**4. Theoretical Approach to Persistence and Extinction**

We studied persistence and extinction using different analytical techniques on systems (1) and (2). We introduce the conditions of persistence and extinction depending on the carrying capacity parameter. Therefore, we have four cases as follows:

In two dimensions, we use the Kolmogorov analysis to find the conditions of persistence and extinction.

For Holling type I, the persistence condition is as follows:

$$0 < \frac{\mu}{c\alpha} < k \tag{19}$$

However, if condition (19) is not satisfied to become as follows:

$$\frac{\mu}{\epsilon \alpha} \ge k \tag{20}$$

Then, the predator tends to be extinct. For Holling type II, the persistence condition is as follows:

$$0 < \frac{u}{e\kappa - uh\kappa} < k \tag{21}$$

However, if

$$\frac{u}{e\alpha - uh\alpha} \ge k \tag{22}$$

Then, the predator tends to be extinct.

In three dimensions, some theorems must be proven for finding the persistence and extinction conditions of system (2) with Holling type I and those of system (2) with Holling type II in the case of nonperiodic solutions. However, the persistence conditions of the case of periodic solutions cannot be derived theoretically according to Freedman and Waltman [24], so we used the numerical simulations to show the probability of persistence and extinction cases.

**Theorem 6.** *The equilibrium point E*ˆ = (*x*ˆ, *y*ˆ, 0) = ( *k*(*u*+*e*1) *<sup>e</sup>*1+*ke*1*<sup>α</sup>* , *ke*1*α*−*<sup>u</sup> <sup>e</sup>*1*α*+*e*1*α*2*k* , 0) *is unstable in the z-direction (i.e., orthogonal to the x* − *y plane), if the following condition is satisfied:*

$$w + c\_2 \mathfrak{H} < \mathfrak{e}\_2 \mathfrak{E} \mathfrak{k} \tag{23}$$

**Proof.** The variational matrix of equilibrium point *E*ˆ = (*x*ˆ, *y*ˆ, 0) is computed as follows:

$$
\mathcal{V} = \begin{pmatrix}
 c\_1 a \hat{\mathfrak{y}} & -c\_1 a \hat{\mathfrak{y}} & -c\_1 \hat{\mathfrak{y}} \\
 0 & 0 & -w + c\_2 \beta\mathfrak{k} - c\_2 \hat{\mathfrak{y}}
\end{pmatrix},
$$

.

From *V*ˆ and by using the Routh–Hurwitz criterion, equilibrium point *E*ˆ is locally asymptotically stable, provided the following conditions hold:

$$
\varepsilon w + c\_2 \mathfrak{z} \rhd c\_2 \mathfrak{z} \mathfrak{x} \tag{22}
$$

The equilibrium point *E*ˆ is stable in the *x* − *y* plane if condition (24) is satisfied, so *E*ˆ is unstable in the z-direction (i.e., orthogonal to the *x* − *y* plane) if condition (24) is not satisfied, which produces condition (23).

**Theorem 7.** *The equilibrium point E*' = (*x*', 0, '*z*) = *<sup>k</sup>*(*w*+*e*2) *<sup>e</sup>*2+*ke*2*β* , 0, *ke*2*β*−*<sup>w</sup> <sup>e</sup>*2*β*+*e*2*β*<sup>2</sup>*<sup>k</sup> is unstable in the y-direction (i.e., orthogonal to the x* − *z plane), if the following condition is satisfied:*

$$u + c\_1 \widetilde{z} \prec\_1 e\_1 a \widetilde{x} \tag{23}$$

**Proof.** Following the same process, we prove this theorem along with Theorem 6, so the variational matrix of equilibrium point *E*' is as follows:

$$
\dot{V} = \begin{pmatrix}
0 & -u + e\_1 a \tilde{\chi} - c\_1 \tilde{\chi} & 0 \\
e\_2 \beta \tilde{\chi} & -c\_2 \tilde{\chi} & -c\_2 \beta \tilde{\chi}
\end{pmatrix},
$$

.

From *V* ' and using the Routh–Hurwitz criterion, equilibrium point *E* ' is locally asymptotically stable, provided the following condition holds:

$$
\mu + c\_1 \tilde{z} \succ e\_1 \mathfrak{a} \tilde{\mathfrak{X}}\tag{26}
$$

.

The equilibrium point *E* ' is stable in the *x* − *z* plane if condition (26) is satisfied, so *E* is unstable in the y-direction (i.e., orthogonal to the *x* − *z* plane) if condition (26) is not satisfied, which produces condition (25).

**Theorem 8.** *System (2) with Holling type I is persistent if the following conditions hold:*

$$k \ge \frac{ue\_2\beta - c\_1w}{e\_1a\beta w - ue\_2\beta^2 + e\_1c\_2a\beta - c\_1c\_2\beta} \tag{27}$$

$$k \ge \frac{w\varepsilon\_1\alpha - c\_2\mu}{c\_2\alpha\beta\mu - w\varepsilon\_1\beta^2 + c\_1c\_2\alpha\beta - c\_2\varepsilon\_1\alpha} \tag{28}$$

**Proof.** As functions J, *Fi*; *i* = 1, 2 of system (2) are continuous in the positive volume *<sup>R</sup>*3+ = {(*<sup>x</sup>*, *y*, *z*) : *x* ≥ 0, *y* ≥ 0, *z* ≥ <sup>0</sup>}, the system is bounded with positive initial conditions because the prey is bounded, where *k* > 0 and the growth of predators depends on the prey. The conditions *<sup>L</sup>*1(*x*', 0, '*z*) > 0 and *<sup>L</sup>*2(*x*ˆ, *y*ˆ, 0) > 0 are exactly needed to make the equilibrium points unstable in the orthogonal of the other coordinate planes (Theorems 6 and 7). System (2) has a nonperiodic solution only (i.e., no limit cycles) through Theorem 2. 

To complete the proof, the following hypotheses are satisfied with Freedman's and Waltman's theorem.

\*\*Hypotesis 1\*\* (H1).\*\*  $\frac{\partial f}{\partial y} = -a < 0$ ;  $\frac{\partial f}{\partial z} = -\beta < 0$ ;  $\frac{\partial L\_1}{\partial x} = c\_1 a > 0$ ;  $\frac{\partial L\_2}{\partial x} = c\_2 \beta > 0$ ,  $L\_1(0, y, z) = -u - c\_1 a y - c\_1 z < 0$ ;  $L\_2(0, y, z) = -w - c\_2 \beta z - c\_2 y < 0$ ,  $\frac{\partial L\_1}{\partial y} = -c\_1 a \le 0$ ;  $\frac{\partial L\_1}{\partial z} = -c\_1 \le 0$ ;  $\frac{\partial L\_2}{\partial y} = -c\_2 \le 0$ ;  $\frac{\partial L\_2}{\partial z} = -c\_2 \beta \le 0$ 

**Hypothesis 2 (H2).** *If the predator is absent, then the prey species x growths to carrying capacity, i.e., J*(0, 0, 0) = 1 > 0*, ∂J ∂x*(*<sup>x</sup>*, *y*, *z*) = −1*k*≤ 0*,* ∃ *k* > 0 *J*(*k*, 0, 0) = 0, *J*(*k*, 0, 0) = 0*.*

**Hypothesis 3 (H3).** *There are no equilibrium points on the y or z coordinate axes and no equilibrium point in the y–z plane.*

**Hypothesis 4 (H4).** *The predator y and the predator z can survive on the prey, there exist points E*´ = (*x*´, *y*´, 0) *and* ()\*+ *E* = (()\*+ *x* , 0, ()\*+ *z* )*, such that J*(*x*´, *y*´, 0) = *<sup>L</sup>*1(*x*´, *y*´, 0) = 0 *and J*(()\*+ *x* , 0, ()\*+ *z* ) = *<sup>L</sup>*2(..*x*, 0, ..*z*) = 0*, x*´, *y*´, ()\*+ *x ,* ()\*+ *z* > 0 *and x*´ < *k,* ()\*+ *x* < *k.*

**Corollary 2.** *The first predator y is extinct of system (2) with Holling type I if the following condition is satisfied:*

$$k < \frac{ue\_2\beta - c\_1w}{e\_1a\beta w - ue\_2\beta^2 + e\_1e\_2a\beta - c\_1e\_2\beta} \tag{29}$$

**Corollary 3.** *The second predator z is extinct of system (2) with Holling type I if the following condition is satisfied:*

$$k < \frac{w\varepsilon\_1\alpha - c\_2\mu}{c\_2\alpha\beta\mu - w\varepsilon\_1\beta^2 + c\_1c\_2\alpha\beta - c\_2c\_1\alpha} \tag{30}$$

As such, we used the same technique to find the persistence and extinction of system (2) with Holling type II.

The equilibrium point *E* ´ = (*x*´, *y*´, 0) of system (2) with Holling type II is obtained through the positive root of the following quadratic equation:

$$\dot{\mathbf{x}}^2 + \left(\frac{1}{h\_1} - \frac{u}{e\_1} - \frac{1}{k} + \frac{1}{h\_1 a}\right)\dot{\mathbf{x}} - \left(\frac{1}{h\_1 a} + \frac{u}{e\_1 h\_1 a}\right) = 0\tag{31}$$

and

$$\check{y} = \frac{1}{a} \left( 1 - \frac{\check{x}}{k} \right) (1 + h\_1 a \check{x}) \tag{32}$$

**Theorem 9.** *The equilibrium point E*´ = (*x*´, *y*´, 0) *is unstable in the z-direction (i.e., orthogonal to the x* − *y plane) if the following condition is satisfied:*

$$
\delta w + c\_2 \circ < \frac{c\_2 \beta \pounds}{1 + h\_2 \beta \pounds} \tag{33}
$$

**Proof.** The variational matrix of equilibrium point *E* ´ = (*x*´, *y*´, 0) is computed as follows:

$$
\dot{V} = \begin{pmatrix}
\dot{\mathfrak{x}} \left( -\frac{1}{k} + \frac{h\_1 a^2 \dot{\mathfrak{y}}}{\left( \frac{1}{l} + h\_1 a \dot{\mathfrak{x}} \right)^2} \right) & -\frac{\dot{\mathfrak{x}} \alpha}{1 + h\_1 a \dot{\mathfrak{x}}} & -\frac{\dot{\mathfrak{x}} \beta}{1 + h\_2 \beta \dot{\mathfrak{x}}} \\
\frac{c\_1 a \dot{\mathfrak{y}}}{\left( 1 + h\_1 a \dot{\mathfrak{x}} \right)^2} & -\frac{c\_1 a \dot{\mathfrak{y}}}{1 + h\_1 a \dot{\mathfrak{x}}} & -c\_1 \dot{\mathfrak{y}} \\
0 & 0 & -\omega + \frac{c\_2 \beta \dot{\mathfrak{x}}}{1 + h\_2 \beta \dot{\mathfrak{x}}} - c\_2 \dot{\mathfrak{y}}
\end{pmatrix}
$$

.

From *V* ´ and using the Routh–Hurwitz criterion, equilibrium point *E* ´ is locally asymptotically stable, provided the following conditions hold:

$$
\Delta w + c\_2 \sharp > \frac{c\_2 \beta \pounds}{1 + h\_2 \beta \pounds} \tag{34}
$$

The equilibrium point *E* ´ is stable in the *x* − *y* plane if condition (34) is satisfied, so *E* ´ is unstable in the z-direction (i.e., orthogonal to the *x* − *y* plane) if condition (34) is not satisfied, which produces condition (33).

The equilibrium point *E* = (()\*+ *x* , 0, ()\*+ *z* ) of system (2) with Holling type II is obtained through the positive root of the quadratic equation as follows:

()\*+

$$
\widehat{\mathbf{x}}^2 + \left(\frac{1}{h\_2} - \frac{w}{c\_2} - \frac{1}{k} + \frac{1}{h\_2 \beta}\right) \widehat{\mathbf{x}} - \left(\frac{1}{h\_2 \beta} + \frac{w}{c\_2 h\_2 \beta}\right) = 0\tag{35}
$$

and

$$
\widehat{z^\*} = \frac{1}{\beta} \left( 1 - \frac{\widehat{\underline{\chi}}}{k} \right) \left( 1 + h\_2 \beta \,\widehat{\underline{\chi}} \right) \tag{36}
$$

**Theorem 10.** *The equilibrium point* ()\*+ *E* = (()\*+ *x* , 0, ()\*+ *z* ) *is unstable in the y-direction (i.e., orthogonal to the x* − *z plane) if the following condition is satisfied:*

$$\|u + c\_1\|\widehat{\tilde{z}} < \frac{\varepsilon\_1 a \widehat{\tilde{x}}}{1 + h\_1 a \widehat{\tilde{x}}}\tag{37}$$

**Proof.** Following the same process, we proved this theorem with Theorem 9, so the variational matrix of equilibrium point ()\*+ *E* is as follows:

$$
\ddot{V} = \begin{pmatrix}
\widehat{\bf x} \left( -\frac{1}{k} + \frac{h\_2 \beta^2 \widehat{\bf x}}{\left( 1 + h\_2 \beta \widehat{\bf x} \right)^2} \right) & -\frac{\widehat{\bf x} \cdot \widehat{\bf x}}{1 + h\_1 \widehat{\bf x} \cdot \widehat{\bf x}} & -\frac{\widehat{\bf x} \cdot \widehat{\bf \beta}}{1 + h\_2 \beta \widehat{\bf x}} \\
& 0 & -u + \frac{e\_1 u \widehat{\bf X}}{1 + h\_1 u \widehat{\bf X}} - c\_1 \widehat{\widetilde z} & 0 \\
& \frac{e\_2 \beta \widehat{\bf x}}{\left( 1 + h\_2 \beta \widehat{\bf x} \right)^2} & -c\_2 \widehat{\widetilde z} & -\frac{e\_2 \beta \widehat{\bf x}}{1 + h\_1 u \widehat{\bf x}} \\
\end{pmatrix}
$$

.

..

From *V* and using the Routh-Hurwitz criterion, equilibrium point *E* is locally asymptotically stable, provided the following condition holds:

$$\|u + c\_1\|\stackrel{\sim}{z} > \frac{\mathfrak{e}\_1 a \stackrel{\sim}{\widehat{X}}}{1 + h\_1 a \stackrel{\sim}{\widehat{X}}}\tag{38}$$

..

The equilibrium point ()\*+ *E* is stable in the *x* − *z* plane if condition (38) is satisfied, so .. *E* is unstable in the y-direction (i.e., orthogonal to the *x* − *z* plane) if condition (38) is not satisfied, which produces condition (37).

We introduce the persistence conditions of system (2) with Holling type II in the nonperiodic dynamic system through the following theorem:

**Theorem 11.** *System (2) with Holling type II is persistent if the following conditions hold:*

$$-\mu + \frac{c\_1 a \ddot{\bar{x}}}{1 + h\_1 a \ddot{\bar{x}}} - c\_1 \'' \hat{\bar{z}} \ge 0 \tag{39}$$

$$-w + \frac{c\_2 \beta \pounds}{1 + h\_2 \beta \pounds} - c\_2 \circlearrowleft 0 \tag{40}$$

**Proof.** As functions J, *Fi*; *i* = 1, 2 of system (2) are continuous in the positive volume *<sup>R</sup>*3+ = {(*<sup>x</sup>*, *y*, *z*) : *x* ≥ 0, *y* ≥ 0, *z* ≥ <sup>0</sup>}, the system is bounded with positive initial conditions because the prey is bounded, where *k* > 0 and the growth of predators depends on the prey. The conditions *<sup>L</sup>*1()\*+ *x* , 0, ()\*+ *z* > 0 and *<sup>L</sup>*2(*x*ˆ, *y*ˆ, 0) > 0 are exactly needed to make the equilibrium points become unstable in the orthogonal of the other coordinate planes (Theorems 9 and 10). System (2) has a nonperiodic solution (i.e., no limit cycles) through Theorem 4. 

To complete the proof, the following hypotheses are satisfied with Freedman's and Waltman's theorem.

We use *y*1 ≡ *y* and *y*2 ≡ *z* to simplify the notations.

**Hypothesis 5 (H5).** *∂J ∂yi* < 0*, ∂Li ∂x* > 0*, Li*(0, *y*, *z*) < 0*, ∂Li <sup>∂</sup>yj* ≤ 0 *i*, *j* = 1, 2*.*

**Hypothesis 6 (H6).** *in the absence of a predator, the prey species x growths to carrying capacity, i.e., J*(0, 0, 0) = 1 > 0*, ∂J ∂x* (*<sup>x</sup>*, *y*, *z*) = −1*k* ≤ 0*,* ∃ *k* > 0 *J*(*k*, 0, 0) = 0, *J*(*k*, 0, 0) = 0*.*

**Hypothesis 7 (H7).** *There are no equilibrium points on the y or z coordinate axes and no equilibrium point in the y–z plane.*

**Hypothesis 8 (H8).** *The predator y and the predator z can survive on the prey, there exist points E*´ = (*x*´, *y*´, 0) *and* ()\*+ *E* = (()\*+ *x* , 0, ()\*+ *z* )*, such that J*(*x*´, *y*´, 0) = *<sup>L</sup>*1(*x*´, *y*´, 0) = 0 *and J*(()\*+ *x* , 0, ()\*+ *z* ) = *<sup>L</sup>*2(..*x*, 0, ..*z*) = 0*, x*´, *y*´, ()\*+ *x ,* ()\*+ *z* > 0 *and x*´ < *k,* ()\*+ *x* < *k.*

**Corollary 4.** *The first predator y is extinct of system (2) with Holling type II if the following condition is satisfied:*

$$-\mu + \frac{\mathfrak{c}\_1 \mathfrak{a} \widehat{\widetilde{X}}}{1 + h\_1 \widehat{\mathfrak{a}} \widehat{\widetilde{X}}} - \mathfrak{c}\_1 \widehat{\widetilde{z}} < 0 \tag{41}$$

**Corollary 5.** *The second predator z is extinct of system (2) with Holling type II if the following condition is satisfied:*

$$-w + \frac{c\_2 \beta \pounds}{1 + h\_2 \beta \pounds} - c\_2 \circ < 0 \tag{42}$$

However, the persistence and extinction conditions of system (2) with Holling type II are not written in terms of the carrying capacity parameter (k), because writing the persistence conditions in this term is difficult where *x*´ and ..*x* are obtained through the positive solutions of quadratic Equations (31)–(36), which involve the carrying capacity parameter (k).

In this section, we obtained the persistence and extinction conditions of systems (1) and (2) based on carrying capacity. In two dimensions, we applied Kolmogorov analysis to find persistence and extinction conditions (19)–(22) of system (1) and (2) with Holling type I and II, respectively. In three dimensions, we applied the Freedman and Waltman method [24] to obtain persistence and extinction conditions (27)–(30) of system (2) with Holling type I, and persistence and extinction conditions (39)–(42) of system (2) with Holling type II in the case of nonperiodic solutions.

Some experimental studies found that the carrying capacity has an important influence on persistence and extinction in experimental populations, as shown by Griffen and Drake [8].
