+𝛽₃+𝛽₆+𝛽₇+𝛽₀, where 𝛽₀ is taken sufficiently large such that the unique solution of (2.39) satisfies ‖(𝑢^∗₁,𝑢^∗₂)^𝑇‖=|𝑢^∗₁|+|𝑢^∗₂|<𝛽₀. Clearly, 𝛽 is independent of 𝜆.
We now take Ω=𝑢₁(𝑡),𝑢₂(𝑡)^𝑇∈𝑋∶𝑢₁(𝑡),𝑢₂(𝑡)^𝑇<𝛽. (2.41) This satisfies condition (i) in Lemma 2.1. When (𝑢₁(𝑡),𝑢₂(𝑡))^𝑇∈𝜕Ω∩Ker𝐿=𝜕Ω∩𝑅², (𝑢₁(𝑡),𝑢₂(𝑡))^𝑇 is a constant vector in 𝑅² with |𝑢₁|+|𝑢₂|=𝛽. Thus, we have 𝑢𝑄𝑁¹𝑢₂=𝑟−𝑔𝑒^𝑢1−1𝜔^𝜔₀𝑐(𝑡)𝑒^𝑢2ℎ(𝑡)𝑒^𝑢2+𝑒^𝑢1d𝑡𝑑−𝑓𝑒^{𝑢2−𝑢1}≠00. (2.42) This proves that condition (ii) in Lemma 2.1 is satisfied.
Taking 𝐽=𝐼∶Im𝑄→Ker𝐿,(𝑢₁,𝑢₂)^𝑇→(𝑢₁,𝑢₂)^𝑇, a direct calculation shows that {}−deg𝐽𝑄𝑁,Ω∩Ker𝐿,0=sgndet𝑔𝑒^𝑢∗1+1𝜔𝑒^𝑢∗1𝑒^𝑢∗2𝜔₀𝑐(𝑡)ℎ(𝑡)𝑒^𝑢∗2+𝑒^𝑢∗121d𝑡−𝜔𝑒^𝑢∗1𝑒^𝑢∗2𝜔₀𝑐(𝑡)ℎ(𝑡)𝑒^𝑢∗2+𝑒^𝑢∗12d𝑡𝑓𝑒^{𝑢∗2−𝑢∗1}−𝑓𝑒^{𝑢∗2−𝑢∗1}=sgn𝑓𝑔𝑒^𝑢∗2≠0. (2.43) 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 𝑡∈𝑅. Consequently, we can obtain the following corollary.

Corollary 2.4. Assume that (H₁), (H₄) hold, and 𝜎∈𝐶²(𝑅,𝑅), 𝜎′(𝑡)<1. Then the following delay ratio-dependent predator-prey model with Holling-Tanner functional response 𝑥^′(𝑡)(𝑡𝑟(𝑡)(𝑡)𝑥((𝑡−𝑐(𝑡)𝑥(𝑡)𝑦(𝑡)=𝑥)[−𝑎𝑡−𝜎))]ℎ(𝑡)𝑦(𝑡)(𝑡),𝑦+𝑥_′(𝑡)(𝑡𝑑(𝑡)(𝑡)𝑦((𝑡=𝑦)−𝑓𝑡−𝜏))𝑥(𝑡−𝜏(𝑡)) (2.44) 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−𝑐(𝑡)𝑥(𝑡)𝑦(𝑡)𝑡−𝜎))ℎ(𝑡)𝑦(𝑡)(𝑡),𝑦+𝑥_′(𝑡)(𝑡𝑑(𝑡)(𝑡)𝑦=𝑦)−𝑓𝑡−𝜏₁𝑡,𝑥(𝑡),𝑦(𝑡)𝑥𝑡−𝜏₁(𝑡)(𝑡,𝑡,𝑥,𝑦) (2.45) 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).