Bifurcation Analysis of a Generalist Predator-Prey Model with Holling Type II Harvesting

Abstract

In this paper, we consider a generalist predator–prey model with nonlinear harvesting, which has at most eight non-negative equilibria. We prove that the double positive equilibrium is a cusp of codimension up to 3; therefore, the system exhibits a cusp-type degenerate Bogdanov–Takens bifurcation of the same codimension. The elementary antisaddle equilibrium can act as a weak focus of the order of no more than two, giving rise to a degenerate Hopf bifurcation of codimension up to two. These high-codimension bifurcations identify organizing centers in parameter space, indicating regions where the ecosystem is highly sensitive and prone to abrupt regime shifts. Our results indicate that the generalist predator can induce a richer bifurcation phenomenon and more complex dynamics and can drive the system to certain desired stable states.

1. Introduction

The Lotka–Volterra predator–prey model, originally proposed by Lotka [1] and Volterra [2], is one of the most well-known models and has served as the foundational framework for many subsequent models across a wide range of disciplines. Bazykin [3] incorporated competition among predators for resources other than prey and proposed a system

$$\begin{array}{c}{\displaystyle \dot{x}=ax-bxy-\epsilon x^2,}\hfill \\ {\displaystyle \dot{y}=-cy+dxy-\mu y^2,}\hfill \end{array}$$

(1)

which exhibits relatively simple stability properties; that is, when the natural mortality rate is relatively small, the prey and predator will coexist; otherwise, the predator will go extinct. Based on system (1), Kuznetsov [4] incorporated the Holling type II functional response to model the change in prey density due to predation per unit time. For system (1), Lu and Huang [5] provided a rigorous theoretical proof for the existence of a focus-type degenerate Bogdanov–Takens bifurcation of codimension 3, as well as for the specific codimension associated with the Hopf bifurcation. Furthermore, many scholars have discussed the influence of various functional responses and factors on the dynamics of predator–prey models [6,7,8,9,10].

In classical theoretical ecology, the Lotka–Volterra model and its immediate descendants often depict predators as specialists, relying exclusively on a single prey species for survival. However, there exists a vast class of generalist predators that are prevalent in natural ecosystems, which consume a variety of prey types and often supplement their diet with alternative food sources, such as other prey species or even plant material. This dietary flexibility fundamentally alters the dynamics of the predator–prey interaction [11,12,13]. The study of generalist predator–prey systems is therefore crucial for understanding complex food web dynamics, ecosystem stability, and the outcomes of biological control strategies, providing a more realistic and nuanced perspective on species’ interactions in the wild. Magal et al. [14] proposed a host–parasitoid model with Holling type II functional response and discussed the dynamical behavior using a bifurcation theory approach, while Xiang et al. [15] and Yuan et al. [16] elaborated further on the complete nonlinear dynamics and bifurcations of the system. Sen et al. [17] found that a prey-predator model with a generalist predator exhibits rich and complex dynamics. Refs. [18,19,20] also considered predator–prey systems with a generalist predator and showed that generalist predation can induce richer bifurcation phenomenon and more complex dynamics.

Note that the classic prey–predator model describes a natural oscillation. However, to translate this theory into practical tools for managing real-world resources, such as fisheries, forests, or pest control, we must introduce the concept of harvesting. This addition is necessary to answer several critical questions: What is the maximum sustainable yield we can extract without causing population extinction? How does our intervention alter the natural balance? By incorporating harvesting, the model evolves from a mere description of nature into a vital framework for informed and responsible resource management, ensuring long-term ecological and economic stability [21,22,23]. Hu and Gao [24] presented a qualitative analysis of a predator–prey system with the nonlinear Michaelis–Menten-type predator harvesting. Mortuja, Chaube and Kumar [25] analyzed the dynamics of a predator–prey system with nonlinear prey harvesting. Bifurcation behavior in a modified Leslie–Gower model with nonlinear prey harvesting was explored in the work of Gupta and Chandra [26]. He and Li [27] discussed the bifurcation of a Holling–Tanner predator–prey model incorporating both anti-predator behavior and harvesting. Ref. [28] proposed a Leslie–Gower predator–prey model incorporating a generalist predator and harvesting, revealing that harvesting has the potential to disrupt the ecosystem’s equilibrium. These results can enable policymakers to determine the maximum sustainable yield, ensuring long-term resource availability while maintaining ecological balance.

Holling type II harvesting reveals how the nonlinear exploitation behavior of humans, as predators with limited capacity, profoundly reshapes ecosystem dynamics and creates unique collapse risks. Unlike the linear type, type II harvesting more realistically reflects real-world production functions constrained by factors such as handling time, storage, or market limitations, whose efficiency remains relatively high at low resource densities but saturates at high densities. This leads to a critical consequence: the resource population level corresponding to the maximum sustainable yield is typically higher than predicted by traditional models. Managing resources based on traditional targets can easily trigger hidden over-exploitation. Therefore, this model explains sudden ecological threshold phenomena such as fishery collapses and serves as a warning for resource management to adopt a precautionary principle, maintaining higher population baselines to avoid triggering irreversible regime shifts.

While Bogdanov–Takens and Hopf bifurcations have been studied in related models, the codimension and the specific organizing center structure for a generalist predator with Holling II harvesting remain unexplored. Therefore, building on the preceding analysis to incorporate a generalist predator with intrinsic logistic growth in the absence of prey, we consider the following generalist predator–prey model with nonlinear harvesting

$$\begin{array}{c}{\displaystyle \dot{x}=rx\left({\displaystyle 1-\frac{x}{K}}\right)-bxy,}\hfill \\ {\displaystyle \dot{y}=y(\alpha +\beta x-\delta y)-\frac{dEy}{m_{1}E+m_{2}y},}\hfill \end{array}$$

(2)

where $x\left(t\right)$ and $y\left(t\right)$ are the densities of prey and predator at time t, respectively; $r>0$ and $K>0$ are the intrinsic growth rate and carrying capacity of the prey, respectively; $b>0$ is the maximum rate of predation; $\alpha >0$ is the intrinsic growth rate of the predator who has an alternative food source; ${\displaystyle \frac{\beta }{b}}>0$ describes the efficiency of the predator in converting the consumed prey into the predator; $\delta >0$ is the mortality rate of the predator due to the intraspecific competition among predators; $d>0$ is the catchability coefficient of the predator species; $E>0$ is the effort applied to harvest individuals; and $m_1$ and $m_2$ are suitable positive constants.

For simplicity, making

$$\overline{x}={\displaystyle \frac{x}{K}},\overline{y}={\displaystyle \frac{by}{r}},\overline{t}=rt,$$

with the bars dropped, system (2) reduces to

$$\begin{array}{c}{\displaystyle \dot{x}=x(1-x-y),}\hfill \\ {\displaystyle \dot{y}=y(c+nx-ay)-\frac{hy}{q+y},}\hfill \end{array}$$

(3)

where

$$c={\displaystyle \frac{\alpha }{r}},n={\displaystyle \frac{\beta K}{r}},a={\displaystyle \frac{\delta }{b}},h={\displaystyle \frac{dbE}{m_2{r}^2}},q={\displaystyle \frac{b{m}_{1}E}{m_{2}r}},$$

whose biological meanings are listed in

Table 1. Biological interpretation of parameters (after nondimensionalization).

Parameter	Biological Meaning
c	Net intrinsic growth rate of predator (due to alternative food)
n	Prey-dependence strength
a	Intraspecific competition strength among predators
h	Nonlinear harvesting intensity parameter
q	Harvesting saturation parameter

.

The remainder of this paper is organized as follows. Section 2 analyzes the existence and types of equilibria in system (3). Section 3 investigates the degenerate Bogdanov–Takens bifurcation of codimension 3 and degenerate Hopf bifurcation of codimension 2. The impact of the generalist predator on the prey–predator system is then explored in Section 4 through numerical bifurcation diagrams and phase portraits. Finally, Section 5 provides a concluding discussion.

2. Equilibria and Their Types

Lemma 1.

All the solutions of system (3) are positive and bounded for $t\ge 0$.

Proof. 

Assume that $\left(x\right(t),y(t\left)\right)$ is any solution to system (3) with $x\left(0\right)>0$ and $y\left(0\right)>0$. Then, we can obtain $x\left(t\right)=x\left(0\right)exp\left\{{\int }_0^t(1-x\left(s\right)-y\left(s\right))\mathrm{d}s\right\}>0$ and $y\left(t\right)=y\left(0\right)exp\{{\int }_0^t(\alpha +\beta x\left(s\right)-\delta y\left(s\right)-{\displaystyle \frac{cE}{m_{1}E+m_{2}y\left(s\right)}}\mathrm{d}s\}>0$. From the first equation of system (3), we have $\dot{x}<x(1-x)$. According to Lemma 2.2 of Chen [29], there is $\underset{t\to +\infty }{lim\; sup}x\left(t\right)<1$. Substituting this into the second equation of system (3), we can similarly prove that $\underset{t\to +\infty }{lim\; sup}y\left(t\right)<{\displaystyle \frac{c+n}{a}}$. □

Let

$$\mathrm{\Omega }=\left\{(x,y)|0\le x\le 1,0\le y\le {\displaystyle \frac{c+n}{a}}\right\}.$$

The Jacobian matrix of system (3), evaluated at a non-negative equilibrium $E(x,y)$ is expressed as

$$\begin{array}{c}{J}_E=\left(\begin{array}{cc}1-2x-y& -x\\ ny& {\displaystyle c+nx-2ay-\frac{hq}{{(q+y)}^2}}\end{array}\right),\hfill \end{array}$$

(4)

and

$$\begin{array}{c}\mathrm{Det}\left(J_E\right)=(1-2x-y)\left({\displaystyle c+nx-2ay-\frac{hq}{{(q+y)}^2}}\right)+nxy,\hfill \\ {\displaystyle \mathrm{Tr}\left(J_E\right)=1+c+(n-2)x-(2a+1)y-\frac{hq}{{(q+y)}^2}.}\hfill \end{array}$$

2.1. Boundary Equilibria and Their Types

In this subsection, we discuss the existence and types of boundary equilibria, defined as equilibria on the non-negative x-axis or y-axis. When $y=0$, a straightforward calculation shows that $E_{00}(0,0)$ and $E_{10}(1,0)$ are always boundary equilibria of system (3).

Theorem 1.

For $E_{00}(0,0)$, we have the following:

(1) 

When $h<cq$, $E_{00}$ is an unstable node; when $h>cq$, $E_{00}$ is a saddle.

(2) 

When $h=cq$ and $c\ne aq$, $E_{00}$ is a saddle-node.

(3) 

When $h=cq$ and $c=aq$, $E_{00}$ is a saddle.

Proof. 

Equation (4) yields the following two eigenvalues for the Jacobian matrix $J_{E_{00}}$: 1 and ${\displaystyle \frac{cq-h}{q}}$. Hence, $E_{00}$ is an unstable node if $h<cq$, and a saddle if $h>cq$.

When $h=cq$, system (3) becomes

$$\begin{array}{c}\dot{x}=x-x^2-xy,\hfill \\ {\displaystyle \dot{y}=\frac{c-aq}{q}{y}^2-nxy-\frac{c}{q^2}{y}^3+{o\left(\right|x,y|}^3).}\hfill \end{array}$$

(5)

By Theorem 7.1 of [30] (i.e., a saddle-node bifurcation criterion), $E_{00}$ is a saddle-node if $c\ne aq$.

When $h=cq$ and $c=aq$, by the center manifold theorem, substituting $x=m_1{y}^2+m_2{y}^3+{o\left(\right|y|}^3)$ into $\dot{x}=0$, we can obtain $m_1=m_2=0$. From the second equation of system (5) with $x=o\left(\right|y{|}^3)$, we have

$$\dot{y}=-{\displaystyle \frac{c}{q^2}}{y}^3+{o\left(\right|y|}^3).$$

By Theorem 7.1 of [30] (i.e., a saddle-node bifurcation criterion), $E_{00}$ is a saddle if $h=cq$ and $c=aq$. □

Let

$$\begin{array}{c}{\displaystyle h_0=\frac{{(aq+c)}^2}{4a},h_1=(c-a)(q+1),h_2=(c+n)q,}\hfill \\ c_0=a(q+2),c_1=a(q+2)+n(q+1).\hfill \end{array}$$

Theorem 2.

For $E_{10}(1,0)$, we have the following:

(1) 

When $h>h_2$(or $h<h_2)$, $E_{10}$ is a stable node (or a saddle).

(2) 

When $h=h_2$ and $q\ne {\displaystyle \frac{c+n}{a+n}}$, $E_{10}$ is a saddle-node.

(3) 

When $h=h_2$ and $q={\displaystyle \frac{c+n}{a+n}}$, $E_{10}$ is a stable node.

Proof. 

Equation (4) yields the following two eigenvalues for the Jacobian matrix $J_{E_{10}}$: $-1$ and ${\displaystyle \frac{h_2-h}{q}}$. So $E_{10}$ is a stable node if $h>h_2$, and a saddle if $h<h_2$.

When $h=h_2$, by successive transformations,

$$\begin{array}{c}x=x_1+1,y=y_1;\hfill \\ x_1=-x_2-y_2,y_1=y_2,\hfill \end{array}$$

system (3) can be rewritten as

$$\begin{array}{c}{\displaystyle {\dot{x}}_2=-x_2+x_2^2+(n+1)x_2{y}_2+\frac{aq+nq-c-n}{q}{y}_2^2+\frac{c+n}{q^2}{y}_2^3+o\left(\right|x_2,y_2{|}^3),}\hfill \\ {\displaystyle {\dot{y}}_2=-\frac{aq+nq-c-n}{q}{y}_2^2-n{x}_2{y}_2-\frac{c+n}{q^2}{y}_2^3+o\left(\right|x_2,y_2{|}^3).}\hfill \end{array}$$

(6)

By Theorem 7.1 of [30] (i.e., a saddle-node bifurcation criterion), $E_{10}$ is a saddle-node if $q\ne {\displaystyle \frac{c+n}{a+n}}$.

When $h=h_2$ and $q={\displaystyle \frac{c+n}{a+n}}$, by the center manifold theorem, substituting $x_2=m_3{y}_2^2+m_4{y}_2^3+o\left(\right|y_2{|}^3)$ into $\dot{x_2}=0$, we can obtain

$$m_3=0and{m}_4={\displaystyle \frac{a+n}{q}}.$$

Then by substituting $x_2={\displaystyle \frac{a+n}{q}}{y}_2^3+o\left(\right|y_2{|}^3)$ into the second equation of (6), we have

$$\dot{y_2}=-{\displaystyle \frac{a+n}{q}}{y}_2^3+o\left(\right|y_2{|}^3).$$

By Theorem 7.1 of [30] (i.e., a saddle-node bifurcation criterion) again, $E_{10}$ is a stable node if $h=h_2$ and $q={\displaystyle \frac{c+n}{a+n}}$. □

Setting $x=0$ in the second equation of (3) yields

$$f_0\left(y\right)=a{y}^2+(aq-c)y+h-cq,$$

whose discriminant is ${\mathrm{\Delta }}_0={(aq+c)}^2-4ah.$ Let

$$\begin{array}{c}{\displaystyle y_{01}=\frac{c-aq-\sqrt{{\mathrm{\Delta }}_0}}{2a},y_{02}=\frac{c-aq+\sqrt{{\mathrm{\Delta }}_0}}{2a}.}\hfill \end{array}$$

Lemma 2.

(1) 

If $h<cq$, or $h=cq$ with $c>aq$, system (3) has a boundary equilibrium $E_{02}(0,y_{02})$;

(2) 

if $cq<h<h_0$ and $c>aq$, system (3) has two boundary equilibria $E_{01}(0,y_{01})$ and $E_{02}(0,y_{02})$;

(3) 

if $h=h_0$ and $c>aq$, system (3) has a boundary equilibrium ${\tilde{E}}_{\ast }(0,{\displaystyle \frac{c-aq}{2a}})$.

Theorem 3.

Supposing that the conditions of Lemma 2 (2) hold, we obtain the stability of $E_{01}$ as follows:

(1) 

When $c\le c_0$; or $c>c_0$ and $h<h_1$, $E_{01}$ is an unstable node;

(2) 

when $c>c_0$ and $h>h_1$, $E_{01}$ is a saddle;

(3) 

when $c>c_0$, $h=h_1$ and $c\ne c_1$, $E_{01}$ is a saddle-node;

(4) 

when $c>c_0$, $h=h_1$ and $c=c_1$, $E_{01}$ is a saddle.

Proof. 

At $E(0,y)$, the Jacobian matrix of system (3) is

$$\begin{array}{c}{J}_E=\left(\begin{array}{cc}1-y& 0\\ ny& {\displaystyle y\left(\frac{h}{{(q+y)}^2}-a\right)}\end{array}\right).\hfill \end{array}$$

(7)

Easily ${\displaystyle \frac{h}{{(q+y_{01})}^2}}-a>0$. When $c\le c_0$, or $c>c_0$ and $h<h_1$, we have $1-y_{01}>0$, which implies $E_{01}$ is an unstable node. When $c>c_0$ and $h>h_1$, there is $1-y_{01}<0$, thus $E_{01}$ is a saddle.

When $c>c_0$ and $h=h_1$, by successively transforming

$$\begin{array}{c}x=x_1,y=y_1+y_{01};\hfill \\ {\displaystyle x_1=x_2,y_1=-x_2+\frac{1}{n}{y}_2,d\tau =\frac{c-c_0}{q+1}dt,}\hfill \end{array}$$

denoting $\tau$ by t, system (3) is

$$\begin{array}{ccc}{\dot{x}}_2\hfill & =& {\displaystyle \frac{(c_1-c)(q+1)}{{(c_0-c)}^2}{x}_2^2-\frac{{(q+1)}^2}{{(c_0-c)}^2}{x}_2{y}_2+o\left(\right|x_2,y_2{|}^2),}\hfill \\ {\dot{y}}_2\hfill & =& {\displaystyle y_2+\frac{(a-c)n^2+(c_1-c)(q+1)n}{{(c_0-c)}^2}{x}_2^2+\frac{(a_0+c-{(q+1)}^2)n}{{(c_0-c)}^2}{x}_2{y}_2}\hfill \\ \hfill & & {\displaystyle +\frac{a_0+a}{{(c_0-c)}^2}{y}_2^2+o\left(\right|x_2,y_2{|}^2),}\hfill \end{array}$$

(8)

where $a_0=a{q}^2+3aq-cq$. By Theorem 7.1 of [30] (i.e., a saddle-node bifurcation criterion), $E_{01}$ is a saddle-node if $c\ne c_1$.

When $c>c_0$, $h=h_1$ and $c=c_1$, by the center manifold theorem, substituting $y_2=m_5{x}_2^2+m_6{x}_2^3+o\left(\right|x_2{|}^3)$ into $\dot{y_2}=0$, we can obtain

$$m_5={\displaystyle \frac{a+n}{q+1}}\mathrm{and}{m}_6={\displaystyle \frac{nq+a+n+aq-2a^2-3an-n^2}{n{(q+1)}^2}}.$$

Then, by substituting $y_2=m_5{x}_2^2+m_6{x}_2^3+o\left(\right|x_2{|}^3)$ into the first equation of (8), we have

$$\dot{x_2}=-{\displaystyle \frac{a+n}{n^2(q+1)}}{x}_2^3+o\left(\right|y_2{|}^3).$$

By Theorem 7.1 of [30] (i.e., a saddle-node bifurcation criterion), $E_{01}$ is a saddle. □

Theorem 4.

Assume that the conditions of Lemma 2 (1) or (2) hold, we obtain the stability of $E_{02}$ as follows.

(1) 

When $c\ge c_0$, or $a<c<c_0$ and $h>h_1$, $E_{02}$ is a stable node;

(2) 

when $c\le a$, or $a<c<c_0$ and $h<h_1$, $E_{02}$ is a saddle;

(3) 

when $a<c<c_0$ and $h=h_1$, $E_{02}$ is a saddle-node.

Proof. 

From (7), one has ${\displaystyle \frac{h}{{(q+y_{02})}^2}}-a<0$. If $c\ge c_0$, or $a<c<c_0$ and $h>h_1$, which implies $1-y_{01}<0$, then $E_{02}$ is a stable node. If $c\le a$, or $a<c<c_0$ and $h<h_1$—that is, $1-y_{01}>0$—then $E_{02}$ is a saddle.

When $a<c<c_0$ and $h=h_1$, similarly to the proof of Theorem 3 (3), we can prove that $E_{02}$ is a saddle-node. □

Theorem 5.

Assume that $h=h_0$ and $c>aq$.

(1) 

If $c\ne c_0$, ${\tilde{E}}_{\ast }$ is a saddle-node.

(2) 

if $c=c_0$, ${\tilde{E}}_{\ast }$ is a nilpotent singularity including a hyperbolic sector and an elliptic sector.

Proof. 

Make transformations successively

$$\begin{array}{c}{\displaystyle x=x_1,y=y_1+\frac{c-aq}{2a};}\hfill \\ {\displaystyle x_1=\frac{c_0-c}{2a}{y}_2,y_1=\frac{c-c_0}{2a}{x}_2+\frac{n(c-aq)}{2a}{y}_2,d\tau =\frac{c_0-c}{2a}dt.}\hfill \end{array}$$

Letting $\tau$ remain as t, system (3) becomes

$$\begin{array}{c}{\displaystyle {\dot{x}}_2=\frac{a(c-aq)}{aq+c}{x}_2^2+\frac{n(2an+c-c_0)(a^2{q}^2-c^2)-a{n}^2{(aq-c)}^3}{{(c_0-c)}^2(aq+c)}{y}_2^2}\hfill \\ {\displaystyle -\frac{2an(aq+c-{(aq-c)}^2)}{(c_0-c)(aq+c)}{x}_2{y}_2+o\left(\right|x_2,y_2{|}^2),}\hfill \\ {\displaystyle {\dot{y}}_2=y_2+x_2{y}_2+\frac{anq-aq-cn-2a+c}{c_0-c}{y}_2^2+o\left(\right|x_2,y_2{|}^2).}\hfill \end{array}$$

By Theorem 7.1 of [30] (i.e., a saddle-node bifurcation criterion), ${\tilde{E}}_{\ast }$ is a saddle-node if $c\ne c_0$.

When $c=c_0$, by successive transformations

$$\begin{array}{c}{\displaystyle x=x_1,y=y_1+\frac{c-aq}{2a};}\hfill \\ x_1=y_2,y_1=x_2,d\tau =ndt,\hfill \end{array}$$

still denoting $\tau$ by t, system (3) is

$$\begin{array}{c}{\displaystyle {\dot{x}}_2=y_2-\frac{a}{n(q+1)}{x}_2^2+x_2{y}_2+o\left(\right|x_2,y_2{|}^2)\triangleq y_2+P(x_2,y_2),}\hfill \\ {\displaystyle {\dot{y}}_2=-\frac{1}{n}{x}_2{y}_2-\frac{1}{n}{y}_2^2+o\left(\right|x_2,y_2{|}^2)\triangleq Q(x_2,y_2).}\hfill \end{array}$$

By the center manifold theorem, substituting $y_2=m_7{x}_2^2+m_8{x}_2^3+o\left(\right|x_2{|}^3)$ into $\dot{x_2}=0$, we can obtain

$$m_7={\displaystyle \frac{a}{n(q+1)}}and{m}_8=-{\displaystyle \frac{a}{n{(q+1)}^2}}.$$

Then, by substituting $y_2=m_7{x}_2^2+m_8{x}_2^3+o\left(\right|x_2{|}^3)$ into $Q(x_2,y_2)$, we obtain

$$\begin{array}{c}{\displaystyle Q(x_2,y_2\left(x_2\right))=-\frac{a}{n^2(q+1)}{x}_2^3+o\left(\right|x_2{|}^3),}\hfill \\ {\displaystyle P_{x_2}^{\prime }(x_2,y_2\left(x_2\right))+Q_{y_2}^{\prime }(x_2,y_2\left(x_2\right))=-\frac{q+2a+1}{n(q+1)}{x}_2+o\left(\right|x_2\left|\right).}\hfill \end{array}$$

By Theorem 7.2 of [30] (i.e., a saddle-node bifurcation criterion), we can easily verify that ${\tilde{E}}_{\ast }$ is a nilpotent singularity including a hyperbolic sector and an elliptic sector. □

2.2. Positive Equilibria and Their Types

In this section, we study the types of positive equilibria of (3), which satisfies equations

$$\begin{array}{c}1-x-y=0,\hfill \\ {\displaystyle (c+nx-ay)-\frac{h}{q+y}=0.}\hfill \end{array}$$

(9)

Substitute $y=1-x$ into the second equation of (9), then let

$$f\left(x\right)=(a+n)x^2-(aq+nq+2a-c+n)x+h-(q+1)(c-a).$$

The discriminant of $f\left(x\right)=0$ is

$$\mathrm{\Delta }={(aq+nq+c+n)}^2-4(a+n)h,$$

and

$$\begin{array}{c}{f}^{\prime }\left(x\right)=2(a+n)x-(a+n)q-2a-n+c.\hfill \end{array}$$

(10)

For any positive equilibrium $E(x,y)$, we can obtain

$$\begin{array}{c}{\displaystyle \mathrm{Det}\left(J_E\right)=\frac{x(x-1)}{q+1-x}{f}^{\prime }\left(x\right).}\hfill \end{array}$$

(11)

Let

$$\begin{array}{c}{\displaystyle h_3=\frac{{(aq+nq+c+n)}^2}{4(a+n)},c_3=aq+nq-n,c_4=aq+nq+a,}\hfill \\ {\displaystyle x_{11}=\frac{c_1-c-\sqrt{\mathrm{\Delta }}}{2(a+n)},x_{12}=\frac{c_1-c+\sqrt{\mathrm{\Delta }}}{2(a+n)},x_{\ast }=\frac{c_1-c}{2(a+n)}.}\hfill \end{array}$$

Lemma 3.

System (3) has, at most, two positive equilibria. Moreover,

(1) 

if $h_2<h<h_1$ with $c>c_4$, or $h=h_1$ with $c_4<c<c_1$, system (3) has a unique positive equilibrium $E_{12}(x_{12},1-x_{12})$, which is a hyperbolic saddle;

(2) 

if $h_1<h<h_2$ and $a<c<c_4$, or $h=h_2$ with $max\{0,c_3\}<c<c_4$, system (3) has a unique positive equilibrium $E_{11}(x_{11},1-x_{11})$, which is an elementary and antisaddle one;

(3) 

if $max\{h_1,h_2\}<h<h_3$ and $max\{0,c_3\}<c<c_1$, system (3) has two positive equilibria $E_{11}(x_{11},1-x_{11})$ and $E_{12}(x_{12},1-x_{12})$;

(4) 

if $h=h_3$ and $max\{0,c_3\}<c<c_1$, system (3) has a unique positive equilibrium $E_{\ast }(x_{\ast },1-x_{\ast })$, which is degenerate.

Proof. 

From (10), (11) and the derivative property of $f\left(x\right)$, we can easily verify that $\mathrm{Det}\left(J_{E_{11}}\right)>0$, $\mathrm{Det}\left(J_{E_{12}}\right)<0$ and $\mathrm{Det}\left(J_{E_{\ast }}\right)=0$. Thus, $E_{11}$ is an elementary antisaddle, $E_{12}$ is a hyperbolic saddle, and $E_{\ast }$ is a degenerate equilibrium. □

Next, we discuss the stability of the degenerate equilibrium $E_{\ast }$. Let

$$\begin{array}{c}{\displaystyle c_{\ast }=\frac{anq+n^{2}q+aq+nq+2a+n-n^2}{1+n},q_{\ast }=\frac{2a+n-1}{{(1+n)}^2},q_1=\frac{c+n}{a+n}.}\hfill \end{array}$$

Theorem 6.

Assume that $h=h_3$ and $max\{0,c_3\}<c<c_1$. Moreover,

(1) 

if $c_{\ast }\le 0$ or $0<c_{\ast }<c$, $E_{\ast }$ is a saddle-node;

(2) 

if $0<c<c_{\ast }$, $E_{\ast }$ is a saddle-node.

Proof. 

When $h=h_3$, making transformations successively,

$$\begin{array}{c}x=x_1+x_{\ast },y=y_1+1-x_{\ast };\hfill \\ {\displaystyle x_1=\frac{c-c_1}{2(a+n)}{x}_2+\frac{c-c_1}{2(a+n)}{y}_2,y_1=\frac{c_1-c}{2(a+n)}{x}_2+\frac{n(c-c_3)}{2(a+n)}{y}_2;}\hfill \\ {\displaystyle d\tau =\frac{(n+1)(c-c_{\ast })}{2(a+n)}dt,}\hfill \end{array}$$

still denoting $\tau$ by t, system (3) becomes

$$\begin{array}{c}{\displaystyle {\dot{x}}_2=\frac{(a+n)(c-c_3){(c_1-c)}^2}{{(c_{\ast }-c)}^2{(n+1)}^2(aq+nq+c+n)}{x}_2^2+c_{11}{x}_2{y}_2+c_{02}{y}_2^2+o\left(\right|x_2,y_2{|}^2),}\hfill \\ {\dot{y}}_2=y_2+d_{20}{x}_2^2+d_{11}{x}_2{y}_2+d_{02}{y}_2^2+o\left(\right|x_2,y_2{|}^2),\hfill \end{array}$$

(12)

where $c_{11}$, $c_{02}$, $d_{20}$, $d_{11}$ and $d_{02}$ are listed in Appendix A. By Theorem 7.1 of [30] (i.e., a saddle-node bifurcation criterion), $E_{\ast }$ is a saddle-node if $c\ne c_{\ast }$. □

Theorem 7.

Assume that $h=h_3$ and $q_{\ast }>0$. Moreover,

(1) 

$E_{\ast }$ is a cusp of codimension 2, if $c=c_{\ast }>0$ and $q\ne q_{\ast }$;

(2) 

$E_{\ast }$ is a cusp of codimension 3, if $c=c_{\ast }>0$ and $q=q_{\ast }$.

Proof. 

When $c=c_{\ast }>0$, by successive transformations,

$$\begin{array}{c}x=x_1+x_{\ast },y=y_1+1-x_{\ast };\hfill \\ {\displaystyle x_1=-\frac{n}{1+n}{x}_2,y_1=\frac{n}{1+n}{x}_2+y_2,}\hfill \end{array}$$

system (3) becomes

$$\begin{array}{c}{\dot{x}}_2=y_2-x_2{y}_2+o\left(\right|x_2,y_2{|}^2),\hfill \\ {\displaystyle {\dot{y}}_2=-\frac{n^2(a+n)}{{(1+n)}^2(nq+q+1)}{x}_2^2+\frac{n(n^{2}q+2nq-2a-n+q+1)}{(1+n)(nq+q+1)}{x}_2{y}_2}\hfill \\ {\displaystyle +\frac{n^{2}q+nq-a}{nq+q+1}{y}_2^2+o\left(\right|x_2,y_2{|}^2).}\hfill \end{array}$$

(13)

By Lemma 3.1 of Huang, Gong and Ruan [31], near the origin, system (13) can be reduced to

$$\begin{array}{c}\dot{X}=Y+{o\left(\right|X,Y|}^3),\hfill \\ \dot{Y}=D{X}^2+\check{E}XY+{o\left(\right|X,Y|}^3),\hfill \end{array}$$

where

$${\displaystyle D=-\frac{n^2(a+n)}{{(1+n)}^2(nq+q+1)},\check{E}=\frac{n(n^{2}q+2nq-2a-n+q+1)}{(1+n)(nq+q+1)}.}$$

Therefore, if $c=c_{\ast }>0$ and $q\ne q_{\ast }$, then $\check{E}\ne 0$, which implies $E_{\ast }$ is a cusp of codimension two.

Assume that $c=c_{\ast }>0$ and $q=q_{\ast }$, then system (13) can be rewritten as

$$\begin{array}{ccc}{\dot{x}}_2\hfill & =& y_2-x_2{y}_2+o\left(\right|x_2,y_2{|}^4),\hfill \\ {\dot{y}}_2\hfill & =& {\displaystyle -\frac{n^2}{2(1+n)}{x}_2^2+\frac{n-1}{2}{y}_2^2-\frac{n^{3}l}{1+n}{x}_2^3-3n^{2}l{x}_2^2{y}_2-3n(n+1)l{x}_2{y}_2^2}\hfill \\ \hfill & & {\displaystyle -{(1+n)}^{2}l{y}_2^3+\frac{n^{4}l}{2(a+n)}{x}_2^4+\frac{2n^3(1+n)l}{a+n}{x}_2^3{y}_2+\frac{3n^2{(1+n)}^{2}l}{a+n}{x}_2^2{y}_2^2}\hfill \\ \hfill & & {\displaystyle +\frac{2n{(1+n)}^{3}l}{a+n}{x}_2{y}_2^3+\frac{{(1+n)}^{4}l}{2(a+n)}{y}_2^4+o\left(\right|x_2,y_2{|}^4),}\hfill \end{array}$$

(14)

where $l={\displaystyle \frac{2a+n-1}{4(a+n)}}$.

Denote

$$x_2=x_3+{\displaystyle \frac{n-3}{4}}{x}_3^2,y_2=y_3+{\displaystyle \frac{n-1}{2}}{x}_3{y}_3,$$

then system (14) is

$$\begin{array}{cc}\hfill {\dot{x}}_3& =y_3+{\tilde{a}}_{21}{x}_3^2{y}_3+{\tilde{a}}_{40}{x}_3^4+{\tilde{a}}_{31}{x}_3^3{y}_3+{\tilde{a}}_{50}{x}_3^5+{\tilde{a}}_{41}{x}_3^4{y}_3+o\left(\right|x_3,y_3{|}^5),\hfill \\ \hfill {\dot{y}}_3& ={\tilde{b}}_{20}{x}_3^2+{\tilde{b}}_{30}{x}_3^3+{\tilde{b}}_{21}{x}_3^2{y}_3+{\tilde{b}}_{12}{x}_3{y}_3^2+{\tilde{b}}_{40}{x}_3^4+{\tilde{b}}_{31}{x}_3^3{y}_3+{\tilde{b}}_{22}{x}_3^2{y}_3^2\hfill \\ \hfill & +{\tilde{b}}_{50}{x}_3^5+{\tilde{b}}_{41}{x}_3^4{y}_3+{\tilde{b}}_{32}{x}_3^3{y}_3^2+o\left(\right|x_3,y_3{|}^5),\hfill \end{array}$$

(15)

where ${\tilde{b}}_{20}=-{\displaystyle \frac{n^2}{2(1+n)}}\ne 0$, and the other coefficients are omitted here.

From Lemma 2.4 of Lu, Huang and Wang [18], system (15) near the origin is equivalent to

$$\begin{array}{c}\dot{X}=Y+{o\left(\right|X,Y|}^4),\hfill \\ \dot{Y}=X^2+M{X}^{3}Y+{o\left(\right|X,Y|}^4),\hfill \end{array}$$

where

$$M=-{\displaystyle \frac{(2a+n-1)n^5}{16{(a+n)}^2}}<0,$$

thus $E_{\ast }$ is a cusp of codimension 3. □

Based on the above conclusions, we summarize the quantity and stability of positive equilibria of system (3) in

Table 2. The quantity and stability of positive equilibria of system (3).

c	Conditions of Parameters	Positive Equilibrium
>$c_4$	$h_2<h<h_1$	$E_{12}$: saddle
	$h=h_1$ with $c<c_1$	$E_{12}$: saddle
<$c_4$	$h_1<h<h_2$ with $c>a$	$E_{11}$: antisaddle
	$h=h_2$ with $c>max\{0,c_3\}$	$E_{11}$: antisaddle
<$c_1$	$max\{h_1,h_2\}<h<h_3$ with $c>max\{0,c_3\}$	$E_{11}$: antisaddle; $E_{12}$: saddle
	$h=h_3$, $c>max\{0,c_3\}$ and $c\ne c_{\ast }$	$E_{\ast }$: saddle-node
	$h=h_3$, $c>max\{0,c_3\}$ and $c=c_{\ast }$	$E_{\ast }$: cusp

.

3. Bifurcations

Next, we discuss bifurcations of the system, such as Bogdanov–Takens bifurcation and Hopf bifurcation.

3.1. Bogdanov–Takens Bifurcation of Codimension 3

According to Theorem 7, the occurrence of a degenerate Bogdanov–Takens bifurcation of codimension 3 is possible for system (3) near $E_{\ast }$, if $h=h_3$, $c=c_{\ast }>0$ and $q=q_{\ast }>0$. Now, select c, h and q as bifurcation parameters. Consider a system

$$\begin{array}{c}{\displaystyle \dot{x}=x(1-x-y),}\hfill \\ {\displaystyle \dot{y}=y(c+{\lambda }_1+nx-ay)-\frac{(h+{\lambda }_2)y}{q+{\lambda }_3+y},}\hfill \end{array}$$

(16)

where $\lambda =({\lambda }_1,{\lambda }_2,{\lambda }_3)$ is a parameters vector lying in a small neighborhood of the origin.

Theorem 8.

If $h=h_3$, $c=c_{\ast }>0$ and $q=q_{\ast }>0$, then system (3) undergoes a degenerate Bogdanov–Takens bifurcation of codimension 3 around $E_{\ast }$.

Proof. 

In a manner analogous to the transformations employed by [13,32], we reformulate system (16) as

$$\begin{array}{c}\dot{u}=v,\hfill \\ \dot{v}={\mu }_1+{\mu }_{2}v+u^2+{\mu }_{3}uv-u^{3}v+{o\left(\right|u,v|}^4),\hfill \end{array}$$

with ${\mu }_i(i=1,2,3)$ being functions of ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, a and q, and their explicit forms are omitted. Using Maple, it is found that

$${\displaystyle \left|\frac{D({\mu }_1,{\mu }_2,{\mu }_3)}{D({\lambda }_1,{\lambda }_2,{\lambda }_2)}\right|=\frac{{(1+n)}^{36/5}{(2a+n-1)}^{4/5}}{8n^{7/5}{(a+n)}^{23/5}}\ne 0,\mathrm{if}{q}_{\ast }>0.}$$

Therefore, according to [33], it can be shown that a degenerate Bogdanov–Takens bifurcation of codimension 3 takes place in system (3) around $E_{\ast }$, if $h=h_3$, $c=c_{\ast }>0$ and $q=q_{\ast }>0$. □

3.2. Hopf Bifurcation

From the proof of Lemma 3, we have $\mathrm{Det}\left(J_{E_{11}}\right)>0$. This means that, when $\mathrm{Tr}\left(J_{E_{11}}\right)=0$, a Hopf bifurcation may occur in system (3) near the equilibrium $E_{11}$. To simplify the subsequent analysis, the coordinate $x_{11}$ is hereafter denoted by z. From $f\left(z\right)=0$ and $\mathrm{Tr}\left(J_{E_z}\right)=0$, we can obtain

$$\begin{array}{c}{\displaystyle a=\frac{(1-z)(c-(1-n\left)z\right)-qz}{(1-z)(q+2(1-z\left)\right)},h=\frac{{(q+1-z)}^2(nz+c+z)}{q+2(1-z)}.}\hfill \end{array}$$

(17)

From $a>0$, $h>0$ and $\mathrm{Det}\left(J_{E_z}\right)>0$, we let

$$\begin{array}{c}\mathcal{M}=\left\{{\displaystyle (c,n,q,z)\in {\mathbb{R}}_{+}^4|q<\frac{c-(1-n)z}{z},c>(1-n)z\mathrm{and}n>\frac{z}{1-z}}\right\}.\hfill \end{array}$$

(18)

To derive the focal values near $E_{11}(z,1-z)$, we implement the following sequence of coordinate transformations:

$$\begin{array}{c}\left(\mathrm{i}\right)\mathrm{d}t=(q+y)\mathrm{d}{\tau }_0;\hfill \\ \left(\mathrm{ii}\right)x=x_1+z,y=y_1+1-z;\hfill \\ {\displaystyle \left(\mathrm{iii}\right)x_1=\frac{z}{(z-1)n}{x}_2-\frac{\sqrt{D_0}}{(q+1-z)(z-1)n}{y}_2,y_1=x_2,\mathrm{d}\tau =\sqrt{D_0}\mathrm{d}{\tau }_0,}\hfill \end{array}$$

where

$$D_0=z{(q+1-z)}^2(n-z-nz).$$

For notational simplicity, we continue to use t to represent $\tau$, upon which system (3) becomes

$$\begin{array}{cc}\hfill {\dot{x}}_2& =y_2+a_{20}{x}_2^2+a_{11}{x}_2{y}_2+a_{30}{x}_2^3+a_{21}{x}_2^2{y}_2,\hfill \\ \hfill {\dot{y}}_2& =-x_2+b_{20}{x}_2^2+b_{11}{x}_2{y}_2+b_{02}{y}_2^2+b_{30}{x}_2^3+b_{21}{x}_2^2{y}_2+b_{12}{x}_2{y}_2^2,\hfill \end{array}$$

(19)

where the expressions of the coefficients are listed in Appendix B.

At the equilibrium $E_{11}$, we present the first- and second-order Lyapunov coefficients according to [30] (i.e., a saddle-node bifurcation criterion):

$$\begin{array}{c}{\displaystyle L_1=\frac{z(nz+c+z)(q+1-z)l_1}{4{(q-2z+1)}^2{D}_0^{3/2}},}\hfill \\ {\displaystyle L_2=-\frac{z(nz+c+z)(q+1-z)l_2}{24n^2{(z-1)}^2{(q-2z+z)}^4{D}_0^{5/2}},}\hfill \end{array}$$

where

$$\begin{array}{c}{l}_1=(2nz-2n+2z-1)q^2-3{(z-1)}^2(n+1)q+2{(z-1)}^2(c+n-1),\hfill \end{array}$$

the lengthy expression for $l_2$ is omitted.

Obviously, the signs of $L_1$ and $L_2$ are determined by those of $l_1$ and $l_2$, respectively. In the following, we let $V(l_1,l_2)$ be the set of common zeros of $l_1$ and $l_2$, and $\mathrm{res}(l_1,l_2,x)$ denote the Sylvester resultant of the two polynomials $l_1$ and $l_2$ with respect to the variable x.

Lemma 4.

Assuming that (17) and (18) hold, then $V(l_1,l_2)\cap \mathcal{M}=Ø$.

Proof. 

By Maple software, we can obtain

$$\begin{array}{c}\mathrm{res}(l_1,l_2,c)=-4nqz{(z-1)}^7{(q-2z+2)}^3(nz-n+z)R_1\hfill \end{array}$$

with

$$R_1=3(1-z)(2q-z+1)n^2+(7z^2-10qz+9q-16z+9)n-4(q+1-z)z.$$

We can easily verify that $R_1>0$ for $n>{\displaystyle \frac{z}{1-z}}$. Thus, $\mathrm{res}(l_1,l_2,c)<0$ in $\mathcal{M}$, which implies that $V(l_1,l_2)\cap \mathcal{M}=Ø$. □

Now, we give some examples to discuss the signs of $l_1$ and $l_2$.

$$\begin{array}{c}{l}_1{|}_{(c,n,q,z)=(1,2,0.4,0.2)}=-\frac{44}{125}<0,l_1{|}_{(c,n,q,z)=(1,2,0.2,0.2)}=\frac{157}{125}>0,\hfill \\ l_1{|}_{(c,n,q,z)=(1,2,\frac{32\sqrt{11}-72}{95},0.2)}=0,l_2{|}_{(c,n,q,z)=(1,2,\frac{32\sqrt{11}-72}{95},0.2)}\approx 18.6683>0.\hfill \end{array}$$

Thus, system (3) can exhibit supercritical Hopf bifurcation, subcritical Hopf bifurcation, and degenerate Hopf bifurcation of codimension 2. Therefore, according to Lemma 4, we can obtain the following theorem.

Theorem 9.

Supposing that (17) and (18) hold, then $E_{11}$ is a weak focus of order at most 2. Moreover,

(1) 

if $l_1<0$, then $E_{11}$ is a weak focus of order 1, which can result in a supercritical Hopf bifurcation and give rise to a stable limit cycle around $E_{11}$;

(2) 

if $l_1>0$, then $E_{11}$ is a weak focus of order 1, which can result in a subcritical Hopf bifurcation and give rise to an unstable limit cycle around $E_{11}$;

(3) 

if $l_1=0$, then $E_{11}$ is a weak focus of order 2, which can result in a degenerate Hopf bifurcation and give rise to two limit cycles around $E_{11}$.

4. Numerical Simulations

In this section, we show the bifurcation phenomena of system (3) by some numerical simulations using Matcont and discuss the influence of a generalist predator on the dynamic behaviors of system (3).

We fix $(n,a,q)=(0.62,1.2,0.8)$, then we present the bifurcation diagram in $(h,c)$-plane in Figure 1, where the degenerate Hopf ($GH$) and Bogdanov–Takens ($BT$) points are indicated. Solid curves correspond to Hopf (blue), saddle-node (black), saddle-node of limit cycles (green), and homoclinic (red) bifurcations.

The parameter plane is partitioned into distinct regions by the above curves, indicating that even minor variations in $(h,c)$ can lead to qualitative changes in the long-term behavior of the system. Particularly near the $GH$ and $BT$ points, the system exhibits its most complex and sensitive dynamics, where abrupt transitions and intricate behaviors are most likely to occur. Therefore, we further fix $h=3.7$ and give the bifurcation diagram in $(c,x)$-plane in Figure 2, whose dynamical behaviors are listed in detail in

Table 3. Dynamical behaviors in Figure 2.

c	${\mathit{E}}_{11}$	${\mathit{E}}_{12}$	${\mathit{E}}_{\mathbf{\ast }}$	Closed Orbits
$(0,3.113904)$	-	-	-	No (see Figure 3a)
3.113904	-	-	cusp	No (see Figure 3b)
$(3.113904,3.1408439)$	unstable focus	saddle	-	No (see Figure 3c)
$(3.1408439,3.1414929)$	unstable focus	saddle	-	A stable limit cycle (see Figure 3d)
$(3.1414929,3.1415268)$	stable focus	saddle	-	Two limit cycle (see Figure 3e)
$(3.1415268,3.2555556)$	stable focus	saddle	-	No (see Figure 3f)
$(3.2555556,4.005)$	-	saddle	-	No (see Figure 3g)

.

To further verify the above conclusions, we will now fix c to the following specific values and present the corresponding phase diagrams. For different types of equilibrium stability, we have the following conclusions. (1) Monostability: In Figure 3a–c, only the boundary equilibrium $E(1,0)$ is stable, which indicates that all populations will eventually be attracted to $E(1,0)$. This means the prey will persist at a constant level while the predator eventually becomes extinct. (2) Bistability: In Figure 3d, the unique limit cycle and $E(1,0)$ are stable. That is, system (3) exhibits two distinct basins of attraction, which determined by the initial position. This indicates initial states lying inside the region bounded by the two stable manifolds are attracted to the limit cycle, indicating that in such cases, both prey and predator populations will ultimately coexist on a periodic orbit. Conversely, trajectories starting from any point outside this region will eventually stabilize at $E(1,0)$, which indicates the persistence of the prey and the extinction of the predator. In Figure 3f, both populations will stabilize at $E(1,0)$ and $E_{11}$. Thus, if the initial state lies within the region bounded by the two stable manifolds, all populations converge asymptotically to $E_{11}$; that is, both the prey and predator will coexist at constant levels; otherwise, all populations will eventually stabilize at $E(1,0)$. In Figure 3g, the boundary equilibria $E(1,0)$ and $E_{02}$ are stable; that is, if the initial state lies above the two stable manifolds, the prey population will eventually go extinct; otherwise, the predator population will be driven to extinction. (3) Tristability: In Figure 3e, the outer limit cycle, $E_{11}$ and $E(1,0)$ are stable. That is, system (3) exhibits three distinct basins of attraction, which are determined by the initial position: If the initial state is interior to the inner limit cycle, all populations converge asymptotically to $E_{11}$; if it lies in the region bounded by the inner limit cycle and the two stable manifolds, they converge to the outer limit cycle; that is, the prey and predator species will eventually achieve coexistence through periodic oscillations; otherwise, convergence occurs to the boundary equilibrium $E(1,0)$; that is, the prey will persist while the predator eventually becomes extinct. Thus, guided by practical ecological needs, it is essential to continuously monitor populations and prudently select suitable values of c and initial conditions to ensure the system remains outside the basin of attraction that leads to extinction. This is crucial for achieving the long-term sustainability of the ecosystem.

5. Conclusions

We investigate a predator–prey model featuring a generalist predator and nonlinear harvesting. It can be shown that the system has no more than four boundary equilibria and no more than two positive equilibria. $E_{00}(0,0)$ is unstable from Theorem 1; $E_{10}(1,0)$ is unstable under the condition that $h<h_2$, and is otherwise stable in the first quadrant from Theorem 2; $E_{01}$ is unstable if it exists from Theorem 3; $E_{02}$ is unstable when $c\le a$, or $a<c<c_0$ and $h<h_1$, and is otherwise stable in the first quadrant from Theorem 4, if it exists.

It follows from Theorems 7 and 8 that the unique positive equilibrium $E_{\ast }$ is a cusp of codimension up to 3 and the occurrence of a codimension-3 cusp-type degenerate Bogdanov–Takens bifurcation occurs in system (3) around $E_{\ast }$ when $h=h_3$, $c=c_{\ast }$ and $q=q_{\ast }$. This point acts a bifurcation organizing center, which means in its neighborhood, multiple critical bifurcation curves (saddle-node, Hopf, homoclinic) converge. That is, the system is in a highly sensitive state. Ecologically, small changes around $(h,c,q)=(h_3,c_{\ast },q_{\ast })$, such as variations in harvesting intensity, the availability of alternative food sources, or predation efficiency, can lead to dramatic shifts in population dynamics. For example, the system may transition abruptly from stable coexistence to predator extinction, exhibit multiple possible stable states (multistability), or experience the emergence or disappearance of limit cycles, which corresponds to the onset or cessation of periodic population fluctuations (as shown numerically). This has direct implications for resource management, warning against operating near such sensitive thresholds.

According to Theorem 9, a codimension-2 degenerate Hopf bifurcation can be observed in system (3) around $E_{11}$, which can possibly exist when $min\{h_1,h_2\}<h\le h_3$ and is a weak focus of the order of no more than two. This indicates that the system operates near a higher-order critical point and exhibits high sensitivity to parameter variations. Even minor parameter perturbations can induce strong dynamic responses, resulting in distinct stable states. Further, the system governs not just the birth of a limit cycle, but the potential for bistable oscillations (two limit cycles of different stability), which leads to the possibility of alternative stable periodic states, where populations can exhibit different amplitudes of fluctuation depending on initial conditions, a crucial concept for predicting population resilience.

The above conclusions indicate that when $min\{h_1,h_2\}<h\le h_3$, the prey and predator populations may stabilize at a positive equilibrium point or a limit cycle. Otherwise, when h increases further, the system will stabilize at the boundary equilibria $(1,0)$ or $E_{02}$. This means when the harvesting intensity is maintained at an appropriate level, the prey and predator populations can coexist sustainably. Otherwise, if overharvesting occurs, either the prey or the predator will become extinct.