Hopf bifurcation, stability switches and chaos in a prey-predator system with three stage structure and two time delays

1.   Introduction

In the real world, many species have two distinctive stages---immature and mature, of life in their lives. A delayed single-specie model with two stages is introduced by Aiello and Freedman ^[1,2] in 1990. A single-specie model with stage-structured is considered by Wang and Chen ^[3] in 1997, and found that there exists a stable periodic solution in that model. The single-specie model with two stage-structured have been received much attentions and summarized by Liu et al. ^[4]. In these papers, the authors assume that the species have two different stages---immature and mature, and only the mature member can reproduce themselves. But, some species go through three different life stages---immature, mature and old. A single-specie model with delay and three different life history stages and cannibalism has investigated by Gao ^[5], and shown that there would be a stability switches for the positive equilibrium when time delays are increased from zero. A nonautonomous predator-prey system (1.1)

with three-stage-structured and time delay has considered by Yang and Shi ^[6], and the conditions for the existence of the positive periodic solution are obtained.

Time delays play an important role in population dynamics, which can cause the loss of stability of the equilibrium, bifurcate various types of periodic solutions, unbounded solutions and even chaotic solutions. Time delay is common in biodynamic systems^[7], and harmful delays can cause fluctuation(period solution) in population density, and which would make the system subject to chaotic oscillation, unstable oscillation and extinct ^{[8,9,10,11,12,13,14,15]}, even the time delay is very small.

Recently, a prey-predator model (1.2)

with three stage structure and time delay is studied in ^[16]. The conditions for the positive equilibrium occurring local and global Hopf bifurcation are obtained. And the properties (direction, stability, etc) of the local Hopf bifurcation are analyzed. Furthermore, a prey-predator system (1.3)

with three stage structure and predator density dependent delay has been considered in ^[17,18], by choosing time delay as a bifurcation parameter, the local and global Hopf bifurcation are investigated. The authors focus on the existence of global Hopf bifurcation in systems (1.2) and (1.3), by using the global Hopf bifurcating theorem for general functional differential equations which introduced by Wu ^[19]. Meanwhile, the harsh conditions for the positive equilibrium occurring local Hopf bifurcation are obtained, i.e. there are only a pair of pure imaginary roots for the characteristic equation about the positive equilibrium.

Note that, the sufficient conditions for the existence of local Hopf bifurcation of systems (1.2) and (1.3) are $C_3^1:f\eta < KE^2$, $C_3^2:f\eta > KE^2$, respectively, where $K, E, \eta, f$ are positive. $K$ is the rate of conversing prey into predator and $E$ is the predation coefficient for predator population. $\eta$ is the density dependent coefficient for prey populations, reflecting the competition effect between prey populations; and $f$ is the density dependent coefficient for predator population, reflecting the competition effect between predator populations; respectively. But, the conditions $C_3^1$ and $C_3^2$ cannot hold at the same time. Then, one of them holds for any parameter values of the system exclude the special case $f\eta = K{{E}^{2}}$, if both digestion delay and density dependent delay considered in a new model. Therefore, there would be a natural Hopf bifurcation for the system with two different time delays $\tau_1$ and $\tau_2$ without any conditions for the values of the parameters. And, how does the dynamic behavior go when $\tau_1 = \tau_2 = \tau$? Does there exist a bifurcating periodic solution, stability switches or other complex dynamic behaviors, if there exist at least a pair of pure imaginary roots for the characteristic equation about the positive equilibrium?

Motivation by aforementioned observations, we consider the following prey-predator model with three stage structure and two time delays:

where $x'_1 (t), x'_2 (t), x'_3 (t)$ are the change of density of the prey population in the three stages of immature, mature and old, and $y'(t)$ is the change of density of the predator population at time $t$, respectively. All of the parameters are positive. For prey population, $\alpha$ is the birth rate; $\gamma_1, \theta_1, b$ are the death rate of the immature, mature and old stages; $\Omega$ and $a$ are the maturity rate and ageing rate, respectively. For predator population, $d$ is the death rate; $\tau _1$ and $\tau _2$ are digestion delay ^[16] and density dependent delay ^[17,18], respectively. The delays ${{\tau }_{1}}$ and ${{\tau }_{2}}$ in system (1.4) can be regarded as a digestion time(or conversion time) and density dependent time of the predators. For ${{\tau }_{1}}$, when the predator catches the prey at time $t$, it needs ${{\tau }_{1}}$ time to convert the energy of the prey into its own energy. For ${{\tau }_{2}}$, the competition between predator populations has a time delay ${{\tau }_{2}}$, as in classical delayed Logistic equation $x'(t) = rx(t)[1- x(t- \tau)/K]$. That is to say, the change rate of the predators ${y}'(t)$ depends on the number of immature preys and of predators present at some previous time ${{x}_{1}}(t-{{\tau }_{1}})$ and $y(t-{{\tau }_{2}})$, respectively.

From the third equation of system (1.4), which is a linear nonhomogeneous equation about $x_3(t)$, then the asymptotic behavior of $x_3(t)$ is dependent on $x_2(t)$. Therefore, we only need to consider the following subsystem

where $\gamma = \gamma _1 + \Omega, \theta = \theta _1 + a$. And the initial conditions for system (1.5) are

The organization of this paper is as follows. We consider the stability of the equilibrium point and the existence of Hopf bifurcation, by choosing time delays as a bifurcation parameter in five different cases, firstly. And, in section 2, the conditions for the positive equilibrium occurring local Hopf bifurcation are obtained in each case. Secondly, in section 3, some numerical examples are given to support the theoretical results, which show that the delayed system considered has not only periodic oscillation, stability switches but also chaotic oscillation, even unbounded oscillation under some parameter sets of values. Finally, in section 4, delays induced Hopf bifurcation, stability switches, complicated dynamic behaviors of the system are analyzed in detail.

2.   Local stability analysis and Hopf bifurcation

2.1.   Local stability analysis

For system (1.5), if condition $C_1 :\alpha \Omega - \gamma \theta > 0$ holds, there're two boundary equilibrium $E_0 = (0, 0, 0), E_1 ({\hat{x}}_{1}, {\hat{x}}_{2}, 0)$; and if condition $C_2 :KEx_1^* - d > 0$ holds, a unique positive equilibrium $E_2 (x_1^*, x_2^*, y^*)$ exists, where

Let $X(t) = ({{x}_{1}}(t), {{x}_{2}}(t), y(t))$, and $\bar{E} = ({{\bar{x}}_{1}}, {{\bar{x}}_{2}}, \bar{y})$ be any arbitrary equilibrium. The linearized equation about $\bar{E}$ is

where

and the characteristic equation about it is given by

Note that, $\bar{y} = 0$ for the boundary equilibrium $E_0$ and $E_1$, then the characteristic equation about $E_0$ and $E_1$ are same as in ^[16,17]. Therefore, we obtain following lemma.

Lemma 2.1. (i) If $\gamma \theta > \alpha \Omega$ then $E_0$ is local stable. And, if $\gamma \theta < \alpha \Omega$ then $E_0$ is unstable and $E_1$ exists.

(ii) If $KE{\hat x_1} < d$ then $E_1$ is local stable. And if $KE{\hat x_1} > d$ then $E_1$ is unstable and $E_2$ exists.

2.2.   Existence of local Hopf bifurcation

From (2.2), one obtain the characteristic equation about the positive equilibrium $E_2$:

where

When $\tau_1 = \tau_2 = 0$, (2.3) becomes to

where

By Routh-Hurwits criterion, all roots of (2.4) have negative real parts, since

Meanwhile, $E_2$ is local stable. We investigate the Hopf bifurcation about ${{E}_{2}}$ in following five cases.

2.2.1.   The case $\tau_1 > 0, \tau_2\equiv 0$

The equation (2.3) is

where

Suppose $\lambda = \rm{i}\omega (\omega > 0)$ is a pure imaginary root of (2.5) and separating the real and imaginary parts, one obtain

and

That is

where

If condition $C_3^1: f\eta < KE^2$ holds, from (2.7) we know that (2.6) has at least one positive root. Without loss of generality, we assume that (2.6) has three different positive roots, denoted by $\omega _k = \sqrt {\varpi_k} (k = 1, 2, 3)$. And, one have

Thus

and the direction of $\tau _{1k}^{(n)}$ passing through the imaginary axis ^[20] when $\omega = {{\omega }_{k}}$ is determined by

Then ${\rm{sign}}\left({\Delta _{{\tau _{\rm{1}}}}^{\rm{k}}} \right) \ne {\rm{0}}$, since ${\varpi _k}(k = 1, 2, 3)$ are three distinct positive roots of (2.6). Therefore, system (1.5) undergoes a local Hopf bifurcation at ${{E}_{2}}$ when $\tau_1 = \tau _{1k}^{(n)}$, by the Hopf bifurcation theorem for functional differential equations ^[21]. Furthermore, system (1.5) undergoes a local Hopf bifurcation at ${{E}_{2}}$ and $\rm{sign}\left(\Delta _{{{\tau }_{1}}}^{1} \right) = 1$, if ${{f}_{11}} > 0$ and condition $C_3^1:f\eta < KE^2$ hold. Then, (2.6) has a unique positive root $\omega_1$, and $\tau_1 = \tau _{1}^{(n)}(n = 0, 1, 2, \cdots)$ corresponding to $\omega_1$.

Define

when ${\tau _1} \in {S_{{\tau _1}}}$, $E_2$ is local stable. Note that, if (2.6) have more than one positive roots, there would be finite stability switches when time delay $\tau_1$ passing through the critical points $\tau_1 = \tau _{1k}^{(n)}(k = 1, 2, 3;n = 0, 1, 2, \cdots)$ and $[0, {\tau _{10}}) \subseteq {S_{{\tau _1}}}$. If (2.6) has only one positive root, there is no stability switches when time delay $\tau_1$ passing through the critical points $\tau_1 = \tau _{1}^{(n)}(n = 1, 2, \cdots)$ and ${S_{{\tau _1}}} = [0, \tau _{1}^{(0)})$.

Theorem 2.1 (i) Suppose (2.6) has at least one positive roots denoted by ${{\varpi }_{k}}(1\le k\le 3)$. There exists a nonempty set ${{S}_{{{\tau }_{1}}}}$ and $[0, {{\tau }_{10}})\subseteq {{S}_{{{\tau }_{1}}}}$, when $\tau_1 \in {{S}_{{{\tau }_{1}}}}$ the positive equilibrium $E_2$ of system (1.5) is local stable. There is a Hopf bifurcation for system (1.5) at $E_2$ when $\tau_1 = \tau _{1k}^{(n)} (k = 1, 2, 3;n = 0, 1, 2, \cdots)$.

(ii) Suppose (2.6) has only one positive root denoted by ${{\varpi }_{1}}$. There exists a nonempty set ${{S}_{{{\tau }_{1}}}}$ and ${{S}_{{{\tau }_{1}}}} = [0, \tau _{1}^{(0)})$, when $\tau_1 \in {{S}_{{{\tau }_{1}}}}$ the positive equilibrium $E_2$ of system (1.5) is local stable and unstable when $\tau_1 > \tau _{1}^{(0)}$. There is a Hopf bifurcation for system (1.5) at $E_2$ when $\tau_1 = \tau _{1}^{(n)} (n = 0, 1, 2, \cdots)$.

Note 2.1 If ${{f}_{11}} > 0$ and condition $C_3^1:f\eta < KE^2$ hold, then (2.6) have only one positive root, and this is a special case of Theorem 2.1 (ii). The local and global Hopf bifurcation in this special situation have been considered in ^[16]. Meanwhile, theorem 2.1 generalizes the result about local Hopf bifurcation in ^[16].

2.2.2.   The case ${{\tau }_{1}}\equiv 0, {{\tau }_{2}} > 0$

The equation (2.3) becomes to

where

Suppose $\lambda = \rm{i}\omega (\omega > 0)$ is a pure imaginary root of (2.9), similar to the case 2.2.1, one have

where

From (2.11) we know that (2.10) has at least one positive root, if condition $C_3^2: f\eta > K{{E}^{2}}$ hold. Without loss of generality, we assume that (2.10) has three distinct positive roots, denoted by $\omega _k = \sqrt {\varpi_k} (k = 1, 2, 3)$ and we obtain

Thus

and the direction of $\tau _{2k}^{(n)}$ passing through the imaginary axis ^[20] when $\omega = {{\omega }_{k}}$ is determined by

System (1.5) undergoes a Hopf bifurcation at ${{E}_{2}}$ when ${{\tau }_{2}} = \tau _{2k}^{(n)}$ since ${\rm{sign}}\left({\Delta _{{\tau _2}}^k}\right)\neq 0$. Furthermore, if ${{f}_{21}} > 0, {{f}_{22}} > 0$ and condition $C_3^2: f\eta > K{{E}^{2}}$ hold, then (2.10) has a unique positive root $\omega_1$, and ${{\tau }_{2}} = \tau _{2}^{(n)}(n = 0, 1, 2, \cdots)$ corresponding to $\omega_1$. There is a Hopf bifurcation at ${{E}_{2}}$ since $\rm{sign}\left(\Delta _{{{\tau }_{2}}}^{1} \right) = 1$.

Define

Theorem 2.2 (i) Suppose (2.10) has at least one positive roots denoted by ${{\varpi }_{k}}(1\le k\le 3)$. There exists a nonempty set ${{S}_{{{\tau }_{2}}}}$ and $[0, {{\tau }_{20}})\subseteq {{S}_{{{\tau }_{2}}}}$, when ${{\tau }_{2}}\in {{S}_{{{\tau }_{2}}}}$ the positive equilibrium $E_2$ of system (1.5) is local stable. There is a Hopf bifurcation for system (1.5) at $E_2$ when ${{\tau }_{2}} = \tau _{2k}^{(n)}(k = 1, 2, 3;n = 0, 1, 2, \cdots)$.

(ii) Suppose (2.10) has only one positive root denoted by ${{\varpi }_{1}}$. There exists a nonempty set ${{S}_{{{\tau }_{2}}}}$ and ${{S}_{{{\tau }_{2}}}} = [0, \tau _{2}^{(0)})$, when ${{\tau }_{2}}\in {{S}_{{{\tau }_{2}}}}$ the positive equilibrium $E_2$ of (1.5) is local stable and unstable when ${{\tau }_{2}} > \tau _{2}^{(0)}$. There is a Hopf bifurcation for system (1.5) at $E_2$ when ${{\tau }_{2}} = \tau _{2}^{(n)}(n = 0, 1, 2, \cdots)$.

Note 2.2 If ${{f}_{21}} > 0, f_{22} > 0$ and condition $C_3^2:f\eta > KE^2$ hold, then (2.10) has only one positive root, and this is a special case of Theorem 2.2 (ii). The local and global Hopf bifurcation in this special situation have been considered in ^[17,18]. Meanwhile, theorem 2.2 generalizes the result about local Hopf bifurcation in ^[17].

2.2.3.   The case $\tau_1=\tau_2=\tau > 0$

The equation (2.3) is

where

Suppose $\lambda = \rm{i}\omega (\omega > 0)$ is a pure imaginary root of (2.13), similar to the case 2.2.1, we have

where

(2.14) has at least one positive root since ${{f}_{30}} < 0$. Without loss of generality, we assume that (2.14) has three different positive roots, denoted by $\omega _k = \sqrt {\varpi_k} (k = 1, 2, 3)$ and we get

Thus

and the direction of $\tau _{k}^{(n)}$ passing through the imaginary axis ^[20] when $\omega = {{\omega }_{k}}$ is determined by

System (1.5) undergoes a Hopf bifurcation at ${{E}_{2}}$ when $\tau = \tau _{k}^{(n)}$. Furthermore, if ${{f}_{31}} > 0, {{f}_{32}} > 0$ hold, then (2.14) has a unique positive root $\omega_1$, and $\tau = {{\tau }^{(n)}}(n = 0, 1, 2, \cdots)$ corresponding to $\omega_1$. There is a Hopf bifurcation at the positive equilibrium ${{E}_{2}}$ since $\rm{sign}\left(\Delta _{\tau }^{1} \right) = 1$.

Define

Theorem 2.3 (i) Suppose (2.14) has at least one positive roots denoted by ${{\varpi }_{k}}(1\le k\le 3)$. There exists a nonempty set ${{S}_{\tau }}$ and $[0, {{\tau }_{0}})\subseteq {{S}_{\tau }}$, when $\tau \in {{S}_{\tau }}$ the positive equilibrium $E_2$ of system (1.5) is local stable. There is a Hopf bifurcation for system (1.5) at $E_2$ when $\tau = \tau _{k}^{(n)}(k = 1, 2, 3;n = 0, 1, 2, \cdots)$.

(ii) Suppose (2.14) has only one positive root denoted by ${{\varpi }_{1}}$. There exists a nonempty set ${{S}_{\tau }}$ and ${{S}_{\tau }} = [0, {{\tau }^{(0)}})$, when $\tau \in {{S}_{\tau }}$ the positive equilibrium $E_2$ of system (1.5) is local stable and unstable when $\tau > {{\tau }^{(0)}}$. There is a Hopf bifurcation for system (1.5) at $E_2$ when $\tau = {{\tau }^{(n)}}(n = 0, 1, 2, \cdots)$.

Note 2.3 If ${{f}_{32}} > 0, {{f}_{31}} > 0$ hold, then (2.14) has only one positive root, and this is a special case of Theorem 2.3 (ii).

2.2.4.   The case $\tau_1 > 0$ and fixed ${\tau _2} \in {S_{{\tau _2}}}$

The characteristic equation about ${{E}_{2}}$ becomes to

Suppose $\lambda = \rm{i}\omega (\omega > 0)$ is a pure imaginary root of (2.16), similar to the case 2.2.1, one have

where

And

where

Assumed that condition $C_3^1: f\eta < KE^2$ holds, then

and ${{F}_{{{\tau }_{1}}({{\tau }_{2}})}}(+\infty) = +\infty$. Therefore, (2.17) has at least one positive root. Without loss of generality, we assume that (2.17) has $N_1(N_1 \in \mathbb{N}^+)$ different positive roots, denoted by ${{\omega }_{k}} = \sqrt{{{\varpi }_{k}}}(k = 1, 2, \cdots, N_1)$ and we have

Thus

and the direction of $\tau _{1k}^{(n)}({{\tau }_{2}})$ passing through the imaginary axis ^[20] when $\omega = {{\omega }_{k}}$ is determined by

Then $\text{sign}\left(\Delta _{{{\tau }_{1}}({{\tau }_{2}})}^{k} \right)\ne 0$, since ${{\omega }_{k}}(k = 1, 2, \cdots, N_1)$ are $N_1$ distinct positive roots of (2.17). And, system (1.5) undergos a Hopf bifurcation at ${{E}_{2}}$ when ${{\tau }_{1}} = \tau _{1k}^{(n)}({{\tau }_{2}})$.

Define

when ${{\tau }_{1}}\in {{S}_{{{\tau }_{1}}({{\tau }_{2}})}}$ the positive equilibrium $E_2$ is local stable. Note that, if (2.17) has more than one positive root, there would be finite stability switches when time delay $\tau_1$ passing through the critical points

and $[0, {{\tau }_{10}}({{\tau }_{2}}))\subseteq {{S}_{{{\tau }_{1}}({{\tau }_{2}})}}$. If ${{f}_{4i}} > 0(i = 1, 2, \cdots, 5)$ and condition $C_3^1: f\eta < K{{E}^{2}}$ hold, (2.17) has only one positive root, there is no stability switches when time delay $\tau_1$ passing through the critical points ${{\tau }_{1}} = \tau _{1}^{(n)}({{\tau }_{2}})(n = 1, 2, \cdots)$ and ${{S}_{{{\tau }_{1}}({{\tau }_{2}})}} = [0, \tau _{1}^{(0)}({{\tau }_{2}}))$.

Theorem 2.4 (i) Suppose (2.17) has at least one positive roots denoted by ${{\omega }_{k}}(1\le k\le N_1)$. There exists a nonempty set ${{S}_{{{\tau }_{1}}({{\tau }_{2}})}}$ and $[0, {{\tau }_{10}}({{\tau }_{2}}))\subseteq {{S}_{{{\tau }_{1}}({{\tau }_{2}})}}$, when ${{\tau }_{1}}\in {{S}_{{{\tau }_{1}}({{\tau }_{2}})}}$ the positive equilibrium $E_2$ of (1.5) is local stable, system (1.5) can undergoes a Hopf bifurcation at the positive equilibrium $E_2$ when

(ii) Suppose (2.17) has only one positive root denoted by ${{\omega }_{1}}$. There exists a nonempty set ${{S}_{{{\tau }_{1}}({{\tau }_{2}})}}$ and ${{S}_{{{\tau }_{1}}({{\tau }_{2}})}} = [0, \tau _{1}^{(0)}({{\tau }_{2}}))$, when ${{\tau }_{1}}({{\tau }_{2}})\in {{S}_{{{\tau }_{1}}({{\tau }_{2}})}}$ the positive equilibrium $E_2$ of (1.5) is local stable and unstable when ${{\tau }_{1}} > \tau _{1}^{(0)}({{\tau }_{2}})$, system (1.5) can undergoes a Hopf bifurcation at the positive equilibrium $E_2$ when ${{\tau }_{1}} = \tau _{1}^{(n)}({{\tau }_{2}})(n = 0, 1, 2, \cdots)$.

Note 2.4 If ${{f}_{4i}} > 0(i = 1, 2, \cdots, 5)$ and condition $C_3^1: f\eta < K{{E}^{2}}$ hold, then (2.17) has only one positive root, and this is a special case of Theorem 2.4 (ii).

2.2.5.   The case $\tau_2 > 0$ and fixed ${{\tau }_{1}}\in {{S}_{{{\tau }_{1}}}}$

The characteristic equation about $E_2$ is given by

Suppose $\lambda = \text{i}\omega (\omega > 0)$ is a pure imaginary root of (2.20), similar to the case 2.2.1, we have

where

And

where

Assumed that condition $C_3^2: f\eta > KE^2$ hold, then

and ${{F}_{{{\tau }_{2}}({{\tau }_{1}})}}(+\infty) = +\infty$, therefore, (2.21) has at least one positive root. Without loss of generality, we assume that (2.21) has $N_2(N_2 \in \mathbb{N}^+)$ distinct positive roots, denoted by ${{\omega }_{k}} = \sqrt{{{\varpi }_{k}}}(k = 1, 2, \cdots, N_2)$ and we have

Thus

and the direction of $\tau _{2k}^{(n)}({{\tau }_{1}})$ passing through the imaginary axis ^[20] when $\omega = {{\omega }_{k}}$ is determined by

Then $\text{sign}\left(\Delta _{{{\tau }_{2}}({{\tau }_{1}})}^{k} \right)\ne 0$, since ${{\omega }_{k}}(k = 1, 2, \cdots, N_2)$ are $N_2$ distinct positive roots of (2.21). System (1.5) undergoes a Hopf bifurcation at ${{E}_{2}}$ when ${{\tau }_{2}} = \tau _{2k}^{(n)}({{\tau }_{1}})$.

Define

when ${{\tau }_{2}}\in {{S}_{{{\tau }_{2}}({{\tau }_{1}})}}$ the positive equilibrium $E_2$ is local stable. Note that, if (2.21) has more than one positive root, there would be finite stability switches when time delay $\tau_2$ passing through the critical points

and $[0, {{\tau }_{20}}({{\tau }_{1}}))\subseteq {{S}_{{{\tau }_{2}}({{\tau }_{1}})}}$. If ${{f}_{5i}} > 0(i = 1, 2, \cdots, 5)$ and condition $C_3^2: f\eta > K{{E}^{2}}$ hold, (2.21) has only one positive root, there is no stability switches when time delay $\tau_1$ passing through the critical points ${{\tau }_{2}} = \tau _{2}^{(n)}({{\tau }_{1}})(n = 1, 2, \cdots)$ and ${{S}_{{{\tau }_{2}}({{\tau }_{1}})}} = [0, \tau _{2}^{(0)}({{\tau }_{1}}))$.

Theorem 2.5 (i) Suppose (2.21) has at least one positive roots denoted by ${{\omega }_{k}}(1\le k\le N_2)$. There exists a nonempty set ${{S}_{{{\tau }_{2}}({{\tau }_{1}})}}$ and $[0, {{\tau }_{20}}({{\tau }_{1}}))\subseteq {{S}_{{{\tau }_{2}}({{\tau }_{1}})}}$, when ${{\tau }_{2}}\in {{S}_{{{\tau }_{2}}({{\tau }_{1}})}}$ the positive equilibrium $E_2$ of system (1.5) is local stable. There is a Hopf bifurcation at $E_2$ when

(ii) Suppose (2.21) has only one positive root denoted by ${{\omega }_{1}}$. There exists a nonempty set ${{S}_{{{\tau }_{2}}({{\tau }_{1}})}}$ and ${{S}_{{{\tau }_{2}}({{\tau}_{1}})}} = [0, \tau _{2}^{(0)}({{\tau }_{1}}))$, when ${{\tau }_{2}}({{\tau }_{1}})\in {{S}_{{{\tau }_{2}}({{\tau }_{1}})}}$ the positive equilibrium $E_2$ of (1.5) is local stable and unstable when ${{\tau }_{2}} > \tau _{2}^{(0)}({{\tau }_{1}})$. There is a Hopf bifurcation at $E_2$ when ${{\tau }_{2}} = \tau _{2}^{(n)}({{\tau }_{1}})(n = 0, 1, 2, \cdots)$.

Note 2.5 If ${{f}_{5i}} > 0(i = 1, 2, \cdots, 5)$ and condition $C_3^2: f\eta > K{{E}^{2}}$ hold, then (2.21) has only one positive root, and this is a special case of Theorem 2.5 (ii).

3.   Numerical simulations

3.1.   Example 1

We consider following system

where $\alpha = 2.5, {{\gamma }_{1}} = 0.15, \Omega = 0.9, \eta = 0.2, E = 1.25, {{\theta }_{1}} = 0.2, a = 0.5, K = 0.6, d = 0.1, f = 1.8, X(0) = (4.0, 5.0, 1.3)$.

In case 2.2.1, ${{\tau }_{1}} > 0, {{\tau }_{2}}\equiv 0$, from (2.6) we have $f_{12} = 24.6366, f_{11} = 84.7024, f_{10} = -5.3829$, the unique positive root $\omega = 0.2498$ and

$\text{sign}\left(\Delta _{{{\tau }_{1}}}^{1} \right) = 1$. According to Theorem 2.2.1 (ⅱ),

The positive equilibrium point $E_2$ is local stable when $\tau_1 = 8.3 < \tau_{10}$, and unstable when $\tau_1 = 8.6 > \tau_{10}$ (Figure 1). And increasing time delay $\tau_1$, the prey and predator populations can coexist with stable limit cycles when $\tau_2 = 0$ and $\tau_1 = 8.5, 8.6, 8.7, 8.8, 8.9, 9, 9.3, 10, 11$, $12, 13, 15, 20, 25, 40, 80$, respectively (Figure 2). Then, there is a global Hopf bifurcation when time delay $\tau_1$ far away from the first bifurcating critical point $\tau_{10}$ ^[16], and the amplitudes of period oscillation are increasing with time delay $\tau_1$ increased. By the fast-slow oscillations, too large time delay $\tau_1$ would make the population to be die out, since the populations are very close to zero when time delay $\tau_1$ increase to some critical value (Figure 3).

In case 2.2.2, ${{\tau }_{2}} > 0, {{\tau }_{1}}\equiv 0$, from (2.10) we have $f_{22} = 7.5640, f_{21} = -103.9972, f_{20} = 5.3829$, and there are two positive roots $\omega _1 = 7.0595, \omega _2 = 0.0520$,

$\text{sign}\left(\Delta _{{{\tau }_{2}}}^{1} \right) = 1, \text{sign}\left(\Delta _{{{\tau }_{2}}}^{2} \right) = -1.$ Note that $\tau _{21}^{(0)} < \tau _{21}^{(1)} < \tau _{22}^{(0)}$, there is no stability switches for $\tau_2$ passing through the critical points $\tau _{21}^{(n)}$ and $\tau _{22}^{(n)}$. According to Theorem 2.2.2 (ⅰ),

The positive equilibrium point $E_2$ is local stable when $\tau_2 = 0.66 < \tau_{20}$, and unstable when $\tau_2 = 0.70 > \tau_{20}$ (Figure 4). And increasing time delay $\tau_2$, the prey and predator populations can coexist with stable limit cycles when $\tau_1 = 0$ and $\tau_2 = 0.7, 0.8, 0.9, 1.0, 1.1, 1.2$, respectively (Figure 5), and the amplitudes of period oscillation are increased. And, time delay $\tau_2$ would make the population to be die out, because the populations are very close to zero and then tend to unbounded solutions as time delay $\tau_2 = 1.23$ (Figure 6).

In case 2.2.3, ${{\tau }_{1}} = {{\tau }_{2}} = \tau, f_{32} = 14.7433, f_{31} = -171.3174, f_{30} = -12.0940$, and the unique positive root $\omega = 2.7753$,

$\text{sign}\left(\Delta _{\tau }^{1} \right) = 1$. Then $\tau_{0} = 0.5017$, According to Theorem 2.2.3 (ⅰ),

The positive equilibrium point $E_2$ is local stable when $\tau = 0.48 < \tau_{0}$, and unstable when $\tau = 0.52 > \tau_{0}$ (Figure 7). And increasing time delay $\tau$, the prey and predator populations can coexist with stable limit cycles when $\tau = 0.503, 0.505, 0.508, 0.513, 0.518, 0.523$, $0.526, 0.53$, respectively (Figure 8), and the amplitudes of period oscillation are increased. And, time delay $\tau$ would make the population to be die out, because the populations are very close to zero and then tend to unbounded solution as time delay $\tau = 0.536$ (Figure 9).

We plot the stable and unstable regions with ${{\tau }_{1}}\times {{\tau }_{2}} = [0, 10]\times [0, 1.4]$ (Figure 10) by using the publicly available Matlab package Trace-DDE ^[22], which by the pseudospectral method for the computation of characteristic roots of delay differential equations introduced in ^[23,24]. From Figure 10, we see that, if one fixed $\tau_2$ about $0.55$, there would be stability switches when $\tau_1$ increasing from $0$ to $10$. Let ${{\tau }_{2}} = 0.52\in {{S}_{{{\tau }_{2}}}}$, in case 2.2.4, from the Figure 11 we see that ${{F}_{{{\tau }_{1}}({{\tau }_{2}})}} = 0$ have three positive roots

and

Note that,

then

and

When ${{\tau }_{2}} = 0.52, {{\tau }_{1}}\in {{S}_{{{\tau }_{1}}({{\tau }_{2}})}}$, the positive equilibrium point $E_2$ is local stable, where ${S}_{{{\tau }_{1}}({{\tau }_{2}})}$ composed of four an increasing intervals. There are four times stability switches when time delay $\tau_1$ crossing ${S_{{\tau _1}({\tau _2})}}$. And continuously increasing time delay $\tau_1$, the prey and predator populations coexist with period oscillation, quasi-period oscillation, even chaotic oscillation when $\tau_1 = 8, 9, 11, 14, 15, 19, 37$, and tend to unbounded oscillation for $\tau_1 = 38$ (Figure 12).

Let ${{\tau }_{1}} = 3.1\in \left(\tau _{11}^{(1)}, \tau _{12}^{(1)} \right)$ (unstable region). We investigate the effect time delay $\tau_2$ on system (3.1). The bifurcation diagrams of time delay $\tau_2$ over $[0.52, 0.68]$ show that system (3.1) has rich dynamics (Figure 13), including (1) periodic oscillating, (2) period-doubling bifurcations, and (3) chaos, and the solution tend to unbounded oscillation for $\tau_2 = 0.69$ at time $t = 210$ (Figure 14).

Furthermore, increasing $\tau_2$ from $0.52$ to $1.4$, then $\tau _{11}^{(n)}$ decreased and $\tau _{12}^{(n)}$ increased, and the stability switches disappear one by one when $\tau _{12}^{(n)} > \tau _{11}^{(n+1)}$ for $n = 2, 1, 0$ (Figure 15). We plot the stable and unstable regions (Figures 16 and 17) by choose $f = 0.6, 0.8, 1.2, 1.5, 1.9, 2.2, 2.4, 2.9$ respectively, and remained other parameters in example 1. By increasing the values of parameter $f$, the stable and unstable regions showing that $\tau _{11}^{(0)}$ increased and $\tau _{21}^{(0)}$ decreased, and the stable regions changed more and more complexity, which is a connect region from the view of topology. If we increasing the values of parameter $f$ and choose $\tau_2$ less than and closed to the first critical point $\tau _{21}^{(0)}$, then there would be more and more stability switches by increasing time delay $\tau_1$ from $0$ to $15$.

3.2.   Example 2

We consider following system

where $\alpha = 2.6, {{\gamma }_{1}} = 0.2, \Omega = 0.9, \eta = 0.3, E = 1.1, {{\theta }_{1}} = 0.15, a = 0.65, k = 0.8, d = 0.15, X(0) = (4.0, 5.0, 1.3)$. We consider the case 2.2.2 with different value of parameter $f$.

Let $f = 0.095$, from (2.10) we have ${{\omega }_{1}} = 0.3760, {{\omega }_{2}} = 0.2963$ and

then

and

Let $f = 0.11$, from (2.10) we have ${{\omega }_{1}} = 0.4037, {{\omega }_{2}} = 0.2957$ and

then

and

Let $f = 0.16$, from (2.10) we have ${{\omega }_{1}} = 0.4874, {{\omega }_{2}} = 0.2972$ and

then

Let $f = 0.35$, from (2.10) we have ${{\omega }_{1}} = 0.7658, {{\omega }_{2}} = 0.2976$ and

then

From above numerical analysis, we see that, the times of stability switches are decreased from four to one by increasing the values of parameter $f$ from $0.095$ to $0.35$; and the first critical point $\tau _{21}^{(0)}$ also decreased (Figure 18). From Figure 19, we see the stable regions changed more and more simple by increasing the values of parameter $f$, and the stable regions from four parts to three parts, and to two parts, finally to one connect region.

4.   Conclusion

We have considered a prey-predator system with three stage structure and two delays, and analyzed the stability of the equilibrium point, obtained the conditions for the positive equilibrium $E_2$ occurring Hopf bifurcation by analyzing the characteristic equation in five cases. From the numerical examples and analysis, we know that the time delays would make the system subject to period oscillation, quasi-period oscillation, chaotic oscillation, finite stability switches, even unbounded oscillation and extinct. That is to say, time delays are important factors to affect the dynamic behaviors of the system.

4.1.   Delays induced Hopf bifurcation

From the analysis in section 2, we know that $f_{30} < 0$ in (2.14) for case 2.2.3, then (2.14) has at least one positive root, and there is a natural Hopf bifurcation for system (1.5) without any conditions according to theorem 2.2.3 (ⅰ). If condition $C_3^1: f\eta < K{{E}^{2}}$ holds then $f_{10} < 0$ in (2.6) for case 2.2.1, and that (2.6) has at least one positive root, and there is a Hopf bifurcation for system (1.5) according to theorem 2.2.1 (ⅰ). Similarly, if condition $C_3^2: f\eta > K{{E}^{2}}$ holds then $f_{20} < 0$ in (2.10) for case 2.2.2, and that (2.10) has at least one positive root, there is a Hopf bifurcation for system (1.5) according to theorem 2.2.2 (ⅰ). Note that conditions $C_3^1: f\eta < K{{E}^{2}}$ and $C_3^2: f\eta > K{{E}^{2}}$ cannot hold at the same time, but one of them can hold for any parameter values of the system exclude the special case $f\eta = K{{E}^{2}}$. Therefore, there is a Hopf bifurcation for system (1.5) with only one time delay $\tau_1$ or $\tau_2$. And then, one of ${S_{{\tau _1}}}$ and ${S_{{\tau _2}}}$ is nonempty set. So, there is a natural Hopf bifurcation for system (1.5), and large time delays would make the positive point $E_2$ eventually unstable. These are harmful delays for system (1.5).

4.2.   Delays induced stability switches

From the analysis in section 2, we know that there would be finite stability switches for system (1.5) when the equation has more than one positive roots $\omega_k(k > 1)$. From example 1 in case $2.2.2$, only time delay $\tau_2$, there are two positive roots and two critical delay sequences $\tau _{21}^{(n)}$ and $\tau _{22}^{(n)}$. But, there is no stability switches since $\tau _{21}^{(0)} < \tau _{21}^{(1)} < \tau _{22}^{(0)}$. From example 1 in case $2.2.4$ fixed $\tau_2 = 0.52\in {S_{{\tau _2}}}$, there are three positive roots and three critical delay sequences $\tau _{11}^{(n)}, \tau _{12}^{(n)}$ and $\tau _{13}^{(0)}$. Note that

and

there are four times stability switches when time delay $\tau_1$ increasing from $0$ to infinity. From the stable and unstable regions in example 1 (Figures 16 and 17), we see that, there is no stability switches on $\tau_1$-axis or $\tau_2$-axis, but there are several times stability switches on $\tau_2$-axis in example 2 for some suitable parameter values (Figure 19). And, the stability regions in examples 1 and 2 are two different types in view of topology. The former is a connected region varying the parameter $f$ from $0.6$ to $2.9$, and the latter from four parts to three parts, to two parts and to one connected region varying the parameter $f$ from $0.095$ to $0.35$. That is to say, parameter $f$ would change the stability switches times for some suitable parameter values of the system.

4.3.   Delays induced complicated dynamic behaviors

The numerical simulations show that delayed system (1.5) has complicated dynamic behaviors (Figures 12, 13 and 14) when we change the time delays and far away from the first bifurcating critical time delay point, including periodic oscillating, quasi-periodic oscillating, period-doubling bifurcations, chaos, and those behaviors undiscovered if the system (1.5) has only one time delay ^[16,17,18]. That is to say, time delays are important factor to affect the complex dynamic behaviors of the system, since the positive equilibrium point $E_2$ of the system (1.5) is global asymptotically stable in the absence of time delays ^[16]. When time delay far away from the first critical point and increased, large time delays would make system (1.5) extinct (unbounded oscillation) undergoing a series of fast-slow oscillations or chaotic oscillations which make the prey and predator populations very closed to zero, and destroyed the permanence of it. And these are not found in ^[16,17,18]. All of the analysis show that the time delays would destroy the stability of the system, and induced complicated dynamic behaviors, even make the system die out.

All in all, time delays induced Hopf bifurcation, stability switches, and complicated dynamic behaviors for system (1.5), and make the system (1.5) subject to period oscillations and finite times stability switches via local Hopf bifurcation, and quasi-period oscillations, period-doubling bifurcations, chaotic oscillations and unbounded oscillations. Harmful time delays destroy the stability of the system, even make the system die out. How to control the bifurcation, unbounded oscillations and even chaos, arising from the multiple time delays system? The impulsive control strategies and the time-varying control strategies would be considered ^[25,26], which could both improve the stability of the system and control periodic and chaotic oscillations effectively. We will continue to study these problems in the future.

Acknowledgments

We are very grateful to anonymous referees for their valuable comments and suggestions.

This work was supported by the Natural Science Foundation of Guizhou Province (No. [2016]1135), the Joint Natural Science Foundation of Guizhou Province (Nos. LH[2014]7437 and LKQS[2013]01).

Conflict of interest

The author declares that there is no conflict of interest.