Double Periodic Solutions for a Ratio-Dependent Predator-Prey System with Harvesting Terms on Time Scales

Abstract

We investigate a nonautonomous ratio-dependent predator-prey model with Beddington-DeAngelis functional response and multiple harvesting (or exploited) terms on time scales. By means of using a continuation theorem based on coincidence degree theory, we obtain sufficient criteria for the existence of at least two periodic solutions for the system. Moreover, when the time scale 𝕋 is chosen as ℝ or ℤ, the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the methods are unified to provide the existence of the desired solutions for the continuous differential equations and discrete difference equations.

1. Introduction

In recent years, the existence of periodic solutions for the predator-prey model has been widely studied. Models with harvesting (or exploited) terms are often considered (see, e.g., [1–4]). Generally, the model with harvesting (or exploited) terms is described as follows: ̇̇𝑥=𝑥𝑓(𝑥,𝑦)−ℎ,𝑦=𝑦𝑔(𝑥,𝑦)−𝑘, (1.1) where 𝑥 and 𝑦 are functions of time representing densities of prey and predator, respectively; ℎ and 𝑘 are harvesting (or exploited) terms standing for the harvests (see [5]). Considering the inclusion of the effect of changing environment, Zhang and Hou [6] considered the following model of ordinary differential equations with Holling-type II functional response and harvesting (or exploited) terms: 𝑥̇(𝑡)=𝑥(𝑡)(𝑎(𝑡)−𝑏(𝑡)𝑥(𝑡)−𝑐(𝑡)𝑦(𝑡)𝑚(𝑡)𝑦(𝑡)+𝑥(𝑡))−ℎ₁̇(𝑡),𝑦(𝑡)=𝑦(𝑡)(−𝑑(𝑡)+𝑓(𝑡)𝑥(𝑡)𝑚(𝑡)𝑦(𝑡)+𝑥(𝑡))−ℎ₂(𝑡), (1.2) where the parameters in system (1.2) are continuous positive 𝜔-periodic functions. Authors discussed the existence of positive periodic solutions of system (1.2) in the region 𝐷={(𝑥,𝑦)∣𝑥>0,𝑦>0}.

On the other hand, the theory of calculus on time scales unifies continuous and discrete analysis, many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on general time scales can reveal such discrepancies and help avoid proving results twice-once for differential equations and once again for difference equations. The two main features of the calculus on time scales are unification and extension. To prove a result for a dynamic equation on a time scale is not only related to the set of real numbers or set of integers but those pertaining to more general time scales.

The principle aim of this paper is to systematically unify the existence of multiple periodic solutions of population models modelled by ordinary differential equations and their discrete analogues in form of difference equations and to extend these results to more general time scales. The approach is based on a continuation theorem in coincidence degree, which has been widely applied to deal with the existence of periodic solutions of differential equations and difference equations. Therefore, we consider the following ratio-dependent predator-prey system with Beddington-DeAngelis functional response and harvesting terms on time scales 𝕋: 𝑧^Δ₁(𝑡)=𝑎(𝑡)−𝑏(𝑡)e^𝑧1(𝑡)−𝑐₁(𝑡)e^𝑧2(𝑡)e^𝑧2(𝑡)+𝛼(𝑡)e^𝑧1(𝑡)+𝛽(𝑡)e^2𝑧2(𝑡)−ℎ₁(𝑡)e^{−𝑧1(𝑡)},𝑧^Δ₂𝑐(𝑡)=−𝑑(𝑡)+²(𝑡)e^𝑧1(𝑡)e^𝑧2(𝑡)+𝛼(𝑡)e^𝑧1(𝑡)+𝛽(𝑡)e^2𝑧2(𝑡)−ℎ₂(𝑡)e^{−𝑧2(𝑡)}, (1.3) where 𝑧₁(𝑡) and 𝑧₂(𝑡) represent the prey and the predator population, respectively; 𝑎(𝑡),𝑏(𝑡),𝑐_𝑖(𝑡)(𝑖=1,2),𝑑(𝑡),ℎ_𝑖(𝑡)(𝑖=1,2), and 𝛼(𝑡),𝛽(𝑡) are all rd-continuous positive 𝜔-periodic functions denoting the prey intrinsic growth rate, death rate, capture rate, conversion rate of predator, death rate of predator, harvesting rate, and Beddington-DeAngelis functional response parameters, respectively.

Remark 1.1. In (1.3), set 𝑥_𝑖(𝑡)=exp{𝑧_𝑖(𝑡)},𝑖=1,2. If 𝕋=ℝ, then (1.3) reduces to the ratio-dependent predator-prey system with Beddington-DeAngelis functional response and harvesting terms governed by the ordinary differential equations 𝑥^′₁(𝑡)=𝑥₁(𝑡)(𝑎(𝑡)−𝑏(𝑡)𝑥₁𝑐(𝑡)−¹(𝑡)𝑥₂(𝑡)𝑥₂(𝑡)+𝛼(𝑡)𝑥₁(𝑡)+𝛽(𝑡)𝑥²₂(𝑡))−ℎ₁𝑥(𝑡),_′2(𝑡)=𝑥₂𝑐(𝑡)(−𝑑(𝑡)+²(𝑡)𝑥₁(𝑡)𝑥₂(𝑡)+𝛼(𝑡)𝑥₁(𝑡)+𝛽(𝑡)𝑥²₂(𝑡))−ℎ₂(𝑡). (1.4) If 𝕋=ℤ, then (1.3) is reformulated as 𝑥₁(𝑘+1)=𝑥₁(𝑘)(𝑎(𝑘)−𝑏(𝑘)𝑥₁𝑐(𝑘)−¹(𝑘)𝑥₂(𝑘)𝑥₂(𝑘)+𝛼(𝑘)𝑥₁(𝑘)+𝛽(𝑘)𝑥²₂(𝑘))−ℎ₁𝑥(𝑘),₂(𝑘+1)=𝑥₂𝑐(𝑘)(−𝑑(𝑘)+²(𝑘)𝑥₁(𝑘)𝑥₂(𝑘)+𝛼(𝑘)𝑥₁(𝑘)+𝛽(𝑘)𝑥²₂(𝑘))−ℎ₂(𝑘), (1.5) which is the discrete time ratio-dependent predator-prey system with Beddington-DeAngelis functional response and harvesting terms and is also a discrete analogue of (1.3).

2. Preliminaries

A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ. Throughout this paper, we assume that the time scale 𝕋 is unbounded above and below, such as ℝ, ℤ, and ⋃_𝑘∈ℤ[2𝑘,2𝑘+1]. The following definitions and lemmas can be found in [7].

Definition 2.1 (see [8]). One says that a time scale 𝕋 is periodic if there exists 𝑝>0 such that if 𝑡∈𝕋, then 𝑡±𝑝∈𝕋. For 𝕋≠ℝ, the smallest positive 𝑝 is called the period of the time scale.

Definition 2.2 (see [8]). Let 𝕋≠ℝ be a periodic time scale with period 𝑝. One says that the function 𝑓∶𝕋→ℝ is periodic with period 𝜔 if there exists a natural number 𝑛 such that 𝜔=𝑛𝑝, 𝑓(𝑡+𝜔)=𝑓(𝑡) for all 𝑡∈𝕋, and 𝜔 is the smallest number such that 𝑓(𝑡+𝜔)=𝑓(𝑡). If 𝕋=ℝ, one says that 𝑓 is periodic with period 𝜔>0 if 𝜔 is the smallest positive number such that 𝑓(𝑡+𝜔)=𝑓(𝑡) for all 𝑡∈𝕋.

Notation 2.3. To facilitate the discussion below, we now introduce some notation to be used throughout this paper. Let 𝕋 be 𝜔-periodic; that is, 𝑡∈𝕋 implies 𝑡+𝜔∈𝕋, 𝜅=min{[0,+∞)∩𝕋},𝐼_𝜔𝑓=[𝜅,𝜅+𝜔]∩𝕋,_𝑀=sup_𝑡∈𝕋𝑓(𝑡),𝑓^𝐿=inf_𝑡∈𝕋𝑓(𝑡), (2.1) where 𝑓∈𝐶_rd(𝕋,ℝ) is an 𝜔-periodic function.

Notation 2.4. Let 𝑋, 𝑍 be two Banach spaces, let 𝐿∶Dom𝐿⊂𝑋→𝑍 be a linear mapping, and let 𝑁∶𝑋→𝑍 be a continuous mapping. If 𝐿 is a Fredholm mapping of index zero and there exist continuous projectors 𝑃∶𝑋→𝑋 and 𝑄∶𝑍→𝑍 such that Im𝑃=Ker𝐿, Ker𝑄=Im𝐿=Im(𝐼−𝑄), then the restriction 𝐿|_{Dom𝐿∩Ker𝑃}∶(𝐼−𝑃)𝑋→Im𝐿 is invertible. Denote the inverse of that map by 𝐾_𝑃. If Ω is an open bounded subset of 𝑋, the mapping 𝑁 will be called 𝐿-compact on Ω if 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 2.5 (Continuation theorem [9]). Let 𝑋, 𝑍 be two Banach spaces, and let 𝐿 be a Fredholm mapping of index zero. Assume that 𝑁∶Ω→𝑍 is 𝐿-compact on Ω with Ω being open bounded in 𝑋. Furthermore assume the following:(a)for each 𝜆∈(0,1), 𝑥∈𝜕Ω∩Dom𝐿, 𝐿𝑥≠𝜆𝑁𝑥;(b)for each 𝑥∈𝜕Ω∩Ker𝐿, 𝑄𝑁𝑥≠0;(c)deg{𝐽𝑄𝑁,Ω∩Ker𝐿,0}≠0.Then the operator equation 𝐿𝑥=𝑁𝑥 has at least one solution in Dom𝐿∩Ω.

3. Existence of Periodic Solutions

Our main result on the existence of two positive periodic solutions for system (1.3) is stated in the following theorem.

Theorem 3.1. Assume that the following holds:(i)𝑎^𝐿−𝑐^𝑀₁>2𝑏^𝑀ℎ^𝑀₁,(ii)𝑐^𝑀₂>𝑑^𝐿𝛼^𝐿,(iii)(𝑑^𝐿𝛼^𝐿−𝑐^𝑀₂)ℎ^𝐿₁+ℎ^𝐿₂𝑎^𝑀>2𝑎^𝑀𝑑^𝐿ℎ^𝐿₁𝛼^𝐿ℎ^𝐿₂, then system (1.3) has at least two positive 𝜔-periodic solutions.

Proof. Let 𝑋=𝑍={𝑧=(𝑧₁,𝑧₂)^T∈𝐶_rd(𝕋,ℝ²)∣𝑧_𝑖(𝑡+𝜔)=𝑧_𝑖(𝑡),𝑖=1,2,∀𝑡∈𝕋}, (3.1) and define ‖𝑧‖=‖(𝑧₁,𝑧₂)^T‖=²_𝑖=1max_{𝑡∈𝐼𝜔}|𝑧_𝑖(𝑡)|,𝑧∈𝑋(or𝑍), (3.2) where |⋅| is the Euclidean norm. Then 𝑋 and 𝑍 are both Banach spaces with the above norm ‖⋅‖. Let 𝑓𝑁𝑧(𝑡)=¹𝑓(𝑡)₂=(𝑡)𝑎(𝑡)−𝑏(𝑡)e^𝑧1(𝑡)−𝑐₁(𝑡)e^𝑧2(𝑡)e^𝑧2(𝑡)+𝛼(𝑡)e^𝑧1(𝑡)+𝛽(𝑡)e^2𝑧2(𝑡)−ℎ₁(𝑡)e^{−𝑧1(𝑡)}𝑐−𝑑(𝑡)+²(𝑡)e^𝑧1(𝑡)e^𝑧2(𝑡)+𝛼(𝑡)e^𝑧1(𝑡)+𝛽(𝑡)e^2𝑧2(𝑡)−ℎ₂(𝑡)e^{−𝑧2(𝑡)},𝐿𝑧(𝑡)=𝑧^Δ1(𝑡),𝑃𝑧(𝑡)=𝑄𝑧(𝑡)=𝜔_𝐼𝜔𝑧(𝑡)Δ𝑡,𝑧∈𝑋. (3.3) Then Ker𝐿={𝑧∈𝑋∣𝑧=(𝑧₁(𝑡),𝑧₂(𝑡))^T∈ℝ²for𝑡∈𝕋},Im𝐿={(𝑧₁,𝑧₂)^T∈𝑍∣_𝐼𝜔𝑧_𝑖(𝑡)Δ𝑡=0,𝑖=1,2,for𝑡∈𝕋}, (3.4) and dimKer𝐿=codimIm𝐿=2. Since Im𝐿 is closed in 𝑍, then 𝐿 is a Fredholm mapping of index zero. It is easy to show that 𝑃, 𝑄 are continuous projectors such that Im𝑃=Ker𝐿 and Ker𝑄=Im𝐿=Im(𝐼−𝑄). Furthermore, the generalized inverse (to 𝐿) 𝐾_𝑃∶Im𝐿→Ker𝑃∩Dom𝐿 exists and is given by 𝐾_𝑃𝑧=𝐾_𝑃𝑧₁𝑧₂=^𝑡𝜅𝑧₁1(𝑠)Δ𝑠−𝜔_𝜅^{𝜅+𝜔𝑡}_𝜅𝑧₁(𝑠)Δ𝑠Δ𝑡_𝑡𝜅𝑧₂1(𝑠)Δ𝑠−𝜔_𝜅^{𝜅+𝜔𝑡}_𝜅𝑧₂(𝑠)Δ𝑠Δ𝑡. (3.5) Thus 𝑧𝑄𝑁𝑧=𝑄𝑁¹𝑧₂=1𝜔_𝜅^𝜅+𝜔𝑓₁1(𝑡)Δ𝑡𝜔_𝜅^𝜅+𝜔𝑓₂,𝐾(𝑡)Δ𝑡_𝑃(𝐼−𝑄)𝑁𝑧=^𝑡𝜅𝑓₁1(𝑠)Δ𝑠−𝜔_𝜅^{𝜅+𝜔𝑡}_𝜅𝑓₁1(𝑠)Δ𝑠Δ𝑡+(2−𝑡𝜔)_𝜅^𝜅+𝜔(𝑡−𝜅)𝑓₁(𝑠)Δ𝑠_𝑡𝜅𝑓₂1(𝑠)Δ𝑠−𝜔_𝜅^{𝜅+𝜔𝑡}_𝜅𝑓₂1(𝑠)Δ𝑠Δ𝑡+(2−𝑡𝜔)_𝜅^𝜅+𝜔(𝑡−𝜅)𝑓₂.(𝑠)Δ𝑡 (3.6)
Obviously, 𝑄𝑁∶𝑋→𝑍, 𝐾_𝑃(𝐼−𝑄)𝑁∶𝑋→𝑋 are continuous. Since 𝑋 is a Banach space, using the Arzela-Ascoli theorem, it is easy to show that 𝐾_𝑃(𝐼−𝑄)𝑁(Ω) is compact for any open bounded set Ω⊂𝑋. Moreover, 𝑄𝑁(Ω) is bounded, thus, 𝑁 is 𝐿-compact on Ω for any open bounded set Ω⊂𝑋. Corresponding to the operator equation 𝐿𝑧=𝜆𝑁𝑧, 𝜆∈(0,1), we have 𝑧^Δ_𝑖(𝑡)=𝜆𝑓_𝑖(𝑡),𝑖=1,2. (3.7)
Suppose that (𝑧₁,𝑧₂)^T∈𝑋 is a solution of (3.7) for certain 𝜆∈(0,1). Then there exist 𝜉_𝑖,𝜂_𝑖∈𝐼_𝜔,𝑖=1,2, such that 𝑧_𝑖(𝜉_𝑖)=min_{𝑡∈𝐼𝜔}{𝑧_𝑖(𝑡)},𝑧_𝑖(𝜂_𝑖)=max_{𝑡∈𝐼𝜔}{𝑧_𝑖(𝑡)},𝑖=1,2. (3.8) It is clear that 𝑧^Δ_𝑖(𝜉_𝑖)=0,𝑧^Δ_𝑖(𝜂_𝑖)=0,𝑖=1,2. From this and system (3.7), we have 𝑓₁(𝜉₁𝑓)=0,₂(𝜉₂)=0, (3.9a)(3.9b) It follows from (3.9a) that 𝑓₁(𝜂₁𝑓)=0,₂(𝜂₂)=0. (3.10a)(3.10b) From (3.9b), we obtain 𝑧₁(𝜉₁𝑎)<ln^𝑀𝑏^𝐿. (3.11) which leads to 𝑑^𝐿≤𝑑(𝜉₂𝑐)<²(𝜉₂)e^𝑧1(𝜉2)e^𝑧2(𝜉2)+𝛼(𝜉₂)e^𝑧1(𝜉2)+𝛽(𝜉₂)e^{2𝑧2(𝜉2)}<𝑐^𝑀₂𝑎^𝑀𝑏^𝐿(e^𝑧2(𝜉2)+𝛽^𝐿e^{2𝑧2(𝜉2)}), (3.12) Equation (3.10a) yields 𝑧₂(𝜉₂)<ln−𝑏^𝐿𝑑^𝐿+(𝑏^𝐿𝑑^𝐿)²+4𝑏^𝐿𝑑^𝐿𝛽^𝐿𝑐^𝑀₂𝑎^𝑀2𝑏^𝐿𝑑^𝐿𝛽^𝐿=∶ln𝜌. (3.13) Equation (3.10b) deduces 𝑧₁(𝜂₁ℎ)>ln^𝐿1𝑎^𝑀. (3.14) which implies 𝑐^𝑀₂𝛼^𝐿>𝑐₂(𝜂₂)e^𝑧1(𝜂2)e^𝑧2(𝜂2)+𝛼(𝜂₂)e^𝑧1(𝜂2)+𝛽(𝜂₂)e^{2𝑧2(𝜂2)}>ℎ^𝐿₂e^𝑧1(𝜂2), (3.15) From (3.9a), we also have 𝑧₂(𝜂₂𝛼)>ln^𝐿ℎ^𝐿₂𝑐^𝑀₂. (3.16) which implies that 𝑏(𝜉₁)e^{2𝑧1(𝜉1)}+𝑐₁(𝜉₁)e^{𝑧1(𝜉1)+𝑧2(𝜉1)}e^𝑧2(𝜉1)+𝛼(𝜉₁)e^𝑧1(𝜉1)+𝛽(𝜉₁)e^{2𝑧2(𝜉1)}+ℎ₁(𝜉₁)−𝑎(𝜉₁)e^𝑧1(𝜉1)=0, (3.17) Set 𝑏^𝑀e^{2𝑧1(𝜉1)}+(𝑐^𝑀₁−𝑎^𝐿)e^𝑧1(𝜉1)+ℎ^𝑀₁>0. (3.18) then 𝑢_±=(𝑎^𝐿−𝑐^𝑀₁)±(𝑐^𝑀₁−𝑎^𝐿)²−4𝑏^𝑀ℎ^𝑀₁2𝑏^𝑀, (3.19) From (3.10a), a parallel argument to (3.20) gives 𝑧₁(𝜉₁)>ln𝑢₊or𝑧₁(𝜉₁)<ln𝑢₋. (3.20) From (3.9b), we also obtain 𝑧₁(𝜂₁)>ln𝑢₊or𝑧₁(𝜂₁)<ln𝑢₋. (3.21) which implies 𝑐₂(𝜉₂)e^𝑧1(𝜉2)e^𝑧2(𝜉2)+𝛼(𝜉₂)e^𝑧1(𝜉2)−𝑑(𝜉₂)−ℎ₂(𝜉₂)e^{−𝑧2(𝜉2)}≥0, (3.22) Set 𝑎^𝑀𝑑^𝐿e^{2𝑧2(𝜉2)}+[(𝑑^𝐿𝛼^𝐿−𝑐^𝑀₂)ℎ^𝐿₁+ℎ^𝐿₂𝑎^𝑀]e^𝑧2(𝜉2)+ℎ^𝐿₁ℎ^𝐿₂𝛼^𝐿≤0. (3.23) then 𝑣_±=(𝑐^𝑀₂−𝑑^𝐿𝛼^𝐿)ℎ^𝐿₁−ℎ^𝐿₂𝑎^𝑀±((𝑐^𝑀₂−𝑑^𝐿𝛼^𝐿)ℎ^𝐿₁−ℎ^𝐿₂𝑎^𝑀)²−4𝑎^𝑀𝑑^𝐿ℎ^𝐿₁ℎ^𝐿₂𝛼^𝐿2𝑎^𝑀𝑑^𝐿, (3.24) From (3.11), (3.14), (3.20), and (3.21), we obtain for all ln𝑣₋<𝑧₂(𝜉₂)<ln𝑣₊orln𝑣₊<𝑧₂(𝜂₂)<ln𝑣₋.(3.25), 𝑡∈𝐼_𝜔 or ℎln^𝐿1𝑎^𝑀<𝑧₁(𝑡)<ln𝑢₋ (3.26) From (3.13), (3.16), and (3.25), we obtain that for all ln𝑢₊<𝑧₁𝑎(𝑡)<ln^𝑀𝑏^𝐿.(3.27), 𝑡∈𝐼_𝜔 Obviously, 𝐴∶=max{ln𝑣₋𝛼,ln^𝐿ℎ^𝐿₂𝑐^𝑀₂}<𝑧₂(𝑡)<min{ln𝑣₊,ln𝜌}=∶𝐵.(3.28), ln(ℎ^𝐿₁/𝑎^𝑀), ln(𝑎^𝑀/𝑏^𝐿), ln𝑢_±, 𝐴 are independent of 𝐵. Now take 𝜆 then Ω₁={𝑧=(𝑧₁,𝑧₂)^T∈𝑋∣𝑧₁ℎ(𝑡)∈(ln^𝐿1𝑎^𝑀,ln𝑢₋),𝑧₂Ω(𝑡)∈(𝐴,𝐵)},₂={𝑧=(𝑧₁,𝑧₂)^T∈𝑋∣𝑧₁(𝑡)∈(ln𝑢₊𝑎,ln^𝑀𝑏^𝐿),𝑧₂(𝑡)∈(𝐴,𝐵)},(3.29) and Ω₁ are bounded open subsets of Ω₂, and 𝑋. Thus Ω₁∩Ω₂=∅ and Ω₁ satisfy the requirement (a) in Lemma 2.5.
Now we show that (b) of Lemma 2.5 holds; that is, we need to prove when Ω₂, 𝑧=(𝑧₁,𝑧₂)^T∈𝜕Ω_𝑖∩Ker𝐿=𝜕Ω_𝑖∩ℝ², 𝑄𝑁𝑧≠(0,0)^T. If it is not true, then when 𝑖=1,2, 𝑧∈𝜕Ω_𝑖∩Ker𝐿=𝜕Ω_𝑖∩ℝ², constant vector 𝑖=1,2 with 𝑧, 𝑧∈𝜕Ω_𝑖, satisfies 𝑖=1,2. Thus there exist two points 𝑄𝑁𝑧=(0,0)^T such that 𝑡_𝑖∈𝐼_𝜔(𝑖=1,2) From this and following the arguments of (3.26)–(3.28), we have 𝑎(𝑡₁)−𝑏(𝑡₁)e^𝑧1−𝑐₁(𝑡₁)e^𝑧2e^𝑧2+𝛼(𝑡₁)e^𝑧1+𝛽(𝑡₁)e^2𝑧2−ℎ₁(𝑡₁)e^−𝑧1=0,−𝑑(𝑡₂𝑐)+²(𝑡₂)e^𝑧1e^𝑧2+𝛼(𝑡₂)e^𝑧1+𝛽(𝑡₂)e^2𝑧2−ℎ₂(𝑡₂)e^−𝑧2=0. (3.30) Thus ℎln^𝐿1𝑎^𝑀<𝑧₁<ln𝑢₋,𝐴<𝑧₂<𝐵orln𝑢₊<𝑧₁𝑎<ln^𝑀𝑏^𝐿,𝐴<𝑧₂<𝐵.(3.31), or 𝑧∈Ω₁∩ℝ². This contradicts the fact that 𝑧∈Ω₂∩ℝ², 𝑧∈𝜕Ω_𝑖∩ℝ². Hence condition (b) in Lemma 2.5 holds.
Finally, we show that (c) in Lemma 2.5 is valid. Noticing that the system of algebraic equations 𝑖=1,2 has two distinct solutions 𝑎(𝑡₁)−𝑏(𝑡₁)e^𝑥−ℎ₁(𝑡₁)e^𝑥=0,−𝑑(𝑡₂𝑐)+²(𝑡₂)e^𝑥e^𝑦+𝛼(𝑡₂)e^𝑥+𝛽(𝑡₂)e^2𝑦−ℎ₂(𝑡₂)e^𝑦=0(3.32) and (𝑥^∗₁,𝑦^∗₁) since the conditions hold in Theorem 3.1, and we have (𝑥^∗₂,𝑦^∗₂) where (𝑥^∗₁,𝑦^∗₁)=(ln𝑥₋,ln𝑦₋),(𝑥^∗₂,𝑦^∗₂)=(ln𝑥₊,ln𝑦₊), (3.33) It is easy to verify that 𝑥_±=𝑎(𝑡₁)±(𝑎(𝑡₁))²−4𝑏(𝑡₁)ℎ₁(𝑡₁)2𝑏(𝑡₁),𝑦_±=12𝑎(𝑡₂)𝑑(𝑡₂){(𝑐₂(𝑡₂)−𝑑(𝑡₂)𝛼(𝑡₂))ℎ₁(𝑡₂)−ℎ₂(𝑡₂)𝑎(𝑡₂)±((𝑐₂(𝑡₂)−𝑑(𝑡₂)𝛼(𝑡₂))ℎ₁(𝑡₂)−ℎ₂(𝑡₂)𝑎(𝑡₂))²−4𝑎(𝑡₂)𝑑(𝑡₂)ℎ₁(𝑡₂)ℎ₂(𝑡₂)𝛼(𝑡₂)}. (3.34) Therefore, ℎln^𝐿1𝑎^𝑀<ln𝑥₋<ln𝑢₋<ln𝑢₊<ln𝑥₊𝑎<ln^𝑀𝑏^𝐿,𝛼ln^𝐿ℎ^𝐿₁𝑐^𝑀₂<ln𝑦₋<ln𝑣₋<ln𝑣₊<ln𝑦₊<ln𝜌.(3.35), (𝑥^∗₁,𝑦^∗₁)∈Ω₁.
Define the homotopy (𝑥^∗₂,𝑦^∗₂)∈Ω₂ by 𝐻_𝜇(𝑧₁,𝑧₂)∶Dom𝐿×[0,1]→ℝ² where 𝐻_𝜇(𝑧₁,𝑧₂)=𝜇𝑄𝑁(𝑧₁,𝑧₂)+(1−𝜇)𝐺(𝑧₁,𝑧₂),for𝜇∈[0,1], (3.36) and 𝐺𝑧₁𝑧₂=𝑎(𝑡₁)−𝑏(𝑡₁ℎ)𝑢−¹(𝑡₁)𝑢−𝑑(𝑡₂𝑐)+²(𝑡₂)𝑢𝑣+𝛼(𝑡₂)𝑢+𝛽(𝑡₂)𝑣²−ℎ₂(𝑡₂)𝑣,(3.37) is a parameter, 𝜇∈[0,1], 𝑢=e^𝑧1, and 𝑣=e^𝑧2 is a constant vector in (𝑢,𝑣)^T. It is easy to show that ℝ². In fact, if there are a certain 0∉𝐻_𝜇(𝜕Ω_𝑖∩Ker𝐿)(𝑖=1,2) and a certain 𝑧^∗=(𝑧^∗₁,𝑧^∗₂) such that 𝜇^∗∈[0,1], where 𝐻_𝜇∗(𝑧^∗₁,𝑧^∗₂)=0, then there exist 𝑧^∗_𝑖∈𝜕Ω_𝑖∩Ker𝐿(𝑖=1,2) such that 𝑡^∗_𝑖∈𝐼_𝜔(𝑖=1,2) By carrying out similar arguments as above, we also obtain conclusions as same as (3.26)–(3.28), that is, 𝑎(𝑡^∗₁)−𝑏(𝑡^∗₁)e^𝑧∗1−𝜇^∗𝑐₁(𝑡^∗₁)e^𝑧∗2e^𝑧∗2+𝛼(𝑡^∗₁)e^𝑧∗1+𝛽(𝑡^∗₁)e^2𝑧∗2−ℎ₁(𝑡^∗₁)e^−𝑧∗1=0,−𝑑(𝑡^∗₂𝑐)+²(𝑡^∗₂)e^𝑧∗1e^𝑧∗2+𝛼(𝑡^∗₂)e^𝑧∗1+𝛽(𝑡^∗₂)e^2𝑧∗2−ℎ₂(𝑡^∗₂)e^−𝑧∗2=0. (3.38) This contradicts the fact that ℎln^𝐿1𝑎^𝑀<𝑧^∗₁<ln𝑢₋,𝐴<𝑧^∗₂<𝐵orln𝑢₊<𝑧^∗₁𝑎<ln^𝑀𝑏^𝐿,𝐴<𝑧^∗₂<𝐵.(3.39).
Note that 𝑧^∗_𝑖∈𝜕Ω_𝑖∩Ker𝐿(𝑖=1,2) (identical mapping), since 𝐽=𝐼, according to the invariance property of homotopy, direct calculation produces Im𝑄=Ker𝐿 where deg{𝐽𝑄𝑁(𝑧₁,𝑧₂)^T,Ω_𝑖∩Ker𝐿,(0,0)^T}=deg{𝐺(𝑧₁,𝑧₂)^T,Ω_𝑖∩Ker𝐿,(0,0)^T}=sign−𝑏(𝑡₁ℎ)+¹(𝑡₁)𝑢^∗_𝑖²0𝑐₂(𝑡₂)(𝑣^∗_𝑖+𝛽(𝑡₂)𝑣^∗_𝑖²)(𝑣^∗_𝑖+𝛼(𝑡₂)𝑢^∗_𝑖+𝛽(𝑡₂)𝑣^∗_𝑖²)²ℎ₂(𝑡₂)𝑣^∗_𝑖²−𝑐₂(𝑡₂)𝑢^∗_𝑖(1+2𝛽(𝑡₂)𝑣^∗_𝑖)(𝑣^∗_𝑖+𝛼(𝑡₂)𝑢^∗_𝑖+𝛽(𝑡₂)𝑣^∗_𝑖²)²=sign−𝑏(𝑡₁)𝑢^∗_𝑖+ℎ₁(𝑡₁)𝑢^∗_𝑖0𝑐₂(𝑡₂)𝑣^∗_𝑖(𝑣^∗_𝑖+𝛽(𝑡₂)𝑣^∗_𝑖²)(𝑣^∗_𝑖+𝛼(𝑡₂)𝑢^∗_𝑖+𝛽(𝑡₂)𝑣^∗_𝑖²)²ℎ₂(𝑡₂)𝑣^∗_𝑖−𝑐₂(𝑡₂)𝑢^∗_𝑖𝑣^∗_𝑖(1+2𝛽(𝑡₂)𝑣^∗_𝑖)(𝑣^∗_𝑖+𝛼(𝑡₂)𝑢^∗_𝑖+𝛽(𝑡₂)𝑣^∗_𝑖²)²=sign[(−𝑏(𝑡₁)𝑢^∗_𝑖+ℎ₁(𝑡₁)𝑢^∗_𝑖ℎ)(²(𝑡₂)𝑣^∗_𝑖−𝑐₂(𝑡₂)𝑢^∗_𝑖𝑣^∗_𝑖(1+2𝛽(𝑡₂)𝑣^∗_𝑖)(𝑣^∗_𝑖+𝛼(𝑡₂)𝑢^∗_𝑖+𝛽(𝑡₂)𝑣^∗_𝑖²)²)]=sign[(𝑎(𝑡₁)−2𝑏(𝑡₁)𝑢^∗_𝑖ℎ)(²(𝑡₂)𝑣^∗_𝑖−(𝑣^∗_𝑖𝑑(𝑡₂)+ℎ₂(𝑡₂))(1+2𝛽(𝑡₂)𝑣^∗_𝑖)𝑐₂(𝑡₂)𝑢^∗_𝑖(ℎ₂(𝑡₂)𝑣^∗_𝑖+𝑑(𝑡₂)))],(3.40) is the Brouwer degree, and deg{⋅,⋅,⋅}, 𝑢^∗_𝑖 are positive solution for (3.32) in 𝑣^∗_𝑖, respectively. And Ω_𝑖 otherwise, we have ℎ₂(𝑡₂)𝑣^∗_𝑖−(𝑣^∗_𝑖𝑑(𝑡₂)+ℎ₂(𝑡₂))(1+2𝛽(𝑡₂)𝑣^∗_𝑖)𝑐₂(𝑡₂)𝑢^∗_𝑖(ℎ₂(𝑡₂)𝑣^∗_𝑖+𝑑(𝑡₂))≠0,(3.41) After test and verification, it is not possible. Thus 𝑐₂(𝑡₂)𝑢^∗_𝑖ℎ₂(𝑡₂)=(1+2𝛽(𝑡₂)𝑣^∗_𝑖)(𝑣^∗_𝑖𝑑(𝑡₂)+ℎ₂(𝑡₂))².
By now we have proved that deg{𝐽𝑄𝑁(𝑧₁,𝑧₂)^T,Ω₁∩Ker𝐿,(0,0)^T}=deg{𝐺(𝑧₁,𝑧₂)^T,Ω₁∩Ker𝐿,(0,0)^T}=sign[(𝑎(𝑡₁)−2𝑏(𝑡₁)𝑥₋ℎ)(²(𝑡₂)𝑦₋−(𝑦₋𝑑(𝑡₂)+ℎ₂(𝑡₂))(1+2𝛽(𝑡₂)𝑦₋)𝑐₂(𝑡₂)𝑥₋(ℎ₂(𝑡₂)𝑦₋+𝑑(𝑡₂)))]=−1,deg{𝐽𝑄𝑁(𝑧₁,𝑧₂)^T,Ω₂∩Ker𝐿,(0,0)^T}=deg{𝐺(𝑧₁,𝑧₂)^T,Ω₂∩Ker𝐿,(0,0)^T}=sign[(𝑎(𝑡₁)−2𝑏(𝑡₁)𝑥₊ℎ)(²(𝑡₂)𝑦₊−(𝑦₊𝑑(𝑡₂)+ℎ₂(𝑡₂))(1+2𝛽(𝑡₂)𝑦₊)𝑐₂(𝑡₂)𝑥₊(ℎ₂(𝑡₂)𝑦₊+𝑑(𝑡₂)))]=1.(3.42) verifies all requirements of Lemma 2.5. Therefore, system (1.3) has at least two Ω_𝑖(𝑖=1,2)-periodic solutions in 𝜔, respectively. The proof is complete.

Corollary 3.2. If the conditions in Theorem 3.1 hold, then both the corresponding continuous model (1.4) and the discrete model (1.5) have at least two Dom𝐿∩Ω_𝑖(𝑖=1,2)-periodic solutions.

Remark 3.3. If 𝜔 in system (1.3), then the system is a ratio-dependent predator-prey model with Holling-type functional response and harvesting terms on time scales: 𝛽(𝑡)≡0 Therefore, we have the following results.

Corollary 3.4. If the conditions in Theorem 3.1 hold, then system (3.43) has at least two 𝑧^Δ₁(𝑡)=𝑎(𝑡)−𝑏(𝑡)e^𝑧1(𝑡)−𝑐₁(𝑡)e^𝑧2(𝑡)e^𝑧2(𝑡)+𝛼(𝑡)e^𝑧1(𝑡)−ℎ₁(𝑡)e^{−𝑧1(𝑡)},𝑧^Δ₂𝑐(𝑡)=−𝑑(𝑡)+²(𝑡)e^𝑧1(𝑡)e^𝑧2(𝑡)+𝛼(𝑡)e^𝑧1(𝑡)−ℎ₂(𝑡)e^{−𝑧2(𝑡)}.(3.43)-periodic solutions. Specifically, both the corresponding continuous model and the discrete model of the system (3.43) have at least two 𝜔-periodic solutions.

Remark 3.5. If 𝜔, ℎ₁(𝑡)≡0 in system (1.3), then the system is a general ratio-dependent predator-prey model with Beddington-DeAngelis functional response on time scales: ℎ₂(𝑡)≡0 Using Lemma 2.5, we can obtain another important conclusion as follows.

Theorem 3.6. Assume that the following holds:(i)𝑧^Δ₁(𝑡)=𝑎(𝑡)−𝑏(𝑡)e^𝑧1(𝑡)−𝑐₁(𝑡)e^𝑧2(𝑡)e^𝑧2(𝑡)+𝛼(𝑡)e^𝑧1(𝑡)+𝛽(𝑡)e^2𝑧2(𝑡),𝑧^Δ₂𝑐(𝑡)=−𝑑(𝑡)+²(𝑡)e^𝑧1(𝑡)e^𝑧2(𝑡)+𝛼(𝑡)e^𝑧1(𝑡)+𝛽(𝑡)e^2𝑧2(𝑡).(3.44),(ii)𝑎^𝐿>𝑐^𝑀₁,(iii)𝑐^𝑀₂>𝑑^𝐿𝛼^𝐿, then system (3.44) has at least a positive 𝑎^𝐿>𝑐^𝑀₁/𝛽^𝐿-periodic solutions.

Corollary 3.7. If the conditions in Theorem 3.6 hold, then both the corresponding continuous model and the discrete model of the system (3.44) have at least a positive 𝜔-periodic solutions.

Acknowledgments

The author is very grateful to his supervisor Professor M. Fan for his many valuable comments and suggestions which greatly improved the presentation of this paper. In addition, he would like to express his gratitude to the reviewers and editor for their valuable comments and careful reading and the time and efforts he/she has spent in the review. This work is supported by the Foundation for Subjects Development of Harbin University (no. HXK200716).