Hopf and Bogdanov–Takens Bifurcations of a Delayed Bazykin Model

Abstract

In this work, the Hopf and Bogdanov–Takens bifurcations of a delayed Bazykin predator–prey model with predator intraspecific interactions and ratio-dependent functional response are studied. Sufficient conditions for the existence of Hopf bifurcation are established. In the Bogdanov–Takens bifurcation, the dynamics near the nonhyperbolic equilibrium can be reduced to the study of the dynamics of the corresponding normal form restricted to the associated two-dimensional center manifold. Some numerical simulations, such as the distribution of eigenvalues, the bifurcation diagrams of Hopf and Bogdanov–Takens bifurcations and phase portraits, are given to illustrate the theoretical criteria. The theoretical and numerical simulation results illustrate that there is supercritical Hopf bifurcation and subcritical Bogdanov–Takens bifurcation in this model. We show that, the dynamics of prey and predator are very sensitive to parameters and delay perturbations which can play a great role in controlling and regulating the number of biological populations.

Appendix: Stability of \(E_0\) and \(E_1\)

In this Appendix, we will discuss the stability of the equilibria $E_0$ and $E_1$.

Lemma A.1

The equilibrium $E_0(0,0)$ is unstable.

Proof

We can evaluate (2.3) directly at $E_0(0,0)$

$$\begin{aligned} J\mid _{(0,0)} = \begin{pmatrix} 1 &{} 0\\ 0 &{} -\alpha \end{pmatrix}. \end{aligned}$$

The eigenvalues are $\lambda _1=1>0$, $\lambda _2= -\alpha <0$. Obviously, $E_0$ is a saddle which is unstable. Furthermore, the stable manifold of $E_0$ is spanned by $\vec {v} = (0,1)$ and the unstable manifold is spanned by $\vec {w} = (1,0)$. $\square$

Remark 8

The equilibrium $E_0(0,0)$ is often called the extinction equilibrium. The fundamental importance for a biological population is its avoidance of extinction. It will threaten the model population with extinction if the extinction equilibrium is locally asymptotically stable, whereas it will open the possibility of population persistence if the extinction equilibrium is unstable [54]. Hence, the extinction equilibrium $E_0(0,0)$ is a saddle which can guarantee persistence of the species.

Lemma A.2

For the equilibrium $E_1(1,0)$,

• (I) It is a stable node if $C<1$.

• (II) It is an unstable saddle if $C>1$.

• (III) It is non-hyperbolic if $C=1$.

Proof

We can evaluate (2.3) directly at $E_1(1,0)$

$$\begin{aligned} J\mid _{(1,0)} = \begin{pmatrix} -1 &{} -Q \\ 0 &{} \alpha (C-1) \end{pmatrix}. \end{aligned}$$

The eigenvalues are $\lambda _1=-1$ and $\lambda _2= \alpha (C-1)$. Hence

(I) If $C<1$, we have $\lambda _1 <0$, $\lambda _2 <0$, then $E_1$ is stable.

(II) If $C>1$, we can obtain $\lambda _2 >0$, then $E_1$ is a saddle which is unstable. The stable manifold of the saddle is spanned by $\vec {v} = (1,0)$. Hence, all trajectories flow (x(t), y(t)) with initial values located in $\{(x,y)|0\le x <1,y=0\}$ will toward equilibrium $E_1$.

(III) If $C=1$, we can obtain $\lambda _1 <0$, $\lambda _2 =0$, then $E_1$ is non-hyperbolic.

To further determine the nature of non-hyperbolic equilibrium, we will analyze the stability of $E_1$ of the delay-free model when $C=1$ since time delay cannot affect the stability of $E_1$. When $C=1$, the delay-free model is

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{x} = x(1-x-\frac{Qy}{x+y}),\\ \dot{y} = \alpha y\left( \frac{x}{x+y}-1-\beta y\right) . \end{array}\right. } \end{aligned}$$

(A.1)

First, we transfer the equilibrium $E_1$ of model (A.1) to the origin by the transformation $(u_1,v_1) = (x-1,y)$ and expand it in a Taylor series

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{u}_1 = -u_1 - Qv_1 - u_1^2 + Qv_1^2 - Qu_1v_1^2 - Qv_1^3 + 2Qu_1v_1^3 + Qu_1^2v_1^2 + Qv_1^4 + h_1(u_1,v_1),\\ \dot{v}_1 = - \alpha (1+\beta )v_1^2 + \alpha u_1v_1^2 + \alpha v_1^3 - 2\alpha u_1 v_1^3 - \alpha u_1^2v_1^2 - \alpha v_1^4 + h_2(u_1,v_1), \end{array}\right. } \end{aligned}$$

(A.2)

where $h_1(u_1,v_1)$ and $h_2(u_1,v_1)$ are power series in $(u_1,v_1)$ with terms $u_1^iv_1^j$ satisfying $i+j\ge 5$.

Let

$$\begin{aligned} T = \begin{pmatrix} 1 &{} 1 \\ 0 &{} -\frac{1}{Q} \end{pmatrix} \end{aligned}$$

be the matrix that transforms the matrix J into Jordan canonical form. Then, under the transformation

$$\begin{aligned} \begin{pmatrix} u_1 \\ v_1 \end{pmatrix} = T \begin{pmatrix} u_2 \\ v_2 \end{pmatrix}, \end{aligned}$$

map (A.2) becomes

$$\begin{aligned} \left\{ \begin{array}{r@{~}l} \dot{u}_2 = &{}-u_2 - u_2^2 - 2u_2v_2 - \frac{\alpha (\beta + 1)+Q-1}{Q}v_2^2 + \frac{\alpha -1}{Q}u_2v_2^2 + \frac{(\alpha -1)(Q-1)}{Q^2}v_2^3 - \frac{\alpha - 1}{Q}u_2^2v_2^2\\ &{} -\frac{2(\alpha -1)(Q-1)}{Q^2}u_2v_2^3-\frac{(\alpha - 1)(Q-1)^2}{Q^3}v_2^4 + h_3(u_2,v_2), \\ \dot{v}_2 = &{} \frac{\alpha (\beta + 1)}{Q}v_2^2 - \frac{\alpha }{Q}u_2v_2^2 - \frac{\alpha (Q-1)}{Q^2}v_2^3 + \frac{\alpha }{Q}u_2^2v_2^2 + \frac{2\alpha (Q-1)}{Q^2}u_2v_2^3 + \frac{\alpha (Q-1)^2}{Q^3}v_2^4 + h_4(u_2,v_2), \end{array} \right. \end{aligned}$$

(A.3)

where $h_3(u_2,v_2)$ and $h_4(u_2,v_2)$ are power series in $(u_2,v_2)$ with terms $u_2^iv_2^j$ satisfying $i+j\ge 5$.

Letting $\tau = -t$, $x = v_2$, $y = u_2$ and rewriting $\tau$ as t, then model (A.3) becomes

$$\begin{aligned} \left\{ \begin{array}{r@{~}l} \dot{x} = &{} -\frac{\alpha (\beta + 1)}{Q}x^2 + \frac{\alpha }{Q}x^2 y + \frac{\alpha (Q-1)}{Q^2}x^3 - \frac{\alpha }{Q}x^2y^2 - \frac{2\alpha (Q-1)}{Q^2}x^3 y - \frac{\alpha (Q-1)^2}{Q^3}x^4 + h_5(x,y),\\ \dot{y} = &{}y + y^2 + 2xy + \frac{\alpha (\beta + 1)+Q-1}{Q}x^2 - \frac{\alpha -1}{Q}x^2 y - \frac{(\alpha -1)(Q-1)}{Q^2}x^3 + \frac{\alpha - 1}{Q}x^2y^2\\ &{} + \frac{2(\alpha -1)(Q-1)}{Q^2}x^3 y + \frac{(\alpha - 1)(Q-1)^2}{Q^3}x^4 + h_6(x,y), \end{array} \right. \end{aligned}$$

(A.4)

where $h_5(x,y) = - h_4(y,x)$, $h_6(x,y) = -h_3(y,x)$.

Using the notations of Theorem 7.1 in Chapter 2 in [55], we obtain $m=2$ and $a_m = -\frac{\alpha (\beta + 1)}{Q} < 0$, then the origin of (A.4) is a saddle node. That is, a neighborhood of the origin of (A.4) is divided into two parts by two separatrices that tend to the origin along the positive and negative of y-axes. One part is a parabolic sector, and the other part consists of two hyperbolic sectors. Since $a_m<0$, then the parabolic sector is on the left halfplane. Combining all of the above transformations, we know that the parabolic sector for $E_1$ of (2.1) is on the upper halfplane of the x-axis and the flow will move toward $E_1$ when the initial point is selected in the region $\Omega = \{(x,y)|0\le x \le 1,y>0\}$. Hence, $E_1$ is a non-hyperbolic saddle-node which includes a stable parabolic sector lies in the domain $\Omega$, see Fig. 8(a). $\square$

Remark 9

In fact, we know that model (2.1) will undergo a static bifurcation if the eigenvalue passes through the imaginary axis along the real axis. When $C=1$, $\lambda = 0$ is one of the eigenvalues of $E_1$. There is a transcritical bifurcation at the bifurcation point $C=1$. Here we only give the result of the numerical continuation and omit the process of deriving the normal form of transcritical bifurcation. The corresponding bifurcation diagram is shown in Fig. 8b. Since we need $0\le x \le 1$, the red dotted line in Fig. 8b only represents unstable equilibria in the mathematical sense, while the red solid line represents biologically unstable equilibria. The blue line stands for stable equilibria.

Remark 10

The boundary equilibrium $E_1(1,0)$ is called the predator extinction equilibrium. Under certain condition(such as $C<1$, see Fig. 8b), the predator extinction equilibrium is stable which means that the prey population persists while the predator population becomes extinct. Unless an apex predator invade an ecosystem with no predators, we do not want predators to go extinct. That is, in generally, such stability should be avoided for the sake of ecosystem persistence. On the other hand, from the proof of Lemma A.2 and Fig. 8b we can see that when $C > 1$, $E_1$ is an unstable saddle. The biological significance is that when the energy conversion efficiency c is greater than the death rate of predators $\mu _0$(see the expression of C), predators can avoid extinction.

Fig. 8

a $E_1(1,0)$ is a non-hyperbolic saddle-node which includes a stable parabolic sector lies in the domain $\Omega$ when $Q = 1.3$, $\alpha = 0.8$, $\beta = 1.3$, $\tau = 5$ and $C = 1$. The parabolic sector for $E_1$ of (2.1) is on the upper halfplane of the x-axis. b The transcritical bifurcation diagram with C variation and other parameters are the same as (a)

As can be seen from the above, the stability of $E_0$ and $E_1$ is also independent of the time delay.