Basin Transition and Alternative States: Role of Multi-species Herbivores-Induced Volatile in Plant–Insect Interactions

Abstract

A simple model on volatile organic compound (VOC)-mediated plant–insect interactions is proposed and examined here, when two different classes of herbivorous insects competing for a common resource (plant) in the presence of a specialist carnivorous enemy, which only predates one of the herbivore species. We, particularly, emphasize the impact of VOCs on plant’s growth fitness. The system experiences several local and global bifurcations with emergent alternative states for variations in recruitment factors and predation rate. Basin transitions and basin of attractions have provided detail descriptions on the selectivity of the alternative states, when only one of the herbivore species can survive depending on the choice of initial population densities of the interacting species and how it provides a steady growth in plant. Additionally, our results support the concept of competitive exclusion principle in an indirect interspecific competition between the two herbivore types for the common resource, plant.

Appendices

Appendix A: Positivity and Boundedness

We first show that solutions of system (1) are positive for all $t\ge 0$. Integrating first three equations from (1) with the initial conditions, we get

$$\begin{aligned} x(t)&= x_0 \exp \left[ \int ^t_0 \left( r(1-\frac{x(\theta )}{k})-\frac{c_1y_1(\theta )}{a_1+x(\theta )}\right) d\theta \right] \ge 0 \\ y_1(t)&= y_{10} \exp \left[ \int ^t_0 \left( -d_1+\frac{p_1c_1x(\theta )}{a_1+x(\theta )} -\frac{m_1z(\theta )}{a_3+y_1(\theta )}\right) d\theta \right] \ge 0 \\ y_2(t)&= y_{20} \exp \left[ \int ^t_0\left( -d_2+\frac{p_2c_2x(\theta )}{a_2+x(\theta )}\right) d\theta \right] \ge 0. \end{aligned}$$

From the last equation of system (1), note that

$$\begin{aligned} \frac{\mathrm{d}z}{\mathrm{d}t}\ge -d_3z+\frac{m_1n_1y_1z}{a_3+y_1}.] \end{aligned}$$

Integrating the above inequality, we get

$$\begin{aligned} z(t)= z_0\exp [\int ^t_0\{-d_3+\frac{m_1n_1y_1(\theta )}{a_3+y_1(\theta )}\}d\theta ]\ge 0. \end{aligned}$$

Hence, all solutions of (1) with $(x_0, y_{10}, y_{20}, z_0)\in R^4_+$ remain in for $t\ge 0$.

To show, all solutions of system (1) are bounded, let, ($x(t),~y_1(t),~y_2(t),z(t)$) be any solution of system (1) with positive initial conditions ( $x_0,y_{10},y_{20},z_0$). As $\frac{\mathrm{d}x}{\mathrm{d}t}\le rx(1-\frac{x}{k})$, by a standard comparison theorem we have,

$$\begin{aligned} \lim _{t\rightarrow \infty }\sup {x(t)}\le K_1, \text{ where }~K_1=\max \{x_0,k \}. \end{aligned}$$

Next to show, $y_1(t)$ is bounded we consider a function $W_1=x+\frac{y_1}{p_1}$ and by taking the time derivative along the solutions of system (1), we get

$$\begin{aligned} \frac{\mathrm{d}W_1}{\mathrm{d}t}+m_{11}W_1\le (r+1)K_1 \text{ where } m_{11}=\min \{1,d_1,d_2\}. \end{aligned}$$

Applying a theorem in differential inequalities (Birkhoff and Rota 1989), we have

$$\begin{aligned}&0 \le W_1\le (r+1)\frac{K_1}{m_{11}}+\frac{W_1(x_0,y_{10})}{e^{m_{11}t}},\\&\quad \text {and} ~\text {as}~ t\rightarrow \infty ,~~0\le W_1(t)\le (r+1)\frac{K_1}{m_{11}}=\gamma _{11} ~\text {(say)}. \end{aligned}$$

Therefore, $y_1\le p_1\gamma _{11}$. Similarly, we can show that $y_2\le p_2\gamma _{11}$. Again, we formulate a function $W=x+\frac{1}{p_1}y_1+\frac{1}{p_2}y_2+\frac{1}{n_1p_1}z$. Now we have $0\le W(t)\le \frac{1}{\bar{m}}[(r+1)K_1+\frac{K_1\gamma _{11}}{n_1p_1}(\mu _1p_1+\mu _2p_2)]=\gamma _2$ (say), where $\bar{m}=\min \{1,d_1,d_2,d_3\}$.

Appendix B: Existence of Equilibria

System (1) admits five equilibrium points:

(i) Population-free equilibrium, $E_0(0,0,0,0)$;

(ii) $E_k(k,0,0,0)$;

(iii) $E_1(\hat{x},0,\hat{y_2},\hat{z})$, where $\hat{x}=\frac{a_2d_2}{p_2c_2-d_2},~\hat{y_2}=r(1-\frac{\hat{x}}{k})\frac{(a_2+\hat{x})}{c_2}, \hat{z}=\frac{\mu _2 \hat{x}\hat{y_2}}{d_3}$, equilibrium $E_1$ exists if $c_2 > \frac{d_2}{p_2}$;

(iv) $E_2(\bar{x}, \bar{y_1},0,\bar{z})$, where $\bar{x}$ is the positive real roots of the equation:

$$-r\mu _1 m_1x^4+G_2x^3+G_3x^2+G_4x+G_5=0,$$

where $G_2=\mu _1m_1r(k-a_1)-(\mu _1m_1a_1+(m_1n_1-d_3)(p_1c_1-d_1))r,$

$G_3=\mu _1m_1rka_1+r(k-a_1)(\mu _1m_1a_1+(m_1n_1-d_3)(p_1c_1-d_1)))+(m_1n_1-d_3)d_1a_1r,$

$G_4=(\mu _1m_1a_1+(m_1n_1-d_3)(p_1c_1-d_1))rka_1-r(k-a_1)(m_1n_1-d_3)d_1a_1$ $~~~~~~~~~ -d_3a_3c_1k(p_1c_1-d_1),$

$G_5=d_3a_3c_1d_1a_1k-d_1a_1rka_1(m_1n_1-d_3).$

Therefore, we have $\bar{y_1}=\frac{r}{c_1}(1-\frac{\bar{x}}{k})(a_1+x)$, and $\bar{z}=\frac{a_3+\bar{y_1}}{m_1}(\frac{p_1c_1\bar{x}}{a_1+\bar{x}}-d_1)$.

v) $E^*(x^*,y_1^*,y_2^*,z^*)$, when

$$\begin{aligned}&x^*=\frac{a_2d_2}{p_2c_2-d_2},\\&y_1^*=\frac{c_2kd_3a_3H-\mu _2 m_1 x^* r(k-x^*)(a_2+x^*) }{k[c_2H(m_1 n_1 -d_3)+ \mu _1 m_1 x^* c_2 -\mu _2 c_1m_1x^*]},\\&y_2^*=[r(1-\frac{x^*}{k})-\frac{c_1y_1^*}{a_1+x^*}]\frac{a_2+x^*}{c_2},\\&z^*=\frac{H}{m_1}(a_3+y_1^*) \end{aligned}$$

where $H=\frac{p_1 c_1 x^*}{a_1+x^*}-d_1$, $m_1^{**}=\frac{d_3c_2H}{c_2Hn_1+(\mu _1c_2-\mu _2c_1)x^*}$ and $m_1^*=\frac{c_2kd_3a_3H}{\mu _2x^*r(k-x^*)(a_2+x^*)}$.

$x^*$, $y_1^*$, $y_2^*$ and $z^*$ are feasible if

(i) $p_2c_2>d_2$, (ii) $m_1^{**}<m_1<m_1^*$ and (iii) $0\le d_1 < \frac{p_1c_1d_2a_2}{p_2c_2a_1-d_2a_1+d_2a_2}$ hold simultaneously.

Appendix C: Stability and Bifurcation Analysis

Variational matrix of system (1) at any arbitrary point $(x, y_1, y_2, z)$ is

$$\begin{aligned}&V(x, y_1, y_2, z)=[v_{ij}]_{4\times 4}, \\&\text {where},~ v_{11}=r-\frac{2rx}{k}-\frac{a_1 c_1 y_1}{(a_1+x)^2}-\frac{a_2 c_2 y_2}{(a_2+x)^2},~ v_{12}=-\frac{c_1x}{a_1+x}, \\&v_{13}=-\frac{c_2x}{a_2+x}, ~v_{21}=\frac{a_1 c_1 p_1 y_1}{(a_1+x)^2},\\&v_{22}=-d_1+\frac{c_1 p_1 x}{a_1+x}-\frac{m_1 z a_3}{(a_3 +y_1))^2}, ~v_{24}= \frac{m_1 y_1}{a_3 +y_1}, v_{31}=\frac{a_2 c_2 p_2 y_2}{(a_2+x)^2}, \\&v_{33}=-d_2+\frac{c_2 p_2 x}{a_2+x}, \\&v_{41}=\mu _1 y_1+\mu _2 y_2, ~v_{42} =\mu _1 x+\frac{m_1 n_1 a_3 z}{(a_3+y_1))^2}, ~v_{43}= \mu _2 x, \\&v_{44}=-d_3+\frac{m_1 n_1 y_1}{a_3+y_1}, ~v_{14}=v_{23}=v_{32}=v_{34}=0. \end{aligned}$$

1.1 Stability Analysis of Equilibrium $E_0$

Variational matrix of system (1) at equilibrium $E_0$ is

$$\begin{aligned} V_0(0,0,0,0)=\left( \begin{array}{cccc} r &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad -d_1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -d_2 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -d_3 \end{array}\right) . \end{aligned}$$

The eigenvalues of $V_0$ is $r~(>0)$, $-d_1~(<0)$, $-d_2~(<0)$ and $-d_3~(<0)$. One eigenvalue is positive. Therefore, equilibrium $E_0$ is unstable always.

1.1.1 Stability Analysis of Equilibrium $E_k$

Variational matrix of system (1) at equilibrium $E_1$ is

$$\begin{aligned}V_1(k,0,0,0)=\left( \begin{array}{cccc} -r &{}\quad -\frac{c_1k}{a_1+k} &{}\quad -\frac{c_2k}{a_2+k} &{}\quad 0 \\ 0 &{}\quad -d_1+\frac{c_1 p_1 k}{a_1+k} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -d_2+\frac{c_2 p_2 k}{a_2+k} &{}\quad 0 \\ 0 &{}\quad \mu _1 k &{}\quad \mu _2 k &{}\quad -d_3 \end{array} \right) . \end{aligned}$$

The eigenvalues of $V_1$ is $-r~(<0)$, $-d_1+\frac{c_1 p_1 k}{a_1+k}$, $-d_2+\frac{c_2 p_2 k}{a_2+k}$ and $-d_3~(<0)$. Two eigenvalues are negative and rest of two eigenvalues are negative, if $d_1>\frac{c_1 p_1 k}{a_1+k}$ and $d_2>\frac{c_2 p_2 k}{a_2+k}$.

Therefore, equilibrium $E_1$ is asymptotically stable, if $d_1>\frac{c_1 p_1 k}{a_1+k}$ or $d_2>\frac{c_2 p_2 k}{a_2+k}$.

1.1.2 Stability Analysis of Equilibrium $E_1$

Variational matrix of system (1) at equilibrium $E_1$ reads as

$$\begin{aligned} V_1(\hat{x},0,\hat{y_2},\hat{z})=\left( \begin{array}{cccc} \hat{a_{11}} &{}\quad \hat{a_{12}} &{}\quad \hat{a_{13}} &{}\quad 0 \\ 0 &{}\quad \hat{a_{22}} &{}\quad 0 &{}\quad 0 \\ \hat{a_{31}} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ \hat{a_{41}} &{}\quad \hat{a_{42}} &{}\quad \hat{a_{43}} &{}\quad \hat{a_{44}} \\ \end{array} \right) , \end{aligned}$$

where $\hat{a_{11}}=-\frac{r \hat{x}}{k}+\frac{c_2 \hat{x} \hat{y_2}}{(a_2+\hat{x})^2},~ \hat{a_{12}}=-\frac{c_1 \hat{x}}{a_1+\hat{x}},~ \hat{a_{13}}=-\frac{c_2 \hat{x}}{a_2+\hat{x}}$,  $\hat{a_{22}}=-d_1+\frac{c_1 p_1 \hat{x}}{a_1+\hat{x}}-\frac{m_1 \hat{z}}{a_3} ,~\hat{a_{31}}= \frac{c_2 a_2 p_2 \hat{y_2}}{(a_2+\hat{x})^2} ,~\hat{a_{41}}=\mu _2 \hat{y_2}$ , $\hat{a_{42}}=\mu _1 \hat{x}+\frac{m_1 n_1 \hat{z}}{a_3} ,~\hat{a_{43}}=\mu _2 \hat{x} ,~\hat{a_{44}}=-d_3.$

The corresponding characteristic equation is $(d_3+\xi _1)(\hat{a_{22}}-\xi _1)(\xi _1^2- \hat{a_{11}}\xi _1-\hat{a_{13}}\hat{a_{31}})=0.$

From Routh–Hurwitz criterion, equilibrium $E_1$ will be locally asymptotically stable if $\hat{a_{22}}<0$ and $\hat{a_{11}}<0,$ and unstable otherwise.

Analysis for transcritical bifurcation:

One of the eigenvalues of the variational matrix $V_1$ at equilibrium $E_1(\hat{x},0,\hat{y_2},\hat{z})$ is

$$\begin{aligned} \hat{a_{22}}=-d_1+\frac{p_1c_1\hat{x}}{a_1+\hat{x}}-\frac{m_1\hat{z}}{a_3}. \end{aligned}$$

If we choose $m_1=\frac{a_3}{\hat{z}}(\frac{p_1c_1a_2d_2}{a_1(p_2c_2 -d_2)+a_2d_2}-d_1)=\hat{m_1} \text{(say) },$ then one eigenvalue of $V_1$ will be zero.

Eigenvectors of $V_1$ and $V_3^T$ with respect to zero eigenvalue are $\hat{U}=(0,\hat{u_2},\hat{u_3},\hat{u_4})^T$ and $\hat{\gamma }_{m_1}=(0,1,0,0)^T$ where $\hat{u_2}=\hat{\theta }$, $\hat{\theta }$ is a real number; $\hat{u_3}=-\frac{\hat{a_{12}}}{\hat{a_{13}}}\hat{\theta }$; and $\hat{u_4}=-\frac{\hat{\theta }}{\hat{a_{13}}\hat{a_{44}}}(\hat{a_{13}}\hat{a_{42}}-\hat{a_{12}}\hat{a_{43}})$.

After some mathematical manipulation, we get $\hat{\gamma }_{m_1}^TF_{m_1}(E_1,\hat{m_1})=0$ which implies that saddle node bifurcation does not occur at $\hat{m_1}$ (Sotomayor’s theorem (Perko 2013).

Further, we get $\hat{\gamma }_{m_1}^T[Df_{m_1}(E_1,\hat{m_1})v_{m_1}] =-\frac{\hat{z}\hat{\theta }}{a_3}\ne 0,$ and $\hat{\gamma }_{m_1}^T[D^2F{m_1}(E_1,\hat{m_1})(v_{m_1},v_{m_1})] =\frac{2\hat{m_1}\hat{\theta }}{a_3}(\frac{\hat{z}\hat{u_2}}{a_3}-\hat{u_4}) \ne 0,$ if $\frac{\hat{z}\hat{u_2}}{a_3}-\hat{u_4} \ne 0.$

Thus, system (1) possesses a transcritical bifurcation at $m_1=\hat{m_1}$, if $\frac{\hat{z}\hat{u_2}}{a_3}-\hat{u_4} \ne 0.$

Through the transcritical bifurcation, system (1) exchanges the stability between the interior ($E^*$) and prey herbivore-free ($E_1$) equilibrium points, when $y_1$ dies out.

1.1.3 Stability Analysis of Equilibrium $E_2$

Variational matrix of system (1) at equilibrium $E_2$ reads as

$$\begin{aligned} V_2(\bar{x},\bar{y_1},0,\bar{z})=\left( \begin{array}{cccc} \bar{a_{11}} &{}\quad \bar{a_{12}} &{}\quad \bar{a_{13}} &{}\quad 0 \\ \bar{a_{21}} &{}\quad \bar{a_{22}} &{}\quad 0 &{}\quad \bar{a_{24}} \\ 0 &{}\quad 0 &{}\quad \bar{a_{33}} &{}\quad 0 \\ \bar{a_{41}} &{}\quad \bar{a_{42}} &{}\quad \bar{a_{43}} &{}\quad \bar{a_{44}} \end{array} \right) , \end{aligned}$$

where

$$\begin{aligned}&\bar{a_{11}}= -\frac{r \bar{x}}{k}+\frac{c_1 \bar{x} \bar{y_1}}{(a_1+\bar{x})^2}, ~\bar{a_{12}}=-\frac{c_1 \bar{x}}{a_1+\bar{x}}, ~\bar{a_{13}}=-\frac{c_2 \bar{x}}{a_2+\bar{x}}, \\&\bar{a_{21}}=-\frac{c_1 p_1 a_1 \bar{y_1}}{(a_1+\bar{x})^2}, ~\bar{a_{22}}= \frac{m_1 \bar{y_1} \bar{z}}{(a_3 +\bar{y_1})^2},\\&\bar{a_{24}}=-\frac{m_1 \bar{y_1}}{a_3 +\bar{y_1}}, ~\bar{a_{33}}=-d_2+\frac{c_2 p_2 \bar{x}}{a_2+\bar{x}}, ~\bar{a_{41}}=\mu _1 \bar{y_1},\\&\bar{a_{42}}=\mu _1\bar{x}+\frac{m_1 n_1 a_3\bar{z}}{(a_3 +\bar{y_1})^2}, ~\bar{a_{43}}=\mu _2 \bar{x}, ~\bar{a_{44}}= -\frac{\mu _1 \bar{x} \bar{y_1}}{\bar{z}}. \end{aligned}$$

The corresponding characteristic equation is

$$\begin{aligned} (\bar{a_{33}}-\xi _2)(\xi _2^3+\bar{C_1}\xi _2^2+\bar{C_2}\xi _2+\bar{C_3})=0, \end{aligned}$$

where $\bar{C_1}$=$- \bar{a_{11}} - \bar{a_{22}} - \bar{a_{44}}$, $\bar{C_2}$=$\bar{a_{11}} \bar{a_{22}} - \bar{a_{12}} \bar{a_{21}} + \bar{a_{11}} \bar{a_{44}} + \bar{a_{22}} \bar{a_{44}} - \bar{a_{24}} \bar{a_{42}}$, $\bar{C_3}$=$\bar{a_{11}} \bar{a_{24}} \bar{a_{42}} - \bar{a_{11}} \bar{a_{22}} \bar{a_{44}} + \bar{a_{12}} \bar{a_{21}} \bar{a_{44}} - \bar{a_{12}} \bar{a_{24}} \bar{a_{41}}$.

Following Routh–Hurwitz criterion, equilibrium $E_2$ will be locally asymptotically stable, if

$$\begin{aligned} \left( d_2>\frac{c_2 p_2 \bar{x}}{a_2+\bar{x}}, ~\bar{C_1}, ~\bar{C_3}, ~\text {and} ~\bar{C_1}\bar{C_2}-\bar{C_3} \right) >0, ~\text {and unstable otherwise.} \end{aligned}$$

Analysis for saddle node bifurcation:

If $\bar{C_3}=0$, then one of the eigenvalue of $V_2$ will be zero. We choose the value of $m_1=\bar{m_1}$ (say) so that $\bar{C_3}$ becomes zero.

Eigenvectors of $V_2$ and $V_2^T$ with respect to zero eigenvalue are $\bar{U}=(\bar{\theta },\bar{u_2},0,\bar{u_4})^T$ and $\bar{\gamma }_{m_1}=(\bar{v_1},\bar{v_2},\bar{v_3},\bar{v_4})^T$ and $\bar{u_2}=-\bar{\theta }\frac{\bar{a_{11}}}{\bar{a_{12}}}$, where $\bar{\theta }$ is a real number.

$\bar{u_4}=-\frac{\bar{\theta } }{\bar{a_{12}}\bar{a_{24}}}(\bar{a_{12}}\bar{a_{21}}-\bar{a_{11}}\bar{a_{22}})$, $\bar{v_2}=-\frac{\bar{a_{44}}}{\bar{a_{24}}}$, $\bar{v_1}=-\frac{1}{\bar{a_{11}}}(\bar{a_{21}}\bar{v_2}+\bar{a_{41}})$, $\bar{v_3}=-\frac{1}{\bar{a_{33}}}(\bar{a_{43}}+\bar{a_{13}}\bar{v_1})$.

After some mathematical manipulation, we get

$$\begin{aligned} \bar{\gamma }_{m_1}^TF_{m_1}(E_2,\bar{m_1})=\frac{\bar{y_1}\bar{z}}{a_3+\bar{y_1}}\{n_1+(a_3+\bar{y_1})\frac{\mu _1\bar{x}\bar{y_1}}{m_1\bar{y_1}\bar{z}}\} \ne 0 \end{aligned}$$

which implies that $E_2$ possesses saddle node bifurcation at $\hat{m_1}$ (Sotomayor’s theorem Perko 2013).

1.1.4 Stability Analysis of $E^*$

Variational matrix of system (1) at $E^*$ reads as

$$\begin{aligned}&V_*(x^*,~y_1^*,~y_2^*,~z^*)=\left( \begin{array}{cccc} a_{11} &{}\quad a_{12} &{}\quad a_{13} &{}\quad 0 \\ a_{21} &{}\quad a_{22} &{}\quad 0 &{}\quad a_{24} \\ a_{31} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ a_{41} &{}\quad a_{42} &{}\quad a_{43} &{}\quad a_{44} \end{array} \right) , \\&\text {where} ~a_{11}=-\frac{r x^*}{K}+\frac{c_1 x^* y_1^*}{(a_1+x^*)^2}+\frac{c_2 x^* y_2^*}{(a_2+x^*)^2} ,~a_{12}=-\frac{c_1 x^*}{a_1+x^*},\\&a_{13}=-\frac{c_2 x^*}{a_2+x^*},~a_{21}=-\frac{c_1 p_1 a_1 y_1^*}{(a_1+x^*)^2},\\&a_{22}=\frac{m_1 y_1^* z^*}{(a_3 +y_1^*)^2}, ~a_{24}=-\frac{m_1 y_1^*}{a_3 +y_1^*},~a_{31}=-\frac{c_2 p_2 a_2 y_2^*}{(a_2+x^*)^2},\\&a_{41}=\mu _1 y_1^* +\mu _2 y_2^*,~a_{42}= \frac{m_1 n_1 a_3 z^*}{(a_3 +y_1^*)^2}+\mu _1 x^*,\\&a_{43}=\mu _2 x^*, ~\text {and}~a_{44}=-d_3+\frac{m_1 n_1 y_1^*}{a_3 +y_1^*}. \end{aligned}$$

So the corresponding characteristic equation will be

$$\begin{aligned} \xi _*^4+ C_1\xi _*^3+C_2\xi _*^2+C_3\xi _*+C_4=0, \end{aligned}$$

where $C_1$=$- a_{11} - a_{22} - a_{44}$, $C_2$=$a_{11}a_{22} - a_{12}a_{21} - a_{13}a_{31} + a_{11}a_{44} + a_{22}a_{44} - a_{24}a_{42}$, $C_3$=$a_{13}a_{22}a_{31} - a_{11}a_{22}a_{44} + a_{11}a_{24}a_{42} + a_{12}a_{21}a_{44} - a_{12}a_{24}a_{41} + a_{13}a_{31}a_{44},$ and $C_4$=$- a_{12}a_{24}a_{31}a_{43} - a_{13}a_{22}a_{31}a_{44} + a_{13}a_{24}a_{31}a_{42}.$

By Routh–Hurwitz criterion, $E^*$ will be locally asymptotically stable, when

$$\begin{aligned} \left( C_1, ~C_3, ~C_4, ~\text {and}~C_1C_2C_3-C_3^2C_1^2C_4\right) >0, ~\text {and unstable otherwise}. \end{aligned}$$

Appendix D: Emergence of Chaos

Further, to check possible occurrence of some complex phenomenon we vary the environmental carrying capacity, which is one of the most vital parameter in ecological context. We draw one parameter bifurcation diagram against k in Fig. 7.

The system (1) undergoes through several critical transitions, such as HB at $k=4.197$, saddle node bifurcation of limit cycle (SNLC) at $k=5.753$, transcritical bifurcation of limit cycle (TRLC) at $k=6.94$, torus bifurcation (TB) at $k=8.536$ and boundary crisis (BC) at $k=8.714$. Yellow, red and green dots represent stable cyclic states, whereas blue for unstable cycles.

Fig. 7

Role of environmental carrying capacity for $m_1=0.9$; other parameters are as in Table 1. Yellow, red and green dots represent stable cyclic state, whereas blue dots for unstable cycles. Critical transition occurred at HB = 4.197, SNLC = 5.753, TRLC = 6.94, TB = 8.536 and BC = 8.714. Chaotic dynamics are magnified in the inset

At lower carrying capacity, system (1) stays in non-prey herbivore-free equilibrium state $E_2$. All populations, except non-prey herbivore, transit from steady state to oscillation after HB which is shown by yellow dots. For higher values of k, an unstable cycle (in blue) coexists with the limit cycle (yellow). At SNLC point, stable cycle in red collides with the unstable cycle (blue) and disappears. Further, with the increase in k, two stable limit cycles (red and green) exchange their stability at TRLC. All population survive at oscillatory state shown in red, whereas prey herbivore goes to extinction and remaining three species maintain the oscillatory state, denoted by green dots.

For larger k, regular cycle (yellow) develops quasi-periodicity via TB, which eventually leads to the onset of irregular dynamics through quasi-periodic route to chaos. The chaotic state coexists with the alternate higher amplitude regular cycle (green). With the further increase in k, apparent chaos disappears at the BC. For a range of $k \in (8.536,8.714)$, the chaotic state coexists with the stable periodic cycle (green). It restores the regularity in temporal dynamics as the higher-amplitude oscillation in green still remains for larger values of carrying capacity.

The system exhibits alternate cyclic states for a range of $5.753<k<8.714$. Before TRLC, the oscillatory state in red coexists with the limit cycles in yellow which exclude non-prey herbivore. After TRLC, principle of competitive exclusion is somewhat followed as either of the two herbivores can survive in one of the alternate cyclic states (shown in green and yellow).