In this section, we focus on analyzing the stability of the equilibria of (2.4). For convenience, we rewrite equation (2.4) as

$$ \begin{gathered} x(n+1)=x(n)\exp\bigl\{ F\bigl(x(n),y(n) \bigr)\bigr\} , \\ y(n+1)=y(n)\exp\bigl\{ G\bigl(x(n),y(n)\bigr)\bigr\} , \end{gathered} $$

(4.1)

where

$$ \begin{gathered} F(x,y)= b-\frac{bx}{\min\{K,(P_{T}-\theta y)/q\}}-\min \biggl\{ p(x), \frac{\hat{f}\theta}{P_{T}-\theta y} \biggr\} y, \\ G(x,y)= \min \biggl\{ \hat{e}xp(x),\frac{P_{T}-\theta y}{\theta}p(x), \frac{\hat{e}\hat{f}\theta x}{P_{T}-\theta y} \biggr\} -d. \end{gathered} $$

To find the equilibrium points of equation (2.4), we solve

$$ \begin{gathered} \text{Producer nullclines:}\quad x\bigl[1-\exp\bigl\{ F(x,y)\bigr\} \bigr]=0, \quad\text{i.e., } x=0 \text{ or } F(x,y)=0, \\ \text{Grazer nullclines:}\quad y\bigl[1-\exp\bigl\{ G(x,y)\bigr\} \bigr]=0, \quad \text{i.e., } y=0 \text{ or } G(x,y)=0. \end{gathered} $$

The Jacobian of (2.4) is

$$ J(x,y)= \left ( \textstyle\begin{array}{c@{\quad}c} \exp\{F(x,y)\}+x\exp\{F(x,y)\}F_{x}(x,y) & x\exp\{F(x,y)\}F_{y}(x,y)\\ y\exp\{G(x,y)\}G_{x}(x,y) & \exp\{G(x,y)\}+y\exp\{G(x,y)\}G_{y}(x,y) \end{array}\displaystyle \right ), $$

(4.2)

where

$$ \begin{gathered} F_{x}(x,y) = \frac{\partial F(x,y)}{\partial x}= \textstyle\begin{cases} -\frac{b}{\min \{K, \frac{P_{T}-\theta y }{q} \}}-p'(x)y, & \text{if } f(x)< \frac{\hat{f}\theta}{Q},\\ -\frac{b}{\min \{K, \frac{P _{T}-\theta y}{q} \}}< 0, & \text{if } f(x) >\frac{\hat{f}\theta}{Q}, \end{cases}\displaystyle \\ F_{y}(x,y)=\frac{\partial F(x,y)}{\partial y}= \textstyle\begin{cases} -p(x)< 0, & \text{if } f(x)< \frac{\hat{f}\theta}{Q}, K< \frac {P_{T}-\theta y}{q},\\ -\frac{P_{T} \hat{f} \theta}{(P_{T}-\theta y)^{2}}< 0, & \text{if } f(x)>\frac{\hat{f}\theta}{Q}, K< \frac{P_{T}-\theta y}{q},\\ -\frac{bq\theta x}{(P_{T}-\theta y)^{2}}-p(x)< 0, & \text{if } f(x)< \frac {\hat{f}\theta}{Q}, K>\frac{P_{T}-\theta y}{q},\\ -\frac{bq\theta x}{(P_{T}-\theta y)^{2}}-\frac{P_{T} \hat{f} \theta }{(P_{T}-\theta y)^{2}}< 0, & \text{if } f(x)>\frac{\hat{f}\theta}{Q}, K>\frac{P_{T}-\theta y}{q}, \end{cases}\displaystyle \\ G_{x}(x,y)=\frac{\partial G(x,y)}{\partial x}= \textstyle\begin{cases} \hat{e}[p(x)+xp'(x)]>0, & \text{if } \hat{e}f(x)< \frac{Q}{\theta }f(x),\hat{e}f(x)< \hat{e}\hat{f}\frac{\theta}{Q},\\ \frac{P_{T}-\theta y}{\theta}p'(x)< 0, & \text{if } \frac{Q}{\theta }f(x)< \hat{e}f(x), \frac{Q}{\theta}f(x)< \hat{e}\hat{f}\frac {\theta}{Q},\\ \frac{\hat{e}\hat{f}\theta}{P_{T}-\theta y}>0, & \text{if } \hat {e}\hat{f}\frac{\theta}{Q}< \hat{e}f(x), \hat{e}\hat{f}\frac {\theta}{Q}< \frac{Q}{\theta}f(x), \end{cases}\displaystyle \\ G_{y}(x,y)=\frac{\partial G(x,y)}{\partial y}= \textstyle\begin{cases} 0, & \text{if } \hat{e}f(x)< \frac{Q}{\theta}f(x),\hat{e}f(x)< \hat {e}\hat{f}\frac{\theta}{Q},\\ -p(x)< 0, & \text{if } \frac{Q}{\theta}f(x)< \hat{e}f(x), \frac {Q}{\theta}f(x)< \hat{e}\hat{f}\frac{\theta}{Q},\\ \frac{\hat{e}\hat{f}\theta^{2} x}{(P_{T}-\theta y)^{2}}>0, & \text{if } \hat{e}\hat{f}\frac{\theta}{Q}< \hat{e}f(x), \hat{e}\hat{f}\frac {\theta}{Q}< \frac{Q}{\theta}f(x). \end{cases}\displaystyle \end{gathered} $$

(4.3)

### 4.1 Boundary equilibria

We use the following standard lemma (see Edelstein-Keshet [9], p. 57) to study the stability of equilibrium points of system (2.4).

### Lemma 4.1

(Jury test)

*Let**A**be a*
\(2\times2\)*constant matrix*. *Both characteristic roots of**A**have magnitude less than* 1 *if and only if*

$$\begin{aligned} 2>1+\operatorname{Det}(A)> \bigl\vert \operatorname{Tr}(A) \bigr\vert . \end{aligned}$$

(4.4)

In order to find the possible equilibrium points of system (2.4), we solve the equations

$$x\bigl[1-\exp\bigl\{ F(x,y)\bigr\} \bigr]=0,\qquad y\bigl[1-\exp\bigl\{ G(x,y)\bigr\} \bigr]=0. $$

It is easy to see that the equilibria of equation (2.4) are exactly the same as those of equation (2.1). The only boundary equilibrium points are \(E_{0}=(0,0)\) and \(E_{1}=(k,0)\).

The Jacobian matrix (4.2) at the origin \(E_{0}\) turns out to be

$$J(E_{0})= \left ( \textstyle\begin{array}{c@{\quad}c} e^{b} & 0 \\ 0 & e^{-d} \end{array}\displaystyle \right ). $$

It is clear that the magnitude of characteristic root \(e^{-d}\) is less than 1 while the magnitude of \(e^{b}\) is larger than 1. Consequently, \(E_{0}\) is always unstable.

The Jacobian matrix (4.2) at \(E_{1}\) becomes

$$J(E_{1})= \left ( \textstyle\begin{array}{c@{\quad}c} 1-b & kF_{y}(k,0) \\ 0 & \exp\{G(k,0)\} \end{array}\displaystyle \right ), $$

where

$$G(k,0)=\min \biggl\{ \hat{e}kp(k),\frac{P_{T}}{\theta}p(k),\frac{\hat {e}\hat{f}\theta k}{P_{T}} \biggr\} -d. $$

From Lemma 4.1, we obtain the following theorem.

### Theorem 4.1

*For* (2.4), \(E_{0}\)*is always unstable*. \(E_{1}\)*is locally asymptotically stable* (*LAS*) *if*

$$0< b< 2 \quad\textit{and} \quad\min \biggl\{ \hat{e}kp(k),\frac{P_{T}}{k\theta }p(k), \frac{\hat{e}\hat{f}\theta k}{P_{T}} \biggr\} < d; $$

*it is unstable if*

$$b>2 \quad\textit{or}\quad \min \biggl\{ \hat{e}kp(k),\frac{P_{T}}{k\theta }p(k), \frac{\hat{e}\hat{f}\theta k}{P_{T}} \biggr\} >d. $$

### Proof

If we let \(\lambda_{1}\) and \(\lambda_{2}\) be the characteristic roots of \(J(E_{1})\), then the condition

$$0< b< 2 \quad\text{and} \quad\min \biggl\{ \hat{e}kp(k),\frac{P_{T}}{k\theta }p(k), \frac{\hat{e}\hat{f}\theta k}{P_{T}} \biggr\} < d $$

ensures that \(|\lambda_{i}|<1\), \(i=1,2\), while the condition

$$b>2 \quad\text{or}\quad \min \biggl\{ \hat{e}kp(k),\frac{P_{T}}{k\theta }p(k), \frac{\hat{e}\hat{f}\theta k}{P_{T}} \biggr\} < d $$

implies \(|\lambda_{1}|>1\) or \(|\lambda_{2}|>1\). □

### 4.2 Internal equilibria

From [21], we claim that equation (2.4) could have multiple interior equilibria because both equations (2.1) and (2.4) have the same equilibria. Now we assume that \(E^{*}(x^{*},y^{*})\) is such an equilibrium and discuss its local stability.

The Jacobian (4.2) matrix at the positive equilibria \(E^{*}\) becomes

$$\begin{aligned} J\bigl(E^{*}\bigr)= \left ( \textstyle\begin{array}{c@{\quad}c} 1+x^{*}F_{x} & x^{*}F_{y} \\ y^{*}G_{x} & 1+y^{*}G_{y} \end{array}\displaystyle \right ), \end{aligned}$$

where

$$ F_{x}=F_{x}\bigl(x^{*},y^{*}\bigr),\qquad F_{y}=F_{y} \bigl(x^{*},y^{*}\bigr),\quad G_{x}=G_{x}\bigl(x^{*},y^{*}\bigr),\qquad G_{y}=G_{y}\bigl(x^{*},y^{*}\bigr). $$

Its trace and determinant are

$$\begin{aligned}& \operatorname{Tr}\bigl(J\bigl(E^{*}\bigr)\bigr)=2+x^{*}F_{x}+y^{*}G_{y}, \\& \begin{aligned}\operatorname{Det}\bigl(J\bigl(E^{*}\bigr)\bigr)&=1+x^{*}F_{x}+y^{*}G_{y}+x^{*}y^{*}[F_{x}G_{y}-F_{y}G_{x}] \\ &=\operatorname{Tr}\bigl(J\bigl(E^{*}\bigr)\bigr)-1+x^{*}y^{*}[F_{x}G_{y}-F_{y}G_{x}], \end{aligned} \end{aligned}$$

respectively. By referenced [21], the phase plane is divided into three biologically significant regions by the two lines \(\hat{e}=\frac{Q}{\theta}\) and \(f(x)=\frac{\hat{f}\theta}{Q}\) to investigate the interior equilibrium \(E^{*}\) (see Fig. 1). We have the following theorem on the local asymptotically stability of \(E^{*}\).

### Theorem 4.2

*In Region I* (*i*.*e*., \(\hat{e}<\frac{Q}{\theta}\)*and*
\(f(x)<\frac{\hat {f}\theta}{Q}\)), *the following are true*:

*If the producer’s nullcline is increasing at*
\(E^{*}\) (*i*.*e*., \(F_{x}>0\)), *then*
\(E^{*}\)*is unstable*.

*If the producer’s nullcline is decreasing at*
\(E^{*}\) (*i*.*e*., \(F_{x}<0\)), *and*

$$ \frac{1}{2}x^{*}y^{*}F_{y}G_{x}-2< x^{*}F_{x}< x^{*}y^{*}F_{y}G_{x}, $$

*then*
\(E^{*}\)*is LAS*.

*If the producer’s nullcline is decreasing at*
\(E^{*}\) (*i*.*e*., \(F_{x}<0\)), *and*

$$ F_{x}>y^{*}F_{y}G_{x} \quad\textit{or}\quad x^{*}F_{x}< \frac{1}{2}x^{*}y^{*}F_{y}G_{x}-2, $$

*then*
\(E^{*}\)*is unstable*.

*In Region II* (*i*.*e*., \(\hat{e}>\frac{Q}{\theta}\)*and*
\(f(x)<\frac{\hat {f}\theta}{Q}\)), *the following are true*:

*If the slope of the producer’s nullcline at*
\(E^{*}\)*is greater than the grazer’s* (*i*.*e*., \(-G_{x}/G_{y}<-F_{x}/F_{y}\)), *then*
\(E^{*}\)*is unstable*.

*If the slope of the grazer’s nullcline at*
\(E^{*}\)*is greater than the producer’s* (*i*.*e*., \(-G_{x}/G_{y}>-F_{x}/F_{y}\)), *and*

$$ \frac{1}{2}x^{*}y^{*}[F_{y}G_{x}-F_{x}G_{y}]-2< x^{*}F_{x}+y^{*}G_{y}< x^{*}y^{*}[F_{y}G_{x}-F_{x}G_{y}], $$

*then*
\(E^{*}\)*is LAS*.

*If the slope of the grazer’s nullcline at*
\(E^{*}\)*is greater than the producer’s* (*i*.*e*., \(-G_{x}/G_{y}>-F_{x}/F_{y}\)), *and*

$$ \begin{gathered} x^{*}F_{x}+y^{*}G_{y}< \frac{1}{2}x^{*}y^{*}[F_{y}G_{x}-F_{x}G_{y}]-2 \quad\textit{or} \\ x^{*}F_{x}+y^{*}G_{y}>x^{*}y^{*}[F_{y}G_{x}-F_{x}G_{y}], \end{gathered} $$

*then*
\(E^{*}\)*is unstable*.

*In Region III* (*i*.*e*., \(\hat{e}<\frac{Q}{\theta}\)*and*
\(f(x)>\frac {\hat{f}\theta}{Q}\)), *the following are true*:

*If the slope of the grazer’s nullcline at*
\(E^{*}\)*is greater than the producer’s* (*i*.*e*., \(-G_{x}/G_{y}>-F_{x}/F_{y}\)), *then*
\(E^{*}\)*is unstable*.

*If the slope of the producer’s nullcline at*
\(E^{*}\)*is greater than the grazer’s* (*i*.*e*., \(-G_{x}/G_{y}<-F_{x}/F_{y}\)), *and*

$$ \frac{1}{2}x^{*}y^{*}[F_{y}G_{x}-F_{x}G_{y}]-2< x^{*}F_{x}+y^{*}G_{y}< x^{*}y^{*}[F_{y}G_{x}-F_{x}G_{y}], $$

(4.5)

*then*
\(E_{2}\)*is LAS*.

*If the slope of the producer’s nullcline at*
\(E^{*}\)*is greater than the grazer’s* (*i*.*e*., \(-G_{x}/G_{y}<-F_{x}/F_{y}\)), *and*

$$ \begin{gathered} x^{*}F_{x}+y^{*}G_{y}< \frac{1}{2}x^{*}y^{*}[F_{y}G_{x}-F_{x}G_{y}]-2 \quad\textit{or} \\ x^{*}F_{x}+y^{*}G_{y}>x^{*}y^{*}[F_{y}G_{x}-F_{x}G_{y}], \end{gathered} $$

(4.6)

*then*
\(E^{*}\)*is unstable*.

### Proof

The LAS of \(E^{*}\) in Region I and II is obtained by using the same arguments as those used in the proof of Theorem 4.2 in [12] where the interested reader can find all the details. Therefore, here we focus on the stability of \(E^{*}\) in Region III, \(\hat{e}<\frac {Q}{\theta}\) and \(f(x)>\frac{\hat{f}\theta}{Q}\). Suppose that \(E^{*}\) lies in Region III, then system (4.3) yields that at \(E^{*}, F_{x}<0\), \(F_{y}<0\), \(G_{x}>0\), and \(G_{y}>0\).

Note that

$$\begin{aligned} \operatorname{Det}\bigl(J\bigl(E^{*}\bigr)\bigr)=1+x^{*}F_{x}+y^{*}G_{y}+x^{*}y^{*}F_{y}G_{y} \biggl[-\frac {G_{x}}{G_{y}}- \biggl(\frac{F_{x}}{F_{y}} \biggr) \biggr]. \end{aligned}$$

If the slope of the grazer’s nullcline at \(E^{*}\) is greater than the producer’s, i.e., \(-G_{x}/G_{y}>-F_{x}/F_{y}\), then \(1+\operatorname {Det}(J(E^{*}))<\operatorname{Tr}(J(E^{*}))\), which implies that equation (4.4) does not hold. Hence, \(E^{*}\) is unstable. If the slope of the producer’s nullcline at \(E^{*}\) is greater than the grazer’s, then

$$-\frac{G_{x}}{G_{y}}< -\frac{F_{x}}{F_{y}}< 0. $$

If equation (4.5) holds, then one can easily show that system (4.4) holds. Consequently, \(E^{*}\) is stable. If equation (4.6) is valid, then \(E^{*}\) is unstable since equation (4.4) is not valid. □