Hopf Bifurcation Control in a Delayed Predator-Prey System with Prey Infection and Modified Leslie-Gower Scheme

Abstract

Hopf bifurcation of a delayed predator-prey system with prey infection and the modified Leslie-Gower scheme is investigated. The conditions for the stability and existence of Hopf bifurcation of the system are obtained. The state feedback and parameter perturbation are used for controlling Hopf bifurcation in the system. In addition, direction of Hopf bifurcation and stability of the bifurcated periodic solutions of the controlled system are obtained by using normal form and center manifold theory. Finally, numerical simulation results are presented to show that the hybrid controller is efficient in controlling Hopf bifurcation.

1. Introduction

The dynamics of epidemiological models have been investigated by many scholars [1–7] since Kermack and McKendrick [8] proposed the classical SIR model. Based on the classical SIR model, Chattopadhyay and Arino [9] proposed a predator-prey epidemiological model with disease spreading in the prey, and they studied the boundedness of the solutions and the existence of Hopf bifurcation for the model. In order to study the influence of disease on an environment where two or more interacting species are present, Zhou et al. [10] proposed the following ecoepidemiological system consisting of three species: 𝑑𝑆𝑑𝑡=𝑟𝑆1−𝑆+𝐼𝐾−𝛽𝑆𝐼,𝑑𝐼𝑐𝑑𝑡=𝛽𝑆𝐼−𝑐𝐼−¹𝐼𝑦𝐼+𝐾₁,𝑑𝑦=𝑎𝑑𝑡²−𝑐₂𝑦𝐼+𝐾₂𝑦, (1) where 𝑆, 𝐼, and 𝑦 denote, respectively, the population density of the susceptible prey, the infected prey, and the predator. It is assumed that the predator eats only the infected prey with the modified Leslie-Gower scheme [11–14]. The coefficients 𝑟, 𝐾, 𝛽, 𝑐, 𝑐₁, 𝐾₁, 𝑎₂, 𝑐₂, and 𝐾₂ in system (1) are all positive constants, and their ecological meanings are interpreted as follows. 𝑟 and 𝐾 represent the intrinsic birth rate and the carrying capacity of the prey population in the absence of disease, respectively. 𝛽 represents the transmission coefficient. 𝑐 represents the death rate of the infected prey. 𝑐₁ represents the maximum value of the per capita rate of the infected prey due to the predator. 𝑐₂ represents the maximum value of the per capita rate of the predator due to the infected prey population. 𝐾₁ and 𝐾₂ represent the extent to which environment protection to the infected prey and the predator, respectively. Zhou et al. studied the boundedness, stability, and the permanence of system (1). The effect of the transmission coefficient and the predation rate on the dynamics of the system were also investigated.

However, an important aspect which should be kept in mind while formulating an epidemiological system is the fact that it is often necessary to incorporate time delays into the system in order to reflect the dynamical behaviors of the system depending on the past history of the system, and epidemiological systems with delay have been studied extensively [4, 5, 15–17]. Zhang et al. [5] formulated a delayed predator-prey epidemiological system with disease spreading in predator. Hu and Li [15] considered a delayed predator-prey system with disease in prey, and they studied Hopf bifurcation and the stability of the periodic solutions induced by the time delay. Motivated by the work above, in the present paper, we incorporate the feedback delay of the predator into system (1) and get the following delayed system: 𝑑𝑆𝑑𝑡=𝑟𝑆1−𝑆+𝐼𝐾−𝛽𝑆𝐼,𝑑𝐼𝑐𝑑𝑡=𝛽𝑆𝐼−𝑐𝐼−¹𝐼𝑦𝐼+𝐾₁,𝑑𝑦=𝑎𝑑𝑡²−𝑐₂𝑦(𝑡−𝜏)𝐼(𝑡−𝜏)+𝐾₂𝑦, (2) where 𝜏 is the negative feedback delay of the predator. The main purpose of this paper is to consider the effect of the delay on the dynamics of system (2). We will study the local existence of Hopf bifurcation and the properties of periodic solutions. In addition, in order to delay the onset of Hopf bifurcation, we will incorporate the state feedback and parameter perturbation into system (2).

The initial conditions for system (2) take the following form 𝑆(𝜃)=𝜙₁(𝜃), 𝐼(𝜃)=𝜙₂(𝜃), 𝑦(𝜃)=𝜙₃(𝜃), 𝜃∈[−𝜏,0], where (𝜙₁(𝜃),𝜙₂(𝜃),𝜙₃(𝜃))∈𝐶([−𝜏,0],𝑅³₊) and 𝜙₁(0)>0, 𝜙₂(0)>0, and 𝜙₃(0)>0.

This paper is organized as follows. In Section 2, we will study the stability of the positive equilibrium and the existence of local Hopf bifurcation of system (2). In Section 3, the state feedback and parameter perturbation are incorporated into system (2) to control the Hopf bifurcation. The direction and the stability of the bifurcated periodic solutions are also determined for the controlled system. Some numerical simulations are given to support the theoretical prediction in Section 4.

2. Stability of Positive Equilibrium and Hopf Bifurcation

According to [10], we can know that if the condition (𝐻₁): 𝑐₁𝑎₂𝐾₂<𝑐₂𝐾₁(𝛽𝐾−𝑐) holds, then system (2) has a unique positive equilibrium 𝐸_∗(𝑆_∗,𝐼_∗,𝑦_∗), where 𝐼_∗=−𝑐₂𝐾₁𝛽+𝛽²𝐾/𝑟+𝑐₁𝑎₂−𝑐₂𝛽𝐾−𝑐+Δ₁2𝑐₂𝛽+𝛽²,𝑦𝐾/𝑟_∗=𝑎₂𝐼_∗+𝐾₂𝑐₂,𝑆_∗=1𝛽𝑐𝑐+¹𝑦_∗𝐼_∗+𝐾₁, (3) with Δ₁=𝑐₂𝐾₁𝛽𝛽+²𝐾𝑟+𝑐₁𝑎₂−𝑐₂𝛽𝐾−𝑐²−4𝑐₂𝛽𝛽+²𝐾𝑟𝑐₁𝑎₂𝐾₂−𝑐₂𝐾₁𝛽𝐾−𝑐. (4) The variational matrix at 𝐸_∗ takes the form 𝐽𝐸_∗=𝑎₁₁𝑎₁₂0𝑎₂₁𝑎₂₂𝑎₂₃0𝑏₃₂𝑒^−𝜆𝜏𝑏₃₃𝑒^−𝜆𝜏, (5) where 𝑎₁₁𝑟=−𝐾𝑆_∗,𝑎₁₂=−𝑆_∗𝑟𝛽+𝐾,𝑎₂₁=𝛽𝐼_∗,𝑎₂₂=𝑐₁𝐼_∗𝑦_∗𝐼_∗+𝐾₁²,𝑎₂₃𝑐=−¹𝐼_∗𝐼_∗+𝐾₁,𝑏₃₂=𝑐₂𝑦²_∗𝐼_∗+𝐾₂²,𝑏₃₃𝑐=−²𝑦_∗𝐼_∗+𝐾₂. (6) The characteristic equation corresponding to 𝐽(𝐸_∗) will be 𝜆³+𝐴₂𝜆²+𝐴₁𝜆+𝐵₂𝜆²+𝐵₁𝜆+𝐵₀𝑒^−𝜆𝜏, (7) where 𝐴₂=−𝑎₁₁+𝑎₂₂,𝐴₁=𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁,𝐵₂=−𝑏₃₃,𝐵₁=𝑎₁₁+𝑎₂₂𝑏₃₃−𝑎₂₃𝑏₃₂,𝐵₀=𝑎₁₁𝑎₂₃𝑏₃₂+𝑎₁₂𝑎₂₁−𝑎₁₁𝑎₂₂𝑏₃₃. (8) For 𝜏=0, characteristic equation (7) reduces to 𝜆³+𝐴₂+𝐵₂𝜆²+𝐴₁+𝐵₁𝜆+𝐵₀=0. (9) Obviously, if the condition (𝐻₂): 𝐴₂+𝐵₂>0 and (𝐴₂+𝐵₂)(𝐴₁+𝐵₁)>𝐵₀ holds, then the positive equilibrium 𝐸_∗(𝑆_∗,𝐼_∗,𝑦_∗) is locally asymptotically stable in the absence of delay.

For 𝜏>0, substituting 𝜆=𝑖𝜔(𝜔>0) into (7) and separating the real and imaginary parts, one can get 𝐵₁𝜔sin𝜏𝜔+𝐵₀−𝐵₂𝜔²cos𝜏𝜔=𝐴₂𝜔²,𝐵₁𝜔cos𝜏𝜔−𝐵₀−𝐵₂𝜔²sin𝜏𝜔=𝜔³−𝐴₁𝜔, (10) which leads to 𝜔⁶+𝑝₀𝜔⁴+𝑞₀𝜔²+𝑟₀=0, (11) where 𝑝₀=𝐴²₂−𝐵²₂−2𝐴₁,𝑞₀=𝐴²₁−𝐵²₁+2𝐵₀𝐵₂,𝑟₀=−𝐵²₀. (12) Let 𝑧=𝜔². Equation (11) can be written as ℎ(𝑧)∶=𝑧³+𝑝₀𝑧²+𝑞₀𝑧+𝑟₀=0. (13)

Obviously, 𝑟₀=−𝐵²₀≤0. Discussion about the roots of (13) is similar to that in [15], so we have the following lemma.

Lemma 1. For the polynomial equation (13), since 𝑟₀≤0, one has the following results: (i)if 𝑟₀<0, then (13) has at least one positive root; (ii)if 𝑟₀=0 and 𝑝²₀−3𝑞₀≤0, then (13) has no positive roots; (iii)if 𝑟₀=0 and 𝑝²₀−3𝑞₀>0, then (13) has positive roots if and only if 𝑧^∗₁=((−𝑝₀+𝑝²₀−3𝑞₀)/3)>0 and ℎ(𝑧^∗₁)≤0.

Suppose that the coefficients in ℎ(𝑧) satisfy the following condition (𝐻₃): 𝑟₀<0 or 𝑟₀=0, 𝑝²₀−3𝑞₀>0, 𝑧^∗₁>0, and ℎ(𝑧^∗₁)<0.

If the condition (𝐻₃) holds, then (13) has at least one positive root. Without loss of generality, we assume that (13) has three positive roots that are denoted as 𝑧₁, 𝑧₂, and 𝑧₃. Then, (11) has tree positive roots 𝜔_𝑘=√𝑧_𝑘, 𝑘=1,2,3, and for every fixed 𝜔_𝑘, the corresponding critical value of time delay 𝜏_𝑘^(𝑗) is 𝜏_𝑘^(𝑗)=1𝜔_𝑘𝐴arccos²𝐵₂+𝐵₁𝜔⁴_𝑘−𝐴₁𝐵₁+𝐴₂𝐵₀𝜔²_𝑘𝐵²₂𝜔⁴_𝑘+𝐵²₁−2𝐵₀𝐵₂𝜔²_𝑘+𝐵²₀+2𝑗𝜋𝜔_𝑘,𝑘=1,2,3;𝑗=0,1,2,…. (14) Let 𝜏₀=𝜏_𝑘⁽⁰⁾=min𝜏_𝑘⁽⁰⁾,𝜔₀=𝜔_𝑘0,𝑘=1,2,3. (15)

Next, we give the transversality condition by the following Lemma.

Lemma 2. Suppose that 𝑧₀=𝜔²₀ and ℎ^′(𝑧₀)≠0. Then 𝑑Re𝜆(𝜏₀)/𝑑𝜏≠0.

Proof. Taking the derivative of 𝜆 with respect to 𝜏 in (7), we obtain 3𝜆²+2𝐴₂𝜆+𝐴₁𝑑𝜆+𝑑𝜏2𝐵₂𝜆+𝐵₁𝑒^−𝜆𝜏𝑑𝜆−𝐵𝑑𝜏₂𝜆²+𝐵₁𝜆+𝐵₀𝑒^−𝜆𝜏𝜆+𝜏𝑑𝜆𝑑𝜏=0, (16) which yields 𝑑𝜆𝑑𝜏⁻¹=−3𝜆²+2𝐴₂𝜆+𝐴₁𝜆𝜆³+𝐴₂𝜆²+𝐴₁𝜆+2𝐵₂𝜆+𝐵₁𝜆𝐵₂𝜆²+𝐵₁𝜆+𝐵₀−𝑑𝜏.𝑑𝜆 (17) Hence, a direct calculation shows that 𝜏Re𝑑𝜆₀𝑑𝜏⁻¹=3𝜔⁴₀+2𝐴²₂−2𝐴₁𝜔²₀+𝐴²₁𝜔⁶₀+𝐴²₂−2𝐴₁𝜔⁴₀+𝐴²₁𝜔²₀−2𝐵²₂𝜔²₀+𝐵²₁−2𝐵₀𝐵₂𝐵²₂𝜔⁴₀+𝐵²₁−2𝐵₀𝐵₂𝜔²₀+𝐵²₀. (18) From (11), we have 𝜔⁶₀+𝐴²₂−2𝐴₁𝜔⁴₀+𝐴²₁𝜔²₀=𝐵²₂𝜔⁴₀+𝐵²₁−2𝐵₀𝐵₂𝜔²₀+𝐵²₀. (19) Thus, 𝜏Re𝑑𝜆₀𝑑𝜏⁻¹=3𝜔⁴₀+2𝐴²₂−𝐵²₂−2𝐴₁𝜔²₀+𝐴²₁−𝐵²₁+2𝐵₀𝐵₂𝜔⁶₀+𝐴²₂−2𝐴₁𝜔⁴₀+𝐴²₁𝜔²₀=ℎ^′𝑧₀𝜔⁶₀+𝐴²₂−2𝐴₁𝜔⁴₀+𝐴²₁𝜔²₀. (20)
Obviously, if ℎ^′(𝑧₀)≠0, then Re(𝑑𝜆(𝜏₀)/𝑑𝜏)⁻¹≠0. In addition, sgn(𝑑Re𝜆(𝜏₀)/𝑑𝜏)=sgnRe(𝑑𝜆(𝜏₀)/𝑑𝜏).
Thus, the proof is completed.

By Lemmas 1 and 2 and Corollary  2.4 in [18], we have the following theorem.

Theorem 3. For system (2), if the conditions (𝐻₁)--(𝐻₃) hold, then(i)the positive equilibrium 𝐸_∗(𝑆_∗,𝐼_∗,𝑦_∗) is asymptotically stable for 𝜏∈[0,𝜏₀);(ii)the positive equilibrium 𝐸_∗(𝑆_∗,𝐼_∗,𝑦_∗) is unstable when 𝜏>𝜏₀;(iii)if ℎ^′(𝑧₀)≠0, then system (2) undergoes a Hopf bifurcation at 𝐸_∗(𝑆_∗,𝐼_∗,𝑦_∗) when 𝜏=𝜏₀. That is, system (2) has a branch of periodic solutions bifurcating from the zero solution near 𝜏=𝜏₀.

3. Hopf Bifurcation Control

In this section, we will incorporate the state feedback and parameter perturbation into system (2) in order to delay the onset of Hopf bifurcation in the system or make the bifurcation disappear. Then we get the following system with controller: 𝑑𝑆𝑑𝑡=𝛼𝑟𝑆1−𝑆+𝐼𝐾−𝛽𝑆𝐼+𝛾𝑆,𝑑𝐼𝑐𝑑𝑡=𝛼𝛽𝑆𝐼−𝑐𝐼−¹𝐼𝑦𝐼+𝐾₁,𝑑𝑦𝑎𝑑𝑡=𝛼₂−𝑐₂𝑦(𝑡−𝜏)𝐼(𝑡−𝜏)+𝐾₂𝑦+𝛾₂𝑦, (21) where 𝛼, 𝛾, and 𝛾₂ are parameters, which can control the system to relocate the onset of an inherent bifurcation.

Similar as in Section 2, we can easily get that if the following condition (𝐻₄) holds (𝐻₄): 𝑐₁𝑟𝐾₂(𝑎₂𝛼+𝛾₂)+𝑐𝑐₂𝑟𝛼𝐾₁<𝑐₂𝛽𝐾𝐾₁(𝑟𝛼+𝛾), then, system (21) has a unique positive equilibrium 𝐸^∗(𝑆^∗,𝐼^∗,𝑦^∗) where 𝐼^∗=−𝐵₁−𝐵₂+Δ₂2𝛼𝑐₂𝛽+𝛽²,𝑦𝐾/𝑟_∗=𝑎₂𝛼+𝛾₂𝐼^∗+𝐾₂𝛼𝑐₂𝑆^∗=1𝛽𝑐𝑐+¹𝑦^∗𝐼^∗+𝐾₁, (22) with 𝐵₁=𝛼𝑐₂𝐾₁𝛽𝛽+²𝐾𝑟+𝑐₁𝑎₂𝛼+𝛾₂,𝐵₂=𝛼𝑐₂+𝑐𝛽𝐾−𝑐²𝐾𝛽𝛾𝑟,Δ₂=𝛼𝑐₂𝐾₁𝛽𝛽+²𝐾𝑟+𝑐₁𝑎₂𝛼+𝛾₂−𝛼𝑐₂−𝑐𝛽𝐾−𝑐²𝐾𝛽𝛾𝑟²−4𝛼𝑐₂𝛽𝛽+²𝐾𝑟𝑐₁𝐾₂𝑎₂𝛼+𝛾₂−𝛼𝑐₂𝐾₁−𝑐𝛽𝐾−𝑐²𝑟𝐾𝐾₁𝛽𝛾. (23)

Using Taylor expansion to expand the right-hand side of system (21) at the positive equilibrium 𝐸^∗(𝑆^∗,𝐼^∗,𝑦^∗), we have 𝑑𝑆𝑑𝑡=𝑎^′₁₁𝑆+𝑎^′₁₂𝐼+𝑓₁,𝑑𝐼𝑑𝑡=𝑎^′₂₁𝑆+𝑎^′₂₂𝐼+𝑎^′₂₃+𝑓₂,𝑑𝑦𝑑𝑡=𝑏^′₃₂𝐼(𝑡−𝜏)+𝑏^′₃₃𝑦(𝑡−𝜏)+𝑓₃, (24) where 𝑎^′₁₁=−𝛼𝑟𝑆^∗𝐾,𝑎^′₁₂=−𝛼𝑆^∗𝑟𝛽+𝐾,𝑎^′₂₁=𝛼𝛽𝐼^∗,𝑎^′₂₂=𝛼𝑐₁𝐼^∗𝑦^∗𝐼^∗+𝐾₁²,𝑎₂₃=−𝛼𝑐₁𝐼^∗𝐼^∗+𝐾₁,𝑏^′₃₂=𝛼𝑐₂𝑦^∗2𝐼^∗+𝐾₂²,𝑏^′₃₃=−𝛼𝑐₂𝑦^∗𝐼^∗+𝐾₂,𝑓₁=𝑎₁₃𝑆²+𝑎₁₄𝑓𝑆𝐼,₂=𝑎₂₄𝑆𝐼+𝑎₂₅𝐼²+𝑎₂₆𝐼𝑦+𝑎₂₇𝐼²𝑦+𝑎₂₈𝐼³𝑓+⋯,₃=𝑐₃₁𝑦𝑦(𝑡−𝜏)+𝑐₃₂𝑦𝐼(𝑡−𝜏)+𝑐₃₃𝐼²(𝑡−𝜏)+𝑐₃₄𝐼(𝑡−𝜏)𝑦(𝑡−𝜏)+𝑐₃₅𝐼³(𝑡−𝜏)+𝑐₃₆𝐼²(𝑡−𝜏)𝑦+𝑐₃₇𝐼²(𝑡−𝜏)𝑦(𝑡−𝜏)+𝑐₃₈𝑦𝐼(𝑡−𝜏)𝑦(𝑡−𝜏)+⋯, (25) with 𝑎₁₃=−𝛼𝑟𝐾,𝑎₁₄=−𝛼𝑟𝐾𝑎−𝛼𝛽,₂₄=𝛼𝛽,𝑎₂₅=𝑐₁𝐾₁𝛼𝑦^∗𝐼^∗+𝐾₁³,𝑎₂₆𝑐=−¹𝐾₁𝛼𝐼^∗+𝐾₁²,𝑎₂₇=2𝑐₁𝐾₁𝛼𝐼^∗+𝐾₁³,𝑎₂₈𝑐=−¹𝐾₁𝛼𝑦^∗𝐼^∗+𝐾₁⁴,𝑐₃₁=−𝛼𝑐₂𝐼^∗+𝐾₂,𝑐₃₂=𝛼𝑐₂𝑦^∗𝐼^∗+𝐾₂²,𝑐₃₃=𝛼𝑐₂𝑦^∗2𝐼^∗+𝐾₂³,𝑐₃₄=𝛼𝑐₂𝑦^∗𝐼^∗+𝐾₂²,𝑐₃₅=𝛼𝑐₂𝑦^∗2𝐼^∗+𝐾₂⁴,𝑐₃₆=−𝛼𝑐₂𝑦^∗𝐼^∗+𝐾₂³,𝑐₃₇=−𝛼𝑐₂𝑦^∗𝐼^∗+𝐾₂³,𝑐₃₈=𝛼𝑐₂𝐼^∗+𝐾₂². (26) The linear system of (24) is 𝑑𝑆𝑑𝑡=𝑎^′₁₁𝑆+𝑎^′₁₂𝐼,𝑑𝐼𝑑𝑡=𝑎^′₂₁𝑆+𝑎^′₂₂𝐼+𝑎^′₂₃,𝑑𝑦𝑑𝑡=𝑏^′₃₂𝐼(𝑡−𝜏)+𝑏^′₃₃𝑦(𝑡−𝜏). (27) The characteristic equation of system (27) is 𝜆³+𝐴^′₂𝜆²+𝐴^′₁𝜆+𝐵^′₂𝜆²+𝐵^′₁𝜆+𝐵^′₀𝑒^−𝜆𝜏, (28) where 𝐴^′₂=−𝑎^′₁₁+𝑎^′₂₂,𝐴^′₁=𝑎^′₁₁𝑎^′₂₂−𝑎^′₁₂𝑎^′₂₁,𝐵^′₂=−𝑏^′₃₃,𝐵^′₁=𝑎^′₁₁+𝑎^′₂₂𝑏^′₃₃−𝑎^′₂₃𝑏^′₃₂,𝐵^′₀=𝑎^′₁₁𝑎^′₂₃𝑏^′₃₂+𝑎^′₁₂𝑎^′₂₁−𝑎^′₁₁𝑎^′₂₂𝑏^′₃₃. (29)

Obviously, the characteristic equation of system (27) is similar to (7). As the analysis method is similar to Section 2, we omit the linear stability and Hopf bifurcation analysis of system (21). By the similar computation as in Section 2, we can get that the critical value of time delay 𝜏_𝑘^(𝑗) for system (21) is 𝜏^′_𝑘^(𝑗)=1𝜔^′_𝑘𝑔arccos¹𝜔^′_𝑘𝑔₂𝜔^′_𝑘+2𝑗𝜋𝜔^′_𝑘,𝑘=1,2,3;𝑗=0,1,2,…, (30) with 𝑔₁𝜔^′_𝑘=𝐴^′₂𝐵^′₂+𝐵^′₁𝜔^′_𝑘⁴−𝐴^′₁𝐵^′₁+𝐴^′₂𝐵^′₀𝜔^′_𝑘²,𝑔₂𝜔^′_𝑘=𝐵^′₂²𝜔^′_𝑘⁴+𝐵^′₁²−2𝐵^′₀𝐵^′₂𝜔^′_𝑘²+𝐵^′₀², (31) where 𝜔^′_𝑘 is a positive root of the following equation: 𝜔^′6+𝑝^′₀𝜔^′4+𝑞^′₀𝜔^′2+𝑟^′₀=0, (32) with 𝑝^′₀=𝐴^′₂²−𝐵^′₂²−2𝐴^′₁,𝑞^′₀=𝐴^′₁²−𝐵^′₁²+2𝐵^′₀𝐵^′₂,𝑟^′₀=−𝐵^′₀². (33) Let 𝜏^′₀=𝜏^′_𝑘⁽⁰⁾=min𝜏^′_𝑘⁽⁰⁾,𝜔^′₀=𝜔^′_𝑘0,𝑘=1,2,3. (34)

In the following, we will use the normal form method and center manifold theorem introduced by Hassard et al. [19] to determine the property of the bifurcated periodic solutions of the controlled system (21) at 𝜏^′₀.

Let 𝜏=𝜏^′₀+𝜇,𝜇∈𝑅. Then 𝜇=0 is the Hopf bifurcation value of the controlled system (21). Rescaling the time 𝑡→𝑡/𝜏, then system (21) can be written as 𝑢̇(𝑡)=𝐿_𝜇𝑢_𝑡+𝐹𝜇,𝑢_𝑡, (35) where 𝑢(𝑡)=(𝑆(𝑡),𝐼(𝑡),𝑦(𝑡))^𝑇∈𝐶([−1,0],𝑅³₊) and 𝐿_𝜇 and 𝐹 are given respectively by 𝐿_𝜇𝜙=𝜏^′₀+𝜇𝐴^′𝜙(0)+𝐵^′𝜙(),𝐹=𝜏−1𝜇,𝜙^′₀+𝜇𝐹₁,𝐹₂,𝐹₃^𝑇, (36) with 𝐴^′=𝑎^′₁₁𝑎^′₁₂0𝑎^′₂₁𝑎^′₂₂𝑎^′₂₃000,𝐵^′=0000000𝑏^′₃₂𝑏^′₃₃,𝐹₁=𝑎₁₃𝜙²₁(0)+𝑎₁₄𝜙₁(0)𝜙₂(0),𝐹₂=𝑎₂₄𝜙₁(0)𝜙₂(0)+𝑎₂₅𝜙²₂(0)+𝑎₂₆𝜙₂(0)𝜙₃(0)+𝑎₂₇𝜙²₂(0)𝜙₃(0)+𝑎₂₈𝜙³₂(0)𝐹+⋯,₃=𝑐₃₁𝜙₃(0)𝜙₃(−1)+𝑐₃₂𝜙₂(−1)𝜙₃(0)+𝑐₃₃𝜙²₂(−1)+𝑐₃₄𝜙₂(−1)𝜙₃(−1)+𝑐₃₅𝜙³₂(−1)+𝑐₃₆𝜙²₂(−1)𝜙₃(0)+𝑐₃₇𝜙²₂(−1)𝜙₃(−1)+𝑐₃₈𝜙₂(−1)𝜙₃(0)𝜙₃(−1)+⋯. (37) Thus, by the Riesz representation theorem, there exists a 3×3 matrix function 𝜂(𝜃,𝜇)∶[−1,0]→𝑅³₊ whose elements are of bounded variation such that 𝐿_𝜇𝜙=⁰₋₁𝑑𝜂𝜃,𝜇𝜙(𝜃),𝜙∈𝐶[−1,0],𝑅³₊. (38) In fact, we choose 𝜂𝜃,𝜇=𝜏₀+𝜇𝐴^′𝛿(𝜃)−𝜏₀+𝜇𝐵^′𝛿(𝜃+1). (39) For 𝜙∈𝐶¹[−1,0], we define 𝐴𝜇(𝜃)𝜙=𝑑𝜙∫𝑑𝜃,−1≤𝜃<0,₀₋₁𝜙(𝜃)𝑅𝜇𝐹𝑑𝜂𝜃,𝜇,𝜃=0,𝜙=0,−1≤𝜃<0,𝜇,𝜙,𝜃=0. (40) Then system (21) can be transformed into the following operator equation: 𝑢̇(𝑡)=𝐴𝜇𝑢_𝑡+𝑅𝜇𝑢_𝑡. (41) The adjoint operator 𝐴^∗ of 𝐴(0) is defined by 𝐴^∗𝜑=−𝑑𝜑(𝑠)∫𝑑𝑠,0<𝑠≤1,₀₋₁𝑑𝜂^𝑇(𝜉,0)𝜑(−𝑠),𝑠=0, (42) and the bilinear inner product: 𝜑(𝑠),𝜙(𝜃)=𝜑(0)𝜙(0)−⁰_𝜃=−1^𝜃_𝜉=0𝜑(𝜉−𝜃)𝑑𝜂(𝜃)𝜙(𝜉)𝑑𝜉, (43) with 𝜂(𝜃)=𝜂(𝜃,0).

By the previous discussions, we know that ±𝑖𝜏^′₀𝜔^′₀ are eigenvalues of 𝐴(0) and 𝐴^∗(0). We assume that 𝑞(𝜃)=(1,𝑞₂,𝑞₃)^𝑇𝑒^{𝑖𝜏′0𝜔′0𝜃} is the eigenvector of 𝐴(0) belonging to the eigenvalue +𝑖𝜏^′₀𝜔^′₀, and 𝑞^∗(𝑠)=𝐷(1,𝑞^∗₂,𝑞^∗₃)𝑒^{𝑖𝜏′0𝜔′0} is the eigenvector of 𝐴^∗(0) belonging to −𝑖𝜏^′₀𝜔^′₀. Then we have 𝐴(0)𝑞(𝜃)=𝑖𝜏^′₀𝜔^′₀𝑞(𝜃),𝐴^∗(0)𝑞^∗(𝜃)=−𝑖𝜏^′₀𝜔^′₀𝑞^∗(𝜃). (44)

By a simple computation, we can get 𝑞₂=𝑖𝜔^′₀−𝑎^′₁₁𝑎^′₁₂,𝑞₃=𝑏^′₃₂𝑖𝜔^′₀−𝑎^′₁₁𝑎^′₁₂𝑖𝜔^′₀𝑒^{𝑖𝜏′0𝜔′0}−𝑏^′₃₃,𝑞^∗₂=−𝑖𝜔^′₀+𝑎^′₁₁𝑎^′₂₁,𝑞^∗₃=𝑎^′₂₃𝑖𝜔^′₀+𝑎^′₁₁𝑎^′₂₁𝑖𝜔^′₀+𝑏^′₃₃𝑒^{𝑖𝜏′0𝜔′0}. (45) Then from (43), we can obtain 𝐷=1+𝑞₂𝑞^∗₂+𝑞₃𝑞^∗₃+𝑏^′₃₂𝑞₂+𝑏^′₃₃𝑞₃𝑞^∗₃𝜏^′₀𝑒^{−𝑖𝜏′0𝜔′0𝜃−1}, (46) such that ⟨𝑞^∗,𝑞⟩=1, ⟨𝑞^∗,𝑞⟩=0.

Next, we can get the coefficients determining the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions by the algorithms given in [19]: 𝑔₂₀=2𝜏^′₀𝐷𝑎₁₃+𝑎₁₄𝑞⁽²⁾(0)+𝑞^∗₂𝑎₂₄𝑞⁽²⁾(0)+𝑎₂₅𝑞⁽²⁾(0)²+𝑎₂₆𝑞⁽²⁾(0)𝑞⁽³⁾(0)+𝑞^∗₃𝑐₃₁𝑞⁽³⁾(0)𝑞⁽³⁾(−1)+𝑐₃₂𝑞⁽²⁾(−1)𝑞⁽³⁾(0)+𝑐₃₃𝑞⁽²⁾(−1)²+𝑐₃₄𝑞⁽²⁾(−1)𝑞⁽³⁾(),𝑔−1₁₁=𝜏^′₀𝐷2𝑎₁₃+𝑎₁₄𝑞⁽¹⁾(0)+𝑞⁽²⁾(0)+𝑞^∗₂2𝑎₂₅𝑞⁽²⁾(0)𝑞⁽²⁾(0)+𝑎₂₆𝑞⁽³⁾(0)𝑞⁽²⁾(0)+𝑞⁽³⁾(0)𝑞⁽²⁾(0)+𝑞^∗₃𝑐₃₁𝑞⁽³⁾(0)𝑞⁽³⁾(−1)+𝑞⁽³⁾(0)𝑞⁽³⁾(−1)+𝑐₃₂𝑞⁽²⁾(−1)𝑞⁽³⁾(0)+𝑞⁽²⁾(−1)𝑞⁽³⁾(0)+2𝑐₃₃𝑞⁽²⁾(−1)𝑞⁽²⁾(−1)+𝑐₃₄𝑞⁽²⁾(−1)𝑞⁽³⁾(−1)+𝑞⁽²⁾(−1)𝑞⁽³⁾(),𝑔−1₀₂=2𝜏^′₀𝐷𝑎₁₃+𝑎₁₄𝑞⁽²⁾(0)+𝑞^∗₂𝑎₂₄𝑞⁽²⁾(0)+𝑎₂₅𝑞⁽²⁾(0)²+𝑎₂₆𝑞⁽²⁾(0)𝑞⁽³⁾(0)+𝑞^∗₃𝑐₃₁𝑞⁽³⁾(0)𝑞⁽³⁾(−1)+𝑐₃₂𝑞⁽²⁾(−1)𝑞⁽³⁾(0)+𝑐₃₃𝑞⁽²⁾(−1)²+𝑐₃₄𝑞⁽²⁾(−1)𝑞⁽³⁾(,𝑔−1)₂₁=2𝜏^′₀𝐷𝑎₁₃2𝑊⁽¹⁾₁₁(0)+𝑊⁽¹⁾₂₀(0)+𝑎₁₄𝑊⁽¹⁾₁₁(0)𝑞⁽²⁾(0)+12𝑊⁽¹⁾₂₀(0)𝑞⁽²⁾(0)+𝑊⁽²⁾₁₁(0)+12𝑊⁽²⁾₂₀(0)+𝑞^∗₂𝑎₂₄𝑊⁽¹⁾₁₁(0)𝑞⁽²⁾(0)+12𝑊⁽¹⁾₂₀(0)𝑞⁽²⁾(0)+𝑊⁽²⁾₁₁(0)+12𝑊⁽²⁾₂₀(0)+𝑎₂₅2𝑊⁽²⁾₁₁(0)𝑞⁽²⁾(0)+𝑊⁽²⁾₂₀(0)𝑞⁽²⁾(0)+𝑎₂₆𝑊⁽²⁾₁₁(0)𝑞⁽³⁾(0)+12𝑊⁽²⁾₂₀(0)𝑞⁽³⁾(0)+𝑊⁽³⁾₁₁(0)𝑞⁽²⁾(0)+12𝑊⁽³⁾₂₀(0)𝑞⁽²⁾(0)+𝑎₂₇𝑞⁽²⁾(0)²𝑞⁽³⁾(0)+2𝑞⁽²⁾(0)𝑞⁽²⁾(0)𝑞⁽³⁾(0)+3𝑎₂₈𝑞⁽²⁾(0)²𝑞⁽²⁾(0)+𝑞^∗₃𝑐₃₁𝑊⁽³⁾₁₁(0)𝑞⁽³⁾()+1−12𝑊⁽³⁾₂₀(0)𝑞⁽³⁾(−1)+𝑊⁽³⁾₁₁(−1)𝑞⁽³⁾(0)+12𝑊⁽³⁾₂₀(−1)𝑞⁽³⁾(0)+𝑐₃₂𝑊⁽²⁾₁₁(−1)𝑞⁽³⁾(0)+12𝑊⁽²⁾₂₀(−1)𝑞⁽³⁾(0)+𝑊⁽³⁾₁₁(0)𝑞⁽²⁾()+1−12𝑊⁽³⁾₂₀(0)𝑞⁽²⁾(−1)+𝑐₃₃2𝑊⁽²⁾₁₁(−1)𝑞⁽²⁾(−1)+𝑊⁽²⁾₂₀(−1)𝑞⁽²⁾(−1)+𝑐₃₄𝑊⁽²⁾₁₁(−1)𝑞⁽³⁾()+1−12𝑊⁽²⁾₂₀(−1)𝑞⁽³⁾(−1)+𝑊⁽³⁾₁₁(−1)𝑞⁽²⁾()+1−12𝑊⁽³⁾₂₀(−1)𝑞⁽²⁾(−1)+3𝑐₃₅𝑞⁽²⁾(−1)²𝑞⁽²⁾(−1)+𝑐₃₆𝑞⁽²⁾(−1)²𝑞⁽³⁾(0)+2𝑞⁽²⁾(−1)𝑞⁽²⁾(−1)𝑞⁽³⁾(0)+𝑐₃₇𝑞⁽²⁾(−1)²𝑞⁽³⁾(−1)+2𝑞⁽²⁾(−1)𝑞⁽²⁾(−1)𝑞⁽³⁾(−1)+𝑐₃₈𝑞⁽²⁾(−1)𝑞⁽³⁾(0)𝑞⁽³⁾(−1)+𝑞⁽²⁾(−1)𝑞⁽³⁾(0)𝑞⁽³⁾()+−1𝑞⁽²⁾(−1)𝑞⁽³⁾(0)𝑞⁽³⁾(−1), (47) with 𝑊₂₀(𝜃)=𝑖𝑔₂₀𝑞(0)𝜏^′₀𝜔^′₀𝑒^{𝑖𝜏′0𝜔′0𝜃}+𝑖𝑔₀₂𝑞(0)3𝜏^′₀𝜔^′₀𝑒^{−𝑖𝜏′0𝜔′0𝜃}+𝐸₁𝑒^{2𝑖𝜏′0𝜔′0𝜃},𝑊₁₁(𝜃)=−𝑖𝑔₁₁𝑞(0)𝜏^′₀𝜔^′₀𝑒^{𝑖𝜏′0𝜔′0𝜃}+𝑖𝑔₁₁𝑞(0)𝜏^′₀𝜔^′₀𝑒^{−𝑖𝜏′0𝜔′0𝜃}+𝐸₂, (48) where 𝐸₁ and 𝐸₂ can be computed as the following equations, respectively, 𝐸₁=22𝑖𝜔^′₀−𝑎^′₁₁−𝑎^′₁₂0−𝑎^′₂₁2𝑖𝜔^′₀−𝑎^′₂₂−𝑎^′₂₃0−𝑏^′₃₂𝑒^{−2𝑖𝜏′0𝜔′0}2𝑖𝜔^′₀−𝑏^′₃₃𝑒^{−2𝑖𝜏′0𝜔′0−1}×𝐺₁₁𝐺₂₁𝐺₃₁,𝐸₂𝑎=−^′11𝑎^′₁₂0𝑎^′₂₁𝑎^′₂₂𝑎^′₂₃0𝑏^′₃₂𝑏^′₃₃⁻¹×𝐻₁₁𝐻₂₁𝐻₃₁, (49) with 𝐺₁₁=𝑎₁₃+𝑎₁₄𝑞⁽²⁾(0),𝐺₂₁=𝑎₂₄𝑞⁽²⁾(0)+𝑎₂₅𝑞⁽²⁾(0)²+𝑎₂₆𝑞⁽²⁾(0)𝑞⁽³⁾(0),𝐺₃₁=𝑐₃₁𝑞⁽³⁾(0)𝑞⁽³⁾(−1)+𝑐₃₂𝑞⁽²⁾(−1)𝑞⁽³⁾(0)+𝑐₃₃𝑞⁽²⁾(−1)²+𝑐₃₄𝑞⁽²⁾(−1)𝑞⁽³⁾(),𝐻−1₁₁=2𝑎₁₃+𝑎₁₄𝑞⁽¹⁾(0)+𝑞⁽²⁾(0),𝐻₂₁=2𝑎₂₅𝑞⁽²⁾(0)𝑞⁽²⁾(0)+𝑎₂₆𝑞⁽³⁾(0)𝑞⁽²⁾(0)+𝑞⁽³⁾(0)𝑞⁽²⁾(0),𝐻₃₁=𝑐₃₁𝑞⁽³⁾(0)𝑞⁽³⁾(−1)+𝑞⁽³⁾(0)𝑞⁽³⁾(−1)+𝑐₃₂𝑞⁽²⁾(−1)𝑞⁽³⁾(0)+𝑞⁽²⁾(−1)𝑞⁽³⁾(0)+2𝑐₃₃𝑞⁽²⁾(−1)𝑞⁽²⁾(−1)+𝑐₃₄𝑞⁽²⁾(−1)𝑞⁽³⁾(−1)+𝑞⁽²⁾(−1)𝑞⁽³⁾(−1). (50) Therefore, we can calculate the following values: 𝐶₁(0)=𝑖2𝜏^′₀𝜔^′₀𝑔₁₁𝑔₂₀−2𝑔₁₁²−𝑔₀₂²3+𝑔₂₁2,𝐶𝛿=−Re₁(0)Re𝜆^′𝜏^′₀,𝜎=2Re𝐶₁(0,𝐶)𝑇=−Im₁(0)+𝜇₂Im𝜆^′𝜏^′₀𝜏^′₀𝜔^′₀. (51) Based on the previous discussion, we can obtain the following results.

Theorem 4. For system (21), when 𝜏=𝜏^′₀, the direction of the Hopf bifurcation and stability of periodic solutions are determined by the formulas (51), and the following results hold.

The Hopf bifurcation is supercritical (subcritical) if 𝛿>0  (𝛿<0); the bifurcating periodic solutions are stable (unstable) if 𝜎<0  (𝜎>0); the period of the bifurcating periodic solution increases (decreases) if 𝑇>0  (𝑇<0).

4. Numerical Simulation Examples

In this section, we give some numerical simulations to illustrate our theoretical analysis in Sections 2 and 3. As an example, we consider the following particular case of system (2): 𝑑𝑆𝑑𝑡=5𝑆1−𝑆+𝐼3−0.5𝑆𝐼,𝑑𝐼𝑑𝑡=0.5𝑆𝐼−0.3𝐼−𝐼𝑦,𝐼+0.6𝑑𝑦=𝑦()𝑑𝑡1−𝑡−𝜏𝐼(𝑡−𝜏)+0.5𝑦. (52) By a simple computation, we have 𝑐₁𝑎₂𝐾₂=0.5, 𝑐₂𝐾₁(𝛽𝐾−𝑐)=0.72. Obviously, 𝑐₁𝑎₂𝐾₂<𝑐₂𝐾₁(𝛽𝐾−𝑐). Namely, the condition (𝐻₁) holds, and we can get that system (52) has an unique positive equilibrium 𝐸_∗(3.4102,0.4537,0.9537). Then, we obtain 𝐴₂+𝐵₂=4.6273>0, (𝐴₂+𝐵₂)(𝐴₁+𝐵₁)=13.3390>𝐵₀=2.1102. Thus, the condition (𝐻₂) holds. Further, we have 𝜔₀=0.7284, 𝜏₀=1.0877, and ℎ^′(𝑧₀)=15.7784>0. That is, the transversality condition is satisfied. By Theorem 3, we can get that the positive equilibrium 𝐸_∗(3.4102,0.4537,0.9537) is locally asymptotically stable for 𝜏∈[0,1.0877), which can be seen from Figure 1, and 𝐸_∗(3.4102,0.4537,0.9537) is unstable when 𝜏>𝜏₀. This property can be illustrated by Figure 2.

(a)

(b)

(c)

(d)

Figure 1 

𝐸∗ is locally asymptotically stable for 𝜏=1.05<𝜏0=1.0877.

(a)

(b)

(c)

(d)

Figure 2 

𝐸∗ is unstable for 𝜏=1.12>𝜏0=1.0877.

Next, we choose 𝛼=0.5, 𝛾=0.3, and 𝛾₂=0.2 to control the Hopf bifurcation, and we get a particular case of system (21): 𝑑𝑆𝑑𝑡=2.5𝑆1−𝑆+𝐼3−0.25𝑆𝐼+0.3𝑆,𝑑𝐼𝑑𝑡=0.25𝑆𝐼−0.15𝐼−0.5𝐼𝑦,𝐼+0.6𝑑𝑦𝑦()𝑑𝑡=0.51−𝑡−𝜏𝐼(𝑡−𝜏)+0.5𝑦+0.2𝑦. (53) Then, we can easily get the unique positive equilibrium of system (53) 𝐸^∗(3.0619,0.2291,1.0206). From the analysis in Section 3, we get 𝜔^′₀=0.2603 and 𝜏^′₀=1.8034. By choosing 𝜏=1.65 and 𝜏=1.92, the dynamical behavior of the controlled system (53) is illustrated in Figures 3 and 4. From the two figures we can see that, when 𝜏=1.65<𝜏^′₀=1.9034, the positive equilibrium 𝐸^∗ is asymptotically stable (see Figure 3). However, once the time delay passes through the critical value 𝜏^′₀, the system loses stability and a Hopf bifurcation occurs (see Figure 4).

(a)

(b)

(c)

(d)

Figure 3 

𝐸∗ is locally asymptotically stable for 𝜏=1.65<𝜏′0=1.8034.

(a)

(b)

(c)

(d)

Figure 4 

𝐸∗ is unstable for 𝜏=1.92>𝜏′0=1.8034.

Comparing Figures 3 and 4 with Figures 1 and 2, it shows that the onset of Hopf bifurcation is delayed when controller has been incorporated into the system, and the critical value of the delay increases from 𝜏₀=1.0877 to 𝜏^′₀=1.8034.

In addition, from (51), we get 𝜇₂=271.5206>0, 𝛽₂=−33.2968<0, and 𝑇₂=113.2090>0. Thus, from Theorem 4, we know that the Hopf bifurcation is supercritical, the bifurcated periodic solutions are stable, and the period of the bifurcated periodic solutions increases. Since the bifurcated periodic solutions are stable, then the species in system (53) can coexist under some conditions in an oscillatory mode from the viewpoint of biology.

5. Conclusions

A delayed predator-prey system with prey infection and the modified Leslie-Gower scheme is investigated. Regarding the negative feedback delay of the predator as a parameter, the local stability of the positive equilibrium and the existence of Hopf bifurcation are analyzed. The results show that, when the delay crosses a critical value, the system will lose its stability and a Hopf bifurcation occurs. To delay the onset of the Hopf bifurcation, we incorporate the state feedback and parameter perturbation into the system, and simulation results show the effectiveness of the controller. In addition, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions for the controlled system are also determined by the normal form theory and the center manifold argument.

Acknowledgments

This work is supported by the National Natural Science Foundation (NNSF) of China under Grant 61273070, a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and Doctor Candidate Foundation of Jiangnan University (JUDCF12030).