On the Nature of Bifurcation in a Ratio-Dependent Predator-Prey Model with Delays

Abstract

A ratio-dependent predator-prey model with two delays is investigated. The conditions which ensure the local stability and the existence of Hopf bifurcation at the positive equilibrium of the system are obtained. It shows that the two different time delays have different effects on the dynamical behavior of the system. An example together with its numerical simulations shows the feasibility of the main results. Finally, main conclusions are included.

1. Introduction

After the seminal models of Volterra and Lotka in the mid-1920s, understanding the dynamics of predator-prey models has been the focus of intense research in recent years. A great deal of excellent and interesting results have been reported. For example, Bhattacharyya and Mukhopadhyay [1] made a detailed discussion on the local and global dynamical behavior of an ecoepidemiological model, Kar and Ghorai [2] analyzed the local stability, global stability, influence of harvesting, and bifurcation of a delayed predator-prey model with harvesting, and Chakraborty et al. [3] studied the bifurcation and control of a bioeconomic model of a delayed prey-predator model. Bhattacharyya and Mukhopadhyay [4] focused on the spatial dynamics of nonlinear prey-predator models with prey migration and predator switching, and Chang and Wei [5] considered the bifurcation nature and optimal control of a diffusive predator-prey system with time delay and prey harvesting. For more related research, one can see [6–26].

In 2011, Wang and Pei [27] investigated the stability and Hopf bifurcation of the following delayed ratio-dependent predator-prey system: 𝑥̇(𝑡)=𝑎𝑥𝑡−𝜏₁−𝑏𝑥²−𝑐𝑥𝑦,𝑦(̇𝑡)=𝑥𝑚𝑦+𝑥𝑡−𝜏₂𝑦𝑡−𝜏₂𝑚𝑦𝑡−𝜏₂+𝑥𝑡−𝜏₂−𝑑𝑦, (1) where 𝑥(𝑡) and 𝑦(𝑡) represent the densities of the prey and the predator population at time 𝑡, respectively. The parameters 𝑎, 𝑘, 𝑐, 𝑚, 𝑓, and 𝑑 are positive constants that stand for prey intrinsic growth rate, carrying capacity, capturing rate, half capturing saturation constant, conversion rate, predator and death rate, respectively. The constants 𝜏₁≥0 and 𝜏₂≥0 denote the time delays due to gestation of the prey and predator, respectively. For more detailed biological meaning of the coefficients of system (1), one can see [27]. Applying the Nyquist criteria and the theory of Hopf bifurcation, Wang and Pei [27] considered the stability of the positive equilibrium and the existence of the local Hopf bifurcation of system (1). By means of the center manifold and normal form theories, they obtained explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solutions.

We would like to point out that Wang and Pei [27] investigated the local stability of system (1) under the assumption 𝜏₁+𝜏₂ in a certain range and focused on the local Hopf bifurcation by choosing the delay 𝜏₂ as bifurcation parameter. A natural problem arising from this is what effect the different delays 𝜏₁ and 𝜏₂ have on the dynamical behavior of system (1). In [27], Wang and Pei did not analyze this aspect. Thus, we think that it is important to deal with the effect of time delay on the dynamics of system (1). To the best of the authors knowledge, there are very few works’ which deal with this topic. In this paper, we will further investigate the stability and bifurcation of model (1) as a complementarity. It will be shown that the two different time delays 𝜏₁ and 𝜏₂ have different effects on the stability and Hopf bifurcation nature of system (1).

The remainder of the paper is organized as follows. In Section 2, we investigate the stability of the positive equilibrium and the occurrence of local Hopf bifurcations. In Section 3, numerical simulations are carried out to illustrate the validity of the main results. Some main conclusions are drawn in Section 4.

2. Stability and Local Hopf Bifurcations

In this section, we shall study the stability of the positive equilibrium and the existence of local Hopf bifurcations.

From [27], we know that if the following conditions  (H1) 𝑑<𝑓<𝑑𝑐,𝑐−𝑚𝑎𝑐>𝑚𝑎or𝑓>𝑑,𝑐≤𝑚𝑎 (2)hold, then system (1) has a unique equilibrium point 𝐸(𝑥^∗,𝑦^∗), where 𝑥^∗=𝑎𝑘𝑙−𝑏₁(𝑘+𝑙)𝑎𝑘𝑙,𝑦^∗=𝑏₁.𝑘𝑙 (3)

Let 𝑥(𝑡)=𝑥(𝑡)−𝑥^∗ and 𝑦(𝑡)=𝑦(𝑡)−𝑦^∗ and still denote 𝑥(𝑡),  𝑦(𝑡) by 𝑥(𝑡), 𝑦(𝑡), respectively. Then (1) reads as 𝑥̇(𝑡)=𝑝₁𝑥+𝑝₂𝑦+𝑝₃𝑥𝑡−𝜏₁,𝑦(̇𝑡)=𝑝₄𝑦+𝑝₅𝑥𝑡−𝜏₂+𝑝₆𝑦𝑡−𝜏₂, (4) where 𝑝₁=𝑐𝑓²−𝑑²−2𝑚𝑎𝑓²𝑚𝑓²,𝑝₂𝑑=−²𝑐𝑓²,𝑝₃𝑝=𝑎,₄𝑝=−𝑑,₅=𝑓−𝑑²,𝑝𝑚𝑓₆=𝑑²𝑓. (5)

The characteristic equation of (4) is given by det𝜆−𝑝₁−𝑝₃𝑒^−𝜆𝜏1−𝑝₂−𝑝₅𝑒^−𝜆𝜏2𝜆−𝑝₄−𝑝₆𝑒^−𝜆𝜏2=0. (6)

That is, 𝜆²−𝑝₁+𝑝₄𝜆−𝑝₃𝜆𝑒^−𝜆𝜏1−𝑝₆𝜆𝑒^−𝜆𝜏2+𝑝₃𝑝₄𝑒^−𝜆𝜏1+𝑝₁𝑝₆−𝑝₂𝑝₅𝑒^−𝜆𝜏2+𝑝₃𝑝₆𝑒^{−𝜆(𝜏1+𝜏2)}+𝑝₁𝑝₄=0. (7)

The following lemma is important for us to analyze the distribution of roots of the transcendental equation (7).

Lemma 1 (see [28]). For the transcendental equation 𝑃𝜆,𝑒^−𝜆𝜏1,…,𝑒^{−𝜆𝜏𝑚}=𝜆^𝑛+𝑝₁⁽⁰⁾𝜆^𝑛−1+⋯+𝑝⁽⁰⁾_𝑛−1𝜆+𝑝_𝑛⁽⁰⁾+𝑝₁⁽¹⁾𝜆^𝑛−1+⋯+𝑝⁽¹⁾_𝑛−1𝜆+𝑝_𝑛⁽¹⁾𝑒^−𝜆𝜏1+⋯+𝑝₁^(𝑚)𝜆^𝑛−1+⋯+𝑝^(𝑚)_𝑛−1𝜆+𝑝_𝑛^(𝑚)𝑒^{−𝜆𝜏𝑚}=0, (8) as (𝜏₁,𝜏₂,𝜏₃,…,𝜏_𝑚) vary, the sum of orders of the zeros of 𝑃(𝜆,𝑒^−𝜆𝜏1,…,𝑒^{−𝜆𝜏𝑚}) in the open right half plane can change, and only a zero appears on or crosses the imaginary axis.

In the sequel, we consider five cases.

Case 1. 𝜏₁=𝜏₂=0. Equation (7) becomes 𝜆²−𝑝₁+𝑝₃+𝑝₄+𝑝₆𝜆²+𝑝₃𝑝₄+𝑝₁𝑝₄+𝑝₁𝑝₆−𝑝₂𝑝₅+𝑝₃𝑝₆=0. (9)
All roots of (9) have a negative real part if the following condition holds: (H2) 𝑝₁+𝑝₃+𝑝₄+𝑝₆𝑝<0,₃𝑝₄+𝑝₁𝑝₄+𝑝₁𝑝₆−𝑝₂𝑝₅+𝑝₃𝑝₆>0. (10)
Then the equilibrium point 𝐸(𝑥^∗,𝑦^∗) is locally asymptotically stable when the conditions (H1) and (H2) are satisfied.

Case 2. 𝜏₁=0, 𝜏₂>0. Equation (7) becomes 𝜆²−𝑝₁+𝑝₄𝜆+𝑝₃𝑝₄+𝑝₁𝑝₄−𝑝₆𝜆+𝑝₂𝑝₅−𝑝₁𝑝₆−𝑝₃𝑝₆𝑒^−𝜆𝜏2=0. (11)
For 𝜔>0, let 𝑖𝜔 be a root of (11). Then it follows that 𝑝₁𝑝₆−𝑝₂𝑝₅+𝑝₃𝑝₆cos𝜔𝜏₂−𝑝₆𝜔sin𝜔𝜏₂=𝜔²−𝑝₃𝑝₄−𝑝₁𝑝₄,𝑝₆𝜔cos𝜔𝜏₂+𝑝₁𝑝₆−𝑝₂𝑝₅+𝑝₃𝑝₆sin𝜔𝜏₂=−𝑝₁+𝑝₃+𝑝₄𝜔 (12) which is equivalent to 𝜔⁴+𝑟₁𝜔²+𝑟₂=0, (13) where 𝑟₁=𝑝₁+𝑝₃+𝑝₄²−2𝑝₃𝑝₄−𝑝₁𝑝₄−𝑝²₆,𝑟₂=𝑝₃𝑝₄−𝑝₁𝑝₄²−𝑝₁𝑝₆−𝑝₂𝑝₅+𝑝₃𝑝₆². (14)
Define Δ₁=𝑟²₁−4𝑟₂. Following Cao and Xiao [29] and Theorem  2.1 in Ge and Yan [30], we have the following result.

Lemma 2. If (H1) holds, then(i)if 𝑟₁<0 and Δ₁=0, then (11) with 𝜏₂=𝜏⁺_2𝑛 has a pair of pure imaginary roots ±𝑖𝜔₊;(ii)if 𝑟₁<0 and Δ₁>0, then (11) with 𝜏₂=𝜏⁺_2𝑛 has two pairs of pure imaginary roots ±𝑖𝜔₊ and ±𝑖𝜔₋, where𝜏^±_2𝑛=1𝜔_±𝜔arccos^2±−𝑝₃𝑝₄−𝑝₁𝑝₄𝑝₁𝑝₆−𝑝₂𝑝₅+𝑝₃𝑝₆−𝑝₆𝑝₁+𝑝₃+𝑝₄𝜔²_±𝑝₁𝑝₆−𝑝₂𝑝₅+𝑝₃𝑝₆²+𝑝₆𝜔_±²+2𝑛𝜋𝜔_± (15)and 𝜔_± satisfies𝜔²₊=−𝑟₁+Δ₁2,𝜔²₋=−𝑟₁−Δ₁2; (16)(iii)if 𝑟₁>0 or Δ₁<0, then all the roots of (11) have negative real parts for 𝜏₂≥0. From Lemma 2, one has the following result.

Theorem 3. Let 𝜏^±_2𝑛 be defined by (15). Under the condition (H1),(i)if 𝑟₁<0 and Δ₁=0, then the trivial solution (𝑥_∗,𝑦_∗)^𝑇 of (11) is asymptotically stable for all 𝜏₂∈[0,𝜏₂₀) and unstable for 𝜏₂>𝜏₂₀. That is, Hopf bifurcation occurs when 𝜏₂=𝜏₂₀;(ii)if 𝑟₁>0 or Δ₁<0, then there are Hopf bifurcations near the trivial solution (𝑥_∗,𝑦_∗)^𝑇 of (11) when 𝜏₂=𝜏⁺_2𝑛 and 𝜏₂=𝜏⁻_2𝑛.

Case 3. 𝜏₁>0, 𝜏₂=0. Equation (7) takes the form 𝜆²−𝑝₁+𝑝₃+𝑝₄+𝑝₆𝜆+𝑝₁𝑝₆−𝑝₂𝑝₅+𝑝₁𝑝₄−𝑝₃𝜆−𝑝₃𝑝₄−𝑝₃𝑝₆𝑒^−𝜆𝜏1=0. (17)
For 𝜂>0, let 𝑖𝜂 be a root of (17). Then it follows that 𝑝₃𝑝₄+𝑝₃𝑝₆cos𝜂𝜏₁−𝑝₃𝜂sin𝜂𝜏₁=𝜂²+𝑝₂𝑝₅−𝑝₁𝑝₆−𝑝₁𝑝₄,𝑝₃𝜂cos𝜂𝜏₁+𝑝₃𝑝₄+𝑝₃𝑝₆sin𝜂𝜏₁=−𝑝₁+𝑝₄+𝑝₆ (18) which is equivalent to 𝜂⁴+𝑠₁𝜂²+𝑠₂=0, (19) where 𝑠₁=𝑝₁+𝑝₄+𝑝₆²+2𝑝₂𝑝₅−𝑝₁𝑝₆−𝑝₁𝑝₄−𝑝²₃,𝑠₂=𝑝₂𝑝₅−𝑝₁𝑝₆−𝑝₁𝑝₄²−𝑝₃𝑝₄+𝑝₃𝑝₆². (20)
Define Δ₂=𝑠²₁−4𝑠₂. Following the Theorem  2.1 in Ge and Yan [30], we have the following result.

Lemma 4. If (H1) holds, then(i)if 𝑠₁<0 and Δ₂=0, then (17) with 𝜏₁=𝜏⁺_1𝑛 has a pair of pure imaginary roots ±𝑖𝜂₊;(ii)if 𝑠₁<0 and Δ₂>0, then (17) with 𝜏₁=𝜏⁺_1𝑛 has two pairs of pure imaginary roots ±𝑖𝜂₊ and ±𝑖𝜂₋, where𝜏^±_1𝑛=1𝜂_±𝜂arccos^2±+𝑝₂𝑝₅−𝑝₁𝑝₆−𝑝₁𝑝₄𝑝₃𝑝₄+𝑝₃𝑝₆−𝑝₃𝑝₁+𝑝₄+𝑝₆𝜂_±𝑝₃𝑝₄+𝑝₃𝑝₆²+𝑝₃𝜂_±²+2𝑛𝜋𝜂_± (21)and 𝜂_± satisfies 𝜂²₊=−𝑠₁+Δ₂2,𝜂²₋=−𝑠₁−Δ₂2; (22)(iii)if 𝑠₁>0 or Δ₂<0, then all the roots of (17) have negative real parts for 𝜏₁≥0.

From Lemma 4, we have the following result.

Theorem 5. Let 𝜏^±_1𝑛 be defined by (21). Under the condition (H1),(i)if 𝑠₁<0 and Δ₂=0, then the trivial solution (𝑥_∗,𝑦_∗)^𝑇 of (17) is asymptotically stable for all 𝜏₁∈[0,𝜏₁₀) and unstable for 𝜏₁>𝜏₁₀. That is, Hopf bifurcation occurs when 𝜏₁=𝜏₁₀;(ii)if 𝑠₁>0 or Δ₂<0, then there are Hopf bifurcations near the trivial solution (𝑥_∗,𝑦_∗)^𝑇 of (17) when 𝜏₁=𝜏⁺_1𝑛 and 𝜏₁=𝜏⁻_1𝑛.

Case 4. 𝜏₁>0, 𝜏₂>0. We consider (7) with 𝜏₂ in its stable interval, by regarding 𝜏₁ as a parameter. Without loss of generality, we consider system (1) under the assumptions (H1) and (H2). Let 𝑖𝜂^∗(𝜂^∗>0) be a root of (7). Then we can obtain 𝜂^∗4+𝑘₁𝜂^∗3+𝑘₂𝜂^∗2+𝑘₃𝜂^∗+𝑘₄=0, (23) where 𝑘₁=2𝑝₆sin𝜂^∗𝜏₂,𝑘₂=𝑝²₆sin²𝜂^∗𝜏₂−2𝑝₁𝑝₆−𝑝₂𝑝₅cos𝜂^∗𝜏₂+𝑝−1+𝑝₄+𝑝₆cos𝜂^∗𝜏₂²,𝑘₃=−2𝑝₆𝑝₁𝑝₆−𝑝₂𝑝₅sin𝜂^∗𝜏₂cos𝜂^∗𝜏₂,𝑘₄=𝑝₆𝑝₁𝑝₆−𝑝₂𝑝₅²+𝑝₁𝑝₆−𝑝₂𝑝₅²sin²𝜂^∗𝜏₂+2𝑝₁+𝑝₄+𝑝₆cos𝜂^∗𝜏₂𝑝₁𝑝₆−𝑝₂𝑝₅sin𝜂^∗𝜏₂−𝑝₃+𝑝₄+𝑝₃𝑝₆cos𝜂^∗𝜏₂²−𝑝₃𝜂+𝑝₃𝑝₆sin𝜂^∗𝜏₂². (24)
Denote 𝐻𝜂^∗=𝜂^∗4+𝑘₁𝜂^∗3+𝑘₂𝜂^∗2+𝑘₃𝜂^∗+𝑘₄. (25)
Assume that  (H3) 𝑘₄<0. (26)
It is easy to check that 𝐻(0)<0 if (H5) holds and lim_{𝜂∗→+∞}𝐻(𝜂^∗)=+∞. We can obtain that (25) has finite positive roots 𝜂^∗₁,𝜂^∗₂,…,𝜂^∗_𝑛. For every fixed 𝜂^∗_𝑖, 𝑖=1,2,3,…,𝑘, there exists a sequence {𝜏^𝑗_1𝑖∣𝑗=1,2,3,…}, such that (25) holds. Let 𝜏₁₀=min𝜏^𝑗_1𝑖∣𝑖=1,2,…,𝑘;𝑗=1,2,…. (27)
When 𝜏₁=𝜏₁₀, (7) has a pair of purely imaginary roots ±𝑖𝜂^∗ for 𝜏₂∈[0,𝜏₂₀).
In the following, we assume that  (H4) 𝑑(Re𝜆)𝑑𝜏_1𝜆=𝑖𝜂^∗≠0. (28)
Thus, by the general Hopf bifurcation theorem for FDEs in Hale [31], we have the following result on the stability and Hopf bifurcation in system (1).

Theorem 6. For system (1), suppose that (H1), (H2), (H3), and (H4) are satisfied and 𝜏₂∈[0,𝜏₂₀). Then the positive equilibrium 𝐸(𝑥^∗,𝑦^∗) is asymptotically stable when 𝜏₁∈[0,𝜏₁₀), and system (1) undergoes a Hopf bifurcation at the positive equilibrium 𝐸(𝑥^∗,𝑦^∗) when 𝜏₁=𝜏₁₀.

Case 5. 𝜏₁>0, 𝜏₂>0. We consider (7) with 𝜏₁ in its stable interval, by regarding 𝜏₂ as a parameter. Without loss of generality, we consider system (1) under the assumptions (H1) and (H2). Let 𝑖𝜔^∗(𝜔^∗>0) be a root of (7). Then we can obtain 𝜔^∗4+𝑡₁𝜔^∗3+𝑡₂𝜔^∗2+𝑡₃𝜔^∗+𝑡₄=0, (29) where 𝑡₁=2𝑝₃sin𝜔^∗𝜏₁,𝑡₂=𝑝²₃sin²𝜔^∗𝜏₁−2𝑝₃𝑝₄cos𝜔^∗𝜏₁−𝑝₁+𝑝₄+𝑝₃cos𝜔^∗𝜏₁²,𝑡₃=2𝑝₃𝑝₄𝑝₁+𝑝₄+𝑝₃sin𝜔^∗𝜏₁sin𝜔^∗𝜏₁−2𝑝²₃𝑝₄sin𝜔^∗𝜏₁cos𝜔^∗𝜏₁,𝑡₄=𝑝₃𝑝₄²sin²𝜔^∗𝜏₂−𝑝₁𝑝₆−𝑝₂𝑝₅+𝑝₃𝑝₆cos𝜔^∗𝜏₁²−𝑝₆𝜔^∗+𝑝₃𝑝₆sin𝜔^∗𝜏₁². (30)
Denote 𝐻_∗𝜔^∗=𝜔^∗4+𝑡₁𝜔^∗3+𝑡₂𝜔^∗2+𝑡₃𝜔^∗+𝑡₄. (31)
Obviously, 𝐻(0)<0 if (H5) holds and lim_{𝜔∗→+∞}𝐻_∗(𝜔^∗)=+∞. We can obtain that (31) has finite positive roots 𝜔^∗₁,𝜔^∗₂,…,𝜔^∗_𝑛. For every fixed 𝜔^∗_𝑖, 𝑖=1,2,3,…,𝑘, there exists a sequence {𝜏^𝑗_2𝑖∣𝑗=1,2,3,…}, such that (31) holds. Let 𝜏₂₀=min𝜏^𝑗_2𝑖∣𝑖=1,2,…,𝑘;𝑗=1,2,…. (32)
When 𝜏₂=𝜏₂₀, (7) has a pair of purely imaginary roots ±𝑖𝜔^∗ for 𝜏₁∈[0,𝜏₁₀).
In the following, we assume that  (H5) 𝑑(Re𝜆)𝑑𝜏_{2𝜆=𝑖𝜔∗}≠0. (33)
In view of the general Hopf bifurcation theorem for FDEs in Hale [31], we have the following result on the stability and Hopf bifurcation in system (1).

Theorem 7. For system (1), assume that (𝐻1), (𝐻2), (𝐻3), and (𝐻5) are satisfied and 𝜏₁∈[0,𝜏₁₀). Then the positive equilibrium 𝐸(𝑥^∗,𝑦^∗) is asymptotically stable when 𝜏₂∈[0,𝜏₂₀), and system (1) undergoes a Hopf bifurcation at the positive equilibrium 𝐸(𝑥^∗,𝑦^∗) when 𝜏₂=𝜏₂₀.

Case 6. 𝜏₁=𝜏₂=𝜏. Equation (7) becomes 𝜆²−𝑝₁+𝑝₄𝜆+𝑝₁𝑝₄−𝑝₂+𝑝₆𝜆−𝑝−3𝑝₄+𝑝₁𝑝₆−𝑝₂𝑝₅𝑒^−𝜆𝜏+𝑝₃𝑝₆𝜆𝑒^−2𝜆𝜏=0, (34) which is equivalent to 𝜆²−𝑝₁+𝑝₄𝜆+𝑝₁𝑝₄𝑒^𝜆𝜏−𝑝₂+𝑝₆𝜆−𝑝₃𝑝₄+𝑝₁𝑝₆−𝑝₂𝑝₅+𝑝₃𝑝₆𝜆𝑒^−𝜆𝜏=0. (35) When 𝜏=0, (35) becomes 𝜆²−𝑝₁+𝑝₃+𝑝₄+𝑝₆𝜆²+𝑝₁𝑝₄+𝑝₃𝑝₄+𝑝₁𝑝₆−𝑝₂𝑝₅+𝑝₃𝑝₆=0. (36)
It is easy to see that if the condition (H2) holds, then all roots of (36) have a negative real part. Then the equilibrium point 𝐸(𝑥^∗,𝑦^∗) is locally asymptotically stable when the conditions (H1) and (H2) are satisfied.
For 𝜃>0, let 𝑖𝜃 be a root of (35). Then it follows that 𝑝₁𝑝₄−𝜃²+𝑝₃𝑝₆cos𝜃𝜏₂+𝑝₃𝑝₆−𝑝₁𝑝₄+𝜃²sin𝜃𝜏=𝑝₂𝑝₅−𝑝₃𝑝₄−𝑝₁𝑝₆,𝑝₁+𝑝₄𝜃cos𝜃𝜏+𝑝₃𝑝₆−𝑝₁𝑝₄+𝜃²𝑝sin𝜃𝜏=−₃+𝑝₆𝜃 (37) which is equivalent to𝑝sin𝜃𝜏=²𝑝₅−𝑝₃𝑝₄−𝑝₁𝑝₆𝑝₁+𝑝₄𝜃+𝑝₃+𝑝₆𝜃𝑝₁𝑝₄−𝜃²+𝑝₃𝑝₆𝑝₁+𝑝₄𝜃²−𝑝₃𝑝₆²+𝑝₁𝑝₄−𝜃²²,𝑝cos𝜃𝜏=³𝑝₄+𝑝₁𝑝₆−𝑝₂𝑝₅𝑝₃𝑝₆−𝑝₁𝑝₄+𝜃²−𝑝₃+𝑝₆𝜃𝑝₁+𝑝₄𝜃𝑝₁+𝑝₄𝜃²−𝑝−3𝑝₆²+𝑝₁𝑝₄−𝜃²². (38)(39)
It follows from sin²𝜃𝜏+cos²𝜃𝜏=1 that 𝑝₂𝑝₅−𝑝₃𝑝₄−𝑝₁𝑝₆𝑝₁+𝑝₄𝜃+𝑝₃+𝑝₆𝜃𝑝₁𝑝₄−𝜃²+𝑝₃𝑝₆²+𝑝₃𝑝₄+𝑝₁𝑝₆−𝑝₂𝑝₅𝑝₃𝑝₆−𝑝₁𝑝₄+𝜃²−𝑝₃+𝑝₆𝜃𝑝₁+𝑝₄𝜃²=𝑝₁+𝑝₄𝜃²−𝑝₃𝑝₆²+𝑝₁𝑝₄−𝜃²²². (40)
Then we have 𝜃⁸+𝑢₃𝜃⁶+𝑢₂𝜃⁴+𝑢₁𝜃²+𝑢₀=0, (41) where 𝑢₀=𝑝₁𝑝₄²−𝑝₃𝑝₆²²−𝑝₂𝑝₅−𝑝₃𝑝₄−𝑝₁𝑝₆𝑝₃𝑝₆−𝑝₁𝑝₄²,𝑢₁=2𝑝₁𝑝₄²−𝑝₃𝑝₆²𝑝₁+𝑝₂²−2𝑝₁𝑝₄²−𝑝₂𝑝₅−𝑝₃𝑝₄−𝑝₁𝑝₆𝑝₁+𝑝₄+𝑝₃+𝑝₆𝑝₁𝑝₄+𝑝₃𝑝₆−2𝑝₂𝑝₅−𝑝₃𝑝₄−𝑝₁𝑝₆𝑝₁+𝑝₄×𝑝₂𝑝₅−𝑝₃𝑝₄−𝑝₁𝑝₆+𝑝₃+𝑝₆𝑝₁+𝑝₄,𝑢₂=2𝑝₁𝑝₄²−𝑝₃𝑝₆²+𝑝₁+𝑝₂²−2𝑝₁𝑝₄²+2𝑝₂𝑝₅−𝑝₃𝑝₄−𝑝₁𝑝₆𝑝₁+𝑝₄+𝑝₃+𝑝₆𝑝₁𝑝₄+𝑝₃𝑝₆𝑝₃+𝑝₆−𝑝−2𝑝₅−𝑝₃𝑝₄−𝑝₁𝑝₆+𝑝₃+𝑝₆𝑝₁+𝑝₄²,𝑢₃=2𝑝₁+𝑝₂²−2𝑝₁𝑝₄−𝑝₃+𝑝₆². (42)
Let 𝑧=𝜃². Then (41) becomes 𝑧⁴+𝑢₃𝑧³+𝑢₂𝑧²+𝑢₁𝑧+𝑢₀=0. (43)
Denote ℎ(𝑧)=𝑧⁴+𝑢₃𝑧³+𝑢₂𝑧²+𝑢₁𝑧+𝑢₀. (44)
Then ℎ^′(𝑧)=4𝑧³+3𝑢₃𝑧²+2𝑢₂𝑧+𝑢₁. (45)
Set 4𝑧³+3𝑢₃𝑧²+2𝑢₂𝑧+𝑢₁=0. (46)
Let 𝑦=𝑧+𝑢₃/4. Then (46) becomes 𝑦³+𝑝₁𝑦+𝑞₁=0, (47) where 𝑝₁=𝑢₂2−3𝑢16²³,𝑞₁=𝑢³₃−𝑢32³𝑢₂8+𝑢₁4. (48)
Define 𝑞𝐷=¹2²+𝑝₁3³√,𝜎=−1+32,𝑦₁=³−𝑞₁2+√𝐷+³−𝑞₁2−√𝑦𝐷,₂=³−𝑞₁2+√𝐷𝜎+³−𝑞₁2−√𝐷𝜎²,𝑦₃=³−𝑞₁2+√𝐷𝜎²+³−𝑞₁2−√𝑧𝐷𝜎,_𝑖=𝑦_𝑖−𝑝₁4,𝑖=1,2,3. (49)
From [32, 33], we have the following result.

Lemma 8. If 𝑢₀<0, then (43) has at least one positive root.

Lemma 9. Assume that 𝑢₀≥0. Then one has the following:(i)if 𝐷≥0, then (43) has positive roots if and only if 𝑧₁>0 and ℎ^′(𝑧₁)<0;(ii)if 𝐷<0, then (43) has positive roots if and only if there exists at least one 𝑧^∗∈{𝑧₁,𝑧₂,𝑧₃} such that 𝑧^∗>0 and ℎ^′(𝑧^∗)≤0.

Without loss of generality, we assume that (43) has four positive roots, defined by 𝑧₁, 𝑧₂, 𝑧₃, 𝑧₄, respectively. Then (41) has four positive roots 𝜃₁=√𝑧₁,𝜃₂=√𝑧₂,𝜃₃=√𝑧₃,𝜃₄=√𝑧₄. (50)

By (39), if we denote𝜏_𝑘^(𝑗)=1𝜃_𝑘𝑝arccos³𝑝₄+𝑝₁𝑝₆−𝑝₂𝑝₅𝑝₃𝑝₆−𝑝₁𝑝₄+𝜃²−𝑝₃+𝑝₆𝜃𝑝₁+𝑝₄𝜃𝑝₁+𝑝₄𝜃²−𝑝−3𝑝₆²+𝑝₁𝑝₄−𝜃²²+2𝑗𝜋, (51)where 𝑘=1,2,3,4; 𝑗=0,1,…, then ±𝑖𝜃_𝑘 are a pair of purely imaginary roots of (35) with 𝜏_𝑘^(𝑗). Define 𝜏₀=𝜏_𝑘⁽⁰⁾₀=min_{𝑘∈{1,2,3,4}}𝜏_𝑘⁽⁰⁾,𝜃₀=𝜃_𝑘0. (52)

Based on above analysis, we have the following result.

Lemma 10. For 𝜏₁=𝜏₂=𝜏, if (H1) and (H2) hold, then all roots of (1) have a negative real part when 𝜏∈[0,𝜏₀), and (1) admits a pair of purely imaginary roots ±𝜃_𝑘𝑖 when 𝜏=𝜏_𝑘^(𝑗)(𝑘=1,2,3,4,𝑗=0,1,2,…).

Let 𝜆(𝜏)=𝛼(𝜏)+𝑖𝜃(𝜏) be a root of (35) near 𝜏=𝜏_𝑘^(𝑗), and let 𝛼(𝜏_𝑘^(𝑗))=0 and 𝜃(𝜏_𝑘^(𝑗))=𝜃_𝑘. Due to functional differential equation theory, for every 𝜏_𝑘^(𝑗), 𝑘=1,2,3,4, 𝑗=0,1,2,…, there exists 𝜀>0 such that 𝜆(𝜏) is continuously differentiable in 𝜏 for |𝜏−𝜏_𝑘^(𝑗)|<𝜀. Substituting 𝜆(𝜏) into the left-hand side of (35) and taking derivative with respect to 𝜏, we have 𝑑𝜆𝑑𝜏⁻¹𝑝=−2𝜆−₁+𝑝₄𝜆𝑒^𝜆𝜏−𝑝₂+𝑝₆+𝑝₃𝑝₆𝑒^−𝜆𝜏𝜆𝜆²−𝑝₁+𝑝₄𝜆+𝑝₁𝑝₄𝑒^𝜆𝜏+𝑝₃𝑝₆𝜆²𝑒^−𝜆𝜏−𝜏𝜆. (53)

We can easily obtain 𝑑(Re𝜆(𝜏))𝑑𝜏⁻¹_{𝜏=𝜏𝑘}^(𝑗)𝑝=−Re2𝜆−₁+𝑝₄𝜆𝑒^𝜆𝜏−𝑝₂+𝑝₆+𝑝₃𝑝₆𝑒^−𝜆𝜏𝜆𝜆²−𝑝₁+𝑝₄𝜆+𝑝₁𝑝₄𝑒^𝜆𝜏+𝑝₃𝑝₆𝜆²𝑒^−𝜆𝜏_{𝜏=𝜏𝑘(𝑗)}𝐴=−Re¹−𝐴₂𝑖𝐵₁+𝐵₂𝑖𝐴=−¹𝐵₁−𝐴₂𝐵₂𝐵²₁+𝐵²₂, (54) where 𝐴₁=𝑝₃𝑝₆−2𝜃²_𝑘cos𝜃_𝑘𝜏_𝑘^(𝑗)+𝑝₁+𝑝₄𝜃_𝑘sin𝜃_𝑘𝜏_𝑘^(𝑗)−𝑝₂+𝑝₆,𝐴₂=𝑝₁+𝑝₄𝜃_𝑘𝜏_𝑘^(𝑗)cos𝜃_𝑘𝜏_𝑘^(𝑗)+2𝜃²_𝑘+𝑝₃𝑝₆sin𝜃_𝑘𝜏_𝑘^(𝑗),𝐵₁=𝑝₁+𝑝₄𝜃²_𝑘cos𝜃_𝑘𝜏_𝑘^(𝑗)−𝑝₁𝑝₄−𝜃²_𝑘𝜃_𝑘sin𝜃_𝑘𝜏_𝑘^(𝑗)−𝑝₃𝑝₆𝜃²_𝑘,𝐵₂=𝑝₁+𝑝₄𝜃²_𝑘sin𝜃_𝑘𝜏_𝑘^(𝑗)+𝑝₁𝑝₄−𝜃²_𝑘𝜃_𝑘cos𝜃_𝑘𝜏_𝑘^(𝑗)+𝑝₃𝑝₆𝜃²_𝑘sin𝜃_𝑘𝜏_𝑘^(𝑗). (55)

Now we assume that  (H6) 𝐴₁𝐵₁≠𝐴₂𝐵₂. (56)

In view of the above analysis and the results of Kuang [33] and Hale [31], we have the following.

Theorem 11. For 𝜏=0, if (𝐻1) and (H2) hold, then the positive equilibrium 𝐸(𝑥^∗,𝑦^∗) of system (1) is asymptotically stable for 𝜏∈[0,𝜏₀). In addition to the conditions (H1) and (H2), one further assumes that (H6) holds. Then system (1) undergoes a Hopf bifurcation at the positive equilibrium 𝐸(𝑥^∗,𝑦^∗) when 𝜏=𝜏_𝑘^(𝑗), 𝑘=1,2,3,4, 𝑗=0,1,2,….

3. Computer Simulations

In this section, we present some numerical results of system (1) to verify the analytical predictions obtained in the previous section. Let us consider the following system: 𝑥̇(𝑡)=2𝑥𝑡−𝜏₁−0.12𝑥²−2𝑥𝑦,𝑦(̇𝑡)=0.77𝑦+𝑥0.205𝑥𝑡−𝜏₂𝑦𝑡−𝜏₂0.77𝑦𝑡−𝜏₂+𝑥𝑡−𝜏₂−0.1𝑦, (57) which has a positive equilibrium 𝐸(5.61,7.02). We can easily obtain that (H1)–(H6) hold true. When 𝜏₁=0, applying MATLAB 7.0, we can get 𝜔₀≈0.5524,𝜏₂₀≈2.3345. The positive equilibrium 𝐸(5.61,7.02) is asymptotically stable for 𝜏₂<𝜏₂₀≈2.3345 and unstable for 𝜏₂>𝜏₂₀≈2.3345 which is shown in Figure 1. When 𝜏₂=𝜏₂₀≈2.3345, (57) undergoes a Hopf bifurcation around the positive equilibrium 𝐸(5.61,7.02). That is, a small amplitude periodic solution occurs near 𝐸(5.61,7.02) when 𝜏₁=0 and 𝜏₂ is close to 𝜏₂₀=2.3345 which is shown in Figure 2. The bifurcation diagram of the case is shown in Figure 3.

(a)

(b)

(c)

(d)

Figure 1 

Trajectory portrait and phase portrait of system (57) with 𝜏1=0, 𝜏2=2.2<𝜏20≈2.3345. The positive equilibrium 𝐸(5.61,7.02) is asymptotically stable. The initial value is (5.6, 7.45).

(a)

(b)

(c)

(d)

Figure 2 

Trajectory portrait and phase portrait of system (57) with 𝜏1=0, 𝜏2=2.5>𝜏20≈2.3345. Hopf bifurcation occurs from the positive equilibrium 𝐸(5.61,7.02). The initial value is (5.6, 7.45).

Figure 3 

Bifurcation diagram with respect to the time delay 𝜏2 for system (57) with 𝜏1=0.

Let 𝜏₂=0.25∈(0,1.8), and choose 𝜏₁ as a parameter. We have 𝜏₁₀≈0.8013. Then the positive equilibrium is asymptotically stable when 𝜏₁∈[0,𝜏₁₀). The Hopf bifurcation value of (57) is 𝜏₁₀≈0.8013 (see Figures 4 and 5) The bifurcation diagram of the case is shown in Figure 6.

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Figure 4 

Trajectory portrait and phase portrait of system (57) with 𝜏2=1.8, 𝜏1=0.75<𝜏10≈0.8013. The positive equilibrium 𝐸(5.61, 7.02) is asymptotically stable. The initial value is (5.6, 7.45).

(a)

(b)

(c)

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Figure 5 

Trajectory portrait and phase portrait of system (57) with 𝜏2=1.8, 𝜏1=0.98>𝜏10≈0.8013. Hopf bifurcation occurs from the positive equilibrium 𝐸(5.61,7.02). The initial value is (5.6, 7.45).

Figure 6 

Bifurcation diagram with respect to the time delay 𝜏1 for system (57) with 𝜏2=1.8.

When 𝜏₂=0, using MATLAB 7.0, we obtain 𝜂₀≈0.9056, 𝜏₁₀≈0.9122. The positive equilibrium 𝐸(5.61,7.02) is asymptotically stable for 𝜏₁<𝜏₁₀≈0.9122 and unstable for 𝜏₁>𝜏₁₀≈0.9122 which is shown in Figure 7. When 𝜏₁=𝜏₁₀≈0.9122, (57) undergoes a Hopf bifurcation at the positive equilibrium 𝐸₀(5.61,7.02). That is, a small amplitude periodic solution occurs around 𝐸(5.61,7.02) when 𝜏₂=0 and 𝜏₁ is close to 𝜏₁₀=0.9122 which is illustrated in Figure 8. The bifurcation diagram of the case is shown in Figure 9.

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(b)

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Figure 7 

Trajectory portrait and phase portrait of system (57) with 𝜏2=0, 𝜏1=0.90<𝜏10≈0.9122. The positive equilibrium 𝐸(5.61,7.02) is asymptotically stable. The initial value is (5.6, 8).

(a)

(b)

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Figure 8 

Trajectory portrait and phase portrait of system (57) with 𝜏2=0, 𝜏1=1.0>𝜏10≈0.9122. Hopf bifurcation occurs from the positive equilibrium 𝐸0(5.61,7.02). The initial value is (5.6, 7.45).

Figure 9 

Bifurcation diagram with respect to the time delay 𝜏1 for system (57) with 𝜏2=0.

Let 𝜏₁=0.5∈(0,0.9122), and choose 𝜏₂ as a parameter. We have 𝜏₂₀≈0.7723. Then the positive equilibrium is asymptotically stable when 𝜏₂∈[0,𝜏₂₀). The Hopf bifurcation value of (57) is 𝜏₂₀≈0.7723(see Figures 10 and 11). The bifurcation diagram of the case is shown in Figure 12.

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(b)

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Figure 10 

Trajectory portrait and phase portrait of system (57) with 𝜏1=0.5, 𝜏2=0.5<𝜏20≈0.7723. The positive equilibrium 𝐸(5.61,7.02) is asymptotically stable. The initial value is (5.6, 7.45).

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(b)

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Figure 11 

Trajectory portrait and phase portrait of system (57) with 𝜏1=0.5, 𝜏2=1.6>𝜏20≈0.7723. Hopf bifurcation occurs from the positive equilibrium 𝐸(5.61,7.02). The initial value is (5.6, 7.45).

Figure 12 

Bifurcation diagram with respect to the time delay 𝜏2 for system (57) with 𝜏1=0.5.

When 𝜏₁=𝜏₂=𝜏, using MATLAB 7.0, we obtain 𝜃₀≈0.8725, 𝜏₀≈1.2032. The positive equilibrium 𝐸(5.61,7.02) is asymptotically stable for 𝜏<𝜏₀≈1.2032 and unstable for 𝜏>𝜏₀≈1.2032 which is shown in Figure 13. When 𝜏=𝜏₀≈1.2032, (57) undergoes a Hopf bifurcation at the positive equilibrium 𝐸₀(5.61,7.02). That is, a small amplitude periodic solution occurs around 𝐸(5.61,7.02) when 𝜏 is close to 𝜏₀≈1.2032 which is illustrated in Figure 14. The bifurcation diagram of the case is shown in Figure 15.

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Figure 13 

Trajectory portrait and phase portrait of system (57) with 𝜏1=0𝜏2=𝜏=1.0<𝜏0≈1.2032. The positive equilibrium 𝐸(5.61,7.02) is asymptotically stable. The initial value is (5.6, 7.45).

(a)

(b)

(c)

(d)

Figure 14 

Trajectory portrait and phase portrait of system (57) with 𝜏1=𝜏2=𝜏=1.25>𝜏0≈1.2032. Hopf bifurcation occurs from the positive equilibrium 𝐸(5.61,7.02). The initial value is (5.6, 7.45).

Figure 15 

Bifurcation diagram with respect to the time delay 𝜏 for system (57) with 𝜏1=𝜏2=𝜏.

4. Conclusions

In this paper, we have investigated local stability of the positive equilibrium 𝐸(𝑥^∗,𝑦^∗) and local Hopf bifurcation of a ratio-dependent predator-prey model with two delays. It is shown that if some conditions hold true and 𝜏₂∈[0,𝜏₂₀), then the positive equilibrium 𝐸(𝑥^∗,𝑦^∗) is asymptotically stable when 𝜏₁∈(0,𝜏₁₀). When the delay 𝜏₁ increases, the positive equilibrium 𝐸(𝑥^∗,𝑦^∗) loses its stability and a sequence of Hopf bifurcations occur at the positive equilibrium 𝐸(𝑥^∗,𝑦^∗). That is, a family of periodic orbits bifurcates from the the positive equilibrium 𝐸(𝑥^∗,𝑦^∗). We also showed that if a certain condition is satisfied and 𝜏₁∈[0,𝜏₁₀), then the positive equilibrium 𝐸(𝑥^∗,𝑦^∗) is asymptotically stable when 𝜏₂∈(0,𝜏₂₀), when the delay 𝜏₂ increases, the positive equilibrium 𝐸(𝑥^∗,𝑦^∗) loses its stability and a sequence of Hopf bifurcations occur at the positive equilibrium 𝐸(𝑥^∗,𝑦^∗). In case 𝜏₁=𝜏₂=𝜏, we have shown that if some conditions are satisfied, and 𝜏∈[0,𝜏₀), then the positive equilibrium 𝐸(𝑥^∗,𝑦^∗) is asymptotically stable when 𝜏∈(0,𝜏₀). When the delay 𝜏 increases, the positive equilibrium 𝐸(𝑥^∗,𝑦^∗) loses its stability and a sequence of Hopf bifurcations occur at the positive equilibrium 𝐸(𝑥^∗,𝑦^∗) which means a family of periodic orbits bifurcates from the the positive equilibrium 𝐸(𝑥^∗,𝑦^∗). Some numerical simulations verifying our theoretical results are carried out. In addition, we must point out that although Ko and Ryu [6] have also investigated the the existence of Hopf bifurcation for system (1) with respect to positive equilibrium 𝐸(𝑥^∗,𝑦^∗), it is assumed that 𝜏₁+𝜏₂ in a certain range and choose the delay 𝜏₂ as bifurcation parameter to consider the Hopf bifurcation nature. But what effect different time delays have on the dynamics of system (1)? Ko and Ryu [6] did not deal with this issue. It is important for us to consider what effect the two different time delays has on the dynamical behavior of system (1). Thus we think that our work generalizes the known results of Ko and Ryu [6]. In addition, we can study the Hopf bifurcation nature of system (1) by regarding the delay 𝜏₂ as bifurcation parameter. We will further focus on the topic elsewhere in the near future.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 11261010 and no. 11101126), the Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), Natural Science and Technology Foundation of Guizhou Province (J[2012]2100), Governor Foundation of Guizhou Province ([2012]53) and Doctoral Foundation of Guizhou University of Finance and Economics (2010).