In this section, we will give the bifurcation diagrams of model (1.2) to confirm the above theoretical analysis and show the new interesting complex dynamical behaviors by numerical simulation. Moreover, different dynamical behaviors caused by different parameters are discussed in the ecological perspective. For equilibrium {E}_{1}(K,0,0), we choose parameter *r*. For equilibrium {E}_{2}(\frac{c}{\beta},\frac{rK}{r+K\beta}(1-\frac{c}{K\beta}),0), we choose four key parameters *r*, *c*, *K*, and *β*. Finally, the key parameters *b* and *k* are selected for positive equilibrium {E}^{\ast}({S}^{\ast},{I}^{\ast},{Y}^{\ast}).

**Example 1** For detecting the dynamical behaviors of model (1.2) impacted by parameter *r* (the intrinsic birth rate of {S}_{t}), we choose b=0.2, c=0.6, d=0.12, k=0.1, m=0.2, \beta =0.05, K=8, and r\in [0.01,4] and initial value ({S}_{0},{I}_{0},{Y}_{0})=(4,0.5,0.1). It is obvious that (b,c,d,k,m,\beta ,K,r)=(0.2,0.6,0.12,0.1,0.2,0.05,8,2)\in M. Then equilibrium {E}_{1}(K,0,0)={E}_{1}(8,0,0) and the flip bifurcation appears (Figure 1).

Figure 1(A) suggests that when 0<r<2 equilibrium {E}_{1}(K,0,0) is local stable and when r=2, {E}_{1}(K,0,0) loses its stability. When r>2 there exists a flip bifurcation. Moreover, a chaotic set is emerged with the increasing of *r*. However, the infected prey and the predator are always in extinction for any value of *r*, which can be seen from Figure 1(B) and (C). This result is agreement of the ecological perspective that when the infected prey is in extinction the predator certainly becomes in extinction.

**Example 2** For detecting the dynamical behaviors of model (1.2) with equilibrium {E}_{2}(\frac{c}{\beta},\frac{rK}{r+K\beta}(1-\frac{c}{K\beta}),0) impacted by parameter *r*, we choose b=0.15, c=0.1, d=0.2, k=0.2, m=0.3, \beta =0.05, K=4, and r\in [0.001,7], and the initial value ({S}_{0},{I}_{0},{Y}_{0})=(2,1,0.5). It is obvious that (b,c,d,k,m,\beta ,K,r)=(0.15,0.1,0.2,0.2,0.3,0.05,4,\frac{80}{19})\in N. Then we have equilibrium {E}_{2}(\frac{c}{\beta},\frac{rK}{r+K\beta}(1-\frac{c}{K\beta}),0)={E}_{2}(2,\frac{2r}{r+0.2},0), and there exists a flip bifurcation (Figure 2).

Figure 2 shows that equilibrium {E}_{2}(2,\frac{2r}{r+0.2},0) is stable when 0.001\le r<\frac{80}{19} and loses stability when r=\frac{80}{19}. Further, when r>\frac{80}{19} there appears a flip bifurcation and chaos at equilibrium {E}_{2}(2,\frac{2r}{r+0.2},0). And a 3-periodic solution of model (1.2) appears when r\approx 6.05. However, {Y}_{t} is in extinction for any value *r* ultimately (Figure 2(C)).

**Example 3** For detecting the dynamical behaviors of model (1.2) with equilibrium {E}_{2}(\frac{c}{\beta},\frac{rK}{r+K\beta}(1-\frac{c}{K\beta}),0) impacted by parameter *c* (the different death rate of infected prey), we choose two subcases of the parameters.

Subcase 1. Choosing (b,d,k,m,r,\beta ,K)=(0.1,0.6,0.1,0.5,0.1,0.2,10) and c\in [0.01,2.5] and the initial value ({S}_{0},{I}_{0},{Y}_{0})=(6,2,1). It is obvious that (b,d,k,m,r,\beta ,K,c)=(0.1,0.6,0.1,0.5,0.1,0.2,10,1)\in P. Then we have equilibrium {E}_{2}(\frac{c}{\beta},\frac{rK}{r+K\beta}(1-\frac{c}{K\beta}),0)={E}_{2}(5c,\frac{5}{21}(2-c),0) and there exists a Hopf bifurcation (Figure 3).

From Figure 3, we find that there exists a Hopf bifurcation and chaos of model (1.2) when 0<c\le 1; When 1<c<2 the number of {S}_{t} is increasing and when c\ge 2 the number of {S}_{t} is stable at 10. The number of {I}_{t} is decreasing when 1<c<2 and {I}_{t} is becoming in extinction ultimately when c\ge 2. The number of {Y}_{t} is always in extinction for any value of *c* ultimately.

Subcase 2. Choosing b=0.2, d=0.2, k=0.2, m=0.5, r=3, \beta =0.2, K=10, and c\in [0.5,2.5], and the initial value ({S}_{0},{I}_{0},{Y}_{0})=(6,2,1). It is obvious that (b,d,k,m,r,\beta ,K,c)=(0.2,0.2,0.2,0.5,3,0.2,10,1),(0.2,0.2,0.2,0.5,3,0.2,10,1.633)\in P,N, respectively. Then we have equilibrium {E}_{2}(\frac{c}{\beta},\frac{rK}{r+K\beta}(1-\frac{c}{K\beta}),0)={E}_{2}(5c,6-3c,0) and there exist two bifurcation values of *c* from the result of Theorem 3. After some computing, we find that when c=1 and c=1.633 the Hopf bifurcation and flip bifurcation at equilibrium {E}_{2}(5c,6-3c,0) appears, respectively (Figure 4).

From Figure 4(A)-(C), we find that there exist a Hopf bifurcation and chaos of model (1.2) when 0<c\le 1. When 1<c<1.633 the number of {S}_{t} is increasing, while the number of {I}_{t} is decreasing. When c\ge 1.633 there appear the flip bifurcation and chaos of model (1.2). {I}_{t} becomes in extinction when c\approx 1.97 (Figure 4(C)). Moreover, a 11-, 16- and 5-periodic solution appears when c\approx 0.779, c\approx 0.81, and c\approx 0.87, respectively (Figure 4(B)). However, {Y}_{t} becomes in extinction because of bk<d (Figure 4(D)).

**Example 4** For detecting the dynamical behaviors of model (1.2) with equilibrium {E}_{2}(\frac{c}{\beta},\frac{rK}{r+K\beta}(1-\frac{c}{K\beta}),0) impacted by parameter *K* (the carrying capacity), we choose b=0.1, c=0.4, d=0.2, k=0.2, m=0.5, r=0.2, \beta =0.2, and K\in [2,8], and the initial values ({S}_{0},{I}_{0},{Y}_{0})=(1,0.5,0.2). It is obvious that (b,c,d,k,m,r,\beta ,K)=(0.1,0.4,0.2,0.2,0.5,0.2,0.2,7)\in P. Then we have equilibrium {E}_{2}(\frac{c}{\beta},\frac{rK}{r+K\beta}(1-\frac{c}{K\beta}),0)={E}_{2}(2,1-\frac{3}{1+K},0), and there exists a Hopf bifurcation (Figure 5).

From Figure 5(A) and (B), we see that equilibrium {E}_{2}(2,1-\frac{3}{1+K},0) is stable when 2\le K<7 and loses stability when K=7. Further, when K>7 there appears a Hopf bifurcation of equilibrium {E}_{2}(2,1-\frac{3}{1+K},0). However, {Y}_{t} is in extinction for any values of *K* ultimately.

**Example 5** For detecting the dynamical behaviors of model (1.2) with equilibrium {E}_{2}(\frac{c}{\beta},\frac{rK}{r+K\beta}(1-\frac{c}{K\beta}),0) impacted by parameter *β* (the transmission coefficient), we choose b=0.15, c=0.5, d=0.2, k=0.2, m=0.3, r=4, K=4, \beta \in [0.125,0.42], and the initial value ({S}_{0},{I}_{0},{Y}_{0})=(2,1,1). It is obvious that (b,c,d,k,m,r,K,\beta )=(0.15,0.5,0.2,0.2,0.3,4,4,0.2083),(0.15,0.5,0.2,0.2,0.3,4,4,0.3750)\in N,P, respectively. Then we have equilibrium {E}_{2}(\frac{c}{\beta},\frac{rK}{r+K\beta}(1-\frac{c}{K\beta}),0)={E}_{2}(\frac{1}{2\beta},\frac{8\beta -1}{2\beta},0). From conclusion (2) of Theorem 3, the two bifurcation values are computed as \beta =0.2083 and \beta =0.3750 from K\beta =\frac{cr(2+c)}{4+cr} and K\beta =1+c, respectively. Further, the interval [0.125,0.42] includes the two bifurcation values. Therefore, there appears two bifurcations: flip bifurcation and a Hopf bifurcation. Figure 6(A) and (B) verifies it. In detail, when *β* changes from 0.125 to 0.45, there appears chaos, flip bifurcation, local stability, Hopf bifurcation, and chaos of {S}_{t} and {I}_{t}. However, {Y}_{t} becomes in extinction because of bk<d (Figure 6(C)). On the other hand, the same results appear for K\in [2,7] with b=0.15, c=0.5, d=0.2, k=0.2, m=0.3, r=4, \beta =0.25, and the initial value ({S}_{0},{I}_{0},{Y}_{0})=(1,0.5,0.2) (Figure 7).

**Example 6** For detecting the dynamical behaviors of model (1.2) with endemic equilibrium {E}^{\ast}({S}^{\ast},{I}^{\ast},{Y}^{\ast}) impacted by parameter *b* (the predation coefficient), we choose c=0.1, d=0.02, k=0.3, m=0.4, r=1.2, \beta =0.25, K=6, and b\in [0.15,0.7], and the initial value ({S}_{0},{I}_{0},{Y}_{0})=(2,1.5,1). It is obvious that the parameters c=0.1, d=0.02, k=0.3, m=0.4, r=1.2, \beta =0.25, K=6, and b=0.28, 0.5725, respectively, satisfy the conditions of conclusion (2) of Theorem 5. Then we have endemic equilibrium {E}^{\ast}({S}^{\ast},{I}^{\ast},{Y}^{\ast})=(\frac{150b-4}{15},\frac{376-600b}{135},\frac{(15b-1)(188-300b)}{27}). Obviously, when 0<b<\frac{188}{300} there exists an endemic equilibrium {E}^{\ast}({S}^{\ast},{I}^{\ast},{Y}^{\ast}) of model (1.2).

From Figure 8 we see that when 0.15<b<{b}_{\ast} there appear chaos and a Hopf bifurcation for model (1.2); when {b}_{\ast}\approx 0.28<b<{b}_{\ast \ast}\approx 0.5725 the endemic equilibrium {E}^{\ast}({S}^{\ast},{I}^{\ast},{Y}^{\ast}) is stable; when {b}_{\ast \ast}\approx 0.5725<b<{b}_{\ast \ast \ast}\approx 0.5785 there appear Hopf bifurcation and chaos. Furthermore, {S}_{t} reaches the *K* value when {b}_{\ast \ast \ast}\approx 0.5785<b, which can be seen from Figure 8(A) and (B). However, {I}_{t} and {Y}_{t} become in extinction when b\approx 0.57 from Figure 8(C)-(E).

**Example 7** For detecting the dynamical behaviors of model (1.2) with endemic equilibrium {E}^{\ast}({S}^{\ast},{I}^{\ast},{Y}^{\ast}) impacted by parameter *k* (the coefficient in conversing prey into predator), we choose b=0.3, c=0.1, d=0.04, m=0.3, r=1.2, \beta =0.25, K=6, and k\in [0.15,1], and the initial value ({S}_{0},{I}_{0},{Y}_{0})=(2,1.5,1). It is obvious that parameters b=0.3, c=0.1, d=0.04, m=0.3, r=1.2, \beta =0.25, K=6, and k=0.34 satisfy the conditions of conclusion (4) of Theorem 5. Then we have endemic equilibrium {E}^{\ast}({S}^{\ast},{I}^{\ast},{Y}^{\ast})=(\frac{66k-8}{15},\frac{392-264k}{135},\frac{(15k-2)(392-264k)}{81}). Obviously, when 0.15\le k\le 1 there exists an endemic equilibrium {E}^{\ast}({S}^{\ast},{I}^{\ast},{Y}^{\ast}).

Figure 9 shows that when 0.15<k<{k}_{\ast}\approx 0.34 there exist chaos and a Hopf bifurcation for {S}_{t}, {I}_{t}, and {Y}_{t}. With the increasing of *k* from {k}_{\ast}\approx 0.34 to 1, the numbers of {S}_{t} and {Y}_{t} are increasing while {I}_{t} is decreasing. And this result agrees with the ecological significance.

For better understanding of the above results, we only provide phase portraits (see Figure 10) of model (1.2) under the condition k=0.2,0.34,0.4 in Example 7. The other cases are similar to Example 7 and we omit them here.

**Remark 3** From Subcase 2 of Example 3, when we choose the same parameters and the initial value, then we vary the parameter *c* we see that there appears a series complex dynamical behavior in equilibrium {E}_{2}(\frac{c}{\beta},\frac{rK}{r+K\beta}(1-\frac{c}{K\beta}),0) of model (1.2), such as chaos → flip bifurcation → local stability → Hopf bifurcation → chaos too.

**Remark 4** From Example 6, when we choose the same parameters and the initial value, then when we vary the parameter *b* we see that there appears a series complex dynamical behaviors in equilibrium {E}^{\ast}({S}^{\ast},{I}^{\ast},{Y}^{\ast}) of model (1.2), such as chaos → Hopf bifurcation → local stability → Hopf bifurcation → chaos too.

**Remark 5** The predation coefficient *b* of predator plays an important role in biological disease prevention and cure which can be seen in Example 6. And this is significant in the actual situation of ecology.

**Open problem 1** Whether there exist some special population models when we fix same parameters and vary one special parameter there appears a series bifurcations and chaos.

**Open problem 2** From the above discussion, whether there exists chaos → flip bifurcation → local stability → flip bifurcation → chaos in some special population model when some parameters are fixed at the same values and one parameter is varied continuously. This will be our future study.