A delayed modified Leslie-Gower predator prey system with nonlinear harvesting is considered. The existence conditions that an equilibrium is Bogdanov-Takens (BT) or triple zero singularity of the system are given. By using the center manifold reduction, the normal form theory, and the formulae developed by Xu and Huang, 2008 and Qiao et al., 2010, the normal forms and the versal unfoldings for this singularity are presented. The Hopf bifurcation of the system at another interior equilibrium is analyzed by taking delay (small or large) as bifurcation parameter.

1. Introduction

For a more detailed study on the properties of the predator prey systems, the multiple bifurcations for some systems (ODE) with more interior equilibria are investigated by many authors, see [1–4] for example. To deal with this type of systems, the more difficult problem is how to obtain the normal form of the system at its degenerate equilibrium, that is, BT bifurcation.

When introducing the time delay into this type of systems, using the methods developed by [5], the authors in [6, 7] have researched the BT bifurcation of some predator prey systems (DDE). Their results show that time delay may have an effect or not on the BT bifurcation.

Recently, papers [8–10] have considered the triple zero bifurcation of some delay differential equations, depending on the parameters in the original system; some interesting bifurcation results are obtained. But we find that there are few results about the triple zero bifurcation for predator prey systems.

Summarizing the above references, we will consider the following predator prey system with Michaelis-Menten type (nonlinear) prey harvesting:

()

where x and y denote the prey and predator populations, respectively. τ represents the negative feedback of the predator’s density. For complete reason, we give the biological meaning of the parameters, one can be seeing them in [11]. r and K are intrinsic growth rate and environmental carrying capacity for the prey, respectively. a₁ is the maximum value of the per capita reduction rate of prey, n measures the extent to which the environment provides protection to prey and predator respectively, s₁ measures the growth rate of the predator species, and s₁a₂ is the maximum value of the per capita reduction rate of predator. p(x) = qEx/(k₁E + k₂x) represents Michaelis-Menten type harvesting, q is the catchability coefficient, E is the effort applied to harvest the prey species, and k₁ and k₂ are suitable constants.

The authors in [12, 13] have studied system (1) without prey harvesting p(x), respectively, in [12], the global stability and persistence of the system are investigated. In [13], by using the Hopf bifurcation theorem and taking the delay as a parameter of bifurcation for small and large cases, the existence of the bifurcated limit cycle around a boundary equilibrium or an interior equilibrium is mainly considered.

For system (1) with τ = 0, the authors in [11] have given detailed analysis about the existence of the multiple bifurcations (including BT bifurcation) depending on the parameters of the system.

For computation simplicity, we first rescale system (1).

Let

, and

; then dropping the bars we obtain

()

where α = 1/r, β = a₂/a₁, m = n/K, ρ = s₁/r, h = qE/rk₂K, and c = k₁E/k₂K.

System (2) with initial conditions is

()

where

and

are all continuous bounded functions in the interval [−τ, 0).

From [11], we know that (2) has interior equilibrium

()

where

and interior equilibrium E^* = (x^*, y^*) if h < h^*, where

, y^* = (m + x^*)/β, and Δ = (α/β + c − 1) ² − 4c(α/β + h/c − 1).

In this paper, for system (2), we will mainly consider the BT and triple zero bifurcations at and the Hopf bifurcation at E^*. It is easy to see that this system is with six parameters which will let our work become more challenging. When dealing with the BT and triple bifurcations of the delay systems, the core problem is to change delay systems as ordinary differential systems (ODEs).

The concrete organization of the paper is as follows: in Section 2, we will give the conditions under which the equilibrium is a BT singularity, and a universal unfolding will be exhibited; in Section 3, when is a triple zero, the universal unfolding will be presented, and in Section 4, some Hopf bifurcation results at E^* will be obtained.

2. Bogdanov-Takens Bifurcation

System (2) also can be written as

()

Linearizing system (5) at

yields the following linear system

()

the corresponding characteristic equation is

()

Evidently, λ = 0 is a double zero eigenvalue if

()

λ = 0 is a triple zero eigenvalue if

()

It is easy to prove that in the above two cases the rest eigenvalues all have negative real parts.

Under the conditions (4) and (8), let

, and let

; then the Taylor expansion system (5) at

()

where i, j, k ≥ 0,

, f⁽¹⁾ = τx(1 − x − αy/(m + x) − h/(c + x)), f⁽²⁾ = τy₀(1 − βy/(m + x)).

In the following, we first give the normal form of the system (10) at the singularity (0,0). Reference [6], we first rewrite system (10) as , here X(t) = (x₁(t), x₂(t)), , and ϕ = (ϕ₁, ϕ₂). By the normal form theory developed by Faria and Magalhaes [5], one can obtain the center manifold of this system at the origin which is two-dimensional and system can be reduced to an ODE in the plane.

Define A₀ to be the infinitesimal generator of system. Consider λ = {0} and let P denote the invariant space of A₀ associated with the eigenvalue λ = 0, using the formal adjoint theory in [5], the phase space C₁ can be decomposed by λ as C₁ = P ⊕ Q. Let Φ and Ψ be the bases for P and P^*, respectively, and be let them normalized such that 〈Ψ, Φ〉 = I, ,and , where Φ and Ψ are 2 × 2 matrices, where .

Next, we will find the Φ(θ) and Ψ(s) based on the techniques developed by [14].

Lemma 1 (see Xu and Huang [14].)The bases of P and their dual space P^* have the following representations:

()

where

, and

, which satisfy

(1)
,
(2)
,
(3)
,
(4)
,
(5)
,
(6)
.

By system (6), we know that

()

By Lemma 1, we have

()

where m₁ = β[2 − (τρ − 2) ²]/2τρ(τρ − 1), m₂ = (2 − τρ)/2(τρ − 1),

()

therefore,

()

Taking a similar theory in [6], system (6) in the center manifold is

()

where

()

Using (10) and (15), system (16) also can be written as

()

where

. Through a series of transformation, system (18) becomes

()

where d₁ = g₁ and d₂ = g₂. If d₁d₂ ≠ 0, we have the following theorem.

Theorem 2. Let (4), (8), and hold. Then, the equilibrium of system (5) is a BT singularity.

Next, we are interested in giving a versal unfolding for system (5) at BT singularity. Choosing h and ρ as bifurcation parameters and incorporating h + λ₁ and ρ + λ₂ to system (5), where λ₁ and λ₂ vary in a small neighborhood of (0,0), we obtain

()

Let

and

. Then, system (20) becomes

()

Then, system (21) can be decomposed as

()

where

()

Then, the normal form of system (21) at

; that is,

()

where

()

Following the normal form formula in Kuznetsov [15], system (24) can be reduced to

()

where

()

Then, system (5) exists in the following bifurcation curves in a small neighborhood of the origin in the (λ₁, λ₂) plane.

Theorem 3. Let (4), (8), and hold. System (5) admits the following bifurcations:

(i)
a saddle-node bifurcation curve ;
(ii)
a Hopf bifurcation curve H = {(λ₁, λ₂); λ₂ = 0, γ₂ < 0};
(iii)
a homoclinic bifurcation curve .

3. Triple-Zero Bifurcation

From Section 2, we know that under the conditions (4) and (9) the equilibrium of system (5) is a triple zero singularity. In the following reference, from the work of [8, 9] we will give the triple zero bifurcation at .

Let

and

; then, system (5) becomes

()

To determine a versal unfolding for the original system (28) at

, we choose τ, ρ, and h as bifurcation parameters, and let them become τ + μ₁, ρ + μ₂, and h + μ₃, respectively, where μ₁, μ₂, and μ₃ vary in a small neighborhood of (0,0, 0); then (28) can be written as

()

Next, we need to find the expressions of Φ(θ) and Ψ(s) based on the techniques developed by [9].

Lemma 4 (see Qiao et al. [9].)The bases of P and their dual space P^* have the following representations:

()

where

, and

, which satisfy

(1)
,
(2)
,
(3)
,
(4)
,
(5)
,
(6)
,
(7)
,
(8)
,
(9)
+ + .

For system (29), one can see that

()

By Lemma 4, we can obtain

()

such that 〈Ψ, Φ〉 = I,

, and

, where J is given by

Using the similar methods as used in [9], system (29) can be rewritten as

()

where μ = (μ₁, μ₂, μ₃) ∈ R³ is a parameter vector, x ∈ R². Let

()

then, we can expand

, x(t − 1), μ) = A₁μ₁x(t) + A₂μ₂x(t) + A₃μ₃x(t) + B₁μ₁x(t − 1) + B₂μ₂x(t − 1) + B₃μ₃x(t − 1) + E₁x₁(t)x(t − 1) + E₂x₂(t)x(t − 1) + F₁x₁(t)x(t) + F₂x₂(t)x(t) + G₁x₁(t − 1)x(t − 1) + G₂x₂(t − 1)x(t − 1) + h.o.t, where

()

Following the formula of Theorem 3.1 in [9], the normal form with versal unfolding of system (29) on the center manifold takes the following form:

()

where

()

Referring [8], we know that if

()

that is,

()

for the unfolding normal form (36) then there exist the following results.

Theorem 5. Let (4), (9), and (39) hold. For the parameters μ₁, μ₂, and μ₃ are sufficiently small,

(i)
system (36) undergoes a transcritical bifurcation at the origin on the curve
()
(ii)
system (36) undergoes a Hopf bifurcation at the origin on the curve
()
(iii)
system (36) undergoes a Hopf-bifurcation at the nontrivial equilibrium point on the curve
()
(iv)
system (36) undergoes a Bogdanov-Takens bifurcation at the origin on the curve
()
(v)
system (36) undergoes a zero-Hopf bifurcation at the origin on the curve
()

4. Hopf Bifurcation

The Jacobian matrix of system (2) at E^* takes the following form:

()

then, the characteristic equation is

()

When τ = 0, by Routh-Hurwitz criterion, all roots of (46) have negative real part if

()

Using the formula in Theorem 2.4 of [13] or Lemma 2.2 of [16], we have the following theorem.

Theorem 6. Let (47) hold. Then, there exist

()

such that the positive equilibrium E^* is asymptotically stable for 0 ⩽ τ < τ₀, moreover, system (2) undergoes Hopf bifurcation at E^* for τ = τ_k, where

and

In the following, using the Hopf bifurcation theorem for a retarded differential system introduced by [17], the Hopf bifurcations at E^* for small delay and large delay are presented.

Using the same methods as the ones used in [13], for small delay, let e^−λτ≃1 − λτ, together with (46), the following bifurcation results can be obtained.

Theorem 7. Let , ρ < αx^*/β(m + x^*), τ_s = A/B and . Then, there exists ϵ_s > 0 such that for each 0 ⩽ ϵ < ϵ_s, system (2) near E^* has a family or periodic solutions γ_s(ϵ) with period T_s = T_s(ϵ) for τ = τ(ϵ) such that γ_s(0) = E^*, T_s(0) = 2π/w_s, and τ(0) = τ_s.

For large delay, by [13] we have the following results.

Theorem 8. Let (47) hold. Then there exists ϵ₀ > 0 such that for each 0 ⩽ ϵ < ϵ₀ system (2) near E^* has a family or periodic solutions γ_l(ϵ) with period T_l = T_l(ϵ) for τ = τ(ϵ) such that γ_l(0) = E^*, T_l(0) = 2π/w₀ and τ(0) = τ₀.

Remark. Because the proofs of Theorems 7 and 8 are the same as the proofs of Theorems 3.4 and 3.5 in [13], we omit them here.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is supported by NSFC (11226142), the Foundation of Henan Educational Committee (2012A110012), the Foundation of Henan Normal University (2011QK04, 2012PL03), and the Scientific Research Foundation for Ph.D. of Henan Normal University (no. 1001).

References

1 Xiao D. M. and Ruan S. G., Codimension two bifurcations in a predator-prey system with group defense, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. (2001) 11, no. 8, 2123–2131, https://doi.org/10.1142/S021812740100336X, MR1859193, ZBL1091.92504.
10.1142/S021812740100336X
Web of Science® Google Scholar
2 Xiao D., Li W., and Han M., Dynamics in a ratio-dependent predator-prey model with predator harvesting, Journal of Mathematical Analysis and Applications. (2006) 324, no. 1, 14–29, https://doi.org/10.1016/j.jmaa.2005.11.048, MR2262452, ZBL1122.34035.
10.1016/j.jmaa.2005.11.048
Web of Science® Google Scholar
3 Wang L. L., Fan Y. H., and Li W. T., Multiple bifurcations in a predator-prey system with monotonic functional response, Applied Mathematics and Computation. (2006) 172, no. 2, 1103–1120, https://doi.org/10.1016/j.amc.2005.03.010, MR2201505, ZBL1102.34031.
10.1016/j.amc.2005.03.010
Web of Science® Google Scholar
4 Li Y. L. and Xiao D. M., Bifurcations of a predator-prey system of Holling and Leslie types, Chaos, Solitons & Fractals. (2007) 34, no. 2, 606–620, https://doi.org/10.1016/j.chaos.2006.03.068, MR2327437, ZBL1156.34029.
10.1016/j.chaos.2006.03.068
Web of Science® Google Scholar
5 Faria T. and Magalhaes L. T., Normal forms for retarded functional-differential equations and applications to Bogdanov-Takens singularity, Journal of Differential Equations. (1995) 122, no. 2, 201–224, https://doi.org/10.1006/jdeq.1995.1145, MR1355889, ZBL0836.34069.
10.1006/jdeq.1995.1145
Web of Science® Google Scholar
6 Xiao D. M. and Ruan S. G., Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response, Journal of Differential Equations. (2001) 176, no. 2, 494–510, https://doi.org/10.1006/jdeq.2000.3982, MR1866284.
10.1006/jdeq.2000.3982
Web of Science® Google Scholar
7 Xia J., Liu Z., Yuan R., and Ruan S., The effects of harvesting and time delay on predator-prey systems with Holling type II functional response, SIAM Journal on Applied Mathematics. (2009) 70, no. 4, 1178–1200, https://doi.org/10.1137/080728512, MR2546358, ZBL1203.34137.
10.1137/080728512
Web of Science® Google Scholar
8 Campbell S. A. and Yuan Y., Zero singularities of codimension two and three in delay differential equations, Nonlinearity. (2008) 21, no. 11, 2671–2691, https://doi.org/10.1088/0951-7715/21/11/010, MR2448236, ZBL1149.92002.
10.1088/0951-7715/21/11/010
Web of Science® Google Scholar
9 Qiao Z. Q., Liu X. B., and Zhu D. M., Bifurcation in delay differential systems with triple-zero singularity, Chinese Annals of Mathematics A. (2010) 31, no. 1, 59–70, MR2662895, https://doi.org/10.1007/s11401-008-0421-2, ZBL1224.34239.
10.1007/s11401-008-0421-2
Google Scholar
10 He X., Li C. D., and Shu Y. L., Triple-zero bifurcation in van der Pol′s oscillator with delayed feedback, Communications in Nonlinear Science and Numerical Simulation. (2012) 17, no. 12, 5229–5239, https://doi.org/10.1016/j.cnsns.2012.05.001, MR2960313, ZBL06160196.
10.1016/j.cnsns.2012.05.001
Web of Science® Google Scholar
11 Gupta R. P. and Chandra P., Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, Journal of Mathematical Analysis and Applications. (2013) 398, no. 1, 278–295, https://doi.org/10.1016/j.jmaa.2012.08.057, MR2984333, ZBL1259.34035.
10.1016/j.jmaa.2012.08.057
Web of Science® Google Scholar
12 Nindjin A. F., Aziz-Alaoui M. A., and Cadivel M., Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay, Nonlinear Analysis: Real World Applications. (2006) 7, no. 5, 1104–1118, https://doi.org/10.1016/j.nonrwa.2005.10.003, MR2260902, ZBL1104.92065.
10.1016/j.nonrwa.2005.10.003
Web of Science® Google Scholar
13 Yafia R., Adnani F. E., and Alaoui H. T., Limit cycle and numerical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Nonlinear Analysis: Real World Applications. (2008) 9, no. 5, 2055–2067, https://doi.org/10.1016/j.nonrwa.2006.12.017, MR2441766, ZBL1156.34342.
10.1016/j.nonrwa.2006.12.017
Web of Science® Google Scholar
14 Xu Y. X. and Huang M. Y., Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity, Journal of Differential Equations. (2008) 244, no. 3, 582–598, https://doi.org/10.1016/j.jde.2007.09.003, MR2384485, ZBL1154.34039.
10.1016/j.jde.2007.09.003
Web of Science® Google Scholar
15 Kuznetsov Y. A., Elements of Applied Bifurcation Theory, 1995, 112, Springer, New York, NY, USA, Applied Mathematical Sciences, MR1344214.
10.1007/978-1-4757-2421-9
Google Scholar
16 Lian F. Y. and Xu Y. T., Hopf bifurcation analysis of a predator-prey system with Holling type IV functional response and time delay, Applied Mathematics and Computation. (2009) 215, no. 4, 1484–1495, https://doi.org/10.1016/j.amc.2009.07.003, MR2571636, ZBL1187.34116.
10.1016/j.amc.2009.07.003
Web of Science® Google Scholar
17 Hale J. K., Theory of Functional Differential Equations, 1997, Springer, New York, NY, USA.
Google Scholar

Citing Literature

All articles

Bogdanov-Takens and Triple Zero Bifurcations of a Delayed Modified Leslie-Gower Predator Prey System

Abstract

1. Introduction

2. Bogdanov-Takens Bifurcation

3. Triple-Zero Bifurcation

4. Hopf Bifurcation

Conflict of Interests

Acknowledgments

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Bogdanov-Takens and Triple Zero Bifurcations of a Delayed Modified Leslie-Gower Predator Prey System

Abstract

1. Introduction

2. Bogdanov-Takens Bifurcation

3. Triple-Zero Bifurcation

4. Hopf Bifurcation

Conflict of Interests

Acknowledgments

References

Citing Literature

References

Related

Information