BOGDANOV-TAKENS BIFURCATION OF CODIMENSION 3 IN A PREDATOR-PREY MODEL WITH CONSTANT-YIELD PREDATOR HARVESTING

Recently, we (J. Huang, Y. Gong and S. Ruan, Discrete Contin. Dynam. Syst. B 18 (2013), 2101-2121) showed that a Leslie-Gower type predator-prey model with constant-yield predator harvesting has a BogdanovTakens singularity (cusp) of codimension 3 for some parameter values. In this paper, we prove analytically that the model undergoes Bogdanov-Takens bifurcation (cusp case) of codimension 3. To confirm the theoretical analysis and results, we also perform numerical simulations for various bifurcation scenarios, including the existence of two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1.

In order to investigate the interaction between krill (prey) and whale (predator) populations in the Southern Ocean, May et al. [21] proposed the following model subject to various harvesting regimes: where x(t) > 0 and y(t) ≥ 0 represent the population densities of the prey and predators at time t ≥ 0, respectively; r 1 and K describe the intrinsic growth rate and carrying capacity of the prey in the absence of predators, respectively; a is the maximum value at which per capita reduction rate of the prey x can attain; r 2 is the intrinsic growth rate of predators; bx takes on the role of a prey-dependent carrying capacity for predators and b is a measure of the quality of the food for predators.H 1 and H 2 describe the effect of harvesting on the prey and predators, respectively.
(a) When H 1 = H 2 = 0, that is, there is no harvesting, system (1) becomes the so-called Leslie-Gower type predator-prey model which has been studied extensively, for example, Hsu and Huang [14].In particular, they showed that the unique positive equilibrium of system (2) is globally asymptotically stable under all biologically admissible parameters.(b) When H 1 = h 1 and H 2 = 0, where h 1 is a positive constant, that is, there is constant harvesting on the prey only, Zhu and Lan [26] and Gong and Huang [13] considered system (1) when only the prey population is harvested at a constant-yield rate They obtained various bifurcations including saddle-node bifurcation, supercritical and subcritical Hopf bifurcations of codimension 1, and repelling Bogdanov-Takens bifurcation of codimension 2. (c) When H 1 = 0 and H 2 = h 2 , where h 2 is a positive constant, that is, there is constant harvesting on the predators only, we (Huang, Gong and Ruan [17]) considered system (1) when only the predator population is harvested at a constantyield rate By the following scaling here are positive constants.The effect of constant-yield predator harvesting on system (5) in the biological-feasible region Ω = {(x, y) : x > 0, y ≥ 0} was studied.The saddle-node bifurcation, repelling and attracting Bogdanov-Takens bifurcations of codimension 2, supercritical and subcritical Hopf bifurcations, and degenerate Hopf bifurcation are shown in model (5) as the values of parameters vary.In particular, it was shown that the model has a Bogdanov-Takens singularity (cusp) of codimension 3.However, the existence of Bogdanov-Takens bifurcation (cusp case) of codimension 3 has not been proved analytically, which is the subject of this paper.
This paper is organized as follows.In section 2, we prove analytically the existence of Bogdanov-Takens bifurcation (cusp case) of codimension 3 for model (5) and describe the bifurcation diagram and bifurcation phenomena.Numerical simulations of various bifurcation cases, including the existence of two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1, are also presented in section 3 to confirm the theoretical analysis.The paper ends with a brief discussion in section 4 about the effect of constant-yield predator harvesting on system (5) and a comparison about different dynamics in systems (2), (3), and (5).
Definition 2.1.The bifurcation that results from unfolding the following normal form of a cusp of codimension 3 is called a Bogdanov-Takens bifurcation (cusp case) of codimension 3. A universal unfolding of the above normal form is given by The following lemma is from Theorem 3.3 (ii) in Huang, Gong and Ruan [17].
1−h , we can find a necessary condition for the existence of higher codimension B-T bifurcations: and the degenerate equilibrium (h, 1 − h) is (2 − √ 3, −1 + √ 3) under the above conditions.
Proof.Firstly, we translate the equilibrium (2 − √ 3, −1 + √ 3) of system ( 6) when r = 0 into the origin and expand system (6) in power series around the origin.Let Then system (6) becomes where in which Next, let Then system (9) can be transformed into where Secondly, following the procedure in Li, Li and Ma [20], we use several steps (I, II, III, IV, V, VI) to transform system (10) to the versal unfolding of Bogdanov-Takens singularity (cusp) of codimension 3, that is system (7).

Bifurcation diagram and numerical simulations.
We describe the bifurcation diagram of system (18) following the bifurcation diagram given in Figure 3 of Dumortier, Roussarie and Sotomayor [10] (see also Zhu, Campbell and Wolkowicz [25], Lamontagne, Coutu and Rousseau [18]) based on a time reversal transformation.System (18) has no equilibria for γ 1 > 0. γ 1 = 0 is a saddle-node bifurcation plane in a neighborhood of the origin, crossing the plane in the direction of decreasing γ 1 , two equilibria are created: a saddle, and a node or focus.The other surfaces of bifurcation are located in the half space γ 1 < 0. The bifurcation diagram has the conical structure in R 3 starting from (γ 1 , γ 2 , γ 3 ) = (0, 0, 0).It can best be shown by drawing its intersection with the half sphere To see the trace of intersection clearly, we draw the projection of the trace onto the (γ 2 , γ 3 )-plane, see Figure 2. Now we summarize the bifurcation phenomena of system (18), which is equivalent to the original system (5).There are three bifurcation curves on S as shown in Figure 2: C : homoclinic bifurcation curve; H : Hopf bifurcation curve; L : saddle-node bifurcation curve of limit cycles.
The curve L is tangent to H at a point h 2 and tangent to C at a point c 2 .The curves H and C have first order contact with the boundary of S at the points b 1 and b 2 .In the neighborhood of b 1 and b 2 , system ( 18) is an unfolding of the cusp singularity of codimension 2. (d) For parameter values in the triangle dh 2 c 2 , there exist exactly two limit cycles: the inner one is unstable and the outer one is stable.These two limit cycles coalesce in a generic way in a saddle-node bifurcation of limit cycles when the curve L is crossed from right to left.On the arc L itself, there exists a unique semistable limit cycle.
In the following, we give some numerical simulations for system (6) to confirm the existence of Bogdanov-Takens bifurcation (cusp case) of codimension 3.In Figure 3, we fix r 1 = 0, r 3 = −0.01.An unstable hyperbolic focus A for r 2 = −0.012 is shown in Figure 3(a); when r 2 increases to r 2 = −0.011,an unstable limit cycle arrounding a stable hyperbolic focus A appears by subcritical Hopf bifurcation (see Figure 3(b)); when r 2 = −0.009999, the coexistence of a stable homoclinic loop and an unstable limit cycle is shown in Figure 3(c), the homoclinic orbit breaks for larger r 2 = −0.0095(see Figure 3(d)).Comparing with Figure 3(b) and (d), we can see that the relative locations of the stable manifold and unstable manifold for the saddle B are reversed, which implies the occurence of a homoclinic bifurcation when r 2 is between r 2 = −0.011and r 2 = −0.0095.In Figure 4, we fix r 1 = −0.1541,r 2 = −0.0234.A stable hyperbolic focus A for r 3 = −0.081 is shown in Figure 4(a); when r 3 increases to r 3 = −0.0799, a stable limit cycle arrounding an unstable hyperbolic focus A appears by supercritical Hopf bifurcation (see Figure 4(b)); when r 3 = −0.07443, a stable homoclinic loop is shown in Figure 4(c), the homoclinic orbit breaks for larger r 3 = −0.073(see Figure 4(d)).Figure 4(b) and (d) show that the relative locations of the stable manifold and unstable manifold for the saddle B are reversed, which implies the occurence of homoclinic bifurcation when r 3 is between r 3 = −0.0799and r 3 = −0.073.parameters (Hsu and Huang [14]), while model (3) with only constant-yield prey harvesting has at most two positive equilibria and exhibits Hopf bifurcation of codimension 1 (Zhu and Lan [26]) and Bogdanov-Takens bifurcation of codimension 2 (Gong and Huang [13]).Combining the results in Huang, Gong and Ruan [17] and in this paper, we can see that the model ( 5) with constant-yield predator harvesting has a Bogdanov-Takens singularity (cusp) of codimension 3 or a weak focus of multiplicity two for some parameter values, respectively, and exhibits saddle-node bifurcation, repelling and attracting Bogdanov-Takens bifurcations, supercritical and subcritical Hopf bifurcations, degenerate Hopf bifurcation, and Bogdanov-Takens (cusp) bifurcation of codimension 3 as the values of parameters vary.Thus the constant-yield predator harvesting in system (5) can cause more complex dynamical behaviors and bifurcation phenomena compared with the unharvested system (2) or system (3) with only constant-yield prey harvesting.In Huang, Gong and Ruan [17], we have shown that the constant-yield predator harvesting h can affect the number and type of equilibria, and the type of bifurcations of the model (5) (see Lemma 2.1 and Theorems 2.2, 2.3, 3.3 and 3.4 in [17]), from Figure 4 in this paper we can also see that the dynamics of the model (5) change dramatically as h changes even slightly.Therefore, our results demonstrate that the dynamical behaviors of model ( 5) are sensitive to the constant-yield predator harvesting and this suggests careful management of resource and harvesting policies in the applied conservation and renewable resource contexts.
Recall that in the original model (1) proposed by May et al. [21], both the prey and predators are subject to harvesting.It will be very interesting (and challenging) to investigate the bifurcations in the model with constant-yield harvesting on both the prey and predators (Beddington and Cooke [1]).

Figure 3 .
Figure 3.The coexistence of a stable homoclinic loop and an unstable limit cycle in system (6) with r 1 = 0, r 3 = −0.01.(a) An unstable hyperbolic focus A for r 2 = −0.012;(b) An unstable limit cycle arrounding a stable focus A for r 2 = −0.011;(c) The coexistence of a stable homoclinic loop and an unstable limit cycle for r 2 = −0.009999;(d) A stable hyperbolic focus A for r 2 = −0.0095.

Figure 5 .
Figure 5.The existence of two limit cycles for system(6).