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Period-doubling and Neimark–Sacker bifurcations of plant–herbivore models

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Abstract

The interaction between plants and herbivores plays a vital role for understanding community dynamics and ecosystem function given that they are the critical link between primary production and food webs. This paper deals with the qualitative nature of two discrete-time plant–herbivore models. In both discrete-time models, function for plant-limitation is of Ricker type, whereas the effect of herbivore on plant population and herbivore population growth rate are proportional to functional responses of type-II and type-III. Furthermore, we discuss the existence of equilibria and parametric conditions of topological classification for these equilibria. Our analysis shows that positive steady states of both discrete-time plant–herbivore models undergo flip and Hopf bifurcations. Moreover, we implement a hybrid control strategy, based on parameter perturbation and state feedback control, for controlling chaos and bifurcations. Finally, we provide some numerical simulations to illustrate theoretical discussion.

Introduction

The mathematical framework for plant–herbivore models is identical to interaction between preys and their predators. In other words, such type models are basically modifications of prey–predator systems [1]. The interaction between plants and herbivores has been investigated by many researchers both in differential and difference equations. Kartal [2] investigated the dynamical behavior of a plant–herbivore model including both differential and difference equations. Kang et al. [3] discussed bistability, bifurcation, and chaos control in a discrete-time plant–herbivore model. Liu et al. [4] investigated stability, limit cycle, Hopf bifurcations, and homoclinic bifurcation for a plant–herbivore model with toxin-determined functional response. Li et al. [5] discussed period-doubling and Hopf bifurcations for a plant–herbivore model incorporating plant toxicity in the functional response of plant–herbivore interactions. Similarly, for some other discussions related to qualitative behavior of plant–herbivore models, we refer the interested reader to [6,7,8,9,10,11,12,13] and references therein.

We consider interaction between plants and herbivores by taking into account nonoverlapping generation. For this, the growth rate for herbivore population is assumed to be proportional to a function of nonlinear type dependent upon their feeding rate [14]. Moreover, we suppose that the growth rate for plant population is inversely proportional to population of herbivore, and the feeding rate of herbivore is dependent on plant density [15]. Furthermore, an intraspecific competition is implemented for controlling the plant population density in the absence of herbivore [14]. Then the general discrete-time plant–herbivore interaction is modeled by the following planar system [16]:

$$ \begin{gathered} U_{n +1} = r U_{n} f ( U_{n} ) g ( V_{n} ), \\ V_{n +1} = c V_{n} h ( U_{n} ), \end{gathered} $$
(1.1)

where \(U_{n}\) and \(V_{n}\) represent the population densities of plant and herbivore at generation n, respectively. Furthermore, \(f ( U_{n} )\) is used for the growth rate of plant, the effect of herbivore population on the plant growth rate is denoted by the function \(g ( V_{n} )\), the saturation function for plant density is represented by \(h ( U_{n} )\), and the population growth rates for plant and herbivore are denoted by r and c, respectively.

Arguing as in [16], we can choose \(f ( U_{n} ) = e^{- a U _{n}}\) as the Ricker growth function [17], where larger values for a give stronger density dependence in the growth rate of plant. Furthermore, we can choose for \(g ( V_{n} )\) and \(h ( U_{n} )\) functional responses of type II as follows:

$$ g ( V_{n} ) = \frac{\alpha }{\beta + V_{n}}, \quad\quad h ( U _{n} ) = \frac{\gamma U_{n}}{\delta + U_{n}}, $$

where α, β, γ, and δ are positive parameters. Then we rewrite system (1.1) as follows:

$$ U_{n +1} = \frac{r \alpha U_{n} e^{- a U_{n}}}{\beta + V_{n}}, \quad\quad V_{n +1} = \frac{c \gamma V_{n} U_{n}}{\delta + U_{n}}. $$
(1.2)

For the dimensionless form of system (1.2), we choose \(x_{n} = a U _{n}\) and \(y_{n} = V_{n} / \beta \). Then it follows that

$$ x_{n +1} = \frac{k x_{n} e^{- x_{n}}}{1+ y_{n}}, \quad\quad y_{n +1} = \frac{p x _{n} y_{n}}{q + x_{n}}, $$
(1.3)

where \(k = \frac{r \alpha }{\beta }\), \(c \gamma = p\), \(a \delta = q\), and \(x_{n}\) and \(y_{n}\) are population densities of plant and herbivore at generation n, respectively.

Similarly, we can implement functional responses of type-III for \(g ( V_{n} )\) and \(h ( U_{n} )\) as follows:

$$ g ( V_{n} ) = \frac{\alpha _{1}}{\beta _{1}^{2} + V_{n} ^{2}}, \quad\quad h ( U_{n} ) = \frac{\gamma _{1} U_{n}^{2}}{\delta _{1}^{2} + U_{n}^{2}}. $$

Due to implementation of functional responses of type-III, we obtain the following plant–herbivore model:

$$ U_{n +1} = \frac{r \alpha _{1} U_{n} e^{- a U_{n}}}{\beta _{1}^{2} + V _{n}^{2}}, \quad\quad V_{n +1} = \frac{c \gamma _{1} V_{n} U_{n}^{2}}{\delta _{1} ^{2} + U_{n}^{2}}. $$
(1.4)

Moreover, implementing the transformations \(x_{n} = a U_{n}\) and \(y_{n} = V_{n} / \beta _{1}\) to system (1.4), we have the following discrete-time plant–herbivore model:

$$ x_{n +1} = \frac{\mu x_{n} e^{- x_{n}}}{1+ y_{n}^{2}}, \quad\quad y_{n +1} = \frac{ \nu x_{n}^{2} y_{n}}{\eta + x_{n}^{2}}, $$
(1.5)

where \(\mu = r \alpha _{1} / \beta _{1}^{2}\), \(c \gamma _{1} = \nu \), and \(a^{2} \delta _{1}^{2} = \eta \).

It is worth pointing out the rationality of considering the two types of models, that is, functional response of type-II in system (1.3) and functional response of type-III in system (1.5). For this, note that the functional response of type-III is connected with switching predators, that is, if there is an option of prey, then the predator prefers the more bumper prey type. Consequently, the functional response of type-III has been observed in experiments. On the other hand, in real life studies the functional response of type-II is frequently observed [18]. Therefore system (1.3) is associated with real-life plant–herbivore interaction, and system (1.5) is a good representative for its experimental study.

In further discussion, we study the qualitative behavior for discrete-time plant–herbivore systems (1.3) and (1.5). The main contributions of this paper are as follows:

  • Keeping in view standard techniques for stability analysis of discrete-time systems, we analyze the local asymptotic behavior of these plant–herbivore models of discrete nature.

  • We investigated that both models undergo flip bifurcation at their positive steady states by implementing bifurcation theory, normal form theory, and center manifold theorem.

  • We study Hopf bifurcation for these models at their positive steady states with bifurcation theory of normal forms.

  • We introduce a chaos control method based on parameter perturbation and introduce state feedback methodology for controlling chaotic and fluctuating behaviors for these models.

  • We present numerical simulations for verification of our theoretical discussions.

In Sects. 2 and 3, we discuss the existence of steady states and their linearized stability for both systems (1.3) and (1.5). In Sect. 4, we investigate period-doubling bifurcations at positive equilibria of systems (1.3) and (1.5). In Sect. 5, we show that the positive steady states of both systems (1.3) and (1.5) undergo Neimark–Sacker bifurcations. Moreover, we implement a hybrid feedback control methodology for controlling chaos and bifurcations for both systems (1.3) and (1.5) in Sect. 6. Finally, in Sect. 7, we present numerical examples to support and illustrate theoretical discussion.

Linearized stability of system (1.3)

The equilibria of system (1.3) can be obtained by solving the following algebraic equations:

$$ x = \frac{kx e^{- x}}{1+ y}, \quad\quad y = \frac{pxy}{q + x}. $$

We can easily obtain the solutions for aforementioned algebraic system as \((0,0)\), \(( \ln ( k ),0 )\), and \(( \frac{q}{p -1}, k \exp ( - \frac{q}{p -1} ) -1 )\). Thus trivial equilibrium \((0,0)\) always exists, boundary equilibrium \(( \ln ( k ),0 )\) exists only for \(k >1\), and a unique positive equilibrium \(( \frac{q}{p -1}, k \exp ( - \frac{q}{p -1} ) -1 )\) for system (1.3) exists if and only if \(p >1\), \(k >1\), and \(0< q <\ln ( k )( p -1)\). For simplicity, the unique positive equilibrium point of system (1.3) is \(( u^{*}, k e^{- u^{*}} -1 )\), where \(u^{*}:= \frac{q}{p -1}\). Moreover, the Jacobian matrix of system (1.3) evaluated at \(( x, y )\) is as follows:

$$ J ( x, y )= \begin{bmatrix} \frac{k e^{- x} (1- x )}{1+ y} & - \frac{k e^{- x} x}{(1+ y )^{2}} \\ \frac{pqy}{( q + x )^{2}} & \frac{px}{q + x} \end{bmatrix}. $$
(2.1)

Taking \(( x, y )=(0,0)\) in (2.1), we obtain the Jacobian matrix \(J (0,0)\) at trivial equilibrium of system (1.3):

$$ J (0,0)= \begin{pmatrix} k & 0 \\ 0 & 0 \end{pmatrix}. $$

Hence, from \(J (0,0)\) it easy to see that the trivial equilibrium \((0,0)\) is a sink if and only if \(0< k <1\), it is saddle point if \(k >1\), and \((0,0)\) is nonhyperbolic at \(k =1\).

Taking into account the relation between roots and coefficients for a quadratic equation, the following lemma gives locality of roots with respect to the unit disk (see also [19,20,21,22,23,24]):

Lemma 2.1

Suppose that \(\mathbb{P}( \kappa )= \kappa ^{2} -A \kappa +B\) with \(\mathbb{P} (1)>0\) and \(\kappa _{1}\), \(\kappa _{2}\) are roots of \(\mathbb{P}( \kappa )=0\). Then, the following results hold: (i) \(\vert \kappa _{1} \vert <1\) and \(\vert \kappa _{2} \vert <1\) if and only if \(\mathbb{P} (-1)>0\) and \(\mathbb{P} (0)<1 \); (ii) \(\vert \kappa _{1} \vert <1\) and \(\vert \kappa _{2} \vert >1\), or \(\vert \kappa _{1} \vert >1\) and \(\vert \kappa _{2} \vert <1\) if and only if \(\mathbb{P} (-1)<0 \); (iii) \(\vert \kappa _{1} \vert >1\) and \(\vert \kappa _{2} \vert >1\) if and only if \(\mathbb{P} (-1)>0\) and \(\mathbb{P} (0)>1 \); (iv) \(\kappa _{1} =-1\) or \(\kappa _{2} =-1\) if and only if \(\mathbb{P} (-1)=0\); and (v) \(\kappa _{1}\) and \(\kappa _{2}\) are conjugate complex numbers with \(\vert \kappa _{1} \vert =1\) and \(\vert \kappa _{2} \vert =1\) if and only if \(A^{2} -4B<0\) and \(\mathbb{P} (0)=1\).

Furthermore, we suppose that \(k >1\). Then the variational matrix \(J ( x, y )\) at the boundary equilibrium \(( x, y )= ( \ln ( k ),0 )\) is computed as follows:

$$ J \bigl( \ln ( k ),0 \bigr) = \begin{bmatrix} 1-\ln ( k ) & -\ln ( k ) \\ 0 & \frac{p \ln ( k )}{q +\ln ( k )} \end{bmatrix}. $$
(2.2)

Lemma 2.2

Assume that \(k>1\). Then the following statements hold:

  1. (i)

    \(( \ln ( k ),0 )\) is a sink if and only if \(1< k < e^{2}\) and \(p \ln ( k )< q +\ln ( k )\).

  2. (ii)

    \(( \ln ( k ),0 )\) is a saddle point if and only if \(1< k < e^{2}\) and \(p \ln ( k )> q +\ln ( k )\), or \(k > e^{2}\) and \(p \ln ( k )< q +\ln ( k )\).

  3. (iii)

    \(( \ln ( k ),0 )\) is a source if and only if \(k > e^{2}\) and \(p \ln ( k )> q +\ln ( k )\).

  4. (iv)

    \(( \ln ( k ),0 )\) is nonhyperbolic if and only if \(k = e^{2}\) or \(p \ln ( k )= q +\ln ( k )\).

Moreover, for \(k \in [1,20]\), \(p \in [0,50]\), and \(q =35\), the topological classification of \(( \ln ( k ),0 )\) is depicted in Fig. 1.

Figure 1
figure1

Topological classification of \(( \ln ( k ),0 )\) for \({k \in [1,20]}\), \(p \in [0,50]\) and \(q =35\)

Next, we assume that \(0< q <( p -1)\ln ( k )\), \(k >1\), and \(p >1\). Then the variational matrix (2.1) computed at positive steady state \(( u^{*}, k e^{- u^{*}} -1 )\) is given as follows:

$$ J \bigl( u^{*}, k e^{- u^{*}} -1 \bigr) = \begin{bmatrix} 1- u^{*} & - \frac{u^{*} e^{u^{*}}}{k} \\ \frac{ ( k e^{- u^{*}} -1 ) ( p -1)}{p u^{*}} & 1 \end{bmatrix}. $$
(2.3)

Moreover, the characteristic polynomial for \(J ( u^{*}, k e^{- u ^{*}} -1 )\) is computed as follows:

$$ F ( \lambda )= \lambda ^{2} - \biggl( 2- \frac{q}{p -1} \biggr) \lambda +2- \frac{e^{\frac{q}{p -1}}}{k} - \frac{1}{p} + \frac{e^{\frac{q}{p -1}}}{kp} - \frac{q}{p -1}. $$
(2.4)

It must be noted that, due to the conditions for positivity of \(( u^{*}, k e^{- u^{*}} -1 )\), we have

$$ F (1)= \frac{ ( k - e^{u^{*}} ) ( p -1)}{kp} >0. $$

Thus we can implement Lemma 2.1 to prove the following results.

Lemma 2.3

Let \(0< q<(p-1)\ln (k)\), \(k>1\), and \(p>1\), then the following hold for the topological classification of equilibrium point \(( u^{*},k e^{- u^{*}} -1 )\) of system (1.3):

  1. (i)

    \(( u^{*},k e^{- u^{*}} -1 )\) is a sink if and only if

    $$ \frac{e^{\frac{q}{p -1}}}{k} + \frac{1}{p} + \frac{2 q}{p -1} < 5+ \frac{e ^{\frac{q}{p -1}}}{kp} $$

    and

    $$ 1+ \frac{e^{\frac{q}{p -1}}}{kp} < \frac{e^{\frac{q}{p -1}}}{k} + \frac{1}{p} + \frac{q}{p-1}. $$
  2. (ii)

    \(( u^{*}, k e^{- u^{*}} -1 )\) is a saddle point if and only if

    $$ \frac{e^{\frac{q}{p -1}}}{k} + \frac{1}{p} + \frac{2 q}{p -1} >5+ \frac{e ^{\frac{q}{p -1}}}{kp}. $$
  3. (iii)

    \(( u^{*}, k e^{- u^{*}} -1 )\) is a source if and only if

    $$ \frac{e^{\frac{q}{p -1}}}{k} + \frac{1}{p} + \frac{2 q}{p -1} < 5+ \frac{e ^{\frac{q}{p -1}}}{kp} $$

    and

    $$ 1+ \frac{e^{\frac{q}{p -1}}}{kp} > \frac{e^{\frac{q}{p -1}}}{k} + \frac{1}{p} + \frac{q}{p -1}. $$
  4. (iv)

    \(( u^{*}, k e^{- u^{*}} -1 )\) is nonhyperbolic with roots \(\lambda _{1} =-1\) or \(\lambda _{2} =-1\) of (2.4) if and only if

    $$ \frac{e^{\frac{q}{p -1}}}{k} + \frac{1}{p} + \frac{2 q}{p -1} =5+ \frac{e ^{\frac{q}{p -1}}}{kp}. $$
  5. (v)

    \(( u^{*}, k e^{- u^{*}} -1 )\) is nonhyperbolic such that complex conjugate roots of (2.4) are with modulus one if and only if

    $$ k = \frac{e^{\frac{q}{p -1}} ( p -1 )^{2}}{1-2 p + p^{2} - pq} $$

    and

    $$ \biggl( 2- \frac{q}{p -1} \biggr)^{2} < 4 \biggl( 2- \frac{e^{\frac{q}{p -1}}}{k} - \frac{1}{p} + \frac{e^{\frac{q}{p -1}}}{kp} - \frac{q}{p -1} \biggr). $$

For \(k \in [1,100]\), \(p \in [ 20,100 ]\), and \(q =80\), the topological classification of \(( u^{*}, k e^{- u^{*}} -1 )\) is shown in Fig. 2.

Figure 2
figure2

Topological classification of \(( u^{*}, k e^{- u^{*}} -1 )\)

Linearized stability of system (1.5)

Suppose that \(( x, y )\) is an arbitrary steady state for system (1.5). Then it solves the following algebraic system:

$$ x = \frac{\mu x e^{- x}}{1+ y^{2}}, \quad\quad y = \frac{\nu x^{2} y}{\eta + x ^{2}}. $$

Simple computation yields the following nonnegative steady states for system (1.5):

$$ (0,0),\quad\quad \bigl( \ln ( \mu ),0 \bigr), \quad\quad \bigl( v^{*}, \sqrt{\mu e ^{- v^{*}} -1} \bigr), $$

where \(v^{*}:= \sqrt{\frac{\eta }{\nu -1}}\), \(\mu >1\), \(\nu >1\), and \(\sqrt{\eta } <\ln ( \mu ) \sqrt{\nu -1}\). Furthermore, the variational matrix for system (1.5) computed at \(( x, y )\) is given as follows:

$$ V ( x, y )= \begin{bmatrix} \frac{\mu e^{- x} (1- x )}{1+ y^{2}} & - \frac{2 \mu xy e^{- x}}{ ( 1+ y^{2} )^{2}} \\ \frac{2 xy \eta \nu }{ ( x^{2} + \eta )^{2}} & \frac{ \nu x^{2}}{\eta + x^{2}} \end{bmatrix}. $$
(3.1)

Then, at \(( x, y ) = ( 0,0 )\), \(V ( x, y )\) reduces to

$$ V (0,0)= \begin{pmatrix} \mu & 0 \\ 0 & 0 \end{pmatrix}. $$

Then from \(V (0,0)\) it follows that \((0,0)\) is a sink if and only if \(0< \mu <1\), saddle if and only if \(\mu >1\), and nonhyperbolic if and only if \(\mu =1\). Moreover, assume that \(\mu >1\). Then the variational matrix \(V ( x, y )\) computed at semitrivial equilibrium \(( x, y )= ( \ln ( \mu ),0 )\) is given as follows:

$$ V \bigl( \ln ( \mu ),0 \bigr) = \begin{bmatrix} 1-\ln ( \mu ) & 0 \\ 0 & \frac{\nu \ln ^{2} ( \mu )}{\eta + \ln ^{2} ( \mu )} \end{bmatrix}. $$
(3.2)

Lemma 3.1

Assume that \(\mu >1\). Then the following statements hold:

  1. (i)

    \(( \ln ( \mu ),0 )\) is a sink if and only if \(1< \mu < e^{2}\) and \(\nu (\ln ( \mu ) )^{2} < \eta +(\ln ( \mu ) )^{2}\).

  2. (ii)

    \(( \ln ( \mu ),0 )\) is a saddle point if and only if \(1< \mu < e^{2}\) and \(\nu (\ln ( \mu ) )^{2} > \eta +(\ln ( \mu ) )^{2}\), or \(\mu > e^{2}\) and \(\nu (\ln ( \mu ) )^{2} < \eta +(\ln ( \mu ) )^{2}\).

  3. (iii)

    \(( \ln ( \mu ),0 )\) is a source if and only if \(\mu > e^{2}\) and \(\nu (\ln ( \mu ) )^{2} > \eta +(\ln ( \mu ) )^{2}\).

  4. (iv)

    \(( \ln ( \mu ),0 )\) is nonhyperbolic if and only if \(\mu = e^{2}\) or \(\nu (\ln ( \mu ) )^{2} = \eta +(\ln ( \mu ) )^{2}\).

Moreover, for \(\mu \in [1,20]\), \(\eta \in [0,50]\), and \(\nu =6\), the topological classification for \(( \ln ( \mu ),0 )\) is depicted in Fig. 3.

Figure 3
figure3

Topological classification of \(( \ln ( \mu ),0 )\)

Moreover, we suppose that \(\sqrt{\eta } <\ln ( \mu ) \sqrt{\nu -1}\), \(\nu >1\), and \(\mu >1\). Then the variational matrix computed at positive steady state \(( v^{*}, \sqrt{\mu e^{- v ^{*}} -1} )\) is given as follows:

$$ V \bigl( v^{*}, \sqrt{\mu e^{- v^{*}} -1} \bigr) = \begin{bmatrix} 1- \frac{\sqrt{\eta }}{\sqrt{\nu -1}} & - \frac{2 \sqrt{e^{\frac{\sqrt{ \eta }}{\sqrt{\nu -1}}}} \sqrt{\eta } \sqrt{\mu - e^{\frac{\sqrt{ \eta }}{\sqrt{\nu -1}}}}}{\mu \sqrt{\nu -1}} \\ \frac{2 \sqrt{\mu - e^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}}} ( \nu -1 )^{3/2}}{\sqrt{e^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}}} \sqrt{ \eta } \nu } & 1 \end{bmatrix}. $$
(3.3)

The characteristic polynomial of \(V ( v^{*}, \sqrt{\mu e^{- v ^{*}} -1} )\) is given by

$$ F ( \lambda )= \lambda ^{2} - \biggl( 2- \frac{\sqrt{\eta }}{\sqrt{ \nu -1}} \biggr) \lambda +5- \frac{\sqrt{\eta }}{\sqrt{\nu -1}} - \frac{4}{\nu } - \frac{4 e^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \nu -1)}{\mu \nu }. $$
(3.4)

Then from (3.4) it follows that

$$ F (1)= \frac{4 ( \mu - e^{ \frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ) ( \nu -1)}{\mu \nu } >0. $$

Therefore Lemma 2.1 can be implemented to prove the following results, which give a topological classification for equilibrium \(( v ^{*}, \sqrt{\mu e^{- v^{*}} -1} )\).

Lemma 3.2

Let \(\sqrt{\eta } <\ln ( \mu ) \sqrt{ \nu -1}\), \(\nu >1\), and \(\mu >1\). Then we have the following results for equilibrium \(( v^{*}, \sqrt{\mu e^{- v ^{*}} -1} )\) of system (1.5):

  1. (i)

    \(( v^{*}, \sqrt{\mu e^{- v^{*}} -1} )\) is a sink if and only if

    $$ \frac{\sqrt{\eta }}{\sqrt{\nu -1}} + \frac{2}{\nu } + \frac{2 e ^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \nu -1)}{\mu \nu } < 2 $$

    and

    $$ 4< \frac{\sqrt{\eta }}{\sqrt{\nu -1}} + \frac{4}{\nu } + \frac{4 e ^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \nu -1)}{\mu \nu }. $$
  2. (ii)

    \(( v^{*}, \sqrt{\mu e^{- v^{*}} -1} )\) is a saddle point if and only if

    $$ 4> \frac{\sqrt{\eta }}{\sqrt{\nu -1}} + \frac{4}{\nu } + \frac{4 e ^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \nu -1)}{\mu \nu }. $$
  3. (iii)

    \(( v^{*}, \sqrt{\mu e^{- v^{*}} -1} )\) is a source if and only if

    $$ \frac{\sqrt{\eta }}{\sqrt{\nu -1}} + \frac{2}{\nu } + \frac{2 e ^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \nu -1)}{\mu \nu } < 2 $$

    and

    $$ 4> \frac{\sqrt{\eta }}{\sqrt{\nu -1}} + \frac{4}{\nu } + \frac{4 e ^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \nu -1)}{\mu \nu }. $$
  4. (iv)

    \(( v^{*}, \sqrt{\mu e^{- v^{*}} -1} )\) is nonhyperbolic with roots \(\lambda _{1} =-1\) or \(\lambda _{2} =-1\) of (3.4) if and only if

    $$ \frac{\sqrt{\eta }}{\sqrt{\nu -1}} + \frac{2}{\nu } + \frac{2 e ^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \nu -1)}{\mu \nu } =4. $$
  5. (v)

    \(( v^{*}, \sqrt{\mu e^{- v^{*}} -1} )\) is nonhyperbolic such that complex conjugate roots of (3.4) are with modulus one if and only if

    $$ \frac{\sqrt{\eta }}{\sqrt{\nu -1}} + \frac{4}{\nu } + \frac{4 e ^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \nu -1)}{\mu \nu } =4 $$

    and

    $$ \biggl( 2- \frac{\sqrt{\eta }}{\sqrt{\nu -1}} \biggr)^{2} < 4 \biggl( 5- \frac{\sqrt{\eta }}{\sqrt{\nu -1}} - \frac{4}{\nu } - \frac{4 e^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \nu -1)}{\mu \nu } \biggr). $$

Taking \(\mu \in [2,20]\), \(\eta \in [1,20]\), and \(\nu =3\), the classification for positive steady state \(( v^{*}, \sqrt{ \mu e^{- v^{*}} -1} )\) is depicted in Fig. 4.

Figure 4
figure4

Topological classification of \(( v^{*}, \sqrt{\mu e^{- v^{*}} -1} )\)

Period-doubling bifurcation

In this section, we analyze that positive equilibrium points of systems (1.3) and (1.5) undergo period-doubling bifurcation. For this bifurcation theory, normal forms and center manifold theorem are implemented for the existence and direction of such type of bifurcation. Recently, period-doubling bifurcation related to discrete-time models has been investigated by many authors [19,20,21,22,23,24].

First, we discuss emergence of flip bifurcation at positive equilibrium point \(( u^{*}, k e^{- u^{*}} -1 )\) of system (1.3). For this, it follows from part (v) of Lemma 2.3 that \(( u^{*}, k e^{- u^{*}} -1 )\) is nonhyperbolic with root one of (2.4), say, \(\delta _{1} =-1\), if the following condition is satisfied:

$$ k = \frac{e^{\frac{q}{p -1}} ( p -1 )^{2}}{(5 p -1)( p -1)-2 pq},\quad\quad (5 p -1) ( p -1)>2 pq. $$
(4.1)

Then, the second root for (2.4), say, \(\delta _{2}\), satisfies \(\vert \delta _{2} \vert \neq 1\) if the following inequalities are satisfied:

$$ q \neq 4( p -1), \quad\quad q \neq 2( p -1). $$
(4.2)

Moreover, under the assumptions that \(p >1\) and \(q >0\), we consider the following set:

$$ \mathbb{S}_{1} = \biggl\{ ( p, q, k ): k = \frac{e^{\frac{q}{p -1}} ( p -1 )^{2}}{(5 p -1)( p -1)-2 pq}, (5 p -1) ( p -1)>2 pq, q \neq 2( p -1),4( p -1) \biggr\} . $$

Furthermore, let \(( p, q, k )\in \mathbb{S}_{1}\). Then the positive steady-state \(( u^{*}, k e^{- u^{*}} -1 )\) of system (1.3) undergoes flip bifurcation such that k is taken as bifurcation parameter, and it varies in a small neighborhood of \(k_{1}\) given by

$$ k_{1}:= \frac{e^{\frac{q}{p -1}} ( p -1 )^{2}}{(5 p -1)( p -1)-2 pq}. $$

Moreover, system (1.3) is represented equivalently with the following two-dimensional map:

$$ \begin{pmatrix} X \\ Y \end{pmatrix} \rightarrow \begin{pmatrix} \frac{kX e^{- X}}{1+ Y} \\ \frac{pXY}{q + X} \end{pmatrix}. $$
(4.3)

To discuss period-doubling bifurcation for fixed point \(( u^{*}, k e^{- u^{*}} -1 )\) of (4.3), we suppose that \(( p, q, k_{1} ) \in \mathbb{S}_{1}\). Then it follows that

$$ \begin{pmatrix} X \\ Y \end{pmatrix} \rightarrow \begin{pmatrix} \frac{k_{1} X e^{- X}}{1+ Y} \\ \frac{pXY}{q + X} \end{pmatrix}. $$
(4.4)

We take \(k^{*}\) as small perturbation in parameter \(k_{1}\). Then a perturbed map corresponding to (4.3) is given as follows:

$$ \begin{pmatrix} X \\ Y \end{pmatrix} \rightarrow \begin{pmatrix} \frac{( k_{1} + k^{*} ) X e^{- X}}{1+ Y} \\ \frac{pXY}{q + X} \end{pmatrix}. $$
(4.5)

Suppose that \(x = X - a^{*}\) and \(y = Y - b^{*}\) are such that \(a^{*} = \frac{q}{p -1}\) and \(b^{*} =( k_{1} + k^{*} ) e^{- a^{*}} -1\). Then from (4.5) we obtain the following map with fixed point at \((0,0)\):

$$ \begin{pmatrix} x \\ y \end{pmatrix} \rightarrow \begin{pmatrix} \alpha _{11} & \alpha _{12} \\ \alpha _{21} & \alpha _{22} \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} f_{1} ( x, y, k^{*} ) \\ f_{2} ( x, y, k^{*} ) \end{pmatrix}, $$
(4.6)

where

$$\begin{aligned}& \begin{aligned} f_{1} \bigl( x, y, k^{*} \bigr)&= \alpha _{13} x^{2} + \alpha _{14} xy + \alpha _{15} y^{2} + \alpha _{16} x^{3} + \alpha _{17} x^{2} y + \alpha _{18} x y^{2} + \alpha _{19} y^{3} + a_{1} y k^{*} + a_{2} xy k^{*} \\ &\quad {} + a_{3} y^{2} k^{*} + a_{4} y \bigl( k^{*} \bigr)^{2} + O \bigl( \bigl( \vert x \vert + \vert y \vert + k^{*} \bigr)^{4} \bigr), \end{aligned} \\& f_{2} \bigl( x, y, k^{*} \bigr)= \alpha _{23} x^{2} + \alpha _{24} xy + \alpha _{25} x^{3} + \alpha _{26} x^{2} y + b_{1} x k^{*} + b_{2} x^{2} k^{*} + O \bigl( \bigl( \vert x \vert + \vert y \vert + k^{*} \bigr)^{4} \bigr), \\& \alpha _{11} =1- a^{*}, \quad\quad \alpha _{12} =- \frac{a^{*} e^{a^{*}}}{k_{1}},\quad\quad \alpha _{13} = \frac{a^{*} -2}{2}, \quad \quad \alpha _{14} = \frac{( a^{*} -1) e ^{a^{*}}}{k_{1}},\\& \alpha _{15} = \frac{a^{*} e^{2 a^{*}}}{k_{1}^{2}},\quad\quad \alpha _{16} = \frac{3- a^{*}}{6},\quad\quad \alpha _{17} = \frac{e^{a^{*}} (2- a^{*} )}{2 k_{1}}, \quad\quad \alpha _{18} = \frac{e ^{2 a^{*}} (1- a^{*} )}{k_{1}^{2}}, \\& \alpha _{19} =- \frac{a^{*} e^{3 a ^{*}}}{k_{1}^{3}}, \quad\quad a_{1} = \frac{a^{*} e^{a^{*}}}{k_{1}^{2}}, \quad\quad a_{2} = \frac{e^{a^{*}} (1- a^{*} )}{k_{1}^{2}}, \quad\quad a_{3} =- \frac{2 a^{*} e^{2 a^{*}}}{k_{1}^{3}}, \\& a_{4} =- \frac{a^{*} e ^{a^{*}}}{k_{1}^{3}}, \quad\quad \alpha _{21} = \frac{pq ( k_{1} e^{- a^{*}} -1 )}{( q + a^{*} )^{2}}, \quad \quad \alpha _{22} = \frac{p a^{*}}{q + a ^{*}},\quad\quad \alpha _{23} = \frac{pq ( 1- k_{1} e^{- a^{*}} )}{( q + a^{*} )^{3}}, \\& \alpha _{24} = \frac{pq}{( q + a^{*} )^{2}}, \quad\quad \alpha _{25} = \frac{pq ( k_{1} e^{- a^{*}} -1 )}{( q + a^{*} )^{4}}, \quad\quad \alpha _{26} =- \frac{pq}{( q + a^{*} )^{3}},\quad\quad b_{1} = \frac{pq e^{- a^{*}}}{( q + a ^{*} )^{2}}, \\& b_{2} =- \frac{pq e^{- a^{*}}}{( q + a^{*} )^{3}}. \end{aligned}$$

Under the assumption that \(( p, q, k_{1} )\in \mathbb{S}_{1}\), the roots of (2.4) are computed as \(\delta _{1} =-1\) and

$$ \delta _{2}:= \frac{3( p -1)- q}{p -1}. $$

To obtain a normal form of (4.6), we consider the following similarity transformation:

$$ \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \alpha _{12} & \alpha _{12} \\ -1- \alpha _{11} & \delta _{2} - \alpha _{11} \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix}. $$
(4.7)

Then implementing similarity transformation (4.7), we obtain the following normal form:

$$ \begin{pmatrix} u \\ v \end{pmatrix} \rightarrow \begin{pmatrix} -1 & 0 \\ 0 & \delta _{2} \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix} + \begin{pmatrix} f_{3} ( u, v, k^{*} ) \\ f_{4} ( u, v, k^{*} ) \end{pmatrix}, $$
(4.8)

where

$$\begin{aligned}& \begin{aligned} f_{3} \bigl( u, v, k^{*} \bigr)&= \biggl( \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{16}}{\alpha _{12} ( \delta _{2} +1 )} - \frac{ \alpha _{25}}{\delta _{2} +1} \biggr) x^{3} + \biggl( \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{17}}{\alpha _{12} ( \delta _{2} +1 )} - \frac{\alpha _{26}}{\delta _{2} +1} \biggr) x ^{2} y \\ &\quad {} + \biggl( \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{13}}{ \alpha _{12} ( \delta _{2} +1 )} - \frac{b_{2} k^{*} + \alpha _{23}}{\delta _{2} +1} \biggr) x^{2} + \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{18} x y^{2}}{\alpha _{12} ( \delta _{2} +1 )} \\ &\quad {} + \biggl( \frac{ ( \delta _{2} - \alpha _{11} ) ( k ^{*} a_{2} + \alpha _{14} )}{\alpha _{12} ( \delta _{2} +1 )} - \frac{\alpha _{24}}{\delta _{2} +1} \biggr) xy - \frac{b _{1} k^{*} x}{\delta _{2} +1} + \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{19} y^{3}}{\alpha _{12} ( \delta _{2} +1 )} \\ &\quad {} + \frac{ ( \delta _{2} - \alpha _{11} ) ( k^{*} a_{3} + \alpha _{15} ) y^{2}}{\alpha _{12} ( \delta _{2} +1 )} + \frac{ ( \delta _{2} - \alpha _{11} ) ( ( k^{*} )^{2} a_{4} + k^{*} a_{1} ) y}{\alpha _{12} ( \delta _{2} +1 )} \\ &\quad {}+ O \bigl( \bigl( \vert u \vert + \vert v \vert + \bigl\vert k^{*} \bigr\vert \bigr)^{4} \bigr), \end{aligned} \\& \begin{aligned} f_{4} \bigl( u, v, k^{*} \bigr)&= \biggl( \frac{ ( 1+ \alpha _{11} ) \alpha _{16}}{\alpha _{12} ( \delta _{2} +1 )} + \frac{ \alpha _{25}}{\delta _{2} +1} \biggr) x^{3} + \biggl( \frac{ ( 1+ \alpha _{11} ) \alpha _{17}}{\alpha _{12} ( \delta _{2} +1 )} + \frac{\alpha _{26}}{\delta _{2} +1} \biggr) x^{2} y \\ & \quad {} + \biggl( \frac{ ( 1+ \alpha _{11} ) \alpha _{13}}{\alpha _{12} ( \delta _{2} +1 )} + \frac{b_{2} k^{*} + \alpha _{23}}{\delta _{2} +1} \biggr) x^{2} + \frac{ ( 1+ \alpha _{11} ) \alpha _{18} x y^{2}}{\alpha _{12} ( \delta _{2} +1 )} \\ & \quad {} + \biggl( \frac{ ( 1+ \alpha _{11} ) ( k^{*} a_{2} + \alpha _{14} )}{\alpha _{12} ( \delta _{2} +1 )} + \frac{ \alpha _{24}}{\delta _{2} +1} \biggr) xy + \frac{b_{1} k^{*} x}{\delta _{2} +1} + \frac{ ( 1+ \alpha _{11} ) \alpha _{19} y^{3}}{ \alpha _{12} ( \delta _{2} +1 )} \\ &\quad {} + \frac{ ( 1+ \alpha _{11} ) ( k^{*} a_{3} + \alpha _{15} ) y^{2}}{\alpha _{12} ( \delta _{2} +1 )} + \frac{ ( 1+ \alpha _{11} ) ( ( k^{*} )^{2} a_{4} + k^{*} a _{1} ) y}{\alpha _{12} ( \delta _{2} +1 )} \\ &\quad {}+ O \bigl( \bigl( \vert u \vert + \vert v \vert + \bigl\vert k^{*} \bigr\vert \bigr)^{4} \bigr), \end{aligned} \end{aligned}$$

and

$$ x = \alpha _{12} ( u + v ), \quad\quad y =-(1+ \alpha _{11} ) u +( \delta _{2} - \alpha _{11} ) v. $$

For the implementation of center manifold theorem, we suppose that \(W^{c} (0,0,0)\) represents the center manifold for (4.8) evaluated at \((0,0)\) in a small neighborhood of \(k^{*} =0\). Then \(W^{c} (0,0,0)\) is approximated as follows:

$$ W^{c} (0,0,0)= \bigl\{ \bigl( u, v, k^{*} \bigr)\in \mathbb{R}^{3}: v = h_{1} u^{2} + h_{2} u k^{*} + h_{3} \bigl( k^{*} \bigr)^{2} + O \bigl( \bigl( \vert u \vert + \bigl\vert k^{*} \bigr\vert \bigr)^{3} \bigr) \bigr\} , $$

where

$$\begin{aligned}& \begin{aligned} h_{1} &= \biggl( \frac{ ( 1+ \alpha _{11} ) \alpha _{13}}{ \alpha _{12} ( 1- \delta _{2}^{2} )} + \frac{\alpha _{23}}{1- \delta _{2}^{2}} \biggr) \alpha _{12}^{2} - \biggl( \frac{ ( 1+ \alpha _{11} ) \alpha _{14}}{\alpha _{12} ( 1- \delta _{2} ^{2} )} + \frac{\alpha _{24}}{1- \delta _{2}^{2}} \biggr) \alpha _{12} ( 1+ \alpha _{11} ) \\ &\quad {} + \frac{\alpha _{15} ( 1+ \alpha _{11} )^{3}}{\alpha _{12} ( 1- \delta _{2}^{2} )}, \end{aligned} \\& h_{2} = \frac{b_{1} \alpha _{12}}{1- \delta _{2}^{2}} - \frac{ ( 1+ \alpha _{11} )^{2} a_{1}}{\alpha _{12} ( 1- \delta _{2}^{2} )}, h_{3} =0. \end{aligned}$$

The map restricted to the center manifold \(W^{c} (0,0,0)\) is computed as follows:

$$ F: u \rightarrow - u + s_{1} u^{2} + s_{2} u k^{*} + s_{3} u^{2} k ^{*} + s_{4} u \bigl( k^{*} \bigr)^{2} + s_{5} u^{3} + O \bigl( \bigl( \vert u \vert + \bigl\vert k^{*} \bigr\vert \bigr)^{4} \bigr), $$

where

$$\begin{aligned}& \begin{aligned} s_{1} &= \biggl( \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{13}}{\alpha _{12} ( 1+ \delta _{2} )} - \frac{ \alpha _{23}}{1+ \delta _{2}} \biggr) \alpha _{12}^{2} - \biggl( \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{14}}{\alpha _{12} ( 1+ \delta _{2} )} - \frac{\alpha _{24}}{1+ \delta _{2}} \biggr) \alpha _{12} ( 1+ \alpha _{11} ) \\ &\quad {} + \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{15} ( 1+ \alpha _{11} )^{2}}{\alpha _{12} ( 1+ \delta _{2} )}, \end{aligned} \\& s_{2} =- \frac{b_{1} \alpha _{12}}{1+ \delta _{2}} - \frac{ ( \delta _{2} - \alpha _{11} ) a_{1} ( 1+ \alpha _{11} )}{ \alpha _{12} ( 1+ \delta _{2} )}, \\& \begin{aligned} s_{3} &=2 \biggl( \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{13}}{\alpha _{12} ( 1+ \delta _{2} )} - \frac{ \alpha _{23}}{1+ \delta _{2}} \biggr) \alpha _{12}^{2} h_{2} - \frac{b _{2} \alpha _{12}^{2}}{1+ \delta _{2}} \\ &\quad {} + \biggl( \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{14}}{ \alpha _{12} ( 1+ \delta _{2} )} - \frac{\alpha _{24}}{1+ \delta _{2}} \biggr) \alpha _{12} ( \delta _{2} - \alpha _{11} ) h_{2} \\ &\quad {} - \biggl( \biggl( \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{14}}{\alpha _{12} ( \delta _{2} +1 )} - \frac{ \alpha _{24}}{1+ \delta _{2}} \biggr) \alpha _{12} h_{2} + \frac{ ( \delta _{2} - \alpha _{11} ) a_{2}}{1+ \delta _{2}} \biggr) ( 1+ \alpha _{11} ) \\ &\quad {} - \frac{b_{1} \alpha _{12} h_{1}}{\delta _{2} +1} -2 \frac{ ( \delta _{2} - \alpha _{11} )^{2} \alpha _{15} ( 1+ \alpha _{11} ) h_{2}}{\alpha _{12} ( 1+ \delta _{2} )} + \frac{ ( \delta _{2} - \alpha _{11} ) a_{3} ( 1+ \alpha _{11} )^{2}}{\alpha _{12} ( 1+ \delta _{2} )} \\ &\quad {}+ \frac{ ( \delta _{2} - \alpha _{11} )^{2} a_{1} h_{1}}{\alpha _{12} ( 1+ \delta _{2} )}, \end{aligned} \\& s_{4} = \frac{ ( \delta _{2} - \alpha _{11} )^{2} a_{1} h _{2}}{\alpha _{12} ( 1+ \delta _{2} )} - \frac{b_{1} \alpha _{12} h_{2}}{1+ \delta _{2}} - \frac{ ( \delta _{2} - \alpha _{11} ) a_{4} ( 1+ \alpha _{11} )}{\alpha _{12} ( 1+ \delta _{2} )}, \\& \begin{aligned} s_{5} &= \biggl( \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{16}}{\alpha _{12} ( \delta _{2} +1 )} - \frac{ \alpha _{25}}{1+ \delta _{2}} \biggr) \alpha _{12}^{3} - \biggl( \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{17}}{\alpha _{12} ( 1+ \delta _{2} )} - \frac{\alpha _{26}}{\delta _{2} +1} \biggr) \alpha _{12}^{2} ( 1+ \alpha _{11} ) \\ &\quad {} +2 \biggl( \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{13}}{ \alpha _{12} ( 1+ \delta _{2} )} - \frac{\alpha _{23}}{1+ \delta _{2}} \biggr) \alpha _{12}^{2} h_{1} + \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{18} ( 1+ \alpha _{11} ) ^{2}}{1+ \delta _{2}} \\ &\quad {} + \biggl( \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{14}}{ \alpha _{12} ( 1+ \delta _{2} )} - \frac{\alpha _{24}}{ \delta _{2} +1} \biggr) \alpha _{12} h_{1} ( \delta _{2} - \alpha _{11} ) \\ &\quad {}- \biggl( \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{14}}{\alpha _{12} ( 1+ \delta _{2} )} - \frac{ \alpha _{24}}{\delta _{2} +1} \biggr) \alpha _{12} h_{1} ( 1+ \alpha _{11} ) \\ &\quad {} - \frac{ ( \delta _{2} - \alpha _{11} ) \alpha _{19} ( 1+ \alpha _{11} )^{3}}{\alpha _{12} ( \delta _{2} +1 )} -2 \frac{ ( \delta _{2} - \alpha _{11} )^{2} \alpha _{15} ( 1+ \alpha _{11} ) h_{1}}{\alpha _{12} ( \delta _{2} +1 )}. \end{aligned} \end{aligned}$$

Next, we define the following two nonzero real numbers:

$$ l_{1} = \biggl( \frac{\partial ^{2} f_{1}}{\partial u \partial k^{*}} + \frac{1}{2} \frac{\partial F}{\partial k^{*}} \frac{\partial ^{2} F}{ \partial u^{2}} \biggr)_{(0,0)} =- \frac{b_{1} \alpha _{12}}{1+ \delta _{2}} - \frac{ ( \delta _{2} - \alpha _{11} ) a_{1} ( 1+ \alpha _{11} )}{\alpha _{12} ( 1+ \delta _{2} )} \neq 0 $$

and

$$ l_{2} = \biggl( \frac{1}{6} \frac{\partial ^{3} F}{\partial u^{3}} + \biggl( \frac{1}{2} \frac{\partial ^{2} F}{\partial u^{2}} \biggr) ^{2} \biggr)_{(0,0)} = s_{1}^{2} + s_{5} \neq 0. $$

Due to aforementioned computation, we have the following result about period-doubling bifurcation of system (1.3).

Theorem 4.1

If \(l_{1} \neq 0\) and \(l_{2} \neq 0\), then system (1.3) undergoes period-doubling bifurcation at the unique positive equilibrium point when parameter k varies in a small neighborhood of \(k_{1}\). Furthermore, if \(l_{2} >0\), then the period-two orbits that bifurcate from positive equilibrium of (1.3) are stable, and if \(l_{2} <0\), then these orbits are unstable.

At the end of this section, we discuss that the positive equilibrium \(( v^{*}, \sqrt{\mu e^{- v^{*}} -1} )\) of system (1.5) undergoes flip bifurcation. For this, we first assume that

$$ 2 \sqrt{\nu -1} (2 \nu -1)> \sqrt{\eta } \nu $$
(4.9)

and take

$$ \mu = \frac{2 e^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \nu -1 )^{3/2}}{2 \sqrt{\nu -1} (2 \nu -1)- \sqrt{\eta } \nu }. $$
(4.10)

Suppose that (4.9) and (4.10) hold. Then the characteristic polynomial (3.4) has real roots \(\tau _{1}\) and \(\tau _{2}\) with \(\tau _{1} =-1\) and \(\vert \tau _{2} \vert \neq 1\) if the following inequalities are satisfied:

$$ \frac{\sqrt{\eta }}{\sqrt{\nu -1}} \neq 4, \quad\quad \frac{\sqrt{\eta }}{\sqrt{ \nu -1}} \neq 2. $$
(4.11)

Keeping in view (4.9), (4.10), and (4.11), we define the following set:

$$ \mathbb{S}_{2} = \biggl\{ ( \mu , \nu , \eta ): \mu = \frac{2 e^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \nu -1 )^{ \frac{3}{2}}}{2 \sqrt{\nu -1} ( 2 \nu -1 ) - \sqrt{ \eta } \nu } \text{ and (4.9), (4.11) hold} \biggr\} . $$

Suppose that \(( \mu , \nu , \eta )\in \mathbb{S}_{2}\). Then the positive steady-state \(( v^{*}, \sqrt{\mu e^{- v^{*}} -1} )\) of system (1.5) undergoes flip bifurcation when μ is taken as bifurcation parameter and varies in a small neighborhood of \(\mu _{1}\) defined as

$$ \mu _{1}:= \frac{2 e^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \nu -1 )^{3/2}}{2 \sqrt{\nu -1} (2 \nu -1)- \sqrt{\eta } \nu }. $$

In the similar fashion as for system (1.3), we construct a perturbed map corresponding to system (1.5) as follows:

$$ \begin{pmatrix} X \\ Y \end{pmatrix} \rightarrow \begin{pmatrix} \frac{( \mu _{1} + \mu ^{*} ) X e^{- X}}{1+ Y^{2}} \\ \frac{\nu X^{2} Y}{\eta + X^{2}} \end{pmatrix}. $$
(4.12)

We consider the translations \(x = X - v^{*}\) and \(y = Y - \sqrt{( \mu _{1} + \mu ^{*} ) e^{- v^{*}} -1}\), where \(v^{*} = \sqrt{\frac{ \eta }{\nu -1}}\), for the conversion of map (4.12) into the following form having \((0,0)\) as its fixed point:

$$ \begin{pmatrix} x \\ y \end{pmatrix} \rightarrow \begin{pmatrix} \beta _{11} & \beta _{12} \\ \beta _{21} & \beta _{22} \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} g_{1} ( x, y, \mu ^{*} ) \\ g_{2} ( x, y, \mu ^{*} ) \end{pmatrix}, $$
(4.13)

where

$$\begin{aligned} &\begin{aligned} g_{1} \bigl( x, y, \mu ^{*} \bigr)&= \beta _{13} x^{2} + \beta _{14} xy + \beta _{15} y^{2} + \beta _{16} x^{3} + \beta _{17} x^{2} y + \beta _{18} x y^{2} + \beta _{19} y^{3} + c_{1} y \mu ^{*} \\ &\quad {} + c_{2} xy \mu ^{*} + c_{3} y^{2} \mu ^{*} + c_{4} y \bigl( \mu ^{*} \bigr)^{2} + O \bigl( \bigl( \vert x \vert + \vert y \vert + \bigl\vert \mu ^{*} \bigr\vert \bigr)^{4} \bigr), \end{aligned} \\ & \begin{aligned} g_{2} \bigl( x, y, \mu ^{*} \bigr)&= \beta _{23} x^{2} + \beta _{24} xy + \beta _{25} x^{3} + \beta _{26} x^{2} y + d_{1} x \mu ^{*} + d_{2} x^{2} \mu ^{*}\\&\quad{} + O \bigl( \bigl( \vert x \vert + \vert y \vert + \bigl\vert \mu ^{*} \bigr\vert \bigr)^{4} \bigr),\end{aligned} \\ & \beta _{11} =1- v^{*}, \quad\quad \beta _{12} =- \frac{2 v^{*} e^{v^{*}} \sqrt{ \mu _{1} e^{- v^{*}} -1}}{\mu _{1}}, \quad\quad \beta _{13} = \frac{v^{*}}{2} -1, \\ & \beta _{14} = \frac{2 \sqrt{\mu _{1} e^{- v^{*}} -1} ( v^{*} -1 )}{e^{- v^{*}} \mu _{1}}, \quad\quad \beta _{15} = \frac{v^{*} e^{v^{*}} ( 3 \mu _{1} -4 e^{v^{*}} )}{ \mu _{1}^{2}}, \quad\quad \beta _{16} = \frac{3- v^{*}}{6}, \\ & \beta _{17} = \frac{(2- v^{*} ) \sqrt{\mu _{1} e^{- v^{*}} -1}}{e^{- v^{*}} \mu _{1}}, \quad\quad \beta _{18} = \frac{e^{v^{*}} (1- v^{*} )(3 \mu _{1} -4 e^{v^{*}} )}{\mu _{1}^{2}}, \\ & \beta _{19} = \frac{4 v^{*} \sqrt{\mu _{1} e^{- v^{*}} -1} ( 2- \mu _{1} e^{- v^{*}} )}{e^{-3 v^{*}} \mu _{1}^{3}},\quad\quad c _{1} = \frac{v^{*} e^{v^{*}} ( \mu _{1} e^{- v^{*}} -2)}{\mu _{1}^{2} \sqrt{ \mu _{1} e^{- v^{*}} -1}}, \\ & c_{2} = \frac{e^{v^{*}} ( \mu _{1} e^{- v^{*}} -2)(1- v^{*} )}{\mu _{1} ^{2} \sqrt{\mu _{1} e^{- v^{*}} -1}}, \quad\quad c_{3} = \frac{v^{*} e^{v^{*}} (8 e^{v^{*}} -3 \mu _{1} )}{\mu _{1}^{3}}, \\ & c_{4} =- \frac{v^{*} e^{v^{*}} ( 3 e^{-2 v^{*}} \mu _{1}^{2} -12 \mu _{1} e^{- v^{*}} +8 )}{4 \mu _{1}^{3} ( \mu _{1} e^{- v ^{*}} -1 )^{\frac{3}{2}}},\quad\quad \beta _{21} = \frac{2 v^{*} ( \nu -1 )^{2} \sqrt{\mu _{1} e^{- v^{*}} -1}}{\eta \nu }, \quad\quad \beta _{22} =1, \\ & \beta _{23} = \frac{( \nu -1 )^{2} ( \nu -4) \sqrt{\mu _{1} e^{- v ^{*}} -1}}{\eta \nu ^{2}}, \quad\quad \beta _{24} = \frac{2 v^{*} ( \nu -1 )^{2}}{ \eta \nu }, \\ & \beta _{25} =- \frac{4 v^{*} ( \nu -1 )^{3} ( \nu -2) \sqrt{ \mu _{1} e^{- v^{*}} -1}}{\eta ^{2} \nu ^{3}}, \quad\quad \beta _{26} = \frac{( \nu -1 )^{2} ( \nu -4)}{\eta \nu ^{2}}, \\ & d_{1} = \frac{v ^{*} e^{- v^{*}} ( \nu -1 )^{2}}{\eta \nu \sqrt{\mu _{1} e^{- v^{*}} -1}},\quad\quad d_{2} = \frac{e^{- v^{*}} ( \nu -1 )^{2} ( \nu -4)}{2 \eta \nu ^{2} \sqrt{\mu _{1} e^{- v^{*}} -1}}. \end{aligned}$$

Moreover, we take

$$ \tau _{2}:=3- \frac{\sqrt{\eta }}{\sqrt{\nu -1}} $$

and construct the following similarity transformation:

$$ \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \beta _{12} & \beta _{12} \\ -1- \beta _{11} & \tau _{2} - \beta _{11} \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix}. $$
(4.14)

With the implementation of similarity transformation (4.14), we obtain the following normal for map (4.13):

$$ \begin{pmatrix} u \\ v \end{pmatrix} \rightarrow \begin{pmatrix} -1 & 0 \\ 0 & \tau _{2} \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix} + \begin{pmatrix} g_{3} ( u, v, \mu ^{*} ) \\ g_{4} ( u, v, \mu ^{*} ) \end{pmatrix}, $$
(4.15)

where

$$\begin{aligned}& \begin{aligned} g_{3} \bigl( u, v, \mu ^{*} \bigr)&= \biggl( \frac{ ( \tau _{2} - \beta _{11} ) \beta _{16}}{\beta _{12} ( \tau _{2} +1 )} - \frac{ \beta _{25}}{\tau _{2} +1} \biggr) x^{3} + \biggl( \frac{ ( \tau _{2} - \beta _{11} ) \beta _{17}}{\beta _{12} ( \tau _{2} +1 )} - \frac{\beta _{26}}{\tau _{2} +1} \biggr) x^{2} y \\ & \quad {} + \biggl( \frac{ ( \tau _{2} - \beta _{11} ) \beta _{13}}{ \beta _{12} ( \tau _{2} +1 )} - \frac{d_{2} \mu ^{*} + \beta _{23}}{\tau _{2} +1} \biggr) x^{2} + \frac{ ( \tau _{2} - \beta _{11} ) \beta _{18} x y^{2}}{\beta _{12} ( \tau _{2} +1 )} \\ & \quad {} + \biggl( \frac{ ( \tau _{2} - \beta _{11} ) ( \mu ^{*} c_{2} + \beta _{14} )}{\beta _{12} ( \tau _{2} +1 )} - \frac{\beta _{24}}{\tau _{2} +1} \biggr) xy - \frac{d_{1} \mu ^{*} x}{\tau _{2} +1} + \frac{ ( \tau _{2} - \beta _{11} ) \beta _{19} y^{3}}{\beta _{12} ( \tau _{2} +1 )} \\ & \quad {} + \frac{ ( \tau _{2} - \beta _{11} ) ( \mu ^{*} c_{3} + \beta _{15} ) y^{2}}{\beta _{12} ( \tau _{2} +1 )} + \frac{ ( \tau _{2} - \beta _{11} ) ( ( \mu ^{*} )^{2} c _{4} + \mu ^{*} c_{1} ) y}{\beta _{12} ( \tau _{2} +1 )} \\ & \quad {} + O \bigl( \bigl( \vert u \vert + \vert v \vert + \bigl\vert \mu ^{*} \bigr\vert \bigr)^{4} \bigr), \end{aligned} \\& \begin{aligned} g_{4} \bigl( u, v, \mu ^{*} \bigr)&= \biggl( \frac{ ( 1+ \beta _{11} ) \beta _{16}}{\beta _{12} ( \tau _{2} +1 )} + \frac{\beta _{25}}{\tau _{2} +1} \biggr) x^{3} + \biggl( \frac{ ( 1+ \beta _{11} ) \beta _{17}}{\beta _{12} ( \tau _{2} +1 )} + \frac{\beta _{26}}{\tau _{2} +1} \biggr) x^{2} y \\ & \quad {} + \biggl( \frac{ ( 1+ \beta _{11} ) \beta _{13}}{\beta _{12} ( \tau _{2} +1 )} + \frac{d_{2} \mu ^{*} + \beta _{23}}{\tau _{2} +1} \biggr) x^{2} + \frac{ ( 1+ \beta _{11} ) \beta _{18} x y^{2}}{\beta _{12} ( \tau _{2} +1 )} \\ & \quad {} + \biggl( \frac{ ( 1+ \beta _{11} ) ( \mu ^{*} c_{2} + \beta _{14} )}{\beta _{12} ( \tau _{2} +1 )} + \frac{ \beta _{24}}{\tau _{2} +1} \biggr) xy + \frac{d_{1} \mu ^{*} x}{\tau _{2} +1} + \frac{ ( 1+ \beta _{11} ) \beta _{19} y^{3}}{ \beta _{12} ( \tau _{2} +1 )} \\ & \quad {} + \frac{ ( 1+ \beta _{11} ) ( \mu ^{*} c_{3} + \beta _{15} ) y^{2}}{\beta _{12} ( \tau _{2} +1 )} + \frac{ ( 1+ \beta _{11} ) ( ( \mu ^{*} )^{2} c_{4} + \mu ^{*} c_{1} ) y}{\beta _{12} ( \tau _{2} +1 )} \\ & \quad {} + O \bigl( \bigl( \vert u \vert + \vert v \vert + \bigl\vert \mu ^{*} \bigr\vert \bigr)^{4} \bigr), \end{aligned} \end{aligned}$$

and

$$ x = \beta _{12} ( u + v ),\quad\quad y =-(1+ \beta _{11} ) u +( \tau _{2} - \beta _{11} ) v. $$

To apply center manifold theorem to map (4.15), we denote by \(W^{c} (0,0,0)\) the center manifold for (4.15) computed at \((0,0)\) in a small neighborhood of \(\mu ^{*} =0\). Then this center manifold for (4.15) is approximated as

$$ W^{c} (0,0,0)= \bigl\{ \bigl( u, v, \mu ^{*} \bigr)\in \mathbb{R}^{3}: v = p_{1} u^{2} + p_{2} u \mu ^{*} + p_{3} \bigl( \mu ^{*} \bigr)^{2} + O \bigl( \bigl( \vert u \vert + \bigl\vert \mu ^{*} \bigr\vert \bigr)^{3} \bigr) \bigr\} , $$

where

$$\begin{aligned}& \begin{aligned} p_{1} &= \biggl( \frac{ ( 1+ \beta _{11} ) \beta _{13}}{ \beta _{12} ( 1- \tau _{2}^{2} )} + \frac{\beta _{23}}{1- \tau _{2}^{2}} \biggr) \beta _{12}^{2} - \biggl( \frac{ ( 1+ \beta _{11} ) \beta _{14}}{\beta _{12} ( 1- \tau _{2}^{2} )} + \frac{\beta _{24}}{1- \tau _{2}^{2}} \biggr) \beta _{12} ( 1+ \beta _{11} ) \\ &\quad {} + \frac{\beta _{15} ( 1+ \beta _{11} )^{3}}{\beta _{12} ( 1- \tau _{2}^{2} )}, \end{aligned} \\& p_{2} = \frac{d_{1} \beta _{12}}{1- \tau _{2}^{2}} - \frac{ ( 1+ \beta _{11} )^{2} c_{1}}{\beta _{12} ( 1- \tau _{2}^{2} )}, \quad\quad p_{3} =0. \end{aligned}$$

Moreover, the map restricted to the center manifold \(W^{c} (0,0,0)\) is computed as follows:

$$ G: u \rightarrow - u + t_{1} u^{2} + t_{2} u \mu ^{*} + t_{3} u^{2} \mu ^{*} + t_{4} u \bigl( \mu ^{*} \bigr)^{2} + t_{5} u^{3} + O \bigl( \bigl( \vert u \vert + \bigl\vert \mu ^{*} \bigr\vert \bigr)^{4} \bigr), $$

where

$$\begin{aligned}& \begin{aligned} t_{1} &= \biggl( \frac{ ( \tau _{2} - \beta _{11} ) \beta _{13}}{\beta _{12} ( 1+ \tau _{2} )} - \frac{\beta _{23}}{1+ \tau _{2}} \biggr) \beta _{12}^{2} - \biggl( \frac{ ( \tau _{2} - \beta _{11} ) \beta _{14}}{\beta _{12} ( 1+ \tau _{2} )} - \frac{\beta _{24}}{1+ \tau _{2}} \biggr) \beta _{12} ( 1+ \beta _{11} ) \\ &\quad {} + \frac{ ( \tau _{2} - \beta _{11} ) \beta _{15} ( 1+ \beta _{11} )^{2}}{\beta _{12} ( 1+ \tau _{2} )}, \end{aligned} \\ & t_{2} =- \frac{d_{1} \beta _{12}}{1+ \tau _{2}} - \frac{ ( \tau _{2} - \beta _{11} ) c_{1} ( 1+ \beta _{11} )}{ \beta _{12} ( 1+ \tau _{2} )}, \\ & \begin{aligned} t_{3} &=2 \biggl( \frac{ ( \tau _{2} - \beta _{11} ) \beta _{13}}{\beta _{12} ( 1+ \tau _{2} )} - \frac{\beta _{23}}{1+ \tau _{2}} \biggr) \beta _{12}^{2} p_{2} - \frac{d_{2} \beta _{12}^{2}}{1+ \tau _{2}} \\ &\quad {} + \biggl( \frac{ ( \tau _{2} - \beta _{11} ) \beta _{14}}{ \beta _{12} ( 1+ \tau _{2} )} - \frac{\beta _{24}}{1+ \tau _{2}} \biggr) \beta _{12} ( \tau _{2} - \beta _{11} ) p _{2} \\ &\quad {} - \biggl( \biggl( \frac{ ( \tau _{2} - \beta _{11} ) \beta _{14}}{\beta _{12} ( \tau _{2} +1 )} - \frac{\beta _{24}}{1+ \tau _{2}} \biggr) \beta _{12} p_{2} + \frac{ ( \tau _{2} - \beta _{11} ) c_{2}}{1+ \tau _{2}} \biggr) ( 1+ \beta _{11} ) \\ &\quad {} - \frac{d_{1} \beta _{12} p_{1}}{\tau _{2} +1} -2 \frac{ ( \tau _{2} - \beta _{11} )^{2} \beta _{15} ( 1+ \beta _{11} ) p_{2}}{\beta _{12} ( 1+ \tau _{2} )} + \frac{ ( \tau _{2} - \beta _{11} ) c_{3} ( 1+ \beta _{11} )^{2}}{ \beta _{12} ( 1+ \tau _{2} )} \\ &\quad {} + \frac{ ( \tau _{2} - \beta _{11} )^{2} c_{1} p_{1}}{\beta _{12} ( 1+ \tau _{2} )}, \end{aligned} \\ & t_{4} = \frac{ ( \tau _{2} - \beta _{11} )^{2} c_{1} p_{2}}{ \beta _{12} ( 1+ \tau _{2} )} - \frac{d_{1} \beta _{12} p _{2}}{1+ \tau _{2}} - \frac{ ( \tau _{2} - \beta _{11} ) c _{4} ( 1+ \beta _{11} )}{\beta _{12} ( 1+ \tau _{2} )}, \\ & t_{5} = \biggl( \frac{ ( \tau _{2} - \beta _{11} ) \beta _{16}}{\beta _{12} ( \tau _{2} +1 )} - \frac{\beta _{25}}{1+ \tau _{2}} \biggr) \beta _{12}^{3} - \biggl( \frac{ ( \tau _{2} - \beta _{11} ) \beta _{17}}{\beta _{12} ( 1+ \tau _{2} )} - \frac{\beta _{26}}{\tau _{2} +1} \biggr) \beta _{12}^{2} ( 1+ \beta _{11} ) \\ & \hphantom{ t_{5}}\quad {} +2 \biggl( \frac{ ( \tau _{2} - \beta _{11} ) \beta _{13}}{ \beta _{12} ( 1+ \tau _{2} )} - \frac{\beta _{23}}{1+ \tau _{2}} \biggr) \beta _{12}^{2} p_{1} + \frac{ ( \tau _{2} - \beta _{11} ) \beta _{18} ( 1+ \beta _{11} )^{2}}{1+ \tau _{2}} \\ & \hphantom{ t_{5}}\quad {} + \biggl( \frac{ ( \tau _{2} - \beta _{11} ) \beta _{14}}{ \beta _{12} ( 1+ \tau _{2} )} - \frac{\beta _{24}}{\tau _{2} +1} \biggr) \beta _{12} p_{1} ( \tau _{2} - \beta _{11} ) \\ & \hphantom{ t_{5}} \quad {}- \biggl( \frac{ ( \tau _{2} - \beta _{11} ) \beta _{14}}{ \beta _{12} ( 1+ \tau _{2} )} - \frac{\beta _{24}}{\tau _{2} +1} \biggr) \beta _{12} p_{1} ( 1+ \beta _{11} ) \\ & \hphantom{ t_{5}} \quad {} - \frac{ ( \tau _{2} - \beta _{11} ) \beta _{19} ( 1+ \beta _{11} )^{3}}{\beta _{12} ( \tau _{2} +1 )} -2 \frac{ ( \tau _{2} - \beta _{11} )^{2} \beta _{15} ( 1+ \beta _{11} ) p_{1}}{\beta _{12} ( \tau _{2} +1 )}. \end{aligned}$$

Furthermore, we define the following nonzero real numbers:

$$ m_{1} = \biggl( \frac{\partial ^{2} g_{1}}{\partial u \partial \mu ^{*}} + \frac{1}{2} \frac{ \partial G}{\partial \mu ^{*}} \frac{\partial ^{2} G}{\partial u^{2}} \biggr) _{(0,0)} =- \frac{d_{1} \beta _{12}}{1+ \tau _{2}} - \frac{ ( \tau _{2} - \beta _{11} ) c_{1} ( 1+ \beta _{11} )}{ \beta _{12} ( 1+ \tau _{2} )} \neq 0 $$

and

$$ m_{2} = \biggl( \frac{1}{6} \frac{\partial ^{3} G}{\partial u^{3}} + \biggl( \frac{1}{2} \frac{\partial ^{2} G}{\partial u^{2}} \biggr) ^{2} \biggr)_{(0,0)} = t_{1}^{2} + t_{5} \neq 0. $$

Due to aforementioned computation, we have the following result about period-doubling bifurcation of system (1.5).

Theorem 4.2

If \(m_{1} \neq 0\) and \(m_{2} \neq 0\), then system (1.5) undergoes period-doubling bifurcation at the unique positive equilibrium point when the parameter μ varies in a small neighborhood of \(\mu _{1}\). Furthermore, if \(m_{2} >0\), then the period-two orbits that bifurcate from positive equilibrium of (1.5) are stable, and if \(m_{2} <0\), then these orbits are unstable.

Neimark–Sacker bifurcation

In this section, we investigate when positive steady-states of systems (1.3) and (1.5) undergo Neimark–Sacker bifurcation. For this bifurcation, theory of normal forms is implemented for the existence and direction of such a type of bifurcation. Recently, Neimark–Sacker bifurcation related to discrete-time models has been investigated by many authors [19,20,21,22,23,24,25,26,27]. Furthermore, in the case of continuous systems, we refer to [28,29,30,31,32] for some recent discussions related to Hopf bifurcation.

First, we show that the positive equilibrium \(( u^{*}, k e^{- u ^{*}} -1 )\) of system (1.3) undergoes Hopf bifurcation such that k is selected as a bifurcation parameter. For this, we see that the characteristic polynomial (2.4) has complex roots if the following inequality holds:

$$ \biggl( 2- \frac{q}{p -1} \biggr)^{2} < 4 \biggl( 2- \frac{e^{\frac{q}{p -1}}}{k} - \frac{1}{p} + \frac{e^{\frac{q}{p -1}}}{kp} - \frac{q}{p -1} \biggr). $$
(5.1)

Furthermore, we suppose that \(\eta _{1}\) and \(\eta _{2}\) are complex roots of (2.4). Then these roots satisfy \(\vert \eta _{1} \vert = \vert \eta _{2} \vert =1\) whenever the following conditions hold:

$$ k = \frac{e^{\frac{q}{p -1}} ( p -1 )^{2}}{( p -1 )^{2} - pq}, \quad\quad ( p -1 )^{2} > pq. $$
(5.2)

Keeping in view (5.1) and (5.2), we consider the following set:

$$ \mathbb{S}_{3} = \biggl\{ ( p, q, k )\in \mathbb{R} _{+}^{3}: p >1, q >0, k = \frac{e^{\frac{q}{p -1}} ( p -1 )^{2}}{( p -1 )^{2} - pq}, pq < ( p -1 )^{2} \biggr\} . $$

Suppose that \(( p, q, k )\in \mathbb{S}_{3}\). Then the positive equilibrium \(( u^{*}, k e^{- u^{*}} -1 )\) of system (1.3) undergoes Hopf bifurcation whenever k varies in a small neighborhood of \(k_{2}\) given as

$$ k_{2}:= \frac{e^{\frac{q}{p -1}} ( p -1 )^{2}}{( p -1 )^{2} - pq}. $$

Assume that \(( p, q, k )\in \mathbb{S}_{3}\). Then plant–herbivore model (1.3) is represented equivalently by the following two-dimensional map:

$$ \begin{pmatrix} X \\ Y \end{pmatrix} \rightarrow \begin{pmatrix} \frac{k_{2} X e^{- X}}{1+ Y} \\ \frac{pXY}{q + X} \end{pmatrix}. $$
(5.3)

Assume that represents a small perturbation in \(k_{2}\). Then the corresponding perturbed map for (5.3) is expressed as follows:

$$ \begin{pmatrix} X \\ Y \end{pmatrix} \rightarrow \begin{pmatrix} \frac{( k_{2} + \tilde{k} ) X e^{- X}}{1+ Y} \\ \frac{pXY}{q + X} \end{pmatrix}. $$
(5.4)

To translate the positive fixed point of (5.4) at \((0,0)\), we implement the translations \(x = X - u^{*}\) and \(y = Y - ( ( k_{2} + \tilde{k} ) e^{- u^{*}} -1 )\) with \(u^{*} = \frac{q}{p -1}\). Then from (5.4) it follows that

$$ \begin{pmatrix} x \\ y \end{pmatrix} \rightarrow \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} h_{1} ( x, y ) \\ h_{2} ( x, y ) \end{pmatrix}, $$
(5.5)

where

$$\begin{aligned}& h_{1} ( x, y )= m_{13} x^{2} + m_{14} xy + m_{15} y^{2} + m_{16} x ^{3} + m_{17} x^{2} y + m_{18} x y^{2} + m_{19} y^{3} + O \bigl(\bigl( \vert x \vert + \vert y \vert \bigr)^{4} \bigr), \\& h_{2} ( x, y )= m_{23} x^{2} + m_{24} xy + m_{25} x^{3} + m_{26} x ^{2} y + O \bigl(\bigl( \vert x \vert + \vert y \vert \bigr)^{4} \bigr), \\& m_{11} =1- u^{*}, \quad\quad m_{12} =- \frac{u^{*} e^{u^{*}}}{( k_{2} + \tilde{k} )},\quad\quad m_{13} = \frac{u^{*} -2}{2}, \quad\quad m_{14} = \frac{( u^{*} -1) e^{u^{*}}}{( k_{2} + \tilde{k} )},\\& m_{15} = \frac{u^{*} e^{2 u^{*}}}{( k_{2} + \tilde{k} )^{2}}, \quad\quad m_{16} = \frac{3- u^{*}}{6},\quad\quad m_{17} = \frac{e^{u^{*}} (2- u^{*} )}{2( k_{2} + \tilde{k} )},\quad\quad m_{18} = \frac{e^{2 u^{*}} (1- u^{*} )}{( k_{2} + \tilde{k} )^{2}}, \\& m_{19} =- \frac{u^{*} e^{3 u^{*}}}{( k_{2} + \tilde{k} )^{3}}, \quad\quad m_{21} = \frac{pq ( ( k_{2} + \tilde{k} ) e^{- u^{*}} -1 )}{( q + u^{*} )^{2}},\quad\quad m_{22} = \frac{p u^{*}}{q + u^{*}}, \\& m _{23} = \frac{pq ( 1-( k_{2} + \tilde{k} ) e^{- u^{*}} )}{( q + u^{*} )^{3}}, \quad\quad m_{24} = \frac{pq}{( q + u^{*} )^{2}}, \quad\quad m_{25} = \frac{pq ( ( k _{2} + \tilde{k} ) e^{- u^{*}} -1 )}{( q + u^{*} )^{4}},\\& m_{26} =- \frac{pq}{( q + u^{*} )^{3}}. \end{aligned}$$

Moreover, the characteristic equation for the variational matrix of system (5.5) computed at \((0,0)\) is given as follows:

$$ \eta ^{2} - P ( \tilde{k} ) \eta + Q ( \tilde{k} )=0, $$
(5.6)

where

$$ P ( \tilde{k} )=2- \frac{q}{p -1}, $$

and

$$ Q ( \tilde{k} )=2- \frac{e^{\frac{q}{p -1}}}{k_{2} + \tilde{k}} - \frac{1}{p} + \frac{e^{\frac{q}{p -1}}}{p ( k_{2} + \tilde{k} )} - \frac{q}{p -1}. $$

Assume that \(( p, q, k )\in \mathbb{S}_{3}\). Then the complex roots for (5.6) are computed as follows:

$$ \eta _{1} = \frac{P ( \tilde{k} )- i \sqrt{4 Q ( \tilde{k} )- ( P ( \tilde{k} ) )^{2}}}{2} $$

and

$$ \eta _{2} = \frac{P ( \tilde{k} )+ i \sqrt{4 Q ( \tilde{k} )- ( {P}( \tilde{k} ) )^{2}}}{2}. $$

Then it easily follows that

$$ \vert \eta _{1} \vert = \vert \eta _{2} \vert = \sqrt{2- \frac{e^{\frac{q}{p -1}}}{k_{2} + \tilde{k}} - \frac{1}{p} + \frac{e^{\frac{q}{p -1}}}{p ( k_{2} + \tilde{k} )} - \frac{q}{p -1}}. $$

Also, we have that

$$ \biggl( \frac{d \vert \eta _{2} \vert }{d \tilde{k}} \biggr)_{\tilde{k} =0} =\biggl( \frac{d \vert \eta _{1} \vert }{d \tilde{k}} \biggr)_{\tilde{k} =0} = \frac{e^{- \frac{q}{p -1}} ( 1+ p^{2} - p (2+ q ) )^{2}}{2( p -1 )^{3} p} \neq 0. $$

Since \(-2< P (0)<2\) as \(( p, q, k )\in \mathbb{S}_{3}\). Moreover, a simple computation yields that \(P (0)=2- \frac{q}{p -1}\), and we suppose that \(P (0)\neq 0\) and \(P (0)\neq -1\), that is,

$$ q \neq 2( p -1), \quad\quad q \neq 3( p -1). $$
(5.7)

Suppose that (5.7) holds and \(( p, q, k )\in \mathbb{S}_{3}\). Then it follows that \(P (0)\neq \pm 2,0,-1\), that is, \(\eta _{1}^{m}, \eta _{2}^{m} \neq 1\) for all \(m =1,2,3,4\) at \(\tilde{k} =0\). Therefore both roots of (5.6) do not lie in the intersection of the unit circle with the coordinate axes when \(\tilde{k} =0\).

Furthermore, we suppose that \(\kappa = \frac{P (0)}{2}\) and \(\omega = \frac{\sqrt{4 Q (0)-( P (0) )^{2}}}{2}\). Then to convert (5.5) into normal form, we take into account the following similarity transformation:

$$ \begin{pmatrix} x \\ y \end{pmatrix} \rightarrow \begin{pmatrix} m_{12} & 0 \\ \kappa - m_{11} & - \omega \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix}. $$
(5.8)

Due to implementation of similarity transformation (5.8), we can obtain the following normal form for (5.5):

$$ \begin{pmatrix} u \\ v \end{pmatrix} \rightarrow \begin{pmatrix} \kappa & - \omega \\ \omega & \kappa \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix} + \begin{pmatrix} \tilde{f} ( u, v ) \\ \tilde{g} ( u, v ) \end{pmatrix}, $$
(5.9)

where

$$\begin{aligned}& \begin{aligned} \tilde{f} ( u, v )&= \frac{m_{13}}{m_{12}} x^{2} + \frac{m_{14}}{m_{12}} xy + \frac{m_{15}}{m_{12}} y^{2} + \frac{m_{16}}{m _{12}} x^{3} + \frac{m_{17}}{m_{12}} x^{2} y + \frac{m_{18}}{m_{12}} x y^{2} + \frac{m_{19}}{m_{12}} y^{3} \\&\quad{} + O \bigl(\bigl( \vert u \vert + \vert v \vert \bigr)^{4} \bigr),\end{aligned} \\& \begin{aligned} \tilde{g} ( u, v ) &=\biggl( \frac{( \kappa - m_{11} ) m_{13}}{m_{12} \omega } - \frac{m_{23}}{\omega } \biggr) x^{2} +\biggl( \frac{( \kappa - m_{11} ) m_{14}}{m_{12} \omega } - \frac{m_{24}}{\omega } \biggr) xy \\ &\quad{} +\biggl( \frac{( \kappa - m_{11} ) m_{15}}{m_{12} \omega } \biggr) y^{2} +\biggl( \frac{( \kappa - m_{11} ) m_{16}}{m_{12} \omega } - \frac{m_{25}}{ \omega } \biggr) x^{3} \\ &\quad{} +\biggl( \frac{( \kappa - m_{11} ) m_{17}}{m_{12} \omega } - \frac{m_{26}}{ \omega } \biggr) x^{2} y +\biggl( \frac{( \kappa - m_{11} ) m_{18}}{m_{12} \omega } \biggr) x y^{2} \\ &\quad{} +\biggl( \frac{( \kappa - m_{11} ) m_{19}}{m_{12} \omega } \biggr) y^{3} + O \bigl(\bigl( \vert u \vert + \vert v \vert \bigr)^{4} \bigr), \end{aligned} \end{aligned}$$

\(x = m_{12} u\), and \(y =( \kappa - m_{11} ) u - \omega v\). To discuss the direction for Hopf bifurcation, we consider the following first Lyapunov exponent computed as

$$ L =\biggl(\biggl[- \operatorname{Re} \biggl( \frac{(1-2 \eta _{1} ) \eta _{2}^{2}}{1- \eta _{1}} \zeta _{20} \zeta _{11} \biggr)- \frac{1}{2} \vert \zeta _{11} \vert ^{2} - \vert \zeta _{02} \vert ^{2} + \operatorname{Re} ( \eta _{2} \zeta _{21} ) \biggr] \biggr)_{\tilde{k} =0}, $$

where

$$\begin{aligned}& \zeta _{20} = \frac{1}{8} \bigl[ \tilde{f}_{uu} - \tilde{f}_{vv} +2 \tilde{g}_{uv} + i ( \tilde{g}_{uu} - \tilde{g}_{vv} -2 \tilde{f}_{uv} ) \bigr], \\& \zeta _{11} = \frac{1}{4} \bigl[ \tilde{f}_{uu} + \tilde{f}_{vv} + i ( \tilde{g}_{uu} + \tilde{g}_{vv} ) \bigr], \\& \zeta _{02} = \frac{1}{8} \bigl[ \tilde{f}_{uu} - \tilde{f}_{vv} -2 \tilde{g}_{uv} + i ( \tilde{g}_{uu} - \tilde{g}_{vv} +2 \tilde{f}_{uv} ) \bigr], \\& \zeta _{21} = \frac{1}{16} \bigl[ \tilde{f}_{uuu} + \tilde{f}_{uvv} + \tilde{g}_{uuv} + \tilde{g}_{vvv} + i ( \tilde{g}_{uuu} + \tilde{g}_{uvv} - \tilde{f}_{uuv} - \tilde{f}_{vvv} ) \bigr]. \end{aligned}$$

Arguing as in [33,34,35,36,37], we present the following result on the existence and direction of Neimark–Sacker bifurcation at positive steady state of system (1.3).

Theorem 5.1

Suppose that (5.7) holds and \(L\neq 0\). Then the positive equilibrium \(( u^{*},k e^{- u^{*}} -1 )\) of system (1.3) undergoes Hopf bifurcation as the bifurcation parameter k varies in a small neighborhood of \(k_{2} = \frac{e^{ \frac{q}{p-1}} (p-1 )^{2}}{(p-1 )^{2} -pq}\). Furthermore, if \(L<0\), then an attracting invariant closed curve bifurcates from the equilibrium point for \(k> k_{2}\), and if \(L>0\), then a repelling invariant closed curve bifurcates from the equilibrium point for \(k< k_{2}\).

Finally, in this section, we discuss when the positive steady state \(( v^{*}, \sqrt{\mu e^{- v^{*}} -1} )\) of system (1.5) undergoes Hopf bifurcation when μ is taken as a bifurcation parameter.

Suppose that the following parametric conditions hold:

$$ \eta < 4 \sqrt{\eta } \sqrt{\nu -1}, \quad\quad 4( \nu -1 )^{3/2} - \sqrt{ \eta } \nu >0. $$
(5.10)

Furthermore, we assume that

$$ \mu = \frac{4 e^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \nu -1 )^{3/2}}{4( \nu -1 )^{3/2} - \sqrt{\eta } \nu }. $$
(5.11)

Then, keeping in view conditions (5.10) and (5.11), we consider the set

$$ \begin{aligned} \mathbb{S}_{4} = {}&\biggl\{ ( \mu , \nu , \eta ): \nu >1, \mu , \eta >0, \mu = \frac{4 e^{ \frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \nu -1 )^{3/2}}{4( \nu -1 )^{3/2} - \sqrt{\eta } \nu }, \\&\eta < 4 \sqrt{\eta } \sqrt{\nu -1}, 4( \nu -1 )^{3/2} > \sqrt{\eta } \nu \biggr\} .\end{aligned} $$

Moreover, assume that \(( \mu , \nu , \eta )\in \mathbb{S} _{4}\). Then the positive steady state \(( v^{*}, \sqrt{\mu e ^{- v^{*}} -1} )\) of system (1.5) undergoes Hopf bifurcation as μ varies in a small neighborhood of \(\mu _{2}\) given as

$$ \mu _{2}:= \frac{4 e^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \nu -1 )^{3/2}}{4( \nu -1 )^{3/2} - \sqrt{\eta } \nu }. $$

Next, we assume that \(( \mu , \nu , \eta )\in \mathbb{S}_{4}\). Then system (1.5) is represented by the following two-dimensional map:

$$ \begin{pmatrix} X \\ Y \end{pmatrix} \rightarrow \begin{pmatrix} \frac{\mu _{2} X e^{- X}}{1+ Y^{2}} \\ \frac{\nu X^{2} Y}{\eta + X^{2}} \end{pmatrix}. $$
(5.12)

Taking a small perturbation μ̃ in \(\mu _{2}\) corresponding to map (5.12), we have the following perturbed mapping:

$$ \begin{pmatrix} X \\ Y \end{pmatrix} \rightarrow \begin{pmatrix} \frac{( \mu _{2} + \tilde{\mu } ) X e^{- X}}{1+ Y^{2}} \\ \frac{\nu X^{2} Y}{\eta + X^{2}} \end{pmatrix}. $$
(5.13)

Furthermore, taking into account the translations \(x = X - v^{*}\) and \(y = Y - \sqrt{( \mu _{1} + \mu ^{*} ) e^{- v^{*}} -1}\), where \(v^{*} = \sqrt{\frac{\eta }{\nu -1}}\), (5.13) is converted into following map:

$$ \begin{pmatrix} x \\ y \end{pmatrix} \rightarrow \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} k_{1} ( x, y ) \\ k_{2} ( x, y ) \end{pmatrix}, $$
(5.14)

where

$$\begin{aligned}& k_{1} ( x, y )= b_{13} x^{2} + b_{14} xy + b_{15} y^{2} + b_{16} x ^{3} + b_{17} x^{2} y + b_{18} x y^{2} + b_{19} y^{3} + O \bigl(\bigl( \vert x \vert + \vert y \vert \bigr)^{4} \bigr), \\& k_{2} ( x, y )= b_{23} x^{2} + b_{24} xy + b_{25} x^{3} + b_{26} x ^{2} y + O \bigl(\bigl( \vert x \vert + \vert y \vert \bigr)^{4} \bigr),\quad\quad b_{11} =1- v^{*}, \\& b_{12} =- \frac{2 v^{*} e^{v^{*}} \sqrt{( \mu _{2} + \tilde{\mu } ) e ^{- v^{*}} -1}}{\mu _{2} + \tilde{\mu }},\quad\quad b_{13} = \frac{v^{*}}{2} -1,\ \\& b_{14} = \frac{2 \sqrt{( \mu _{2} + \tilde{\mu } ) e^{- v^{*}} -1} ( v^{*} -1 )}{e^{- v^{*}} ( \mu _{2} + \tilde{\mu } )}, \quad\quad b_{15} = \frac{v^{*} e^{v^{*}} ( 3( \mu _{2} + \tilde{\mu } )-4 e ^{v^{*}} )}{( \mu _{2} + \tilde{\mu } )^{2}}, \\& b_{16} = \frac{3- v^{*}}{6}, \quad\quad b_{17} = \frac{(2- v^{*} ) \sqrt{( \mu _{2} + \tilde{\mu } ) e^{- v^{*}} -1}}{e^{- v^{*}} ( \mu _{2} + \tilde{\mu } )}, \\& b_{18} = \frac{e^{v^{*}} (1- v^{*} )(3( \mu _{2} + \tilde{\mu } )-4 e ^{v^{*}} )}{( \mu _{2} + \tilde{\mu } )^{2}}, \quad\quad b_{19} = \frac{4 v^{*} \sqrt{( \mu _{2} + \tilde{\mu } ) e^{- v^{*}} -1} ( 2-( \mu _{2} + \tilde{\mu } ) e^{- v^{*}} )}{e^{-3 v^{*}} ( \mu _{2} + \tilde{\mu } )^{3}}, \\& b_{21} = \frac{2 v^{*} ( \nu -1 )^{2} \sqrt{( \mu _{2} + \tilde{\mu } ) e^{- v^{*}} -1}}{\eta \nu }, \quad\quad b_{22} =1, \\& b_{23} = \frac{( \nu -1 )^{2} ( \nu -4) \sqrt{( \mu _{2} + \tilde{\mu } ) {e} ^{- v^{*}} -1}}{\eta \nu ^{2}}, \quad\quad b_{24} = \frac{2 v^{*} ( \nu -1 )^{2}}{\eta \nu }, \\& b_{25} =- \frac{4 v ^{*} ( \nu -1 )^{3} ( \nu -2) \sqrt{( \mu _{2} + \tilde{\mu } ) e ^{- v^{*}} -1}}{\eta ^{2} \nu ^{3}},\quad\quad b_{26} = \frac{( \nu -1 )^{2} ( \nu -4)}{\eta \nu ^{2}}. \end{aligned}$$

Then the characteristic equation for the variational matrix of (5.14) computed at its equilibrium \((0,0)\) is given as follows:

$$ \eta ^{2} - R ( \tilde{\mu } ) \eta + S ( \tilde{\mu } )=0, $$
(5.15)

where

$$ R ( \tilde{\mu } )=2- \frac{\sqrt{\eta }}{\sqrt{\nu -1}} $$

and

$$ S ( \tilde{\mu } )=5- \frac{\sqrt{\eta }}{\sqrt{\nu -1}} - \frac{4}{ \nu } - \frac{4 e^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \nu -1)}{( \mu _{2} + \tilde{\mu } ) \nu }. $$

Assume that \(( \mu , \nu , \eta )\in \mathbb{S}_{4}\). Then the complex roots for (5.17) are computed as follows:

$$ \eta _{1} = \frac{R ( \tilde{\mu } )- i \sqrt{4 S ( \tilde{\mu } )- ( R ( \tilde{\mu } ) )^{2}}}{2} $$

and

$$ \eta _{2} = \frac{R ( \tilde{\mu } )+ i \sqrt{4 S ( \tilde{\mu } )- ( R ( \tilde{\mu } ) )^{2}}}{2}. $$

Moreover, it follows that

$$ \vert \eta _{1} \vert = \vert \eta _{2} \vert = \sqrt{5- \frac{\sqrt{ \eta }}{\sqrt{\nu -1}} - \frac{4}{\nu } - \frac{4 e^{\frac{\sqrt{ \eta }}{\sqrt{\nu -1}}} ( \nu -1)}{( \mu _{2} + \tilde{\mu } ) \nu }}. $$

Similarly, a simple computation yields that

$$ \biggl( \frac{d \vert \eta _{2} \vert }{d \tilde{\mu }} \biggr)_{\tilde{\mu } =0} =\biggl( \frac{d \vert \eta _{1} \vert }{d \tilde{\mu }} \biggr)_{\tilde{\mu } =0} = \frac{e ^{- \frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ( \sqrt{\eta } \nu -4( \nu -1 )^{3/2} )^{2}}{8( \nu -1 )^{2} \nu } \neq 0. $$

Furthermore, according to the existence condition for bifurcation, we have that \(-2< R (0)<2\). Next, it follows that \(R (0)=2- \frac{\sqrt{ \eta }}{\sqrt{\nu -1}}\). Moreover, we suppose that \(R (0)\neq 0\) and \(R (0)\neq -1\), that is,

$$ \sqrt{\eta } \neq 2 \sqrt{\nu -1}, \quad\quad \sqrt{\eta } \neq 3 \sqrt{ \nu -1}. $$
(5.16)

Assume that \(( \mu , \nu , \eta )\in \mathbb{S}_{4}\) and (5.16) holds. Then it follows that \(R (0)\neq \pm 2,0,-1\), and due to these restrictions, we have that \(\eta _{1}^{m}, \eta _{2}^{m} \neq 1\) for all \(m =1,2,3,4\) at \(\tilde{\mu } =0\). Therefore, at \(\tilde{\mu } =0\), the roots of (5.14) do not lie in the intersection of the unit circle with the coordinate axes.

Furthermore, we take \(\alpha = \frac{R (0)}{2}\) and \(\beta = \frac{\sqrt{4 S (0)-( R (0) )^{2}}}{2}\). Then, to obtain the normal form for (5.14) at \(\tilde{\mu } =0\), we consider the following similarity transformation:

$$ \begin{pmatrix} x \\ y \end{pmatrix} \rightarrow \begin{pmatrix} b_{12} & 0 \\ \alpha - b_{11} & - \beta \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix}. $$
(5.17)

Due to implementation of similarity transformation (5.17), we obtain the following normal form for (5.14):

$$ \begin{pmatrix} u \\ v \end{pmatrix} \rightarrow \begin{pmatrix} \alpha & - \beta \\ \beta & \alpha \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix} + \begin{pmatrix} \tilde{h} ( u, v ) \\ \tilde{k} ( u, v ) \end{pmatrix}, $$
(5.18)

where

$$\begin{aligned}& \tilde{h} ( u, v )= \frac{b_{13}}{b_{12}} x^{2} + \frac{b_{14}}{b_{12}} xy + \frac{b_{15}}{b_{12}} y^{2} + \frac{b_{16}}{b _{12}} x^{3} + \frac{b_{17}}{b_{12}} x^{2} y + \frac{b_{18}}{b_{12}} x y^{2} + \frac{b_{19}}{b_{12}} y^{3} + O \bigl(\bigl( \vert u \vert + \vert v \vert \bigr)^{4} \bigr), \\& \begin{aligned} \tilde{k} ( u, v ) &=\biggl( \frac{( \alpha - b_{11} ) b_{13}}{b_{12} \beta } - \frac{b_{23}}{\beta } \biggr) x^{2} +\biggl( \frac{( \alpha - b_{11} ) b _{14}}{b_{12} \beta } - \frac{b_{24}}{\beta } \biggr) xy \\ &\quad{} +\biggl( \frac{( \alpha - b_{11} ) b_{15}}{b_{12} \beta } \biggr) y^{2} +\biggl( \frac{( \alpha - b_{11} ) b_{16}}{b_{12} \beta } - \frac{b_{25}}{\beta } \biggr) x ^{3} \\ &\quad{} +\biggl( \frac{( \alpha - b_{11} ) b_{17}}{b_{12} \beta } - \frac{b_{26}}{ \beta } \biggr) x^{2} y +\biggl( \frac{( \alpha - b_{11} ) b_{18}}{b_{12} \beta } \biggr) x y^{2} \\ &\quad{} +\biggl( \frac{( \alpha - {b}_{11} ) b_{19}}{b_{12} \beta } \biggr) y ^{3} + O \bigl(\bigl( \vert u \vert + \vert v \vert \bigr)^{4} \bigr), \end{aligned} \end{aligned}$$

\(x = b_{12} u\), and \(y =( \alpha - b_{11} ) u - \beta v\). To discuss the direction for Hopf bifurcation, we consider the following first Lyapunov exponent computed as

$$ \varUpsilon =\biggl(\biggl[- \operatorname{Re} \biggl( \frac{(1-2 \eta _{1} ) \eta _{2}^{2}}{1- \eta _{1}} \zeta _{20} \zeta _{11} \biggr)- \frac{1}{2} \vert \zeta _{11} \vert ^{2} - \vert \zeta _{02} \vert ^{2} + \operatorname{Re} ( \eta _{2} \zeta _{21} ) \biggr] \biggr)_{\tilde{\mu } =0}, $$

where

$$\begin{aligned}& \zeta _{20} = \frac{1}{8} \bigl[ \tilde{h}_{uu} - \tilde{h}_{vv} +2 \tilde{k}_{uv} + i ( \tilde{k}_{uu} - \tilde{k}_{vv} -2 \tilde{h}_{uv} ) \bigr], \\& \zeta _{11} = \frac{1}{4} \bigl[ \tilde{h}_{uu} + \tilde{h}_{vv} + i ( \tilde{k}_{uu} + \tilde{k}_{vv} ) \bigr], \\& \zeta _{02} = \frac{1}{8} \bigl[ \tilde{h}_{uu} - \tilde{h}_{vv} -2 \tilde{k}_{uv} + i ( \tilde{k}_{uu} - \tilde{k}_{vv} +2 \tilde{h}_{uv} ) \bigr], \\& \zeta _{21} = \frac{1}{16} \bigl[ \tilde{h}_{uuu} + \tilde{h}_{uvv} + \tilde{k}_{uuv} + \tilde{k}_{vvv} + i ( \tilde{k}_{uuu} + \tilde{k}_{uvv} - \tilde{h}_{uuv} - \tilde{h}_{vvv} ) \bigr]. \end{aligned}$$

Next, due to aforementioned computation, we state the following theorem, which gives conditions for the existence and direction of Hopf bifurcation for positive equilibrium of system (1.5).

Theorem 5.2

Suppose that (5.16) holds and \(\varUpsilon \neq 0\). Then the positive equilibrium \(( v^{*}, \sqrt{\mu e ^{- v^{*}} -1} )\) of system (1.5) undergoes Hopf bifurcation as the bifurcation parameter μ varies in a small neighborhood of \(\mu _{2} = \frac{4 e^{\frac{\sqrt{\eta }}{\sqrt{ \nu -1}}} ( \nu -1 )^{3/2}}{4( \nu -1 )^{3/2} - \sqrt{\eta } \nu }\). Furthermore, if \(\varUpsilon <0\), then an attracting invariant closed curve bifurcates from the equilibrium point for \(\mu > \mu _{2}\), and if \(\varUpsilon >0\), then a repelling invariant closed curve bifurcates from the equilibrium point for \(\mu < \mu _{2}\).

Chaos control

For controlling random irregular and fluctuating behavior in a biological system, chaos control is considered to be a practical tool for avoiding this chaotic and complex behavior. For further details related to biological meanings of chaos control and its practical use in the real world, we refer to [38].

In this section we implement a simple chaos control technique for both systems (1.3) and (1.5). Moreover, there are various chaos control methods for discrete-time dynamical systems. For further details related to these techniques, we refer to [39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61].

Here we implement a hybrid control method (also see [24, 47]). The hybrid control technique is based on parameter perturbation and a state feedback control method. First, we implement this methodology to system (1.3) as follows:

$$ \begin{gathered} x_{n +1} = \alpha \biggl( \frac{k x_{n} e^{- x_{n}}}{1+ y_{n}} \biggr) + ( 1- \alpha ) x_{n}, \\ y_{n +1} = \alpha \biggl( \frac{p x_{n} y_{n}}{q + x_{n}} \biggr) +(1- \alpha ) y_{n}, \end{gathered} $$
(6.1)

where \(0< \alpha <1\) is a control parameter. Similarly, implementation of a hybrid control strategy to system (1.5) yields:

$$ \begin{gathered} x_{n +1} = \beta \biggl( \frac{\mu x_{n} e^{- x_{n}}}{1+ y_{n}^{2}} \biggr) + ( 1- \beta ) x_{n}, \\ y_{n +1} = \beta \biggl( \frac{\nu x_{n}^{2} y_{n}}{\eta + x_{n} ^{2}} \biggr) +(1- \beta ) y_{n}, \end{gathered} $$
(6.2)

where \(0< \beta <1\) is a control parameter for a hybrid control strategy. Next, system (6.1) is controllable as long as its steady state \(( u^{*}, k e^{- u^{*}} -1 )\) is locally asymptotically stable. Then for particular choice of control parameter α, we can obtain the desired interval for controlling chaos and bifurcation. Furthermore, the variational matrix for the controlled system (6.1) at its positive equilibrium \(( u^{*}, k e^{- u^{*}} -1 )\) is computed as follows:

$$ \begin{bmatrix} 1- \alpha u^{*} & - \frac{\alpha u^{*} e^{u^{*}}}{k} \\ \frac{\alpha ( k e^{- u^{*}} -1 ) ( p -1)}{p u^{*}} & 1 \end{bmatrix}. $$

Then the characteristic polynomial for the aforementioned variational matrix is given by

$$ F ( \eta )= \eta ^{2} - \biggl( 2- \frac{\alpha q}{p -1} \biggr) \eta +1- \frac{q \alpha }{p -1} + \alpha ^{2} - \frac{e^{ \frac{q}{p -1}} \alpha ^{2}}{k} - \frac{\alpha ^{2}}{p} + \frac{e^{\frac{q}{p -1}} \alpha ^{2}}{kp}. $$
(6.3)

Lemma 6.1

Assume that \(0< q<(p-1)\ln (k)\), \(k>1\), and \(p>1\). Then the positive steady state \(( u^{*},k e^{- u^{*}} -1 )\) of the controlled system (6.1) is a sink if and only if

$$ \biggl\vert 2- \frac{\alpha q}{p -1} \biggr\vert < 2- \frac{q \alpha }{p -1} + \alpha ^{2} - \frac{e^{\frac{q}{p -1}} \alpha ^{2}}{k} - \frac{ \alpha ^{2}}{p} + \frac{e^{\frac{q}{p -1}} \alpha ^{2}}{kp} < 2. $$

For \(0< \alpha <1\), \(p \in [1,1000]\), \(k =60\), and \(q =50\), the controllable region for system (6.1) is depicted in Fig. 5 in the -plane.

Figure 5
figure5

Controllable region for system (6.1)

Furthermore, we suppose that \(\sqrt{\eta } <\ln ( \mu ) \sqrt{ \nu -1}\), \(\mu >1\), and \(\nu >1\). Then controlled system (6.2) has a unique positive steady-state \(( v^{*}, \sqrt{\mu e^{- v^{*}} -1} )\), which is similar to the positive equilibrium point of (1.5). Moreover, the variational matrix for controlled system (6.2) is computed as follows:

$$ \begin{bmatrix} 1- \frac{\beta \sqrt{\eta }}{\sqrt{\nu -1}} & - \frac{2 \beta \sqrt{e ^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}}} \sqrt{\eta } \sqrt{ \mu - e^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}}}}{\mu \sqrt{\nu -1}} \\ \frac{2 \beta \sqrt{\mu - e^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}}} ( \nu -1 )^{3/2}}{\sqrt{e ^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}}} \sqrt{\eta } \nu } & 1 \end{bmatrix}. $$

Furthermore, the characteristic polynomial for the aforementioned variational matrix is given by

$$ P ( \eta )= \eta ^{2} - \biggl( 2- \frac{\beta \sqrt{\eta }}{\sqrt{ \nu -1}} \biggr) \eta +1- \frac{\beta \sqrt{\eta }}{ \sqrt{\nu -1}} + \frac{4 \beta ^{2} ( \mu - e^{\frac{\sqrt{ \eta }}{\sqrt{\nu -1}}} ) ( \nu -1)}{\mu \nu }. $$
(6.4)

Lemma 6.2

Suppose that \(\sqrt{\eta } <\ln ( \mu ) \sqrt{ \nu -1}\), \(\mu >1\), and \(\nu >1\). Then the positive steady state \(( v^{*}, \sqrt{\mu e^{- v^{*}} -1} )\) of system (6.2) is locally asymptotically stable if the following condition is satisfied:

$$ \biggl\vert 2- \frac{\beta \sqrt{\eta }}{\sqrt{\nu -1}} \biggr\vert < 2- \frac{\beta \sqrt{\eta }}{\sqrt{\nu -1}} + \frac{4 \beta ^{2} ( \mu - e^{\frac{\sqrt{\eta }}{\sqrt{\nu -1}}} ) ( \nu -1)}{\mu \nu } < 2. $$

For \(0< \beta <1\), \(\nu \in [1,100]\), \(\eta =30\), and \(\mu =40\), the controllable region for system (6.2) is depicted in Fig. 6 in the νβ-plane.

Figure 6
figure6

Controllable region for system (6.2)

Numerical simulation and discussion

Example 7.1

First, we choose \(p=20.1\), \(q=45.1\), and \(k\in [5,20]\). Then system (1.3) undergoes flip bifurcation as bifurcation parameter k varies in the interval \([5,20]\). Moreover, the bifurcation diagram for plant population density \(x_{n}\) is depicted in Fig. 7a, and the corresponding maximum Lyapunov exponents (MLE) are depicted in Fig. 7b.

Figure 7
figure7

Bifurcation diagram and MLE for system (1.3) with \(p =20.1\), \(q =45.1\), \(k \in [ 5,20 ]\), and \(( x_{0}, y_{0} )=(2.36,3.17)\)

Example 7.2

Next, we choose \(p=7.3\), \(q=4.6\), \(k\in [ 5,55 ]\), and initial values \(( x_{0}, y _{0} )=(0.73,5.49)\). Then system (1.3) undergoes Hopf bifurcation as the bifurcation parameter k varies in a small neighborhood of \(k=13.48167321833215\). If we choose \((p,q,k)=(7.3,4.6,13.482)\), then the positive steady state for system (1.3) is given by \((0.730159, 5.49591)\). Furthermore, the characteristic equation for the variational matrix is given by

$$ \eta ^{2} -1.26984126984127 \eta +1=0. $$
(7.1)

The complex conjugate roots for (7.1) are \(\eta _{1} =0.634921+0.772577 i\) and \(\eta _{2} =0.634921-0.772577 i\) with \(\vert \eta _{1} \vert = \vert \eta _{2} \vert =1\). Thus, we have \(( p, q, k )=(7.3,4.6,13.48167321833215) \in \mathbb{S}_{3}\). Moreover, bifurcation diagrams and MLE are depicted in Fig. 8. Taking \(q =4.6\), \(p =7.3\), and \(k =13.46,13.48167,13.6,14,16,19.5\), phase portraits for system (1.3) are depicted in Fig. 9. To apply a hybrid control strategy, we choose \(( p, q, k )=(7.3,4.6,19.5)\). Then controlled system (6.1) takes the following form:

$$ \begin{gathered} x_{n +1} = \alpha \biggl( \frac{19.5 x_{n} e^{- x_{n}}}{1+ y_{n}} \biggr) + ( 1- \alpha ) x_{n}, \\ y_{n +1} = \alpha \biggl( \frac{7.3 x_{n} y_{n}}{4.6+ x_{n}} \biggr) +(1- \alpha ) y_{n}. \end{gathered} $$
(7.2)
Figure 8
figure8

Bifurcation diagrams and MLE for system (1.3) with \(p =7.3\), \(q =4.6\), \(k \in [ 5,55 ]\), and \(( x_{0}, y_{0} )=(0.73,5.49)\)

Figure 9
figure9

Phase portraits of system (1.3) for \(p =7.3\), \(q =4.6\), \(x_{0} =0.73\), \(y_{0} =5.49\) with different values of k

Then system (7.2) has a positive fixed point \((0.730159,8.39573)\). Furthermore, the characteristic equation for the variational matrix of (7.2) is computed as follows:

$$ \eta ^{2} -(2-0.730159 \alpha ) \eta +1-0.730159 \alpha +0.771162 \alpha ^{2} =0. $$
(7.3)

Due to the Jury condition, the equilibrium point \((0.730159,8.39573)\) is a sink if and only if \(0< \alpha <0.946829\). Choosing \(\alpha =0.945\), plots for the controlled system (7.2) are shown in Fig. 10.

Figure 10
figure10

Plots of the controlled system (7.2) with \(\alpha =0.945\) and \(( x_{0}, y_{0} )=(0.73,8.3957)\)

Example 7.3

Now we consider system (1.5) for the numerical verification of flip bifurcation. For this, we choose \(\mu \in [2,22]\), \(\nu =8.5\), \(\eta =55.8\), and the initial conditions \(( x_{0}, y_{0} )=(2.72,0.83)\). Then the population density of plants \(x_{n}\) undergoes flip bifurcation. We can see the bifurcation diagram for \(x_{n}\) and corresponding MLE in Fig. 11.

Figure 11
figure11

Bifurcation diagram and MLE for system (1.5) with \(\nu =8.5\), \(\eta =55.8\), \(\mu \in [ 2,22 ]\), and \(( x_{0}, y_{0} )=(2.72,0.83)\)

Example 7.4

At the end of this section, we verify the existence of Hopf bifurcation for system (1.5) by taking into account some particular parametric values. For such verification, we choose \(\nu =4.1\), \(\eta =1.5\), \(\mu \in [ 2,8 ]\), and \(( x_{0}, y_{0} )=(0.6956,0.5465)\). Then system (1.5) undergoes Hopf bifurcation as the parameter μ varies in a small neighborhood of a particular value \(\mu =2.603801522628164\). Furthermore, if we choose the parametric values \(\mu =2.603801522628164\), \(\nu =4.1\), and \(\eta =1.5\), then the unique positive fixed point for system (1.5) is \((0.695608,0.546535)\). At \((0.695608,0.546535)\) the characteristic equation for system (1.5) is

$$ \eta ^{2} -1.3043916563597475 \eta +1=0. $$
(7.4)

Furthermore, \(\eta _{1} =0.652196+0.758051 i\) and \(\eta _{2} =0.652196-0.758051 i\) are the roots of (7.4) with modulus \(\vert \eta _{1} \vert = \vert \eta _{2} \vert =1\). Therefore it follows that \(( \mu , \nu , \eta )=(2.603801522628164, 4.1,1.5)\in \mathbb{S}_{4}\), and bifurcation diagrams and MLE are depicted in Fig. 12. Moreover, for various values of μ, the phase portraits for system (1.5) are shown in Fig. 13. Lastly, we check the effectiveness of hybrid control strategy for system (1.5). For this purpose, we choose \(( \mu , \nu , \eta )=(8,4.1,1.5)\). Then due to this choice, controlled system (6.2) is given by

$$ \begin{gathered} x_{n +1} = \beta \biggl( \frac{8 x_{n} e^{- x_{n}}}{1+ y_{n}^{2}} \biggr) + ( 1- \beta ) x_{n}, \\ y_{n +1} = \beta \biggl( \frac{4.1 x_{n}^{2} y_{n}}{1.5+ x_{n}^{2}} \biggr) +(1- \beta ) y_{n}. \end{gathered} $$
(7.5)
Figure 12
figure12

Bifurcation diagrams and MLE for system (1.5) with \(\nu =4.1\), \(\eta =1.5\), \(\mu \in [ 2,8 ]\), and \(( x_{0}, y_{0} )=(0.6956,0.5465)\)

Figure 13
figure13

Phase portraits of system (1.5) for \(\nu =4.1\), \(\eta =1.5\), \(x_{0} =0.6956\), \(y_{0} =0.5465\) with different values of μ

Then system (7.5) has a unique positive equilibrium point \((0.695608,1.72921 )\), and the characteristic equation of the Jacobian matrix of (7.5) evaluated at \((0.695608,1.72921)\) is

$$ \eta ^{2} -(2-0.695608 \beta ) \eta +1-0.695608 \beta +2.26643 \beta ^{2} =0. $$
(7.6)

Now, according to the Jury condition, the roots of (7.6) lie inside the open unit disk if and only if

$$ \vert 2-0.695608 \beta \vert < 2-0.695608 \beta +2.26643 \beta ^{2} < 2. $$

Or, equivalently,

$$ 4-1.39122 \beta +2.26643 \beta ^{2} >0, \quad\quad 2.26643 \beta < 0.695608. $$

From aforementioned inequalities it follows that \(0< \beta <0.306918\). Thus the unique positive equilibrium point \((0.695608,1.72921)\) of the controlled system (7.5) is locally asymptotically stable if and only if \(0< \beta <0.306918\). The plots of the controlled system (7.5) are shown in Fig. 14 for \(\beta =0.304\).

Figure 14
figure14

Plots for the controlled system (7.5) with \(\beta =0.304\) and \(( x_{0}, y_{0} )=(0.6956,1.729)\)

Concluding remarks

This paper is concerned with qualitative behavior of two discrete-time plant–herbivore models in exponential forms. The models are proposed by taking into account that the function for plant limitation is of Ricker type, whereas the effect of herbivore on plant population and herbivore population growth rate are proportional to functional responses of type-II and type-III, respectively. The parametric conditions for local asymptotic stability of equilibria of both systems are investigated. Due to implementation of bifurcation theory and center manifold theorem, we obtained that both models undergo Neimark–Sacker bifurcation and period-doubling bifurcation at their positive steady states. Our results show that parameters related to growth rates of plants have strong stability effects or vice versa. To control chaotic behaviors of the systems, a hybrid control strategy is implemented. The effectiveness of this control strategy is illustrated through numerical simulations. Moreover, complex dynamics for both models is exhibited through periodic orbits, quasi-periodic orbits, and chaotic sets and windows. Furthermore, in present discussion of qualitative analysis for plant–herbivore interaction, Holling type-II and III functional responses are implemented with Ricker-type function for plant self-limitation. It is interesting to implement some other choice of plant self-limitation function. In future, we will apply a Beverton–Holt-type function for plant self-limitation.

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Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. G–405–130–39. The authors, therefore, acknowledge with thanks DSR for technical and financial support.

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Correspondence to E. M. Elsayed.

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Keywords

  • Plant–herbivore model
  • Local stability
  • Period-doubling bifurcation
  • Neimark–Sacker bifurcation
  • Chaos control