Multiple stability switches and Hopf bifurcation in an age structured prey-predator system: effects of maturation and cooperative hunting delay

Abstract

The classical prey-predator model assumes that prey species has no life history. But many species whose individuals have a life history with two classes, namely immature and mature. Since, immature prey can be easily captured by the predators because of their short life experiences and less activity, while predators make some extra effort to catch mature prey. So, in this paper, a mathematical model is proposed to study the complex behaviour of a delayed prey-predator system with stage in prey population. We have included a maturation delay which is the time required for immature prey to join mature class and a cooperative hunting delay which is the time taken by predator to cooperate and attack on prey. Stability analysis has been addressed with different pairs of delays. Hopf bifurcation analysis with its directions is also performed. Global stability of predator free equilibrium point has been examined analytically and shown graphically. Both maturity and cooperative hunting delay significantly influence the system dynamics and produce multi-times stability switches. The presence of cooperative hunting delay of predators reduces the maturation time of prey due to which prey becomes mature very fast. Increase in cooperative hunting rate of predators suppress mature prey populations. Numerical simulations have been performed in various ways to support our theoretical findings.

1 Introduction

Table 1 Some ecological example of age based classified prey-predator species and hunting strategy of some predators

In ecology, prey-predator interactions are a ubiquitous topic that has received a great deal of attention. But there, the stages of the species have been completely ignored. Since, the life cycle (from birth to death) of individuals of many species passes through two phases, namely, immature and mature. This type of population division is quite different from the common division like susceptible and infected species, generalist and specialist predator, etc. The birth rate of newly born species depends on the age factor of the species. Prey and predator species do not participate in the reproductive process in the first phase of their life cycle, but they do in the second phase. During each phase of the life stage, an individual evidently exhibits various kind of natural behaviours likes: consumption, competition, reproduction, cooperation, and survival behavior [1,2,3]. However, an immature predators do not reproduce, they are raised by their parents, but they may attack prey or may not attack prey. Therefore, incorporation of stage structure into mathematical ecological model is an interesting and biologically more meaningful formulation technique that provides a tremendous insightful consequences in population dynamics. Recent years have seen a significant increment in the study of stage structured models [2,3,4,5,6,7,8,9,10]. We have listed the name of some prey-predator species in Table 1 that have been studied by classifying them into age-based stages. Devi [7] investigated a predator–prey model incorporating prey refuge and stage structured on prey population. In this work, author observed that when the utilization of refuge mechanism is high enough, the mature prey population is increased which leads the predator’s extermination. Xu et al. [2] examined that when the growth rate of both immature and mature preys and the variation rate from immature to mature preys are sufficiently low, or when the death rate of both preys are large enough, the prey and predator population will be exterminated. Xu [3] reported that a large movement of the predators create a pattern formation dynamics whereas a large movement of mature and immature prey can cause to disappear of spatial patterns. Optimal harvesting of mature prey with impact of harvesting and competition were studied in [8, 9].

Cooperation among individuals of social animals is often observed and spread throughout biological systems. In the context of hunting, two or more predators use such mechanism to hunt prey togetherness to raise their fitness known as hunting cooperation [11]. Therefore, undoubtedly, it can be easily said that the group hunting rate that usually depends on both predator and prey densities increases with predator density. Sometimes, the cooperative behaviour of predator potentially lead to species extinction, but is oftenly beneficial for predator to survival perspective when they would go disappear without hunting [12,13,14]. It has recently renewed interest for both experimental and analytical studies of an ecosystem. Nazmul et al. [15] observed that the cooperative behavior of the predator induces fear in the prey species which reduces birth rate of the prey and multiple stability switches have been occurred due to delay in cooperative hunting. Earlier, Alves and Hilker [12] investigated a new functional response by adding a hunting cooperation term. After their qualitative work, a tremendous interest has been seen regarding the study of cooperative models by several investigators [11,12,13, 16, 17]. Wu and Zhao [17] investigated a diffusive cooperative model and analysed that Turing instability is experienced by the model when hunting cooperation is involved, while in the absence of cooperative hunting, Turing instability does not appear. Ye and Wu [18] noticed that increasing cooperatively hunting rate cause Hopf bifurcation where the system alters its stable state into unstable one and further unstable state to stable one. Mondal et al. [13] observed that when cooperative strength of the predator is too high but the time to capture prey is too short, the prey species is unable to survive in the ecosystem. However, if both cooperative strength of the predator and the time it takes to capture prey are high then both species are comfortable to survive. In predator population, foraging activity becomes particularly more interesting phenomenon therefore many predators adopt cooperative mechanisms for hunting. For example, in Table 1, we have listed some predator’s names that attack prey cooperatively.

A more realistic and logical assumption would be that the transformation from juvenile to mature prey, and predators cooperation for hunting do not happen instantly; rather, there is some time required for maturation and cooperation. Therefore, to capture more realistic features of the ecosystem, many researchers incorporate a well-known factor time delay in the ecological model. In general, delay depicts more complex dynamics and has capability to make the model more trustworthy, robust and realistic [19]. Pal et al. [20] reported a model involving cooperation delay among predators to capture prey and noticed that if long enough time has been spent by predators to form groups to hunt cooperatively then the system represents a chaotic behaviour. Maiti et al. [21] investigated age-structured delay model where predator feeds only mature preys and observed that per capita prey’s consumption rate declines as predator’s population inclines in the presence of gestation delay. Kundu and Maitra [22] studied a maturation delay model in which there is a cooperation between mature and juvenile prey to protect juvenile prey from the predation. They observed that density of all species decline when the mortality of juvenile prey becomes higher than mortality of mature one. An inclusion of both maturation and gestation delay stage structured model has been examined by Banerjee et al. [23]. They analysed that a small maturity time for juvenile prey cause to its extermination and if juvenile preys take sufficiently large time to become adult, all species goes to disappear when gestation delay is absent, whereas in the presence of gestation delay an oscillatory behavior has been seen due to maturation. Thus, a large body of stage-structured mathematical models with delay and/or diffusion parameters have been carried out where time delay and/or diffusions greatly enrich ecological background [2, 3, 24,25,26].

According to the above brief discussion and to the best of our knowledge, it is found that a prey-predator ecological model where stage structure on prey species with two different discrete delays cooperative hunting delay and maturity delay has not been investigated so far. Therefore, the main purpose of this study is to investigate the biological effects of cooperative hunting delay and maturation delay on the stage structured prey-predator model with different pairs of the delays. Also, our aim is to find the stability region with respect to both delay parameters and how these delays generate multi-time stability switches? How the presence of cooperative hunting delay of predators affect the maturation time of prey? Bearing this objective in mind, the mathematical model has been prepared with basic ecological assumptions and the reasons behind it in Sect. 2. Basic mathematical results are explained in Sect. 3. Steady points and linear stability analysis with and without delay have been done in Sect. 4. In Sect. 5, stability and the direction of Hopf bifurcation is analysed. Section 6 contains a brief explanation with computer simulations of the proposed model. Finally, we have briefly summarized all possible significant results of the model in Sect. 7.

2 Frameworking the model

In natural world, the life cycle of many species are categorized into two different stages, immature and mature. Taking into consideration the fact “stage structure” on prey population, we assume that U(t), V(t) and W(t) be the immature prey, mature prey and the predator’s population (in number/unit area), respectively. The birth rate of immature prey depends on the matting and population of mature prey. We consider that immature prey’s growth rate is proportional to the existing mature prey. Therefore, we take linear growth on immature prey, while mature prey subjected to a logistic manner [37]. The parameter b denotes conversion rate of immature prey into mature prey which is proportional to existing immature prey [21]. It is also speculated that there is some time lag $\tau _{1}$ (in days) required to become mature from immature one called maturity delay, $bU(t-\tau _{1})$ is the number of immature prey born at time $t-\tau _{1}$ and survive at time t (in days). Since, immature prey has less capability to escape and combat from the predation due to their unawareness and a little life experience about species interaction in their environment [38]. Therefore, predators can easily attack and eat immature prey, and become saturated which causes the predator’s prey rate to decline. Also, when individual predator consumes immature prey, there is no immature prey limitation because of their linear growth. Therefore, we use Holling type-II function to limit the consumptions of immature preys and involve the saturation effect. Further, we assume that an individual predator does not easily consume mature prey due to their hiding behaviour and avoiding predation, and may cause hunting cooperation to appear. Keeping this in mind, we assume that predators use cooperative hunting technique to capture mature prey species. That technique is modeled by Alves and Hilker [12] and proposed a functional response of the form: $\Phi (V,W)=(a+mW)V$, where $a>0$ and $m>0$ represent capture rate of per predator and cooperative behavior of predators during hunting, respectively. This functional response is a density dependent that depends on the both prey and predator density. Thus, as predator density increases, attacks on prey also increases. We also believe that predators need some time to cooperate with themselves and attack mature prey. So, we modified the functional response using time delay parameter $\tau _{2}$ (in days) as a cooperation delay in hunting and expressed as $\Phi (V,W)=(a+mW(t-\tau _{2}))V$. Under the above speculations we derive a coupled non-linear model as:

$$\begin{aligned} \begin{aligned} \frac{dU}{dt}&=rV-bU(t-\tau _{1})-d_{1}U-\frac{\alpha _{1}UW}{d+U},\\ \frac{dV}{dt}&=sV(1-\frac{V}{K})+bU(t-\tau _{1}) -d_{2}V\\&\quad -(a+mW(t-\tau _{2}))VW,\\ \frac{dW}{dt}&=\frac{\alpha _{2}UW}{d\!+\!U}\!+\!c(a\!+\!mW(t-\!\tau _{2}))VW-d_{3}W, \end{aligned} \end{aligned}$$

(2.1)

where r and s be the growth rate (per day) of U(t) and V(t), respectively, $b>0$, ($0<b<1$) is conversion rate (per day) from immature to mature, $d_{i}>0$ for $i=1,2,3$ are death rate (per day) of U(t), V(t) and W(t), respectively. $K>0$ is environmental carrying capacity (number/unit area) of V(t), $\alpha _{1}$ is intake rate (per day) of predator, $\alpha _{2}>0$, $c>0$ ($0<c<1$) are conversion efficiency (with constants rate) of immature and mature prey into the predator biomass [39], respectively, and $d>0$ is the half saturation constant (per day).

The model (2.1) is subjected with initial conditions $U(\theta )=U_{0}(\theta )\ge 0,\hspace{0.1cm} V(\theta )=V_{0}(\theta )\ge 0,\hspace{0.1cm} W(\theta )=W_{0}(\theta )\ge 0,$

where $\theta \in [-\tau ,0]$ with $\tau =\tau _{1}+\tau _{2}$, $\phi (0)=(U_{0},V_{0},W_{0})>0$, and $\phi \in [-\tau ,0]\rightarrow R^{3}$ with the norm

$||\phi ||=sup_{-\tau \le \theta \le 0}\{(|U_{0}(\theta )|,|V_{0}(\theta )|,|W_{0}(\theta )|)\}.$

3 Fundamental results: positivity and boundedness

In population dynamics, the quantity of species can not be negative. However, species reproduces new child species, but growth of any organism is not always abruptly due to limit number of foods, spaces and other biological factors. Population of any species is always limited throughout environment and may become extinct from the system. Therefore, we compute positive solution and boundedness of the system mathematically.

Without loss of generality, we have from system (2.1) as:

$$\begin{aligned} U(t)&=U_{0}\exp \Big \{\int _{0}^{t} \Big ( \frac{rV(y)}{U(y)}-\frac{bU(y-\tau _{1})}{U(y)}-d_{1}\\&\quad -\frac{\alpha _{1}W(y)}{d+U(y)}\Big )dy\Big \},\\ V(t)&=V_{0}\exp \Big \{\int _{0}^{t} \Big (s\big (1-\frac{V(y)}{K}\big ) +\frac{bU(y-\tau _{1})}{U(y)}\\&\quad -d_{2} -\big (a+mW(y-\tau _{2})\big )W(y)\Big )dy\Big \},\\ W(t)&=W_{0}\exp \Big \{\int _{0}^{t} \Big (\frac{\alpha _{2}U(y)}{d+U(y)}\\&\quad +c\big (a+mW(y-\tau _{2})\big )V(y)-d_{3}\Big )dy\Big \}. \end{aligned}$$

We conclude from above equations that any solution originating with positive initial solution $(U_{0},V_{0},W_{0})$ from an interior of $\mathbb {R}^{3}_{+}$ lies there for all future time.

Now we will show that boundedness of solution of (2.1). Let us define a function $\chi (U,V,W)$ as

$$\begin{aligned} \chi (t)=U(t)+V(t)+\frac{1}{c}W(t). \end{aligned}$$

(3.1)

Differentiating (3.1) w.r.t. t, we obtain

$$\begin{aligned}&\frac{d\chi (t)}{dt}=\frac{dU(t)}{dt}+\frac{dV(t)}{dt}+\frac{1}{c}\frac{dW(t)}{dt},\\&=rV-bU(t-\tau _{1})-d_{1}U-\frac{\alpha _{1}UW}{d+U}+sV\big (1-\frac{V}{K}\big )\\&\quad + bU(t-\tau _{1})-d_{2}V-(a+mW(t-\tau _{2}))VW\\&\quad +\frac{\alpha _{2}UW}{c(d+U)}+(a+mW(t-\tau _{2}))VW-\frac{d_{3}}{c}W,\\&\quad \le -d_{1}U-d_{2}V-\frac{d_{3}}{c}W+sV\big (1-\frac{V}{K}\big )+rV\\&\quad +\big (\frac{\alpha _{2}}{c}-\alpha _{1}\big )\frac{UW}{(d+U)},\\&\quad \le -nU-nV-\frac{n}{c}W+(n-d_{1})U+(n-d_{2})V\\&\quad +rV+sV\big (1-\frac{V}{K}\big )+\frac{(n-d_{3})}{c}W\\&\quad +\big (\frac{\alpha _{2}}{c}-\alpha _{1}\big )\frac{UW}{(d+U)},\\&\quad \le -n\chi +rV+sV\big (1-\frac{V}{K}\big )+(n-d_{1})U\\&\quad +(n-d_{2})V+\frac{(n-d_{3})}{c}W+\big (\frac{\alpha _{2}}{c}-\alpha _{1}\big )\frac{UW}{(d+U)}, \end{aligned}$$

where n is a positive real number. Assuming that ${\textbf {(H1)}}:$ $n=\min \{d_{1},d_{2},d_{3}\}$, ${\textbf {(H2)}}:$

$$\begin{aligned} \alpha _{2}<c\alpha _{1}, \end{aligned}$$

and ${\textbf {(H3)}}:$

$$\begin{aligned} \overline{M}=\max \big \{rV+sV\big (1-\frac{V}{K}\big )\big \}=\frac{K(r+s)^{2}}{4s} \end{aligned}$$

at

$$\begin{aligned} V=\frac{K(r+s)}{4s} \end{aligned}$$

, then we have from the last equation

$$\begin{aligned} \frac{d\chi }{dt}+n\chi \hspace{0.1cm}&\le \hspace{0.1cm}\overline{M}. \end{aligned}$$

It implies that

$$\begin{aligned}&0<\chi (U(t),V(t),W(t))\le \frac{\overline{M}}{n}(1-e^{-t})\\&\qquad +\frac{1}{n}\chi (U_{0},V_{0},W_{0})e^{-t}, \end{aligned}$$

if $t\rightarrow \infty$, then

$$\begin{aligned}{} & {} 0\hspace{0.1cm}<\hspace{0.1cm}\chi \hspace{0.1cm}\\{} & {} \qquad \le \hspace{0.1cm}\frac{\overline{M}}{n}. \end{aligned}$$

Hence, we can say that all solutions of (2.1) are bounded if ${\textbf {(H1)}}$, ${\textbf {(H2)}}$ and ${\textbf {(H3)}}$ satisfies. $\square$

4 Equilibrium points and stability analysis

This section describes all possible feasible equilibrium points of the system (2.1) when $\tau _{1}=\tau _{2}=0$, and discuss their dynamics. There are three steady points like: trivial equilibrium $E_{1}(0,0,0)$ exists always; predator free equilibrium $E_{2}(U_{2*},V_{2*},0)$, where $U_{2*}=rV_{2*}/(b+d_{1})$, $V_{2*}=K\big (s(b+d_{1})+br-d_{2}\big )/s$, and if $K\big (s(b+d_{1})+br\big )>d_{2}$, then $V_{2*}$ exists, and it is analysed that $\lim _{s\rightarrow \infty }V_{2*}(s)=K(b+d_{1})$, i.e. when mature prey’s growth becomes high, their population size goes towards $K(b+d_{1})$ and immature prey’s populations becomes $U_{2*}=rK$. Here, we analysed that if $E_{2}$ exists, then it is unique, therefore we can say that $E_{2}$ will be globally stable if and only if it is stable. Further, a coexistence equilibrium point $E_{3}(U^{*},V^{*},W^{*})$ which is positive, can be obtained by solving the system of equation:

$$\begin{aligned} \begin{aligned}&rV^{*}-bU^{*}-d_{1}U^{*}-\frac{\alpha _{1}U^{*}W^{*}}{d+U^{*}}=0,\\&\qquad sV^{*}\big (1-\frac{V^{*}}{K}\big )+bU^{*}-d_{2}V^{*}\\&\qquad -(a+mW^{*})V^{*}W^{*}=0,\\&\qquad \frac{\alpha _{2}U^{*}W^{*}}{d+U^{*}}+c(a+mW^{*})V^{*}W^{*}-d_{3}W^{*}=0, \end{aligned} \end{aligned}$$

(4.1)

Throughout this paper we will only study the dynamical behavior of $E_{3}(U^{*},V^{*},W^{*})$.

4.1 Stability analysis when $\tau _{1}=\tau _{2}=0$

Linear stability means the system exhibits stable dynamics over a small perturbation about equilibrium. A stable behavior is observed in small interval, so a linear stability can also be called a local stability. Here, we compute a variational matrix of (2.1) about $E_{3}$ as

$$\begin{aligned} J|_{E_{3}}= \begin{pmatrix} j_{11}&{}j_{12}&{}j_{13}\\ j_{21}&{}j_{22}&{}j_{23}\\ j_{31}&{}j_{32}&{}j_{33} \end{pmatrix}, \end{aligned}$$

where entries are defined below

$$\begin{aligned} \displaystyle j_{11}=-b-d_{1}-\frac{d\alpha _{1}W^{*}}{(d+U^{*})^{2}}, \end{aligned}$$

$j_{12}=r$,

$$\begin{aligned} j_{13}=-\frac{\alpha _{1}U^{*}}{d+U^{*}}, \end{aligned}$$

$j_{21}=b$,

$$\begin{aligned} j_{22}=s\left( 1-\frac{2V^{*}}{K}\right) -d_{2}-\left( a+mW^{*}\right) W^{*}, \end{aligned}$$

$j_{23}=-(a+2mW^{*})V^{*}$,

$$\begin{aligned} j_{31}=\frac{d\alpha _{2}W^{*}}{(d+U^{*})^{2}}, \end{aligned}$$

$j_{32}=c(a+mW^{*})W^{*}$,

$j_{33}=cmV^{*}W^{*}$.

The characteristic equation of matrix $J|_{E_{3}}$ is

$$\begin{aligned} \lambda ^{3}+J_{1}\lambda ^{2}+J_{2}\lambda +J_{3}=0, \end{aligned}$$

(4.2)

where

$J_{1}=-j_{11}-j_{22}-j_{33}$,

$J_{2}=j_{22}j_{33}-j_{23}j_{32}+j_{11}j_{33}-j_{13}j_{31}+j_{11}j_{22}-j_{12}j_{21}$,

$J_{3}=-j_{11}(j_{22}j_{33}-j_{23}j_{32})+j_{12}(j_{21}j_{33}-j_{31}j_{23})-j_{13}(j_{21}j_{32}-j_{22}j_{31})$.

All roots of (4.2) will be purely negative real number, or will have negative real parts if $J_{1},J_{3}>0$ and $J_{1}J_{2}-J_{3}>0$. Thus, in absence of delay, the system (2.1) is asymptotically stable around ${E_{3}}$ iff $J_{1},J_{3}>0$ and $J_{1}J_{2}-J_{3}>0$.

4.2 Stability analysis when $\tau _{1},\tau _{2}\ne 0$

The model (2.1) can be redesigned as

$$\begin{aligned} \dot{X}(t)=F\big (X(t),X(t-\tau _{1}),X(t-\tau _{2})\big ), \end{aligned}$$

(4.3)

where

$X(t)=(U(t),V(t),W(t))^{T}$,

$X(t-\tau _{i})=(U(t-\tau _{i}),V(t-\tau _{i}),W(t-\tau _{i}))^{T}$, $i=1,2.$

Let us we perturb the system a small unit around $E_{3}$ such that $U(t)=U^{*}+U'$, $V(t)=V^{*}+V'$, $W(t)=W^{*}+W'$, then linearized system of original system (2.1) can be formulated by

$$\begin{aligned} \dot{Y}(t)=AY(t)+BY(t-\tau _{1})+CY(t-\tau _{2}), \end{aligned}$$

(4.4)

where

$$\begin{aligned}{} & {} A=\frac{\partial F}{\partial X(t)}\Big |_{E_{3}},\quad B=\frac{\partial F}{\partial X(t-\tau _{1})}\Big |_{E_{3}},\\{} & {} C=\frac{\partial F}{\partial X(t-\tau _{2})}\Big |_{E_{3}}, \end{aligned}$$

$Y(t)=(U'(t),V'(t),W'(t))^{T}.$

We have computed the variational matrix of (2.1) at $E_{3}$, given by

$$\begin{aligned} J_{\tau }=A+Be^{-\lambda \tau _{1}}+Ce^{-\lambda \tau _{2}}, \end{aligned}$$

(4.5)

Straight forward calculation provides,

$$\begin{aligned} J_{\tau }=\begin{pmatrix} a_{1}-be^{-\lambda \tau _{1}} &{} j_{12} &{} j_{13}\\ be^{-\lambda \tau _{1}} &{} j_{22} &{} a_{2}+a_{3}e^{-\lambda \tau _{2}}\\ j_{31} &{} j_{32} &{} j_{33}e^{-\lambda \tau _{2}} \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned} a_{1}=-d_{1}-\frac{d\alpha _{1}W^{*}}{(d+U^{*})^{2}}, \end{aligned}$$

$a_{2}=-(a+mW^{*})V^{*}$, $a_{3}=-mV^{*}W^{*}$, and its characteristic equation is calculated as

$$\begin{aligned}&\lambda ^{3}+A_{1}\lambda ^{2}+A_{2}\lambda +A_{3}+(A_{4}\lambda ^{2}+A_{5}\lambda +A_{6})e^{-\lambda \tau _{1}}\nonumber \\&\qquad +(A_{7}\lambda ^{2}+A_{8}\lambda +A_{9})e^{-\lambda \tau _{2}}\nonumber \\&\qquad +(A_{10}\lambda +A_{11})e^{-\lambda (\tau _{1}+\tau _{2})}=0, \end{aligned}$$

(4.6)

where $A_{1}=-(a_{1}+j_{22})$,

$A_{2}=-a_{2}j_{32}-j_{13}j_{31}+a_{1}j_{22}$,

$A_{3}=a_{1}a_{2}j_{32}-a_{2}j_{12}j_{31}+j_{13}j_{22}j_{31}$,

$A_{4}=b$,

$A_{5}=-b(j_{22}+j_{12})$,

$A_{6}=-a_{2}bj_{32}-bj_{13}j_{32}$,

$A_{7}=-j_{33}$,

$A_{8}=j_{22}j_{33}-a_{3}j_{32}+a_{1}j_{33}$,

$A_{9}=-a_{1}j_{22}j_{33}+a_{1}a_{3}j_{32}-a_{3}j_{12}j_{31}$,

$A_{10}=-bj_{33}$,

$A_{11}=bj_{22}j_{33}-ba_{3}j_{32}+bj_{12}j_{33}$.

We have the following cases to analyse the system (2.1):

Case I: When $\tau _{1}=\tau _{2}=0$, the Eq. (4.6) turns into Eq. (4.6) exactly, of which analyzation has been done earlier in Sect. (4.1). Here, we have found $J_{1}=A_{1}+A_{4}+A_{7}$, $J_{2}=A_{2}+A_{5}+A_{8}+A_{10}$, $J_{3}=A_{3}+A_{6}+A_{9}+A_{11}$.

Case II: When $\tau _{1}=0,\tau _{2}>0$, the Eq. (4.6) is transformed into this one,

$$\begin{aligned} \lambda ^{3}+\beta _{1}\lambda ^{2}\!+\!\beta _{2}\lambda \!+\!\beta _{3}\!+\!\big (\beta _{4}\lambda ^{2}\!+\!\beta _{5}\lambda \!+\!\beta _{6}\big )e^{-\lambda \tau _{2}}\!=\!0, \end{aligned}$$

(4.7)

where $\beta _{1}=A_{1}+A_{4}$, $\beta _{2}=A_{2}+A_{5}$, $\beta _{3}=A_{3}+A_{6}$, $\beta _{4}=A_{7}$, $\beta _{5}=A_{8}+A_{10}$, $\beta _{6}=A_{9}+A_{11}$.

Suppose that $i\omega$, where $\omega >0$, be purely imaginary root of the transcendental Eq. (4.7), then it implies that

$$\begin{aligned} \begin{aligned} \beta _{5}\omega \cos (\omega \tau _{2})+(\beta _{4}\omega ^{2}-\beta _{6})\sin (\omega \tau _{2})=\omega ^{3}-\beta _{2}\omega ,\\ (\beta _{6}-\beta _{4}\omega ^{2})\cos (\omega \tau _{2})+\beta _{5}\omega \sin (\omega \tau _{2})=\beta _{1}\omega ^{2}-\beta _{3}. \end{aligned} \end{aligned}$$

(4.8)

After elimination of cos and sin function from (4.8), we get a polynomial equation of degree six that can have at most six real and positive zeros, that equation is,

$$\begin{aligned} \omega ^{6}+b_{1}\omega ^{4}+b_{2}\omega ^{2}+b_{3}=0, \end{aligned}$$

(4.9)

where $b_{1}=\beta _{1}^{2}-2\beta _{2}-\beta _{4}^{2}$, $b_{2}=\beta _{2}^{2}-2\beta _{1}\beta _{3}-\beta _{5}^{2}+2\beta _{4}\beta _{6}$, $b_{3}=\beta _{3}^{2}-\beta _{6}^{2}$.

If we substitute $\omega ^{2}=p$, then Eq. (4.9) is reduced as

$$\begin{aligned} g(p)=p^{3}+b_{1}p^{2}+b_{2}p+b_{3}=0, \end{aligned}$$

(4.10)

which is a three degree polynomial. It has at least one real root and at most three. Thus, without any doubt, it is stipulated that $\omega _{0}=\sqrt{p_{0}}>0$ is a zeros of g(p). We substitute this one into (4.8), that provides critical values of $\tau _{2}$ in this fashion

$$\begin{aligned} \tau _{2j}&=\frac{1}{\omega _{0}}\cos ^{-1}\nonumber \\&\quad \Big [\frac{\beta _{5}\omega _{0}^{2}(\omega _{0}^{2}-\beta _{2})+(\beta _{4}\omega _{0}^{2}-\beta _{6})(\beta _{3}-\beta _{1}\omega _{0}^{2})}{\beta _{5}^{2}\omega _{0}^{2}+(\beta _{4}\omega _{0}^{2}-\beta _{6})^{2}}\Big ]\nonumber \\&\quad +\frac{2\pi j}{\omega _{0}}, \end{aligned}$$

(4.11)

where $j=0,1,2,...$.

Assume that (A1): $g'(\omega _{0}^{2})=3\omega _{0}^{2}+2b_{1}\omega _{0}+b_{2}\ne 0$. Thus, we consider $\lambda (\tau _{2j})=\pm i\omega _{0}$ as purely imaginary roots of (4.7) and do some analytical calculation that yields $Re\Big [\frac{d\lambda }{d\tau _{2}}\Big ]^{-1}_{\lambda =i\omega _{0},\tau =\tau _{2j}}=\frac{g'(\omega _{0}^{2})}{\beta _{5}^{2}\omega _{0}^{2}+(\beta _{4}\omega _{0}^{2}-\beta _{6})^{2}}\ne 0.$ Since, the sign of $Re\Big [\frac{d\lambda }{d\tau _{2}}\Big ]^{-1}_{\lambda =i\omega _{0},\tau =\tau _{2j}},$$Re\Big [\frac{d\lambda }{d\tau _{2}}\Big ]_{\lambda =i\omega _{0},\tau =\tau _{2j}},$ and $\Big [\frac{dRe(\lambda )}{d\tau _{2}}\Big ]_{\lambda =i\omega _{0},\tau =\tau _{2j}}$ are equal, but it depends on the sign of $g'(\omega _{0}^{2})$. Hence, we get transversality condition under (A1), i.e.

$$\begin{aligned} \Big [\frac{dRe(\lambda )}{d\tau _{2}}\Big ]_{\lambda =i\omega _{0},\tau =\tau _{2j}}>0, \end{aligned}$$

when $g'(\omega _{0}^{2})>0$.

Thus, the following result can be stated.

Theorem 1

Assuming that (A1) holds and $\tau _{1}=0$. Then there exists a positive real number $\tau _{20}$ such that the system (2.1) demonstrates asymptotically stable dynamics at $E_{3}$ when $\tau _{2}<\tau _{20}$, unstable dynamics $\tau _{2}>\tau _{20}$, and undergoes into a Hopf-bifurcation when $\tau _{2}=\tau _{20}$, where $\tau _{20}=\min \{\tau _{2j}\}$, $j=0,1,2,...$

Case III: When $\tau _{1}>0, \tau _{2}=0$, the Eq. (4.6) is turned into the given equation

$$\begin{aligned} \lambda ^{3}\!+\!\mu _{1}\lambda ^{2}\!+\!\mu _{2}\lambda \!+\!\mu _{3}\!+\!\big (\mu _{4}\lambda ^{2}\!+\!\mu _{5}\lambda \!+\!\mu _{6}\big )e^{-\lambda \tau _{1}}\!=\!0, \end{aligned}$$

(4.12)

where $\mu _{1}=A_{1}+A_{7}$, $\mu _{2}=A_{2}+A_{8}$, $\mu _{3}=A_{3}+A_{9}$, $\mu _{4}=A_{4}$, $\mu _{5}=A_{5}+A_{10}$, $\mu _{6}=A_{6}+A_{11}$.

Since, stability switches appears when $Re(\lambda )$ vanishes. Therefore, we stipulate that $i\omega$, where $\omega >0$, be purely imaginary eigenvalues of transcendental Eq. (4.12), then it implies that

$$\begin{aligned} \begin{aligned}&\mu _{5}\omega \cos (\omega \tau _{1})+(\mu _{4}\omega ^{2}-\mu _{6})\sin (\omega \tau _{1})=\omega ^{3}-\mu _{2}\omega ,\\&\quad (\mu _{6}\!-\!\mu _{4}\omega ^{2})\cos (\omega \tau _{1})\!+\!\mu _{5}\omega \sin (\omega \tau _{1})\!=\!\mu _{1}\omega ^{2}\!-\!\mu _{3}. \end{aligned} \end{aligned}$$

(4.13)

We remove cos and sin function from (4.13) and get a six degree polynomial of at most six real and positive zeros may exists, which informed that the zeros may cross imaginary axis at most six times. The equation is expressed as,

$$\begin{aligned} \omega ^{6}+b_{10}\omega ^{4}+b_{20}\omega ^{2}+b_{30}=0, \end{aligned}$$

(4.14)

where $b_{10}=\mu _{1}^{2}-2\mu _{2}-\mu _{4}^{2}$, $b_{20}=\mu _{2}^{2}-2\mu _{1}\mu _{3}-\mu _{5}^{2}+2\mu _{4}\mu _{6}$,

$b_{30}=\mu _{3}^{2}-\mu _{6}^{2}$.

If we substitute $\omega ^{2}=q$, then Eq. (4.14) is reduced as

$$\begin{aligned} f(q)=q^{3}+b_{10}q^{2}+b_{20}q+b_{30}=0, \end{aligned}$$

(4.15)

which is three degree polynomial. It has at least one real root and at most three might be possible. Thus, without any doubt, it is stipulated that $\omega _{0}=\sqrt{q_{0}}>0$ is a zeros of f(q). We substitute this one into (4.13), that provides critical values of $\tau _{1}$ in this fashion

$$\begin{aligned} \tau _{1j}&=\frac{1}{\omega _{0}}\cos ^{-1}\nonumber \\&\quad \Big [\frac{\mu _{5}\omega _{0}^{2}(\omega _{0}^{2}-\mu _{2})+(\mu _{4}\omega _{0}^{2}-\mu _{6})(\mu _{3}-\mu _{1}\omega _{0}^{2})}{\mu _{5}^{2}\omega _{0}^{2}+(\mu _{4}\omega _{0}^{2}-\mu _{6})^{2}}\Big ]\nonumber \\&\quad +\frac{2\pi j}{\omega _{0}}, \end{aligned}$$

(4.16)

where $j=0,1,2,...$.

Assume that (A2): $f'(\omega _{0}^{2})=3\omega _{0}^{2}+2b_{10}\omega _{0}+b_{20}\ne 0$. Thus, we consider $\lambda (\tau _{1j})=\pm i\omega _{0}$ as a pair of purely imaginary roots of (4.12) and do some analytical calculation that yields

$$\begin{aligned} Re\Big [\frac{d\lambda }{d\tau _{1}}\Big ]^{-1}_{\lambda =i\omega _{0},\tau =\tau _{1j}}=\frac{f'(\omega _{0}^{2})}{\mu _{5}^{2}\omega _{0}^{2}+(\mu _{4}\omega _{0}^{2}-\mu _{6})^{2}}\ne 0. \end{aligned}$$

Since, the sign of $Re\Big [\!\frac{d\lambda }{d\tau _{1}}\!\Big ]^{-1}_{\lambda =i\omega _{0},\tau =\tau _{1j}},$ $Re\Big [\!\frac{d\lambda }{d\tau _{1}}\!\Big ]_{\lambda =i\omega _{0},\tau =\tau _{1j}},$ and $\Big [\frac{dRe(\lambda )}{d\tau _{1}}\Big ]_{\lambda =i\omega _{0},\tau =\tau _{1j}}$ are equal, but it depends on the sign of $f'(\omega _{0}^{2})$. Hence, we get transversality condition under (A2), i.e.

$$\begin{aligned} \Big [\frac{dRe(\lambda )}{d\tau _{1}}\Big ]_{\lambda =i\omega _{0},\tau =\tau _{1j}}>0 \end{aligned}$$

, when $f'(\omega _{0}^{2})>0$. Thus, the following result can be stated.

Theorem 2

Assuming that (A2) holds and $\tau _{2}=0$. Then there exists a positive real number $\tau _{10}$ such that the system (2.1) demonstrates asymptotically stable dynamics at $E_{3}$ when $\tau _{1}<\tau _{10}$, unstable dynamics $\tau _{1}>\tau _{10}$, and undergoes into a Hopf-bifurcation when $\tau _{1}=\tau _{10}$, where $\tau _{10}=\min \{\tau _{1j}\}$, $j=0,1,2,...$

Case IV: Taking $\tau _{1}\in (0,\tau _{10})$, $\tau _{2}>0$, i.e. we take $\tau _{2}$ as a varying parameter and retain $\tau _{1}$ within a stable range $(0,\tau _{10})$. Let $\lambda =i\omega (\omega >0)$ be a root of (4.6) then the Eq. (4.6) breakdown into the following equation by separating real and imaginary parts,

$$\begin{aligned}&-\omega ^{3}\!+\!A_{2}\omega \!+\!A_{5}\omega \cos (\omega \tau _{1})\!+\!(A_{4}\omega ^{2}\!-\!A_{6})\sin (\omega \tau _{1})\nonumber \\&\quad +\cos (\omega \tau _{2})[A_{8}\omega +A_{10}\omega \cos (\omega \tau _{1})-A_{11}\sin (\omega \tau _{1})]\nonumber \\&\quad +\sin (\omega \tau _{2})[A_{7}\omega ^{2}-A_{9}-A_{10}\omega \sin (\omega \tau _{1})\nonumber \\&\quad -A_{11}\cos (\omega \tau _{1})]=0, \end{aligned}$$

(4.17)

$$\begin{aligned}&\quad -A_{1}\omega ^{2}+A_{3}+A_{5}\omega \sin (\omega \tau _{1})\nonumber \\&\quad +(A_{6}-A_{4}\omega ^{2})\cos (\omega \tau _{1})-\cos (\omega \tau _{2})[A_{7}\omega ^{2}\nonumber \\&\quad -A_{9}-A_{10}\omega \sin (\omega \tau _{1})-A_{11}\cos (\omega \tau _{1})]+\nonumber \\&\quad \sin (\omega \tau _{2})[A_{8}\omega \!+\!A_{10}\omega \cos (\omega \tau _{1})\!-\!A_{11}\sin (\omega \tau _{1})]\!=\!0, \end{aligned}$$

(4.18)

The above Eqs. (4.17) and (4.18) can be further modified as

$$\begin{aligned}&c_{1}+c_{2}\cos (\omega \tau _{1})+c_{3}\sin (\omega \tau _{1})+\cos (\omega \tau _{2})[c_{4}+\nonumber \\&\qquad c_{5}\cos (\omega \tau _{1})-c_{6}\sin (\omega \tau _{1})]+\sin (\omega \tau _{2})[c_{7}-\nonumber \\&\qquad c_{5}\sin (\omega \tau _{1})-c_{6}\cos (\omega \tau _{1})]=0, \end{aligned}$$

(4.19)

$$\begin{aligned}&\qquad c_{8}-c_{3}\cos (\omega \tau _{1})+c_{2}\sin (\omega \tau _{1})-\cos (\omega \tau _{2})[c_{7}-\nonumber \\&\qquad c_{5}\sin (\omega \tau _{1})-c_{6}\cos (\omega \tau _{1})]+\sin (\omega \tau _{2})[c_{4}+\nonumber \\&\qquad c_{5}\cos (\omega \tau _{1})-c_{6}\sin (\omega \tau _{1})]=0, \end{aligned}$$

(4.20)

where $c_{1}=-\omega ^{3}+A_{2}\omega$, $c_{2}=A_{5}\omega$, $c_{3}=A_{4}\omega ^{2}-A_{6}$, $c_{4}=A_{8}\omega$, $c_{5}=A_{10}\omega$, $c_{6}=A_{11}$, $c_{7}=A_{7}\omega ^{2}-A_{9}$, $c_{8}=-A_{1}\omega ^{2}+A_{3}$.

To alienate $\tau _{2}$ from the Eqs. (4.19) and (4.20), squaring and adding these two equation which gives

$$\begin{aligned}&c_{1}^{2}\!+\!c_{2}^{2}\!+\!c_{3}^{2}\!+\!c_{8}^{2}\!-\!c_{4}^{2}\!-\!c_{5}^{2}-c_{6}^{2}-c_{7}^{2}+2(c_{1}c_{2}-c_{3}c_{8}\nonumber \\&\qquad -c_{4}c_{5}+c_{6}c_{7})\cos (\omega \tau _{1})+2(c_{1}c_{3}\!+\!c_{2}c_{8}\!-\!c_{4}c_{6}\nonumber \\&\qquad +c_{5}c_{7})\sin (\omega \tau _{1})=0. \end{aligned}$$

(4.21)

Equation (4.21) demonstrates a transcendental equation in very complicated form. It is very difficult to predict the behaviour of the zeros. Without any detail explanation, it is supposed that $\omega _{0}>0$ is a root of the Eq. (4.21). Further, the Eqs. (4.17) and (4.18) can be re-expressed as

$$\begin{aligned}&C_{1}\cos (\omega \tau _{2})+C_{2}\sin (\omega \tau _{2})=C_{3}, \end{aligned}$$

(4.22)

$$\begin{aligned}&\qquad -C_{2}\cos (\omega \tau _{2})+C_{1}\sin (\omega \tau _{2})=C_{4}, \end{aligned}$$

(4.23)

where $C_{1}=A_{8}\omega +A_{10}\omega \cos (\omega \tau _{1})-A_{11}\sin (\omega \tau _{1})$,

$C_{2}=A_{7}\omega ^{2}-A_{9}-A_{10}\omega \sin (\omega \tau _{1})-A_{11}\cos (\omega \tau _{1})$,

$C_{3}=\omega ^{3}-A_{2}\omega -A_{5}\omega \cos (\omega \tau _{1})-(A_{4}\omega ^{2}-A_{6})\sin (\omega \tau _{1})$,

$C_{4}=A_{1}\omega ^{2}-A_{3}-A_{5}\omega \sin (\omega \tau _{1})-(A_{6}-A_{4}\omega ^{2})\cos (\omega \tau _{1})$.

After some calculations, these two Eqs. (4.22) and (4.23) provides a critical values

$$\begin{aligned} \tau _{2j}'&=\frac{1}{\omega _{0}}\cos ^{-1}\Big [\frac{C_{1}C_{3}-C_{2}C_{4}}{C_{1}^{2}+C_{2}^{2}}\Big ]\nonumber \\&\quad +\frac{2\pi j}{\omega _{0}}, j=0,1,2,... \end{aligned}$$

(4.24)

For confirmation of Hopf-bifurcation, we verify the transversality conditions. So, differentiate Eqs. (4.17) and (4.18) w.r.t. $\tau _{2}$, the mathematical computation yields

$$\begin{aligned} \begin{aligned}&R(\omega )\frac{d(Re(\lambda ))}{d\tau _{2}}+P(\omega )\frac{d\omega }{d\tau _{2}}=Q(\omega ),\\&\qquad -P(\omega )\frac{d(Re(\lambda ))}{d\tau _{2}}+R(\omega )\frac{d\omega }{d\tau _{2}}=S(\omega ), \end{aligned} \end{aligned}$$

(4.25)

where

$$\begin{aligned} P(\omega )&=-3\omega ^{2}+A_{2} +(2A_{4}-\tau _{1}A_{5})\omega \sin (\omega \tau _{1})\\&\quad +(A_{5}+\tau _{1} (A_{4}\omega ^{2}-A_{6}))\cos (\omega \tau _{1})\\&\quad +(2A_{7}-\tau _{2}A_{8})\omega \sin (\omega \tau _{2})\\&\quad +(-A_{8}+\tau _{2}(A_{7}\omega ^{2}-A_{9}))\cos (\omega \tau _{2})\\&\quad -(\tau _{1}+\tau _{2})A_{10}\omega \sin (\omega (\tau _{1}+\tau _{2}))\\&\quad +(A_{10}-A_{11}(\tau _{1}+\tau _{2})) \cos (\omega (\tau _{1}+\tau _{2})),\\ Q(\omega )&=A_{8}\omega ^{2}\sin (\omega \tau _{2})-\omega (A_{7}\omega ^{2}-A_{9}) \cos (\omega \tau _{2})\\&\quad +A_{10}\omega ^{2} \sin (\omega (\tau _{1}+\tau _{2})),\\ R(\omega )&=-2A_{1}\omega +(A_{5}\tau _{1}-2A_{4})\omega \cos (\omega \tau _{1})\\&\quad +(\tau _{1}(A_{6}-A_{4}\omega ^{2}) +A_{5})\sin (\omega \tau _{1})\\&\quad +(\tau _{2}(A_{7}\omega ^{2}-A_{9}) +A_{8})\sin (\omega \tau _{2})\\&\quad + A_{8}\tau _{2}\omega \cos (\omega \tau _{2})+(A_{10}-A_{11}(\tau _{1}+\tau _{2}))\\&\quad \times \sin (\omega (\tau _{1}+\tau _{2})) +A_{10}(\tau _{1}+\tau _{2})\omega \\&\quad \times \cos (\omega (\tau _{1}+\tau _{2})),\\ S(\omega )&=-\omega (A_{7}\omega ^{2}-A_{9})\sin (\omega \tau _{2})\\&\quad +2A_{7}\omega \cos (\omega \tau _{2})-A_{8}\omega ^{2}\cos (\omega (\tau _{2}))\\&\quad -A_{10}\omega ^{2}\cos (\omega (\tau _{1}+\tau _{2})). \end{aligned}$$

Solving Eq. (4.25) to remove

$$\begin{aligned} \frac{d\omega }{d\tau _{2}}, \end{aligned}$$

we obtain

$$\begin{aligned}{} & {} \Big [\frac{d(Re(\lambda ))}{d\tau _{2}}\Big ]_{\lambda =i\omega _{0},\tau _{2} =\tau _{20}'}\\{} & {} \qquad =\frac{R(\omega )Q(\omega )-P(\omega )S(\omega )}{R^{2}(\omega )+P^{2}(\omega )}. \end{aligned}$$

We assume that (A3): $R(\omega )Q(\omega )-P(\omega )S(\omega )\ne 0$.

Theorem 3

Let $\tau _{1}\in (0,\tau _{10})$ and (A3) holds. Then there exists a positive real number $\tau _{20}'$ such that the system (2.1) exhibits asymptotically stable nature at $E_{3}$ when $\tau _{2}<\tau _{20}'$, unstable when $\tau _{2}>\tau _{20}'$, and undergoes into a Hopf-bifurcation when $\tau _{2}=\tau _{20}'$, where $\tau _{20}'=\min \{\tau _{2j}'\}$, $j=0,1,2,...$.

Case V: Taking $\tau _{2}\in (0,\tau _{20})$, $\tau _{1}>0$, i.e. we take $\tau _{1}$ as a varying parameter and retain $\tau _{2}$ within a stable range $(0,\tau _{20})$. Let $\lambda =\nu (\tau _{1})+i\omega (\tau _{1})$ is an characteristic root of (4.6), but the stability switch can be inspected when $Re(\lambda )$ vanishes. Therefore, if we set $\lambda =i\omega (\tau _{1})$, where $\omega >0$, then Eq. (4.6) becomes exactly (4.17) and (4.18), which can be re-arranged as

$$\begin{aligned}&-\omega ^{3}\!+\!A_{2}\omega \!+\!cos(\omega \tau _{1})[A_{5}\omega \!+\!A_{10}\omega \cos (\omega \tau _{2})\!-\!A_{11}\nonumber \\&\qquad \sin (\omega \tau _{2})]+\sin (\omega \tau _{1})[A_{4}\omega ^{2}\nonumber \\&\qquad -A_{6}-A_{11}\cos (\omega \tau _{2})-A_{10}\nonumber \\&\qquad \omega \sin (\omega \tau _{2})]+(A_{7}\omega ^{2}-A_{9})\sin (\omega \tau _{1})=0, \end{aligned}$$

(4.26)

$$\begin{aligned}&\qquad -A_{1}\omega ^{2}+A_{3}+cos(\omega \tau _{1})[A_{6}\nonumber \\&\qquad -A_{4}\omega ^{2}+A_{11}\cos (\omega \tau _{2})+A_{10}\nonumber \\&\qquad \omega \sin (\omega \tau _{2})]+\sin (\omega \tau _{1})[A_{5}\omega +A_{10}\omega \cos (\omega \tau _{2})\nonumber \\&\qquad -A_{11}\sin (\omega \tau _{2})]+A_{8}\omega \sin (\omega \tau _{2})=0, \end{aligned}$$

(4.27)

Above Eqs. (4.26) and (4.27) further reduced as

$$\begin{aligned}&c_{1}+\cos (\omega \tau _{1})[c_{2}+c_{5}\cos (\omega \tau _{2})-c_{6}\nonumber \\&\qquad \sin (\omega \tau _{2})]+\sin (\omega \tau _{1})\nonumber \\&\qquad [c_{3}\!-\!c_{6}\cos (\omega \tau _{2})\!-\!c_{5}\sin (\omega \tau _{2})]\!+\!c_{7}\sin (\omega \tau _{2})\!=\!0, \end{aligned}$$

(4.28)

$$\begin{aligned}&\qquad c_{8}-\cos (\omega \tau _{1})[c_{3}-c_{6}\cos (\omega \tau _{2})-c_{5}\nonumber \\&\qquad \sin (\omega \tau _{2})]+\sin (\omega \tau _{1})\nonumber \\&\qquad [c_{2}\!+\!c_{5}\cos (\omega \tau _{2})\!-\!c_{6}\sin (\omega \tau _{2})]\!+\!c_{4}\sin (\omega \tau _{2})\!=\!0, \end{aligned}$$

(4.29)

To alienate $\tau _{1}$ from the Eqs. (4.28) and (4.29), squaring and adding these two equation which gives

$$\begin{aligned}&c_{2}^{2}+c_{3}^{2}+c_{5}^{2}+c_{6}^{2}-c_{1}^{2}-c_{8}^{2}-(c_{4}^{2}+c_{7}^{2})\sin ^{2}(\omega \tau _{2})+\nonumber \\&\qquad 2(c_{2}c_{5}-c_{3}c_{6})\cos (\omega \tau _{2})-2(c_{2}c_{6}+c_{3}c_{5}+c_{1}c_{7}\nonumber \\&\qquad +c_{4}c_{8})\sin (\omega \tau _{2})=0. \end{aligned}$$

(4.30)

Equation (4.30) represents the complex form of a transcendental equation. Consequently, the behaviour of the zeros can not predictable easily. Ignoring the detail explanation, we only make the sense that $\omega _{0}>0$ is a possible root of the Eq. (4.30). Further, the Eqs. (4.26) and (4.27) can be re-arranged as

$$\begin{aligned} D_{1}\cos (\omega \tau _{1})+D_{2}\sin (\omega \tau _{1})=D_{3}, \end{aligned}$$

(4.31)

$$\begin{aligned} -D_{2}\cos (\omega \tau _{2})+D_{1}\sin (\omega \tau _{2})=D_{4}, \end{aligned}$$

(4.32)

where

$D_{1}=A_{5}\omega +A_{10}\omega \cos (\omega \tau _{2})-A_{11}\sin (\omega \tau _{2})$,

$D_{2}=A_{4}\omega ^{2}-A_{6}-A_{10}\omega \sin (\omega \tau _{2})-A_{11}\cos (\omega \tau _{2})$,

$D_{3}=\omega ^{3}-A_{2}\omega -(A_{7}\omega ^{2}-A_{9})\sin (\omega \tau _{2})$,

$D_{4}=A_{1}\omega ^{2}-A_{3}-A_{8}\omega \sin (\omega \tau _{2})$.

Equations (4.31) and (4.32) leads

$$\begin{aligned} \tau _{1j}'&=\frac{1}{\omega _{0}}\cos ^{-1}\Big [\frac{D_{1}D_{3}-D_{2}D_{4}}{D_{1}^{2}+D_{2}^{2}}\Big ]\nonumber \\&\quad +\frac{2\pi j}{\omega _{0}}, j=0,1,2,... \end{aligned}$$

(4.33)

For confirmation of Hopf-bifurcation, we verify the transversality conditions. So, differentiate Eqs. (4.17) and (4.18) w.r.t. $\tau _{1}$, the mathematical computation yields

$$\begin{aligned} \begin{aligned} R'(\omega )\frac{d(Re(\lambda ))}{d\tau _{1}}+P'(\omega )\frac{d\omega }{d\tau _{1}}=Q'(\omega ),\\ -P'(\omega )\frac{d(Re(\lambda ))}{d\tau _{1}}+R'(\omega )\frac{d\omega }{d\tau _{1}}=S'(\omega ), \end{aligned} \end{aligned}$$

(4.34)

where

$$\begin{aligned} P'(\omega )= & {} -3\omega ^{2}+A_{2}+(2A_{4}-\tau _{1}A_{5})\omega \sin (\omega \tau _{1})\\{} & {} +(A_{5}+\tau _{1}(A_{4}\omega ^{2}-A_{6}))\cos (\omega \tau _{1})\\{} & {} +(2A_{7}-\tau _{2}A_{8})\omega \sin (\omega \tau _{2})\\{} & {} +(A_{8}+\tau _{2}(A_{7}\omega ^{2}-A_{9}))\cos (\omega \tau _{2})\\{} & {} -(\tau _{1}+\tau _{2})A_{10}\omega \times \sin (\omega (\tau _{1}+\tau _{2}))\\{} & {} +(A_{10}-A_{11}(\tau _{1}+\tau _{2}))\cos (\omega (\tau _{1}+\tau _{2})),\\ Q'(\omega )= & {} A_{5}\omega ^{2}\sin (\omega \tau _{1})-\omega (A_{4}\omega ^{2}-A_{6})\\{} & {} \times \cos (\omega \tau _{1})+A_{10}\omega ^{2}\sin (\omega (\tau _{1}+\tau _{2}))\\{} & {} +A_{11}\omega \cos (\omega (\tau _{1}+\tau _{2})),\\ R'(\omega )= & {} -2A_{1}\omega +(A_{5}\tau _{1}-2A_{4})\omega \cos (\omega \tau _{1})\\{} & {} +(\tau _{1}(A_{4}\omega ^{2}- A_{6})+A_{5})\sin (\omega \tau _{1})\\{} & {} +(\tau _{2}(A_{7}\omega ^{2}-A_{9})+A_{8})\sin (\omega \tau _{2})\\{} & {} +(A_{8}\tau _{2}-2A_{7})\omega \cos (\omega \tau _{2})\\{} & {} +(A_{10}-A_{11}(\tau _{1}+\tau _{2}))\sin (\omega (\tau _{1}+\tau _{2}))\\{} & {} +A_{10}(\tau _{1}+\tau _{2})\omega \cos (\omega (\tau _{1}+\tau _{2}))\\ S'(\omega )= & {} -\omega (A_{4}\omega ^{2}-A_{6})\sin (\omega \tau _{1})\\{} & {} -A_{5}\omega ^{2}\cos (\omega \tau _{1})+A_{11}\omega \sin (\omega (\tau _{1}+\tau _{2}))\\{} & {} -A_{10}\omega ^{2}\cos (\omega (\tau _{1}+\tau _{2})). \end{aligned}$$

Solving Eq. (4.34) to remove $\frac{d\omega }{d\tau _{1}},$ we obtain

$$\begin{aligned} \Big [\frac{d(Re(\lambda ))}{d\tau _{1}}\Big ]_{\lambda =i\omega _{0},\tau _{1}\!=\!\tau _{10}'}\!=\!\frac{R'(\omega )Q'(\omega )\!-\!P'(\omega )S'(\omega )}{R'^{2}(\omega )\!+\!P'^{2}(\omega )}. \end{aligned}$$

We assume that (A4): $R'(\omega )Q'(\omega )-P'(\omega )S'(\omega )\ne 0$.

Theorem 4

Let $\tau _{2}\in (0,\tau _{20})$ and (A4) holds. Then there exists a positive real number $\tau _{10}'$ such that the system (2.1) exhibits asymptotically stable nature at $E_{3}$ when $\tau _{1}<\tau _{10}'$, unstable when $\tau _{1}>\tau _{10}'$, and undergoes into a Hopf-bifurcation when $\tau _{1}=\tau _{10}'$, where $\tau _{10}'=\min \{\tau _{1j}'\}$, $j=0,1,2,...$.

5 The stability and direction of bifurcating periodic solution

In the earlier section, some criteria for Hopf-bifurcation have been established for various combinations of delay parameters $\tau _{1}$ and $\tau _{2}$. In the present section, we will systematically derive the stability, periodicity and direction of bifurcated oscillatory solutions concerning $\tau _{1}=\tau _{10}'$ as a critical point of the system (2.1). The above dynamical behaviors obey both the normal form theory and the central manifold theorem, which are discussed by Hassard et al. [40]. We assume that $\tau _{2}^{*}<\tau _{10}'$, where $\tau _{2}^{*}\in (0,\tau _{20}')$. Now, we assume that when $\tau _{1}\in (0,\tau _{10}')$ and $\tau _{2}^{*}\in (0,\tau _{20}^{'})$, then the system shows stable behaviour. Now, the reason behind for $\tau _{2}^{*}<\tau _{10}'$ is that we want to study the effect of maturation delay when cooperative delay has an stabilizing effect. Let us take a small disturbance around $E_{3}$, i.e. $x(t)=U(t)-U^{*}$, $y(t)=V(t)-V^{*}$, $z(t)=W(t)-W^{*}$, and setting $\tau _{1}=\tau _{10}'+\mu$, where $\mu \in \mathbb {R}$ such that Hopf-bifurcation is observed when $\mu =0$. By scaling,

$$\begin{aligned} t\rightarrowtail \frac{t}{\tau _{1}}, \end{aligned}$$

we normalize the delay, then the model (2.1) can be re-expressed as

$$\begin{aligned}{} & {} \dot{X}(t)=\tau _{1}\Big (AX(t)+BX(t-1)\nonumber \\{} & {} \qquad +CX(t-\frac{\tau _{2}^{*}}{\tau _{1}})+g(U,V,W)\Big ), \end{aligned}$$

(5.1)

where $X(t)=(U(t),V(t),W(t))^{T}\in \mathbb {R}^{3}$,

$$\begin{aligned} A= & {} \begin{pmatrix} a_{1} &{}&{} j_{12} &{}&{} j_{13}\\ 0 &{}&{} j_{22} &{}&{} a_{2}\\ j_{31} &{}&{} j_{32} &{}&{} 0 \end{pmatrix},\\ B= & {} \begin{pmatrix} -j_{21} &{}&{} 0 &{}&{} 0\\ j_{21} &{}&{} 0 &{}&{} 0\\ 0 &{}&{} 0 &{}&{} 0 \end{pmatrix},\\ C= & {} \begin{pmatrix} 0 &{}&{} 0 &{}&{} 0\\ 0 &{}&{} 0 &{}&{} a_{3}\\ 0 &{}&{} 0 &{}&{} j_{33} \end{pmatrix},\\ g(x,y,z)= & {} \begin{pmatrix} g_{1}\\ g_{2}\\ g_{3} \end{pmatrix}. \end{aligned}$$

The non-linear terms are given by

$$\begin{aligned} g_{1}= & {} \frac{2d\alpha _{1}W^{*}}{(d+U^{*})^{3}}U^{2}(t)-\frac{d\alpha _{1}}{(d+U^{*})^{2}}U(t)W(t)\\{} & {} -\frac{6d\alpha _{1}W^{*}}{(d+U^{*})^{4}}U^{3}(t)+\cdots ,\\ g_{2}= & {} -\frac{2s}{K}V^{2}(t)-(a+mV^{*})V(t)W(t)\\{} & {} -mV^{*}W(t)W(t-\frac{\tau _{2}^{*}}{\tau _{1}})\\{} & {} -mW^{*}V(t)W(t-\frac{\tau _{2}^{*}}{\tau _{1}})+\cdots ,\\ g_{3}= & {} -\frac{2d\alpha _{2}W^{*}}{(d+U^{*})^{3}}U^{2}(t)+\frac{d\alpha _{2}}{(d+U^{*})^{2}}U(t)W(t)\\{} & {} +c(a+mW^{*}) V(t)W(t)+cmV^{*}W(t)\\{} & {} \times W(t-\frac{\tau _{2}^{*}}{\tau _{1}}) +cmW^{*}V(t)W(t-\frac{\tau _{2}^{*}}{\tau _{1}}) \\{} & {} +\frac{6d\alpha _{2}W^{*}}{(d+U^{*})^{3}}U^{3}(t)+\cdots , \end{aligned}$$

Linearized equation (5.1) can be written around origin as

$$\begin{aligned} \dot{X}(t)=\tau _{1}\Big (AX(t)+BX(t-1)+CX(t-\frac{\tau _{2}^{*}}{\tau _{1}}). \end{aligned}$$

(5.2)

For $\nu =(\nu _{1},\nu _{2},\nu _{3})^{T}\in C([-1,0],\mathbb {R}^{3})$, define

$$\begin{aligned} J_{\mu }(\nu )=(\tau _{1}+\mu )[A\nu (0)+B\nu (-1)+C\nu (-\frac{\tau _{2}^{*}}{\tau _{1}})]. \end{aligned}$$

By the Riesz representation theorem, a matrix $\xi (\beta ,\mu )$ where $(-1\le \beta \le 0)$ of order-3 exists whose entries are function of bounded variations such that

$$\begin{aligned} J_{\mu }(\nu )=\int _{-1}^{0} d\xi (\beta ,\mu )\nu (\beta ), \end{aligned}$$

(5.3)

for $\nu \in C([-1,0],\mathbb {R}^{3})$. Eventually, we can obtain

$$\begin{aligned} \xi (\beta ,\mu )\!=\!\left\{ \begin{aligned} (\tau _{10}'+\mu )(A+B+C), \quad if \hspace{0.1cm} \beta =0,\\ (\tau _{10}'+\mu )(B+C), \quad if\hspace{0.1cm} \beta \in (-\frac{\tau _{2}^{*}}{\tau _{1}},0),\\ (\tau _{10}'+\mu )B, \quad if\hspace{0.1cm} \beta \in (-1,-\frac{\tau _{2}^{*}}{\tau _{1}}),\\ 0, \quad if\hspace{0.1cm} \beta =-1.\\ \end{aligned} \right. \end{aligned}$$

Then Eq. (5.3) is satisfied. We define an operator $M(\mu )$ for $\nu \in C^{1}([-1,0],\mathbb {R}^{3})$ as

$$\begin{aligned} M(\mu )\nu (\beta )=\left\{ \begin{aligned} \frac{d\nu (\beta )}{d\beta }, \quad \beta \in [-1,0),\\ \int _{-1}^{0} d\xi (\theta ,\mu )\mu (\theta ),\quad \beta =0, \end{aligned} \right. \end{aligned}$$

and

$$\begin{aligned} R(\mu )\nu (\beta )=\left\{ \begin{aligned} 0,\hspace{2.3cm} \beta \in [-1,0),\\ F(\mu ,\nu ),\hspace{2.3cm}\beta =0. \end{aligned} \right. \end{aligned}$$

where

$F(\mu ,\nu )=(\tau _{10}'+\mu )\begin{pmatrix} f_{1}\\ f_{2}\\ f_{3}\end{pmatrix}$,

$\nu =(\nu _{1},\nu _{2},\nu _{3})^{T}\in C([-1,0],\mathbb {R}^{3})$, and

$$\begin{aligned} f_{1}= & {} \frac{2d\alpha _{1}W^{*}}{(d+U^{*})^{3}}U^{2}(0)-\frac{d\alpha _{1}}{(d+U^{*})^{2}}U(0)W(0)\\{} & {} -\frac{6d\alpha _{1}W^{*}}{(d+U^{*})^{4}}U^{3}(0)+\cdots ,\\ f_{2}= & {} -\frac{2s}{K}V^{2}(0)-(a+mV^{*})V(0)W(0)\\{} & {} -mV^{*}W(0)W(-\frac{\tau _{2}^{*}}{\tau _{1}})\\{} & {} -mW^{*}V(0)W(-\frac{\tau _{2}^{*}}{\tau _{1}})+\cdots ,\\ f_{3}= & {} -\frac{2d\alpha _{2}W^{*}}{(d+U^{*})^{3}}U^{2}(0)+\frac{d\alpha _{2}}{(d+U^{*})^{2}}U(0)W(0)\\{} & {} +c(a+mW^{*}) V(0)W(0)+cmV^{*}W(0)\\{} & {} \times W(-\frac{\tau _{2}^{*}}{\tau _{1}}) +cmW^{*}V(0)W(-\frac{\tau _{2}^{*}}{\tau _{1}})\\{} & {} +\frac{6d\alpha _{2}W^{*}}{(d+U^{*})^{3}}U^{3}(0)+\cdots , \end{aligned}$$

The original model (2.1) is similar to the given operator equation

$$\begin{aligned} \dot{X}_{t}=M(\mu )X_{t}+R(\mu )X_{t}, \end{aligned}$$

(5.4)

where $X_{t}=X(t+\beta )$ for $\beta \in [-1,0]$.

For $\phi \in C^{1}\big ([0,1],(\mathbb {R}^{3})^{*}\big )$, we define

$$\begin{aligned} M^{*}\phi (s)=\left\{ \begin{aligned} -\frac{d\phi (s)}{ds}, \hspace{2.3cm} s\in [0,1),\\ \int _{-1}^{0} \phi (-\theta )d\xi (\theta ,0), \hspace{1cm}s=0, \end{aligned} \right. \end{aligned}$$

and a bilinear form

$$\begin{aligned}&<\phi (s),\nu (\beta )>\hspace{0.1cm} =\bar{\phi }(0)\nu (0)\nonumber \\&\qquad -\int _{\beta =-1}^{0}\int _{\theta =0}^{\beta }\bar{\phi }(\theta -\beta )d\xi (\beta )\nu (\theta )d\theta , \end{aligned}$$

where $\xi (\beta )=\xi (\beta ,0)$. Then M(0) and $M^{*}$ are adjoint operators. In the earlier section, it has been rationally examined that $\pm i\omega _{0}\tau _{10}'$ are the eigenvalues of M(0) which are also the eigenvalues for $M^{*}$. Let $q(\beta )=\displaystyle (1,\alpha _{0},\beta _{0})^{T}e^{i\omega _{0}\tau _{10}'\beta }$ $(\beta \in [-1,0])$ and $q^{*}(s)={\displaystyle \frac{1}{D}(1,\alpha _{0}^{*},\beta _{0}^{*})e^{-i\omega _{0}\tau _{10}'s}}$ $(s\in [0,1])$are the eigenvectors of M(0) and $M^{*}$ corresponding to $+i\omega _{0}\tau _{10}'$ and $-i\omega _{0}\tau _{10}'$, respectively, where

$<q^{*}(s),q(\beta )>=1$, $<q^{*}(s),\bar{q}(\beta )>=1$,

$$\begin{aligned} \alpha _{0}= & {} \frac{i\omega _{0}-a_{1}+be^{-i\omega _{0}\tau _{10}'}}{j_{12}}\\{} & {} -\frac{j_{13}[j_{12}j_{31}+j_{32}\big (i\omega _{0}-a_{1}+be^{-i\omega _{0}\tau _{10}'}\big )]}{j_{12}j_{32}j_{13}+j_{12}^{2}\big (i\omega _{0}-j_{33}e^{-i\omega _{0}\frac{\tau _{2}^{*}}{\tau _{10}'}}\big )},\\ \beta _{0}= & {} \frac{j_{12}j_{31}+j_{32}\big (i\omega _{0}-a_{1}+be^{-i\omega _{0}\tau _{10}'}}{j_{32}j_{13}+j_{12}\big (i\omega _{0}-j_{33}e^{-i\omega _{0}\frac{\tau _{2}^{*}}{\tau _{10}'}}\big )}.\\ \alpha _{0}^{*}= & {} \frac{j_{12}j_{31}-j_{32}\big (i\omega _{0}+a_{1}-be^{-i\omega _{0}\tau _{10}'}\big )}{j_{32}be^{-i\omega _{0}\tau _{10}'}-j_{31}(i\omega _{0}+j_{22})},\\ \beta _{0}^{*}= & {} \frac{(i\omega _{0}\!+\!j_{22})[j_{32}\big (i\omega _{0}+a_{1}\!-\!be^{-i\omega _{0}\tau _{10}'}\big )\!-\!j_{12}j_{31}]}{j_{32}^{2}be^{-i\omega _{0}\tau _{10}'}\!-\!j_{32}j_{31}(i\omega _{0}+j_{22})}\!-\!\frac{j_{12}}{j_{32}}, \end{aligned}$$

$$\begin{aligned} \bar{D}= & {} \Big [1+\overline{\alpha }_{0}^{*}\alpha _{0}+\overline{\beta }_{0}^{*}\beta _{0}+b\tau _{10}'(\overline{\alpha }_{0}^{*}-1)e^{-i\omega _{0}\tau _{10}'}\\{} & {} +\frac{\tau _{2}^{*}}{\tau _{10}'}mV^{*}W^{*}(c\overline{\beta }_{0}^{*}-\overline{\alpha }_{0}^{*})e^{-i\omega _{0}\frac{\tau _{2}^{*}}{\tau _{10}'}}\Big ]. \end{aligned}$$

We follow the algorithm given in Hassard et al. [40] and use the mathematical calculation of [41] to obtain Hopf-bifurcation properties. Thus, we find

$$\begin{aligned}&h(w,\bar{w})=\frac{\tau _{10}'}{\overline{D}}(1,\alpha _{0}^{*},\beta _{0}^{*}),\nonumber \\&\times \!\begin{pmatrix} &{}O_{11}w^{2}+O_{12}w\bar{w}+O_{13}\bar{w}^{2}+O_{14}w^{2}\bar{w}\\ &{}O_{21}w^{2}+O_{22}w\bar{w}+O_{23}\bar{w}^{2}+O_{24}w^{2}\bar{w}\\ &{}O_{31}w^{2}+O_{32}w\bar{w}+O_{33}\bar{w}^{2}+O_{34}w^{2}\bar{w}\\ \end{pmatrix}\!+\!\cdots , \end{aligned}$$

(5.5)

where

$O_{11}=L_{2000}+\beta _{0}L_{1001}$,

$O_{12}=2L_{2000}+2Re(\beta _{0})L_{1001}$,

$O_{13}=L_{2000}+\overline{\beta }_{0}L_{1001}$,

$$\begin{aligned} O_{14}= & {} L_{2000}\big (2Y^{(1)}_{11}(0)+Y^{(1)}_{20}(0)\big )+L_{1001}\big (Y^{(3)}_{11}(0)\\{} & {} +\beta _{0}Y^{(1)}_{11}(0)+\frac{Y^{(3)}_{20}(0)}{2}+\frac{\bar{\beta }_{0}Y^{(1)}_{20}(0)}{2}\big ),\\ O_{21}= & {} B_{0200}\alpha _{0}^{2}+\alpha _{0}\beta _{0}(B_{0110}+B_{0101}e^{-i\omega _{0}\tau _{2}^{*}}),\\ O_{22}= & {} 2\big [B_{0200}\alpha _{0}\overline{\alpha }_{0}+B_{0110}Re(\alpha _{0}\overline{\beta }_{0})\\{} & {} +B_{0101}Re(\beta _{0}\overline{\alpha }_{0}e^{-i\omega _{0}\tau _{2}^{*}})\\{} & {} +B_{0011}Re(\beta _{0}\overline{\beta }_{0}e^{i\omega _{0}\tau _{2}^{*}}),\\ O_{23}= & {} B_{0200}\overline{\alpha }_{0}^{2}+\overline{\alpha }_{0}\overline{\beta }_{0}(B_{0110}+B_{0101}e^{i\omega _{0}\tau _{2}^{*}})\\{} & {} +B_{0011}\overline{\beta }_{0}^{2}e^{i\omega _{0}\tau _{2}^{*}},\\ O_{24}= & {} \big (2\alpha _{0}B_{0200}+\beta _{0}B_{0110}+\beta _{0}B_{0101}e^{-i\omega _{0}\tau _{2}^{*}}\big ) Y_{11}^{(2)}(0)\\{} & {} +\big (\overline{\alpha }_{0}B_{0200}+\frac{\overline{\beta }_{0}B_{0110}}{2}+\frac{\overline{\beta }_{0}B_{0101}}{2}e^{i\omega _{0}\tau _{2}^{*}}\big ) Y_{20}^{(2)}(0)\\{} & {} +\big (\alpha _{0}B_{0110}+\beta _{0}B_{0011}e^{-i\omega _{0}\tau _{2}^{*}}\big )Y_{11}^{(3)}(0)\\{} & {} +\big (\frac{\overline{\alpha }_{0}B_{0110}}{2}+\frac{\overline{\beta }_{0}B_{0011}}{2}e^{i\omega _{0}\tau _{2}^{*}}\big )Y_{20}^{(3)}(0)\\{} & {} +\big (B_{0101}+\beta _{0}B_{0011}\big )Y_{11}^{(3)}(-\frac{\tau _{2}^{*}}{\tau _{1}})+\big (\frac{\alpha _{0}B_{0101}}{2}\\{} & {} +\frac{\overline{\beta }_{0}B_{0011}}{2}\big )Y_{20}^{(3)}(-\frac{\tau _{2}^{*}}{\tau _{1}}),\\ O_{31}= & {} C_{2000}+C_{1010}\beta _{0}+C_{0110}\alpha _{0}\beta _{0}\\{} & {} +C_{0101}\alpha _{0}\beta _{0}e^{-i\omega _{0}\tau _{2}^{*}}+ C_{0011}\beta _{0}^{2}e^{-i\omega _{0}\tau _{2}^{*}},\\ O_{32}= & {} 2\Big [C_{2000}+C_{1010}Re(\beta _{0})+C_{0110}Re(\alpha _{0}\overline{\beta }_{0})\\{} & {} +C_{0101}Re(\overline{\alpha }_{0}\beta _{0}e^{-i\omega _{0}\tau _{2}^{*}}){+}C_{0011}Re(\beta _{0}\overline{\beta }_{0}e^{i\omega _{0}\tau _{2}^{*}})\Big ],\\ O_{33}= & {} C_{2000}+\overline{\beta }_{0}C_{1010}+\overline{\alpha }_{0}\overline{\beta }_{0}(C_{0110}+C_{0101}e^{i\omega _{0}\tau _{2}^{*}})\\{} & {} +C_{0011}\overline{\beta }_{0}^{2}e^{i\omega _{0}\tau _{2}^{*}},\\ O_{34}= & {} C_{2000}\big (2Y_{11}^{(1)}(0)+Y_{20}^{(1)}(0)\big )+C_{1010}\big (Y_{11}^{(3)}(0)\\{} & {} +\frac{Y_{20}^{(3)}(0)}{2}+\frac{\overline{\beta }_{0}Y_{20}^{(1)}(0)}{2}+\beta _{0}Y_{11}^{(1)}(0)\big )\\{} & {} +C_{0101}\big (Y_{11}^{(3)}(-\frac{\tau _{2}^{*}}{\tau _{1}})+\frac{\alpha _{0}}{2}\\{} & {} Y_{20}^{(3)}(-\frac{\tau _{2}^{*}}{\tau _{1}})+\frac{\overline{\beta }_{0}}{2}Y_{20}^{(1)}(0)e^{i\omega _{0}\tau _{2}^{*}}\\{} & {} +\beta _{0}Y_{11}^{(2)}(0)e^{-i\omega _{0}\tau _{2}^{*}}\big )++C_{0011}\big (\beta _{0}Y_{11}^{(3)}(-\frac{\tau _{2}^{*}}{\tau _{1}})\\{} & {} +\frac{\overline{\beta }_{0}}{2}Y_{20}^{(3)}(-\frac{\tau _{2}^{*}}{\tau _{1}})+\frac{\overline{\beta }_{0}}{2}Y_{20}^{(3)}(0)\\{} & {} e^{i\omega _{0}\tau _{2}^{*}}+\beta _{0}Y_{11}^{(3)}(0)e^{-i\omega _{0}\tau _{2}^{*}}\big ),\\ \end{aligned}$$

and

$$\begin{aligned}{} & {} L_{2000}=\frac{2d\alpha _{1}W^{*}}{(d+U^{*})^{3}},\quad L_{1001}=-\frac{d\alpha _{1}}{(d+U^{*})^{2}},\\{} & {} B_{0200}=-\frac{2s}{K},\quad B_{0110}=-(a+mW^{*}),\\{} & {} B_{0101}=-mW^{*},\quad B_{0011}=-mV^{*},\\{} & {} C_{2000}=-\frac{2d\alpha _{2}W^{*}}{(d+U^{*})^{3}},\quad C_{1010}=\frac{d\alpha _{2}}{(d+U^{*})^{2}},\\{} & {} C_{0110}=c(a+mW^{*}),\quad C_{0101}=cmW^{*},\\{} & {} C_{0011}=cmV^{*}. \end{aligned}$$

We also calculate the following values

$$\begin{aligned} U(0)= & {} w+\bar{w}+Y^{(1)}_{20}(0)\frac{w^{2}}{2}+Y^{(1)}_{11}(0)w\bar{w}\\{} & {} +Y^{(1)}_{02}(0)\frac{\bar{w}^{2}}{2}+Y^{(1)}_{21}(0)\frac{w^{2}\bar{w}}{2}+\cdots ,\\ V(0)= & {} w\alpha _{0}+\bar{w}\bar{\alpha }_{0}+Y^{(2)}_{20}(0)\frac{w^{2}}{2}+Y^{(2)}_{11}(0)w\bar{w}\\{} & {} +Y^{(2)}_{02}(0)\frac{\bar{w}^{2}}{2}+Y^{(2)}_{21}(0)\frac{w^{2}\bar{w}}{2}+\cdots ,\\ W(0)= & {} w\beta _{0}+\bar{w}\bar{\beta }_{0}+Y^{(3)}_{20}(0)\frac{w^{2}}{2}+Y^{(3)}_{11}(0)w\bar{w}\\{} & {} +Y^{(3)}_{02}(0)\frac{\bar{w}^{2}}{2}+Y^{(3)}_{21}(0)\frac{w^{2}\bar{w}}{2}+\cdots ,\\ \end{aligned}$$

$$\begin{aligned} U(-\frac{\tau _{2}^{*}}{\tau _{10}'})= & {} we^{-i\omega _{0}\tau _{2}^{*}}+\bar{w}e^{i\omega _{0}\tau _{2}^{*}}+Y^{(1)}_{20}(-\frac{\tau _{2}^{*}}{\tau _{10}'})\\{} & {} \times \frac{w^{2}}{2}+Y^{(1)}_{11}(-\frac{\tau _{2}^{*}}{\tau _{10}'})w\bar{w}\\{} & {} +Y^{(1)}_{02}(-\frac{\tau _{2}^{*}}{\tau _{10}'})\frac{\bar{w}^{2}}{2} +Y^{(1)}_{21}(-1)\frac{w^{2}\bar{w}}{2}\\{} & {} +\cdots ,\\ V(-\frac{\tau _{2}^{*}}{\tau _{10}'})= & {} w\alpha _{0}e^{-i\omega _{0}\tau _{2}^{*}}+\bar{w}\bar{\alpha }_{0}e^{i\omega _{0}\tau _{2}^{*}}\\{} & {} +Y^{(2)}_{20}(-\frac{\tau _{2}^{*}}{\tau _{10}'})\frac{w^{2}}{2}+Y^{(2)}_{11}(-\frac{\tau _{2}^{*}}{\tau _{10}'})w\bar{w}\\{} & {} +Y^{(2)}_{02}(-\frac{\tau _{2}^{*}}{\tau _{10}'})\frac{\bar{w}^{2}}{2}\\{} & {} +Y^{(2)}_{21}(-\frac{\tau _{2}^{*}}{\tau _{10}'})\frac{w^{2}\bar{w}}{2}\\{} & {} +\cdots ,\\ W(-\frac{\tau _{2}^{*}}{\tau _{10}'})= & {} w\beta _{0}e^{-i\omega _{0}\tau _{2}^{*}}+\bar{w}\bar{\beta }_{0}e^{i\omega _{0}\tau _{2}^{*}}\\{} & {} +Y^{(3)}_{20}(-\frac{\tau _{2}^{*}}{\tau _{10}'})\frac{w^{2}}{2}+ Y^{(3)}_{11}(-\frac{\tau _{2}^{*}}{\tau _{10}'})w\bar{w}\\{} & {} +Y^{(3)}_{02}(-\frac{\tau _{2}^{*}}{\tau _{10}'})\frac{\bar{w}^{2}}{2}+Y^{(3)}_{21}(-\frac{\tau _{2}^{*}}{\tau _{10}'})\frac{w^{2}\bar{w}}{2}\\{} & {} +\cdots ,\\ \end{aligned}$$

Since, we have

$$\begin{aligned} h(w,\bar{w})= & {} h_{20}\frac{w^{2}}{2}+h_{11}w\bar{w}+h_{02}\frac{\bar{w}^{2}}{2} +h_{21}\frac{w^{2}\bar{w}}{2}+\cdots .\nonumber \\ \end{aligned}$$

(5.6)

The following coefficients have been examined from (5.5) and (5.6) as

$$\begin{aligned}&h_{20}=\frac{2\tau _{10}'}{\overline{D}}(O_{11}+\overline{\alpha _{0}^{*}}O_{21}+\overline{\beta _{0}^{*}}O_{31}),\\&h_{11}=\frac{\tau _{10}'}{\overline{D}}(O_{12}+\overline{\alpha _{0}^{*}}O_{22}+\overline{\beta _{0}^{*}}O_{32}),\\&h_{02}=\frac{2\tau _{10}'}{\overline{D}}(O_{13}+\overline{\alpha _{0}^{*}}O_{23}+\overline{\beta _{0}^{*}}O_{33}),\\&h_{21}=\frac{2\tau _{10}'}{\overline{D}}(O_{14}+\overline{\alpha _{0}^{*}}O_{24}+\overline{\beta _{0}^{*}}O_{34}),\\ \end{aligned}$$

Since, $O_{14}$, $O_{24}$, and $O_{34}$ are contained in $h_{21}$, and they are containing $Y_{20}$ and $Y_{11}$, so we still need to compute $Y_{20}^{l}(\beta )$ and $W_{11}^{l}(\beta )$ for $l=1,2,3$. After straightforward calculation, we get the value as

$$\begin{aligned} \begin{aligned} Y_{20}(\beta )&=-\frac{ih_{20}q(0)e^{i\omega _{0}\tau _{10}'\beta }}{\omega _{0}\tau _{10}'}+\frac{i\bar{h}_{02}\bar{q}(0)}{3\omega _{0}\tau _{10}'}e^{-i\omega _{0}\tau _{10}'\beta }\\&\quad +I_{1}e^{2i\omega _{0}\tau _{10}'\beta },\\ Y_{11}(\beta )&{=}-\frac{ih_{11}q(0)e^{i\omega _{0}\tau _{10}'\beta }}{\omega _{0}\tau _{10}'}\!+\!\frac{i\bar{h}_{11}\bar{q}(0)e^{-\!i\omega _{0}\tau _{10}'\beta }}{\omega _{0}\!\tau _{10}'}\!+\!I_{2}, \end{aligned} \end{aligned}$$

where $I_{1}=(I^{(1)}_{1},I^{(2)}_{1},I^{(3)}_{1})^{T}\in \mathbb {R}^{3}$ and $I_{2}=(I^{(1)}_{2},I^{(2)}_{2},I^{(3)}_{2})^{T}\in \mathbb {R}^{3}$ are integral constant, which are examined in Table 2 given below.

Table 2 Values of $I_{1}$ and $I_{2}$

Thus, $h_{21}$ can be determined and expressed by the delay parameters $\tau _{10}'$ and $\tau _{2}^{*}$, and the standard results have been calculated as given below:

$$\begin{aligned} \begin{aligned}&c_{1}(0)=\frac{i(h_{20}h_{11}-2|h_{11}|^{2}-\frac{|h_{20}|^{2}}{3})}{2\omega _{0}\tau _{10}'}+\frac{h_{21}}{2},\\&\mu _{2}=-\frac{Re\{c_{1}(0)\}}{Re\{\frac{d\lambda (\tau _{10}')}{d\tau _{1}}\}},\\&\nu _{2}=2Re\{c_{1}(0)\},\\&T_{2}=-\frac{\Im \{c_{1}(0)\}+\mu _{2}\Im \{\frac{d\lambda (\tau _{10}')}{d\tau _{1}}\}}{\omega _{0}\tau _{10}'}. \end{aligned} \end{aligned}$$

The above values provides a detail explanation of bifurcating oscillating solution at the critical value $\tau _{10}'$ of time delay in the center manifold of the system (2.1). The results can be stated as:

Theorem 5

In the neighborhood of $\mu =0$, the following conditions hold:

• (1) The Hopf-bifurcation is supercritical if $\mu _{2}>0$, and subcritical if $\mu _{2}<0$.

• (2) The bifurcated periodic solution is stable if $\nu _{2}<0$, and unstable if $\nu _{2}>0$.

• (3) The bifurcated periodic solution increases if $T_{2}>0$, and decreases if $T_{2}<0$.

6 Numerical analysis

We will demonstrate some simulation results in this section for graphical visualization of our analytical results using MATLAB software.

Fig. 1

Dynamics of the system (2.1) for the parameter set (6.1) when $\tau _{1},\tau _{2}=0$: a Time series for population U, V and W, b 3-dimension view

Fig. 2

a Supercritical bifurcation at $m^{*}=0.025$, b stable limit cycle at $m=0.05$, for the parameter set (6.1) except m when $\tau _{1},\tau _{2}=0$. Trajectory in magenta color started from (5, 4.3, 4.2), black color started from (7, 5, 3), and green color started from (3, 2.8, 2.4)

Fig. 3

Bifurcation diagrams for the species U, V and W, where m is a bifurcation parameter when $\tau _{1}=0$, $\tau _{2}=0$ and other parameters defined in (6.1)

Fig. 4

a Supercritical bifurcation at $b^{*}=0.056$, b stable limit cycle at $b=0.0398$, for the parameter set (6.1) except b when $\tau _{1},\tau _{2}=0$. Trajectory in magenta, blue, and green colors start from (5, 4.3, 4.2), (7, 5, 3), and (5, 4.7, 3), respectively

Fig. 5

a Bifurcation curve with stability and instability regions with respect to the parameter m and b, when $\tau _{1}=\tau _{2}=0$, b global stability of the predator free equilibrium when $b=0.05$, $d_{1}=0.98$, $\alpha _{1}=1.2$, $d=12$, $d_{2}=1.10$, $a=0.05$, and the rest parameters are defined in (6.1). Yellow and magenta colors dashed line show nullcline of U and V

6.1 Delay free system

Case I: When $\tau _{1},\tau _{2}=0$, then we have chosen the following set of parameter for the system (2.1):

$$\begin{aligned}&r=1.3, \hspace{0.2cm} b=0.25, \hspace{0.2cm} d_{1}\!=\!0.34,\hspace{0.2cm} \alpha _{1}\!=\!1.18,\hspace{0.2cm} d\!=\!20,\nonumber \\&\quad s=1.15,\hspace{0.1cm} K=100, \hspace{0.1cm}d_{2}\!=\!0.38,\hspace{0.1cm} a=0.26,\nonumber \\&\quad m\!=\!0.005,\alpha _{2}=0.32,\hspace{0.1cm} c=0.28,\hspace{0.1cm} d_{3}=0.3, \end{aligned}$$

(6.1)

with initial solutions $U(0)=7$, $V(0)=5$, $W(0)=3$. For the parameters defined in (6.1), there exists a coexistence equilibrium point $E_{3}(4.9872,3.0065,4.1004)$ such that the local stability conditions given in Sect. (4.1) are held, i.e. $J_{1}=1.1771>0$, $J_{2}=0.2905>0$, $J_{3}=0.2898>0$, and $J_{1}J_{2}-J_{3}=0.0521>0$. Hence, the system is local asymptotically stable about $E_{3}$ which is demonstrated in Fig. 1. From this figure, it is noted that the population densities fluctuate initially, and after some finite time, they settle down into a stable state at respective equilibrium. Next, in the bifurcation diagram Fig. 3, if we magnify predator’s cooperative strength m keeping other parameters exactly same as (6.1), then we see that as m increases the fluctuations of populations also increase and the system (2.1) enters into Hopf bifurcations at a Hopf point $m=m^{*}=0.025$. Further increment in the strength of m takes the system towards an unstable dynamics for $m>m^{*}$ with a limit cycle. Since, this limit cycle is stable and coexist with unstable equilibrium point, where the solution trajectories start from different initial conditions converge into a unique limit cycle (shown in black color in Fig. 2b). Therefore, the bifurcation occurs at $m=m^{*}$ is a type of supercritical.

Similarly, if we deduce the value of parameter b fixing all other parameter as (6.1), then the system (2.1) goes into Hopf bifurcation at critical point $b=b^{*}=0.056$. Now, if we further deduce value of b, then the system becomes unstable for $b<b^{*}$ and a periodic solution exists. Thus, we obtain a stable limit cycle which is coexist with an unstable equilibrium point, where the solution trajectories start from different initial conditions converge into a unique limit cycle (shown in blue color). Therefore, the bifurcation appears at $b=b^{*}$ is a supercritical type which is shown in Fig. 4.

In Fig. 5a, a bifurcation curve with respect to the parameters m and b has been plotted for the model (2.1). Each and every points of the curve (shown in red color) are a Hopf points at which the system switches their dynamics. The whole region is divided into instable and stable regions by a Hopf line curve. The upper part of the curve indicates the stable region and the lower part indicates instable region.

Now, if we take $b=0.05$, $d_{1}=0.98$, $\alpha _{1}=1.2$, $d=12$, $d_{2}=1.10$, $a=0.05$ and remaining parameters are fixed as (6.1), then the predator species may destroy. We draw nullcline for the above parameters and obtained a unique predator free equilibrium point $E_{2}(12.41,9.83,0.00)$ in Fig. 5b. This figure also depicts a globally stable behaviour because all solution trajectories start from different initial points move towards the equilibrium $E_{2}$. Mathematically, we have analysed that if $E_{2}$ exists, then it is unique. Therefore, if $E_{2}$ is stable, then it is also globally stable.

6.2 Delayed system

After simulation of the system (2.1) in the absence of delay, we will analyse the dynamical behaviour of the system in the presence of delay in this subsection. In order to see the delay induced stability changes and to clarify some mathematical results performed in previous sections, we follow the following set of parameters given below:

$$\begin{aligned}&r=1.3, \hspace{0.2cm} b=0.25, \hspace{0.2cm} d_{1}\!=\!0.34,\hspace{0.2cm} \alpha _{1}\!=\!1.18,\hspace{0.2cm} d\!=\!20,\nonumber \\&s=1.15,\hspace{0.1cm} K=100, \hspace{0.1cm}d_{2}=0.38,\hspace{0.1cm} a=0.26,\nonumber \\&m=0.01,\alpha _{2}=0.32,\hspace{0.1cm} c=0.28,\hspace{0.1cm} d_{3}=0.3, \end{aligned}$$

(6.2)

with initial solutions $U(0)=7$, $V(0)=5$, $W(0)=3$.

Fig. 6

Equilibrium $E_{3}$ is: a stable when $\tau _{2}=0.6(<\tau _{2*}^{(1)})$, b unstable when $\tau _{2}=1.35(>\tau _{2*}^{(1)})$, for the parameters (6.2) when $\tau _{1}=0$. The solution trajectories shown in blue and magenta colors are started from the initial conditions (7, 5, 3) and (8.5, 6.9, 5.5), respectively

Fig. 7

Bifurcation diagrams for the species U, V and W where $\tau _{2}$ is a bifurcation parameter and $\tau _{1}=0$. The system (2.1) changes their stability six times in the interval [0, 35]

Fig. 8

Dynamics of $E_{3}$ when $\tau _{1}=0$: a stability at $\tau _{2}=0.6,7.86,20.35,33.45$, b instability at $\tau _{2}=1.40,14.23,30.73$, in blue, green and red colors, respectively for the parameters (6.2)

Case II: When $\tau _{1}=0,\tau _{2}>0$, then we compute a positive equilibrium point as $E_{3}(4.7834,2.8488,3.8746)$ which satisfy the locally asymptotically stable conditions, i.e. $J_{1}=1.1610>0$, $J_{2}=0.2937>0$, $J_{3}=0.3030>0$, and $J_{1}J_{2}-J_{3}=0.0380>0$ discussed in Sect. (4.1) for parameters (6.2). Putting $j=0$, then we have determined two critical values of $\omega$ as

$$\begin{aligned} \omega _{0}^{(1)}=0.5067,\hspace{0.1cm}\omega _{0}^{(2)}=0.4654, \end{aligned}$$

and with respect to these $\omega 's$, we also have two critical values of $\tau _{2}$ as

$$\begin{aligned} \tau _{2*}^{(1)}=0.9718,\hspace{0.1cm}\tau _{2*}^{(2)}=4.0809, \end{aligned}$$

respectively, from (4.10) and (4.11). The transversality conditions are also satisfied, i.e.

$Re\Big [\frac{d\lambda }{d\tau _{2}}\Big ]^{-1}_{\lambda =i\omega _{0}^{(1)},\tau =\tau _{2*}^{(1)}}=660.7822>0$, and

$Re\Big [\frac{d\lambda }{d\tau _{2}}\Big ]^{-1}_{\lambda =i\omega _{0}^{(2)},\tau =\tau _{2*}^{(2)}}=577.7240>0$.

Hence, The system (2.1) enters into Hopf bifurcation at first critical value $\tau _{2}=\tau _{2*}^{(1)}$. Therefore, the system exhibits stable behaviour when $\tau _{2}<\tau _{2*}^{(1)}$ and instable behaviour when $\tau _{2}>\tau _{2*}^{(1)}$. The stable and instable dynamics are shown through the time series diagram for $\tau _{2}=0.6(<\tau _{2*}^{(1)})$, and through the phase portrait diagram for $\tau _{2}=1.35(>\tau _{2*}^{(1)})$, respectively in the Fig. 6. In Fig. 6b, the limit cycle is stable, so at $\tau _{2}=\tau _{2*}^{(1)}$ supercritical bifurcation occurs. Furthermore, the dynamical nature of the system switches for the critical values of $\tau _{2}$ at $\tau _{2*}^{(2)}=4.0809$, $\tau _{2*}^{(3)}=13.45$, $\tau _{2*}^{(4)}=17.83$, $\tau _{2*}^{(5)}=25.64$, and $\tau _{2*}^{(6)}=31.48$. There are six times stability switches have been examined from the bifurcation diagram Fig. 7. Thus, the system is stable for $\tau _{2}$ $\in$ $\big [0, \tau _{2*}^{(1)})$ $\bigcup$ $(\tau _{2*}^{(2)}, \tau _{2*}^{(3)})$ $\bigcup$ $(\tau _{2*}^{(4)}, \tau _{2*}^{(5)})$ $\bigcup$ $(\tau _{2*}^{(6)}, 35\big ]$ and unstable for $\tau _{2}$ $\in$ $(\tau _{2*}^{(1)}, \tau _{2*}^{(2)})$ $\bigcup$ $(\tau _{2*}^{(3)}, \tau _{2*}^{(4)})$ $\bigcup$ $(\tau _{2*}^{(5)}, \tau _{2*}^{(6)})$. We have plotted time series and phase plane view to see the stable and instable nature for different values of $\tau _{2}\in [0,35]$ in Fig. 8.

In Table 3, we have shown that how the cooperative delay $\tau _{2}$ affects the conversion rate b. From the table, it is analysed that as $\tau _{2}$ inclines, the length of stable interval of b declines. Further, when cooperative strength becomes high, i.e. $\tau _{2}>0.7$ then it destabilizes the system for all $b>0$.

Case III: When $\tau _{1}>0$, $\tau _{2}=0$, then equilibrium $E_{3}$ is computed and obtained exactly same as Case II. Some mathematical calculation gives the following roots $q=-0.3240$ and $q=0.2915\pm 0.0772i$ of the Eq. (4.15). Therefore, a positive real root $\omega _{0}$ of (4.14) does not exist in this case. Thus, there is no critical value of $\tau _{1}$ for which the system can change its dynamical nature. It means, equilibrium $E_{3}$ depicts a stable dynamics for all $\tau _{1}\ge 0$. This stability nature of the system can be seen through time evaluation diagrams for the different values of $\tau _{1}$ in Fig. 9. Now, if we replace $d_{1}=0.34$ by $d_{1}=0.01$, then the system can changes their nature from a stable behaviour to the limit cycle oscillation via Hopf bifurcation at critical point $\tau _{1}^{*}=9.50$. From the bifurcation diagram Fig. 10, the stability can be clearly observed in the left side of the figure when $\tau _{1}<\tau _{1}^{*}$ and instability in the right side of the figure when $\tau _{1}>\tau _{1}^{*}$. To verify the stability and instability shown in Fig. 10, the phase space and time evaluation diagram have also been shown in Fig. 11 for $\tau _{1}=1,7,11.85$.

In Table 3, we have also computed that how the maturity delay $\tau _{1}$ influences the conversion rate b. From the table, it is concluded that when $\tau _{1}=0$, the model depicts stable nature only in the small range $b\in (0.056,1.53)$. But as maturity delay involve in the system it stabilizes the system to the wide range $b\in (0,16.58)$. Further, as $\tau _{1}$ increases the length of stable interval of b decreases.

Table 3 Effects of $\tau _{1}$ and $\tau _{2}$ on the stability range of b for the parameters (6.2)

Case IV: When $\tau _{1}=0.5\in (0,\hspace{0.1cm}\infty )$, $\tau _{2}>0$, then a little computation yields a stable steady state $E_{3}(4.8896,2.9289,3.9879)$ which is depicted in Fig. 12 for parameters (6.2). The critical points are calculated as

$$\begin{aligned} \omega _{0}=0.4860, \hspace{0.15cm} \tau _{20}^{(1)}=1.3543. \end{aligned}$$

The transversality condition is also fulfilled. Hence, the system (2.1) under goes into Hopf bifurcation at $\tau _{2}=\tau _{20}^{(1)}$, which is a supercritical bifurcation. Further, the stability switches occurs for some different critical points of $\tau _{2}$ at $\tau _{20}^{(2)}=4.08$, $\tau _{20}^{(3)}=14.32$, $\tau _{20}^{(4)}=18.35$, $\tau _{20}^{(5)}=27.36$, and $\tau _{20}^{(6)}=32.63$. This stability behavior can be clearly visualize from the bifurcation figure Fig. 13 in which the stability switches have been appeared six times in the interval $\tau _{2}\in [0,35]$.

Thus, the system is stable for $\tau _{2}$ $\in$ $\big [0, \tau _{20}^{(1)})$ $\bigcup$ $(\tau _{20}^{(2)}, \tau _{20}^{(3)})$ $\bigcup$ $(\tau _{20}^{(4)}, \tau _{20}^{(5)})$ $\bigcup$ $(\tau _{20}^{(6)}, 35\big ]$ and unstable for $\tau _{2}$ $\in$ $(\tau _{20}^{(1)}, \tau _{20}^{(2)})$ $\bigcup$ $(\tau _{20}^{(3)}, \tau _{20}^{(4)})$ $\bigcup$ $(\tau _{20}^{(5)}, \tau _{20}^{(6)})$. We have plotted time series and phase plane view to clearly visualize the stable and instable nature for different values of $\tau _{2}\in [0,35]$ in Fig. 14.

Fig. 9

a, b Stable nature of the system (2.1) at $\tau _{1}=1,4,8,13,19,26,36,40,50,60,70,80,90,100$, where $\tau _{2}=0$, for the parameters (6.2)

Fig. 10

Bifurcation diagrams for the species U, V and W where $\tau _{1}$ is a bifurcation parameter when $d_{1}=0.01$, $\tau _{2}=0$ and other parameters defined in (6.2)

Fig. 11

a Stable behavior for $\tau _{1}=1,7$ $(<\tau _{1}^{*})$, b periodic solution when $\tau _{1}=11.85(>\tau _{1}^{*})$, for the parameters (6.2) when $\tau _{2}=0$

Fig. 12

$E_{3}$ is stable when $\tau _{1}=0.5$, $\tau _{2}=0$ for parameters (6.2) a time series, b phase space of U, V and W, c stable limit oscillation at $\tau _{2}=1.82 (>\tau _{20}^{(1)})$ when $\tau _{1}=0.5$ for the two initial conditions (7, 5, 3) (blue color) and (5.2, 4.8, 2.93) (red color)

Fig. 13

Bifurcation diagrams for the species U, V and W where $\tau _{2}$ is a bifurcation parameter, $\tau _{1}=0.5$ and remaining parameters same as (6.2). The system (2.1) changes their stability six times in the interval [0, 35]

Fig. 14

Dynamics of the model (2.1) for the parameter (6.2) when $\tau _{1}=0.5$: a time series at $\tau _{2}=1.01,8.12,24.65,34.19$, b phase portraits at $\tau _{2}=3.5$ (magenta color), $\tau _{2}=15.3$(blue color), and $\tau _{2}=28.2$ (green color) for the parameters (6.2)

Fig. 15

a Stable nature at $\tau _{2}=0.65\in (0,\hspace{0.1cm}\tau _{2*}^{(1)})$, when $\tau _{1}=0$, b unstable behavior at $\tau _{2}=0.65$, $\tau _{1}=11.67$ for the parameters (6.2)

Case V: When $\tau _{1}>0$, $\tau _{2}=0.65\in (0,\hspace{0.1cm}\tau _{2*}^{(1)})$, then a positive stable equilibrium $E_{3}(4.7837,2.8545,3.8653)$ exists which is demonstrated in Fig. 15a for the parameters (6.2). Now, we increase $\tau _{1}$ to know the impact on the stability of the system. The bifurcation occurs at $\tau _{1}=\tau _{1}^{1*}=10.25$ and the system becomes unstable (Fig. 15b). Again, an instability is changed into a stable behavior and a stable behavior is changed into an unstable behavior via bifurcations at $\tau _{1}=\tau _{1}^{2*}=12.25$ and $\tau _{1}=\tau _{1}^{3*}=21.17$, respectively. Thus, the system demonstrate stable nature for $\tau _{1}\in (0,\tau _{1}^{1*})$ $\bigcup$ $(\tau _{1}^{2*},\tau _{1}^{3*})$, and unstable for $\tau _{1}\in (\tau _{1}^{1*},\tau _{1}^{2*})$ $\bigcup$ $(\tau _{1}^{3*}, 25\big ]$. We have drawn bifurcation diagram in Fig. 16 with respect to $\tau _{1}$ to visualize the effect of $\tau _{1}$ when $\tau _{2}=0.65\in (0,\hspace{0.1cm}\tau _{2*}^{(1)})$.

Fig. 16

Bifurcation diagrams for the species U, V and W where $\tau _{1}$ is a bifurcation parameter, $\tau _{2}=0.65$ and remaining parameters same as (6.2). The system (2.1) changes their behavior three times in the interval [0, 25]

Fig. 17

Region of attraction of the system (2.1) with respect to the parameters $\tau _{1}$ and $\tau _{2}$ for the set of parameters (6.2). Green shaded area represents stable region and red shaded areas represent unstable regions

One important features of a non-linear dynamical system is the region of attraction (ROA). ROA of dynamical system is a subset of a state space where an equilibrium point becomes locally asymptotically stable inside the subset and becomes unstable outside it. Therefore, finding stability and instability regions becomes more helpful to analyse the dynamical behavior of the system very clearly. So, in Fig. 17, we have plotted ROA for the model (2.1) with respect to the parameters $\tau _{1}$ and $\tau _{2}$ and remaining parameter’s values are given in (6.2). From this figure, it is examined that for every value of $\tau _{1}\in [0,1.25]\bigcup [9.30,14]$, there exists a critical value of $\tau _{2}$ where the stability switch occurs and the system enters into one state to another state via bifurcation. But if $\tau _{1}\in (1.25,9.30)$, then stability switch do not occur and the system shows stable behavior for all $\tau _{2}\in [0,5]$. Similarly, when $\tau _{2}\in (0.56,4.48)$, then there exists a critical values of $\tau _{1}$ at which dynamical behavior of the system is changed through bifurcation. But, when $\tau _{2}\in [0,0.56)\bigcup (4.48,5]$, then there exists no critical point of $\tau _{1}$ and the system (2.1) depicts stable dynamics for all $\tau _{1}\in [0,14]$.

7 Discussion and conclusion

The present study describes the global dynamical complexities of a prey-predator model including the concepts of stage structure in prey population and two discrete time delays for the maturation and cooperation. We have examined how cooperative delay ($\tau _{2}$), maturity delay ($\tau _{1}$), predator’s cooperative strength (m) and transformation rate (b) of juvenile prey can generate a richly complex dynamic in the system. The fundamental analytical results like positivity, boundedness and steady points of the model have been determined. The model contains only three equilibrium points namely, trivial ($E_{1}$), predator free ($E_{2}$) and a positive equilibrium ($E_{3}$). It has been examined that the equilibrium $E_{2}$ exists uniquely and depicts the global stable behavior. Ecologically, the prey species survive always in the system while the predator go to extermination. The stability conditions of equilibrium $E_{3}$ is derived in the Sect. 4 with delay and non-delay. Biologically, high abundance of prey species sufficiently saturates the predators due to which stable dynamics appears in the system. Xu [24] analyzed that high density of mature prey balance the prey-predator system and all species coexist. Increasing in cooperative hunting rate (m) of predators suppresses mature prey populations, leading to population declines for all species. Biologically, the high group hunting of predators easily catches prey species due to which population of all species decline and fluctuate. Alves and Hilker [12] suggests that hunting cooperation can be harmful to predators when prey density drastically decreases due to increased predation pressure, which in turn reduces predator intake. Similarly, low value of the conversion rate b (when $b<b^{*}$) of immature prey forces the populations to fluctuate and destabilizes the system. It means when the conversion rate from immature prey individual to mature individual declines per unit time, the number of immature population in the system increases which causes unstable dynamics. Further, Hopf bifurcation of the delayed model is analysed mathematically and numerically with respect to delay parameters. In order to discuss the dynamics like stability switches concerning $\tau _{1}$ and $\tau _{2}$ as a varying parameters, we have established some possible conditions to the existence of Hopf bifurcation with different pairs of delay in Case-II to Case-V.

Through our detail simulation analysis, it is summed up that in each cases, as delay increases, fluctuations of the population densities increase slowly, and beyond a critical value of delay the system exhibits unstable behavior with oscillatory solution. This suggests that time delays can destroy the stability of co-existing equilibria, leading to population fluctuations due to Hopf bifurcation. It is recognizable that in Case-I (for non-delay system) and Case-III there are only one time stability switch occur wherein the system goes into an unstable state from the stable one. It is also noticeable in Case-III that when the parameter $d_{1}$, which is proportional to the death rate of juvenile prey, is high ($d_{1}=0.34$), then maturation delay $\tau _{1}$ does not induce any stability switches, i.e. stable equilibria remains stable, but when $d_{1}$ becomes low ($d_{1}=0.01$), the maturation delay destabilizes the system through the stability switch. Again in Case-II, IV, and V, the parameters $\tau _{1}$ and $\tau _{2}$ induce multiple times stability switches in the system. It means, the dynamical behavior of the system is changed from stable to unstable and vice-versa as cooperative (maturity) delay increases while maturity (cooperative) delay is fixed in the stable interval. The similar types of stability switches has also been observed in due to maturation delay [37]. When maturation time of immature fish population increases, the periodic solution arises. Species extinction and permanence behavior are also depends on the maturation period of immature fish species [27]. Here, an interesting result is noted that the stability switch appears through $\tau _{1}$ is either due to in the presence of $\tau _{2}$, or due to the lower death rate of juvenile prey. Otherwise, in the absence of $\tau _{2}$, or in the high mortality of juvenile prey, the maturation delay does not destabilize the system. Biologically, the presence of cooperative hunting delay of predators reduce the maturation time of prey due to which the prey become mature very fast. So, the rapid maturation of prey is responsible for population fluctuations. The supercritical type Hopf bifurcation is also observed for the first critical value of $\tau _{2}$ and the parameters m, b through a stable bifurcating periodic solutions. A small maturation delay $\tau _{1}$ stabilizes the system while increasing value of $\tau _{1}$ and $\tau _{2}$ lowers the stable range of conversion rate b (see Table 3). The same result has been obtained by [22] for maturation delay parameter. Ecologically, the conversion rate of immature population decreases as maturation delay and group hunting rate increases, i.e. less number of immature prey becomes mature. Graphically, the region of attractions have also been represented in the plane $\tau _{1}-\tau _{2}$ and $m-b$ to clearly visualize stability and instability regions. Finally, from an ecological perspective, the outcomes we observed in this study further enrich the ecological significance and it will help to understand both the cooperating hunting and maturity delay induced dynamics of the proposed prey-predator delayed system. This study shows that the delay in cooperative hunting affects the maturation time of the prey in a very strong way.