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# Bifurcation analysis of a predator-prey system with sex-structure and sexual favoritism

- Shunyi Li
^{1}Email author and - Zuoliang Xiong
^{2}

**2013**:219

https://doi.org/10.1186/1687-1847-2013-219

© Li and Xiong; licensee Springer 2013

**Received: **29 November 2012

**Accepted: **18 June 2013

**Published: **23 July 2013

## Abstract

In this paper, a predator-prey system with sex-structure and sexual favoritism is considered. Firstly, the impact of the sexual favoritism coefficient on the stability of the ordinary differential equation (ODE) model is studied. By choosing sexual favoritism coefficient as a bifurcation parameter, it is shown that a Hopf bifurcation can occur as it passes some critical value, and the stability of the bifurcation is also considered by using an analytical method. Secondly, the impact of the time delay on the stability of the delay differential equation (DDE) model is investigated, where time delay is regarded as a bifurcation parameter. It is found that a Hopf bifurcation can occur as the time delay passes some critical values. Using the normal form theory and center manifold argument, the explicit formulae which determine the stability, direction and other properties of bifurcating periodic solutions are derived. Numerical simulations are performed to support theoretical results and some complex dynamic behaviors are observed, including period-halving bifurcations, period-doubling bifurcations, high-order periodic oscillations, chaotic oscillation, fast-slow oscillation, even unbounded oscillation. Finally, a brief conclusion is given.

**MSC:**34K13, 34K18, 34K60, 37D45, 37N25, 92D25.

## Keywords

- predator-prey system
- time delay
- bifurcation
- chaos
- sex-structure
- sexual favoritism

## 1 Introduction and formulation of the model

Sex ratio means the comparison of male and female individual number in populations. Usually, we assume the sex ratio is $1:1$. However, to some wildlife, the sex ratio of populations will change with the kinds, mate, environment conditions, social behavior, resource, adaptability, heredity, gene structure, *etc.* (see [1–9]). The animal’s sex ratio will change with different animals in the different life history stage. Along with the growth of the age, the male individuals tend to relatively decrease, but there exists a variety of birds for which, on the contrary, the male individuals relatively increase. In isolated populations, males compete locally for mates and resource, sex ratio will affect the dynamic behavior of the population [10]. Sex ratio is a basic dynamic factor for the analysis of populations, and it has important influence on the dynamic state of populations.

*et al.*[11] introduced the following sex-structure model:

where $m(t)$ and $f(t)$ are the male, female individuals of the prey population, $x(t)$ is the predator population. The parameters *a*, *b*, ${b}_{1}$, ${b}_{2}$, ${c}_{1}$, ${c}_{2}$, ${d}_{1}$, ${d}_{2}$, *k* are positive, ${b}_{1}$, ${b}_{2}$, ${d}_{1}$, ${d}_{2}$ are constants of proportionality for male and female prey growth and death (${b}_{2}>{d}_{2}$, where $\beta ={b}_{2}-{d}_{2}$), *a* is the constants of proportionality for a predator, ${c}_{1}$ is the predation coefficient for a predator and ${c}_{1}/{c}_{2}$ ($0<{c}_{1}/{c}_{2}<1$) is the rate of conversing prey into a predator, respectively. The authors obtained the conditions for the equilibrium stability of system (1.1).

Moreover, Boukal *et al.* [12] considered sex-selective predation using several simple predator-prey models, for example, male-biased predation is frequently related to prey traits shaped by sexual selection, predators and parasitoids are attracted by mating signals of their male prey; female-biased predation is often related to prey traits shaped by fecundity selection since it is easier or more rewarding to detect them than prey. The author found that long-term effects of sex-selective predation depend on the interplay of predation bias and prey mating system, given the conclusion that ‘predation on the “less limiting” prey sex can yield a stable predator-prey equilibrium, while predation on the other sex usually destabilizes the dynamics and promotes population collapses’. For the methods, models, data, results, and more details, see [12].

*et al.*[11] introduced the following ODE model:

where *σ* is the sexual favoritism coefficient, $\sigma >1$ means that the predator prefers predating male prey to female prey, $0<\sigma <1$ means that the predator prefers predating female prey to male prey, and $\sigma =1$ means there is no sexual favoritism. The authors obtained the conditions for the equilibrium stability of system (1.2). But, how does the dynamic behavior go when the positive equilibrium loses stability? Does there exist a periodic solution or other rich dynamic behaviors?

*i.e.*, on past values of the state variables. We assume that the reproduction of the predator after predating the prey is not instantaneous and needs some discrete time delay required for gestation of the predator (see [23–25]). Then we formulate the following DDE model:

*τ*($\tau >0$) is the time required for the gestation of the predator. The initial conditions for (1.3) are

where ${\mathbb{R}}_{+}^{3}=\{(m,f,x)\in {\mathbb{R}}^{3},m\ge 0,f\ge 0,x\ge 0\}$.

To the best of our knowledge, few papers focus on the predator-prey system with sex-structure. Recently, Xiong and Zhang [26] have studied a predator-prey model with sex-structure. They obtained the sufficient and realistic conditions for the existence of a positive periodic solution by constructing a *V* functional, and using the result of the existence of positive periodic solutions, the global attractivity of a positive periodic solution was also obtained. Based on system (1.1), Li and Xiong [27] investigated the discrete periodic sex structure model and obtained sufficient and realistic conditions for the existence and global attractivity of a positive periodic solution for it. The pest management strategy of a prey-predator system model with sexual favoritism was considered by Pei *et al.* [28], and the conditions for a global asymptotically stable pest-eradication periodic solution and permanence of the system were established. The local asymptotic stability of system (1.2) has been studied by Liu *et al.* [11]. However, by choosing *σ* as a bifurcation parameter, we obtain Hopf bifurcation conditions for system (1.2). In model (1.3), we introduce time delay due to the gestation of the predator. So, we believe that this is the first time that a predator-prey model with sex-structure and time delay has been formulated and analyzed.

This paper is organized as follows. In Section 2, we first focus on the stability of the equilibrium point and the Hopf bifurcation of ODE system (1.2) by choosing *σ* as a bifurcation parameter. The stability of the bifurcation is also considered by using an analytical method introduced by Kazarinov [29]. In Section 3, we investigate the existence of Hopf bifurcations and the estimation of the length of delay to preserve the stability of DDE system (1.3). By using the normal form theory and center manifold argument introduced by Hassard [30], we derive the explicit formulae for determining the stability, direction, and other properties of bifurcating periodic solutions. Finally, in Section 4, numerical simulations are performed to support the theoretical results. Numerical results show that ODE system (1.2) considered has chaotic behavior under some parameter sets of values and the Hopf bifurcation of DDE system (1.3) is subcritical, and the bifurcating periodic solutions are unstable under certain conditions.

## 2 ODE model (1.2)

### 2.1 Stability of equilibrium and the existence of a Hopf bifurcation

By verifying the characteristic roots of Eq. (2.1) at each equilibrium, it is easily seen that the equilibrium ${E}_{0}$ is always unstable.

According to the above analysis and the result [31], we obtain the following theorem.

**Theorem 2.1** *The positive equilibrium* ${E}^{\ast}$ *is locally asymptotically stable when* $\sigma >1$ *and unstable when* $0<\sigma <1$. *There exists a critical value* ${\sigma}_{0}=1$ *such that a single Hopf bifurcation occurs at* $\sigma ={\sigma}_{0}$ *for decreasing* *σ*, *i*.*e*., *there exists a nontrivial orbitally periodic orbit of system* (1.2) *if* $\sigma \in ({\sigma}_{0}-\epsilon ,{\sigma}_{0})$.

### 2.2 Hopf bifurcation analysis

We deal with the Hopf bifurcation of system (1.2) using an analytical method introduced by Kazarinov [29]. We first introduce some definitions. Suppose that ${\mathbb{C}}^{n}$ is a linear space defined on the complex number field ℂ. For any vectors $x={({x}_{1},{x}_{2},\dots ,{x}_{n})}^{T}$ and $y={({y}_{1},{y}_{2},\dots ,{y}_{n})}^{T}$, where ${x}_{i},{y}_{i}\in \mathbb{C}$ ($i=1,2,\dots ,n$), $\u3008x,y\u3009={\sum}_{i=1}^{n}{\overline{x}}_{i}{y}_{i}$ is the inner product of the vectors *x* and *y*.

*A*only has a pair of pure imaginary eigenvalues ${\lambda}_{1,2}=\pm \alpha \mathrm{i}$, $\alpha >0$, and other eigenvalues are negative, there exists a single Hopf bifurcation. Let $q\in {\mathbb{C}}^{n}$ be a complex eigenvector corresponding to the eigenvalue ${\lambda}_{1}$, then we have $Aq=\mathrm{i}\alpha q$, $A\overline{q}=-\mathrm{i}\alpha \overline{q}$. At the same time, we introduce the adjoint eigenvector $p\in {\mathbb{C}}^{n}$ which satisfies the following conditions:

with ${h}_{jk}\in {\mathbb{C}}^{3}$, ${h}_{jk}={\overline{h}}_{kj}$.

where ${I}_{3}$ is the unit $3\times 3$ matrix.

*first Lyapunov coefficient*${l}_{1}$, which decides, by the analysis of third-order terms at the equilibrium, its stability, if negative, or instability, if positive, is defined by

$w\in \mathbb{C}$, *γ*, *w*, ${l}_{1}$ are smooth continuations of 0, *α* and the first Lyapunov coefficient at the Hopf point [28], respectively. When ${l}_{1}<0$ (${l}_{1}>0$), a family of stable (unstable) periodic orbits can be found on this family of center manifolds which shrink to the equilibrium point at the Hopf point.

## 3 DDE model (1.3)

### 3.1 Existence of a Hopf bifurcation

*τ*, we get

where $\mathrm{\Gamma}={\omega}_{0}^{2}[{({n}_{0}-{n}_{2}{\omega}_{0}^{2})}^{2}+{({n}_{1}{\omega}_{0})}^{2}]>0$. According to the Hopf bifurcation theorem for functional differential equations [32], we have the following result.

**Theorem 3.1**

*Suppose that*$\sigma \in (1,\mathrm{\infty})$. (i)

*There exists a*${\tau}_{0}$

*such that for*$\tau \in [0,{\tau}_{0})$

*the positive equilibrium*${E}^{\ast}$

*of system*(1.3)

*is asymptotically stable and unstable when*$\tau >{\tau}_{0}$.

- (ii)
*System*(1.3)*can undergo a Hopf bifurcation at the positive equilibrium*${E}^{\ast}$*when*$\tau ={\tau}_{n}$ ($n=0,1,2,\dots $),*where*${\tau}_{n}$*is defined by*(3.7).

**Remark 3.1** It must be pointed out that Theorem 3.1 cannot determine the stability and the direction of bifurcating periodic solutions, that is, the periodic solutions may exist either for $\tau >{\tau}_{0}$ or for $\tau <{\tau}_{0}$, near ${\tau}_{0}$. Furthermore, we can investigate the stability of the bifurcating periodic orbits by analyzing higher-order terms according to Hassard *et al.* [30] by using normal form theory and center manifold theorem and prove that the Hopf bifurcation is subcritical and bifurcating periodic solutions are unstable.

### 3.2 Estimation of the length of delay to preserve stability

*Laplace*transform of the system given by (3.1)

and $\overline{m}(s)$, $\overline{f}(s)$, $\overline{x}(s)$ are the *Laplace* transforms of $m(t)$, $f(t)$, $x(t)$, respectively.

*Nyquist*criterion [34], it can be shown that the conditions of local asymptotic stability of ${E}^{\ast}$ given by [34] are

where $H(\lambda )=M(\lambda )+N(\lambda ){e}^{-\lambda \tau}=0$, and ${\eta}_{0}$ is the smallest positive root of Eq. (3.10).

*τ*increases from zero. This can be proved using Butler’s lemma [34], already stated before. In fact, Eqs. (3.9) and (3.10) give

Equations (3.11) and (3.12), if satisfied simultaneously, are sufficient conditions to guarantee stability. We shall utilize them to get an estimate on the length of delay. Our aim is to find an upper bound ${\eta}_{+}$ on ${\eta}_{0}$, independent of *τ*, and then to estimate *τ* so that (3.11) holds for all values of *η*, $0\le \eta \le {\eta}_{+}$, and in particular at $\eta ={\eta}_{0}$.

then, clearly, from (3.13) we have ${\eta}_{0}\le {\eta}_{+}$.

then stability is preserved for $0\le \tau \le {\tau}_{+}$.

### 3.3 Direction and stability of a Hopf bifurcation

In Section 3.1, we have obtained the conditions under which a family of periodic solutions bifurcates from the positive equilibrium of system (1.3) when the delay crosses through the critical values ${\tau}_{n}$. In this subsection, we shall study the direction of these Hopf bifurcations and the stability of bifurcated periodic solutions arising through Hopf bifurcations by applying the normal form theory and center manifold theorem introduced by Hassard *et al.* [30].

*Riesz*representation theorem, there exists a matrix whose components are bounded variation functions $\eta (\theta ,\mu )$ in $[-1,0]$ such that

where $\delta (\theta )$ is a *Dirac* function, then (3.20) is satisfied.

*A*is defined by

where $\eta (\theta )=\eta (\theta ,0)$, we know that $\pm \mathrm{i}{\tau}_{n}{\omega}_{0}$ are eigenvalues of $A(0)$. Thus they are also eigenvalues of ${A}^{\ast}$. To determine the *Poincaré* normal form of the operator *A*, we need to calculate the eigenvector *q* of *A* belonging to the eigenvalue $\mathrm{i}{\omega}_{0}$ and the eigenvector ${q}^{\ast}$ of ${A}^{\ast}$ belonging to the eigenvalue $-\mathrm{i}{\omega}_{0}$.

*D*. From (3.26), we have

*z*and $\overline{z}$ are local coordinates of the center manifold ${C}_{0}$ in the direction of

*q*and ${q}^{\ast}$, respectively. For the solution ${u}_{t}\in {C}_{0}$, since $\mu =0$, we have

**Theorem 3.2**(i) ${\mu}_{2}$

*determines the direction of the Hopf bifurcation*.

*If*${\mu}_{2}>0$ (<0),

*then the Hopf bifurcation is supercritical*(

*subcritical*),

*and the bifurcating periodic solution exists for*$\tau >{\tau}_{0}$ ($\tau <{\tau}_{0}$);

- (ii)
${\beta}_{2}$

*determines the stability of bifurcating periodic solutions*.*If*${\beta}_{2}>0$ (<0),*the bifurcating periodic solutions are unstable*(*stable*); - (iii)
${T}_{2}$

*determines the period of bifurcating periodic solutions*.*If*${T}_{2}>0$ (<0),*the period increases*(*decreases*).

## 4 Numerical simulation

**Example 1**Let ${b}_{1}=3$, ${d}_{1}=0.1$, ${c}_{1}=2$, ${c}_{2}=0.5$, $\beta =2$, $a=0.3$,

*i.e.*, we consider the following ODE system:

*σ*on system (4.1). The bifurcation diagrams of

*σ*over $[0.1,0.6]$ show that system (4.1) has rich dynamics (see Figure 4), including (1) periodic oscillating, (2) period-doubling bifurcations, (3) period-halving bifurcations and (4) chaos. When $0.1\le \sigma <{\sigma}_{1}\approx 0.176$, system (4.1) experiences a

*T*-periodic solution (Figure 5(a)). When $\sigma >{\sigma}_{1}$, the

*T*-periodic solution leads to a 2

*T*-periodic solution, and there is a period-doubling bifurcation leading to chaos when $\sigma >{\sigma}_{2}\approx 0.284$ (Figure 5(b), (c)). When $\sigma >{\sigma}_{3}\approx 0.426$, the chaos suddenly disappears and a 2

*T*-periodic solution appears, and there is a cascade of period-halving bifurcations leading to a

*T*-periodic solution when ${\sigma}_{3}<\sigma \le 0.6$ (Figure 5(c)-(e)).

**Example 2**Let ${b}_{1}=3$, ${d}_{1}=0.4$, $\sigma =2.2$, ${c}_{1}=1.5$, ${c}_{2}=0.4$, $\beta =1.6$, $a=0.5$, we consider the following DDE system:

*τ*crosses ${\tau}_{0}$ to the left; also since ${\beta}_{2}>0$, the bifurcating periodic solution is unstable (see Figure 7).

**Example 3**We consider ODE system (4.1) with time delays:

*t*(see Figure 9).When $\sigma =0.35$, a typical unbounded oscillation solution is observed for a very small time delay $\tau =0.001$ and time $t\approx 101$ (see Figure 10). That is to say, a very small delay would make DDE system (4.3) extinct (unbounded oscillation) undergoing a series of fast-slow oscillations and destroying the permanence of it, if the corresponding ODE system (4.1) is chaotic oscillating when $\sigma =0.35$. All the analysis shows that the time delay would destroy the stability of the system, even make the system die out.

## 5 Conclusion

In this paper, we have investigated a predator-prey system with sex-structure and sexual favoritism. Firstly, the impact of the sexual favoritism coefficient *σ* on the stability of the ODE model is studied. From Theorem 2.1, we know the sexual favoritism coefficient *σ* would determine the stability of ODE system (1.2). In the ecology, sexual favoritism predation could impact population dynamics differently and affect reduced male and female densities in the prey. The numerical simulations show that ODE system (1.2) has complicated dynamic behaviors when we change the parameter *σ*, including periodic oscillating, period-doubling bifurcations, period-halving bifurcations and chaos. That is to say, sexual favoritism coefficient *σ* would be an important factor to affect the dynamic behaviors of the system. Secondly, by analyzing the associated characteristic equation, the impact of the time delay *τ* on the stability of DDE system (1.3) is obtained and the explicit formulae, which determine the stability, direction, and other properties of bifurcating periodic solutions, are also obtained by the Hassard method.

We have obtained estimated length of gestation delay which does not affect the stable coexistence of both predator and prey species at their equilibrium values. From the numerical simulations, we know that the Hopf bifurcation is subcritical and the bifurcating periodic solutions are unstable. It is clear that the larger values of gestation time delay cause fluctuation in population density, and even a very small time delay would make the system subject to unstable oscillation and extinct. These are harmful delays. How to control the bifurcation arising from the DDE system? How can one do this if the time delays make the system subject to unbounded oscillations? The time-varying control strategies and the impulsive control strategies would be considered [35], which could both improve the stability of the system and control the amplitude of the bifurcated periodic solution effectively. We will continue to study these problems in the future.

## Declarations

### Acknowledgements

The authors would like to thank the reviewers and the editor for their valuable suggestions and comments which have led to a much improved paper. This work was supported by the Natural Science Foundation of Guizhou Province (No. [2011]2116), Natural Science Foundation of Jiangxi Province (No. 20122BAB201002).

## Authors’ Affiliations

## References

- Hamilton WD: Extraordinary sex ratios. A sex-ratio theory for sex linkage and inbreeding has new implications in cytogenetics and entomology.
*Science*1967, 156(3774):477-488. 10.1126/science.156.3774.477View ArticleGoogle Scholar - Clark AB: Sex ratio and local resource competition in a prosimian primate.
*Science*1978, 201(4351):163-165. 10.1126/science.201.4351.163View ArticleGoogle Scholar - Wakano JY: Evolution of extraordinary female-biased sex ratios: the optimal schedule of sex ratio in local mate competition.
*J. Theor. Biol.*2005, 237(2):193-202. 10.1016/j.jtbi.2005.04.006MathSciNetView ArticleGoogle Scholar - James WH: Possible constraints on adaptive variation in sex ratio at birth in humans and other primates.
*J. Theor. Biol.*2006, 238(2):383-394. 10.1016/j.jtbi.2005.05.022View ArticleGoogle Scholar - Hu XS, Yeh FC, He F: Sex-ratio distortion driven by migration loads.
*Theor. Popul. Biol.*2007, 72(4):547-559. 10.1016/j.tpb.2007.08.002View ArticleGoogle Scholar - Law PR, Linklater WL: Optimising the sex ratio of translocation for genetic rescue as a function of invested resources.
*Ecol. Model.*2007, 208(2-4):317-341. 10.1016/j.ecolmodel.2007.06.005View ArticleGoogle Scholar - Onagbola EO, Fadamiro HY, Mbata GN: Longevity, fecundity, and progeny sex ratio of Pteromalus cerealellae in relation to diet, host provision, and mating.
*Biol. Control*2007, 40(2):222-229. 10.1016/j.biocontrol.2006.10.010View ArticleGoogle Scholar - Kamimura Y, Abe J, Ito H: The continuous public goods game and the evolution of cooperative sex ratios.
*J. Theor. Biol.*2008, 252(2):277-287. 10.1016/j.jtbi.2008.02.017MathSciNetView ArticleGoogle Scholar - Wang XY, Yang ZQ, Wu H,
*et al*.: Effects of host size on the sex ratio, clutch size, and size of adult Spathius agrili, an ectoparasitoid of emerald ash borer.*Biol. Control*2008, 44(1):7-12. 10.1016/j.biocontrol.2007.10.011View ArticleGoogle Scholar - Charnov EL:
*The Theory of Sex Allocation*. Princeton University Press, Princeton; 1982.Google Scholar - Liu HW, Wang RX, Liu JX: The prey-predator model with sex-structure.
*J. Biomath.*2005, 20(2):179-182. (in Chinese)MathSciNetGoogle Scholar - Boukal DS, Berec L, Křivan V: Does sex-selective predation stabilize or destabilize predator-prey dynamics?
*PLoS ONE*2008., 3(7): Article ID e2687Google Scholar - Long SJ: Attracting and invariant sets of nonlinear neutral differential equations with delays.
*Adv. Differ. Equ.*2012., 2012: Article ID 113Google Scholar - Lu HY, Wang WG: Dynamics of a delayed discrete semi-ratio-dependent predator-prey system with Holling type IV functional response.
*Adv. Differ. Equ.*2011., 2011: Article ID 7Google Scholar - Zhang QH, Yang LH, Liao DX: Existence and globally exponential stability of equilibrium for fuzzy BAM neural networks with distributed delays and impulse.
*Adv. Differ. Equ.*2011., 2011: Article ID 8Google Scholar - Alzabut JO: Existence of periodic solutions for a type of linear difference equations with distributed delay.
*Adv. Differ. Equ.*2012., 2012: Article ID 53Google Scholar - Wu RC, Liu JL: Isochronal function projective synchronization between chaotic and time-delayed chaotic systems.
*Adv. Differ. Equ.*2012., 2012: Article ID 37Google Scholar - Sangapate P: New sufficient conditions for the asymptotic stability of discrete time-delay systems.
*Adv. Differ. Equ.*2012., 2012: Article ID 28Google Scholar - Dong YL, Wei J: Output feedback stabilization of nonlinear discrete-time systems with time-delay.
*Adv. Differ. Equ.*2012., 2012: Article ID 73Google Scholar - Chen HB, Tang HW, Sun JT: Periodic solutions of second-order differential equations with multiple delays.
*Adv. Differ. Equ.*2012., 2012: Article ID 43Google Scholar - Zhao YX, Ma YC: Stability of neutral-type descriptor systems with multiple time-varying delays.
*Adv. Differ. Equ.*2012., 2012: Article ID 15Google Scholar - Song QK, Cao JD: Synchronization of nonidentical chaotic neural networks with leakage delay and mixed time-varying delays.
*Adv. Differ. Equ.*2011., 2011: Article ID 16Google Scholar - Wang WD, Chen LS: A predator-prey system with stage-structure for predator.
*Comput. Math. Appl.*1997, 33(8):83-91.MathSciNetView ArticleGoogle Scholar - Xiao YN, Chen LS: Modeling and analysis of a predator-prey model with disease in the prey.
*Math. Biosci.*2001, 171(1):59-82. 10.1016/S0025-5564(01)00049-9MathSciNetView ArticleGoogle Scholar - Zhao T, Kuang Y, Smith HL: Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems.
*Nonlinear Anal., Theory Methods Appl.*1997, 28(8):1373-1394. 10.1016/0362-546X(95)00230-SMathSciNetView ArticleGoogle Scholar - Xiong XS, Zhang ZQ: Existence and global attractivity of a periodic solution for a prey-predator model with sex-structure.
*Appl. Math. Comput.*2007, 190(2):1213-1224. 10.1016/j.amc.2007.02.005MathSciNetView ArticleGoogle Scholar - Li BW, Xiong XS: Existence and global attractivity of periodic solution for a discrete prey-predator model with sex structure.
*Nonlinear Anal., Real World Appl.*2010, 11(3):1986-2000. 10.1016/j.nonrwa.2009.05.001MathSciNetView ArticleGoogle Scholar - Pei YZ, Yang Y, Li CG,
*et al*.: Pest management of a prey-predator model with sexual favoritism.*Math. Med. Biol.*2009, 26(2):97-115. 10.1093/imammb/dqn025View ArticleGoogle Scholar - Kuznetsov YA:
*Elements of Applied Bifurcation Theory*. 2nd edition. Springer, New York; 1998.Google Scholar - Hassard BD, Kazarinoff ND, Wan YH:
*Theory and Applications of Hopf Bifurcation*. Cambridge University Press, Cambridge; 1981.Google Scholar - Liu WM: Criterion of Hopf bifurcation without using eigenvalues.
*J. Math. Anal. Appl.*1994, 182(1):250-255. 10.1006/jmaa.1994.1079MathSciNetView ArticleGoogle Scholar - Hale JK:
*Theory of Functional Differential Equations*. Springer, New York; 1997.Google Scholar - Freedman HI, Rao VSH: The trade-off between mutual interference and time lags in predator-prey systems.
*Bull. Math. Biol.*1983, 45(6):991-1004.MathSciNetView ArticleGoogle Scholar - Erbe LH, Freedman HI, Rao VSH: Three-species food chain models with mutual interference and time delays.
*Math. Biosci.*1986, 80(1):57-80. 10.1016/0025-5564(86)90067-2MathSciNetView ArticleGoogle Scholar - Zhao YH, Yu XH, Wang L: Bifurcation and control in an inertial two-neuron system with time delays.
*Int. J. Bifurc. Chaos*2012., 22(2): Article ID 1250036Google Scholar

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