A Mathematical Analysis of Wind-Driven Bottom-Up Interactions in a Plant–Ungulate–Wolf Ecosystem

Abstract

Understanding how abiotic influences affect organisms and their interactions within ecological communities has long been of interest to ecologists. The significance of this comprehension has increased given rapid human-induced climate change. Wind speed variation is one of the less researched effects of climate change, as wind speeds worldwide are generally falling. Although wind is a fairly universal component of the environment, relatively little work has been conducted to integrate our knowledge of how wind affects interactions between different species. In this study, the influence of wind on bottom-up trophic interactions in the plant–ungulate–wolf food chain has been investigated through mathematical modeling. Positivity and boundedness of the model have been proven, and the existence of equilibrium points and their stability have been established. Saddle-node, transcritical, and Hopf bifurcations are discussed. Finally, all theoretical results are confirmed by numerical simulations.

1. Introduction

In ecosystems where the interplay of predator–prey dynamics is pivotal for maintaining balance and biodiversity, the intricate nature of these interactions has a significant impact on mathematical biology [1,2]. To effectively delve into these biological phenomena, mathematical modeling proves invaluable for understanding complex interactions and predicting ecosystem dynamics [3,4]. Malthus introduced an early demographic principle at the end of the eighteenth century, arguing that human population tends to grow faster than food supply, leading to resource limitation and population checks [5]. According to this theory, populations would theoretically expand (or contract) exponentially, although this model does not precisely mirror real-world dynamics. Verhulst proposed the logistic growth model [6], which incorporates intra-species competition in proportion to the current population size, along with a constant per-capita population growth rate. In this framework, the population size adjusts relative to its initial size and ultimately stabilizes around the ecosystem’s carrying capacity, whether it surpasses or falls short of that threshold. Biology and ecology researchers examine the intricate characteristics of natural occurrences, necessitating the use of specialized approaches for description and abstraction. Most studies on predator–prey systems only look at the direct devouring of prey by the predator because predation is easy to observe in nature [7]. One of the two-species food chain building block models is the predator–prey interaction model. Despite extensive research into or development of predator–prey interaction models, numerous ecological and modeling issues remain unresolved. A modification that has been made is the addition of a new prey or predator. To comprehend the environment’s coexistence circumstances, species extinction, the presence of stable periodic solutions, or prey–predator cycle, several researchers have presented mathematical models that represent the interaction between two prey and one predator [8,9].

One of the fundamental interspecific relationships in biology and ecology is the predator–prey relationship. It also serves as the building block for complex food chains, food webs, and biochemical network structures [8,10]. Since the groundbreaking study by Aiello and Freedman [11], a discrete time delay is used to create a single-species growth model with a stage structure made up of immature and mature phases. It is demonstrated that a globally asymptotically stable positive equilibrium exists given appropriate conditions. There are some intriguing findings about the dynamics of predator–prey systems in [12]. The dynamical characteristics of a three-species ratio-dependent food-chain model were examined in [13]. For a three-species food-chain system, Cui et al. [14] studied the stability and bifurcation of periodic solutions. Pei et al [15]’s delay three-species ecosystem with Holling functional response was set up, and the system’s dynamical behaviors were investigated. Mbava et al. examined a food-chain model incorporating disease dynamics in species’ interactions in [16]. The majority of the literature currently available on theoretical investigations of "prey-predator-top-predator" systems focuses on situations without stage structure.

Ecological systems are constantly influenced by wind, which can have a wide range of strong consequences. Understanding how wind affects pollination [17], dispersal [18], and its destructive power on the landscape [19] has been the subject of much research. On the other hand, the direct effects of wind on the relationships between creatures within a community have received less attention. This is especially important, as noted by Pryor et al. [20] and McVicar et al. [21], since wind patterns and speeds have changed significantly in recent decades, and the effects of these changes on ecosystems remain unknown. Although there are places where wind speeds have increased—particularly in maritime environments—this phenomenon is frequently referred to as “global stilling” [22]. Wind speeds have dropped by 5 to 15% over the past 30 years in the contiguous United States [20] and globally [22]. It is generally anticipated that these declines will continue into the 21st century. Although it is not hard to envision that shifting winds could impact how animals interact, there is currently no overarching theory or synthesis explaining how wind influences predator–prey relationships. The survival of species is significantly impacted by global warming. The impact of global warming as a mathematical model has been the subject of very few studies. There has not been any prior research on how global warming affects the carrying capacity of predators and prey. In the literature, many findings about “prey-predator-top-predator”-type food-chain models have been described [23,24].

In this paper, we incorporate the wind flow effect in prey–predator species and examine its impact on the dynamic behavior of the model. Section 2 discusses the model. Section 3 discusses theoretical studies that describe the existence of positive solutions and boundedness. The equilibrium and its stability analysis have been studied in Section 4. Local bifurcation analysis was conducted, in which saddle-node, transcritical, and Hopf bifurcations have been studied, in Section 5. In Section 6, we performed a numerical simulation to validate the analytical results. Finally, the results and discussion are included in Section 7.

2. Model Formulation

The plant dynamics in Equation (1) are based on standard ecological modeling principles. Logistic growth, $rp\left(1-{\displaystyle \frac{p}{k}}\right)$, represents density-dependent limitation of plant biomass. Herbivory is modeled using a Holling type II functional response, ${\displaystyle \frac{a_{1}pu}{1+a_{2}p}}$, to capture saturating consumption due to handling and digestive constraints of ungulates. The wind-effect function, ${\displaystyle \frac{1}{1+w}}$, acts as a bounded, decreasing modifier reducing grazing efficiency under adverse wind conditions. This formulation ensures biological realism, boundedness, and analytical tractability. To construct a $puy$ mathematical model representing the dynamics of plants (p), ungulates (u), and wolves (y), we begin by describing the interactions among these classes while incorporating the effect of wind (w) on the system (Figure 1). The ecological processes include the logistic growth of plants, herbivory by ungulates, and predation by wolves, which are modeled through nonlinear functional responses.

2.1. Plant Population Equation

The plant–ungulate–wolf model represents a frequently employed ecological paradigm, commonly utilized to elucidate the interactions among three trophic levels within an ecosystem: plants (producers), ungulates (herbivores), and wolves (predators). This model commonly originates from the tenets of food web dynamics and the Lotka–Volterra equations, which delineate predator–prey and resource–consumer interactions. The plant population follows the logistic growth model influenced by herbivory:

$$\begin{array}{c}\hfill {\displaystyle \frac{dp\left(t\right)}{dt}}=rp\left(1-{\displaystyle \frac{p}{k}}\right)-{\displaystyle \frac{a_{1}pu}{(1+w)(1+a_{2}p)}}.\end{array}$$

(1)

Here, the first term $rp\left(1-{\displaystyle \frac{p}{k}}\right)$ represents logistic growth of the plant biomass with intrinsic growth rate r and carrying capacity k. The second term ${\displaystyle \frac{a_{1}pu}{(1+w)(1+a_{2}p)}}$ models the consumption of plant resources by ungulates, incorporating a Holling type II functional response.

• $a_1$ is the rate at which ungulates consume plants.
• $a_2$ is a parameter related to the handling time or saturation level.
• The factor $(1+w)$ reflects the wind-modulated difficulty in plant consumption.

The half-saturation constant for ungulates is ${\displaystyle \frac{1}{a_2}}$, and it becomes dependent on wind due to its appearance in the denominator.

2.2. Wind-Effect Function

The effect of wind on plant consumption and predator–prey interactions is modeled through the function $\theta \left(w\right)$, defined as $\theta \left(w\right)=1+w$, which satisfies the following assumptions:

1.

$\theta \left(0\right)=1$: in the absence of wind, the factor becomes unity, restoring the original attack rates.

2.

${\theta }^{\prime }\left(w\right)>0$: increasing wind reduces effective consumption rates by increasing the difficulty of foraging and predation.

2.3. Ungulate Population Equation

The dynamics of the ungulate population are given by:

$$\begin{array}{c}\hfill {\displaystyle \frac{du\left(t\right)}{dt}}={\displaystyle \frac{e_1{a}_{1}pu}{(1+w)(1+a_{2}p)}}-{\displaystyle \frac{(1+w)(b_1+my)uy}{1+c_1(1+w)(b_1+my)u}}-d_{1}u,\end{array}$$

(2)

This equation consists of three key terms:

• The first term represents biomass gain from plant consumption with conversion efficiency $e_1$.
• The second term denotes predation by wolves using a Holling type II response, where:

  –

  $b_1$ is the base predation rate by wolves,

  –

  m denotes cooperation among wolves,

  –

  $c_1$ is a saturation constant.

• The third term $d_{1}u$ accounts for natural mortality in ungulates.

The half-saturation constant here is given by ${\displaystyle \frac{1}{1+c_1{b}_1(1+w)}}$, again showing wind dependence.

2.4. Wolf Population Equation

The wolf population is modeled by:

$$\begin{array}{c}\hfill {\displaystyle \frac{dy\left(t\right)}{dt}}={\displaystyle \frac{e_2(1+w)(b_1+my)uy}{1+c_1(1+w)(b_1+my)u}}-d_{2}y.\end{array}$$

(3)

This equation includes:

• Biomass gain from consuming ungulates, scaled by conversion efficiency $e_2$.
• Natural mortality term $d_{2}y$.

The predation rate includes the cooperative effect of wolves ($my$), the wind factor $(1+w)$, and saturation due to prey availability.

2.5. Complete Mathematical Model

By collecting the equations described above, the complete system becomes:

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \frac{dp\left(t\right)}{dt}=rp\left(1-\frac{p}{k}\right)-\frac{a_{1}pu}{(1+w)(1+a_{2}p)}}\hfill \\ {\displaystyle \frac{du\left(t\right)}{dt}=\frac{e_1{a}_{1}pu}{(1+w)(1+a_{2}p)}-\frac{(1+w)(b_1+my)uy}{1+c_1(1+w)(b_1+my)u}-d_{1}u}\hfill \\ {\displaystyle \frac{dy\left(t\right)}{dt}=\frac{e_2(1+w)(b_1+my)uy}{1+c_1(1+w)(b_1+my)u}-d_{2}y}\hfill \end{array}\right\}.\end{array}$$

(4)

2.6. Initial Conditions

The model is equipped with the following positive initial conditions:

$$\begin{array}{c}\hfill p\left(0\right)=p_0>0,u\left(0\right)=u_0>0,y\left(0\right)=y_0>0.\end{array}$$

(5)

2.7. Parameter Assumptions

All parameters involved in the system are assumed to be nonnegative. Their biological roles and meanings are explained clearly in

Table 1. Description of parameters used in the model with typical units.

Parameter	Biological Interpretation	Units
r	Intrinsic growth rate of the plant population	day−1
k	Carrying capacity of the plant population	individuals (or biomass units)
$a_1$	Rate of plant consumption by ungulates	day−1
$a_2$	Handling time-related saturation constant	dimensionless
$e_1$	Efficiency of converting consumed plants into ungulate biomass	dimensionless
$b_1$	Base rate of ungulate predation by wolves	day−1
m	Strength of cooperative hunting among wolves	dimensionless
$c_1$	Saturation effect in wolf predation	dimensionless
$e_2$	Efficiency of converting consumed ungulates into wolf biomass	dimensionless
$d_1$	Death rate of ungulates	day−1
$d_2$	Death rate of wolves	day−1
w	Wind intensity parameter affecting predation and search efficiency	dimensionless

.

3. Well-Posedness of the Model (4)

3.1. Positivity

Theorem 1.

With initial condition (5), every solution of model (4) exists in $[0,\infty )$ and is still positive $\forall t>0$.

Proof. 

As the density of populations at any time t cannot be negative, we must ensure that the positivity property is met by the model system we have suggested (4). The solution of the system (4) with the starting condition (5) exists and is unique on $[0,\tau ]$ where $0\le \tau \le \infty$. The right-hand side of the system (4) is continuous and locally Lipschitz in the first quadrant $Z=(p,u,y):p,u,y\ge 0$. (4) and (5) give us

$$\begin{array}{cc}\hfill p\left(t\right)& =p\left(0\right)e^{\left[{\int }_0^t\{rp\left(q\right)(1-{\displaystyle \frac{p\left(q\right)}{k}})-{\displaystyle \frac{a_{1}p\left(q\right)u\left(q\right)}{(1+w)(1+a_{2}p\left(q\right))}}\}dq\right]}\ge 0.\hfill \end{array}$$

As a result, at any time t, the solutions of the system (4) stay positive in the first quadrant. □

3.2. Boundedness

Theorem 2.

The solutions of the PUY system with nonnegative initial solutions are bounded.

Proof. 

To verify the preceding assertion, we initially select a function $L=p+{\displaystyle \frac{1}{e_1}}u+{\displaystyle \frac{1}{e_1{e}_2}}y$.

So,

$$\begin{array}{c}\hfill {\displaystyle \frac{dL}{dt}}={\displaystyle \frac{dp}{dt}}+{\displaystyle \frac{1}{e_1}}{\displaystyle \frac{du}{dt}}+{\displaystyle \frac{1}{e_1{e}_2}}{\displaystyle \frac{dy}{dt}},\end{array}$$

i.e.,

$$\begin{array}{c}\hfill {\displaystyle \frac{dL}{dt}}=rp\left(1-{\displaystyle \frac{p}{k}}\right)-{\displaystyle \frac{1}{e_1}}{d}_{1}u-{\displaystyle \frac{1}{e_1{e}_2}}{d}_{2}y.\end{array}$$

Using the maximum plant carrying capacity k yields the following:

$$\begin{array}{c}\hfill {\displaystyle \frac{dL}{dt}}+\eta L\le 2rk,\end{array}$$

where $\eta \le min.\{r,{\displaystyle \frac{d_1}{e_1}},{\displaystyle \frac{d_2}{e_1{e}_2}}\}$. Then, the following is obtained by applying Gronwall’s inequality:

$$\begin{array}{c}\hfill 0\le L(p\left(t\right),u\left(t\right),y\left(t\right))\le 2rk\left(1-e^{-\eta t}\right)+L\left(0\right)e^{-\eta t},\end{array}$$

hence,

$$0\le \underset{t\to \infty }{lim}supL\left(t\right)\le 2rk.$$

Thus, $p\left(t\right)$, $u\left(t\right)$, and $y\left(t\right)$ will stay bounded. This means that the population densities of the species involved remain nonnegative and finite at all times. Ecologically, this implies that none of the populations undergo unrealistically large growth due to resource limitations, environmental constraints, and interspecific interactions incorporated in the model. Thus, the system evolves within an ecologically feasible region of the state space. □

4. Existence of Equilibrium Points and Their Stability

The current section is dedicated to examining the equilibrium points that can be derived from our model (4). The system (4) was unable to attain its solutions due to its nonlinearity. However, the equilibrium points and their stability can be used to predict the system’s behavior. In the ecological context, equilibrium points represent possible long-term states of the system. The trivial equilibrium corresponds to the extinction of all populations. Boundary equilibria represent partial extinction or dominance, where one or more species persist while others vanish. The interior equilibrium denotes the coexistence of all populations at positive densities. Stability of an equilibrium implies ecological resilience, meaning that small perturbations in population sizes dissipate over time and the system returns to its original ecological state. In contrast, instability indicates that small disturbances may lead to extinction, dominance by one species, or sustained oscillations in population size.

4.1. Existence of Equilibria

The model (4) admits four equilibrium points, as follows:

1.

$S_1=(0,0,0)$.

2.

$S_2=(k,0,0)$

3.

$S_3=\left({\displaystyle \frac{d_1(1+w)}{e_1{a}_1-d_1{a}_2(1+w)}},{\displaystyle \frac{r{e}_1{a}_1(1+w)(k{e}_1{a}_1-d_1(1+w)(k{a}_2+1))}{k{a}_1(e_1^2{a}_1^2-d_1{a}_2(1+w)(2e_1{a}_1-d_1{a}_2(1+w)))}},0\right)$, for $\overline{p}$ and $\overline{u}$ to be positive, the following condition must be satisfied:

$$\begin{array}{c}\hfill e_1{a}_1>d_1{a}_2(1+w),\end{array}$$

(6)

$$\begin{array}{c}\hfill {\displaystyle \frac{r{e}_1{a}_1(1+w)(k{e}_1{a}_1-d_1(1+w)(k{a}_2+1))}{k{a}_1(e_1^2{a}_1^2-d_1{a}_2(1+w)(2e_1{a}_1-d_1{a}_2(1+w)))}}>0.\end{array}$$

(7)

4.

Coexistence equilibrium point $S_4=(p^{*},u^{*},y^{*})$. One can find it with the following procedures. From the third equation of system (4) we find

$$\begin{array}{c}\hfill u^{*}=f\left(y\right):={\displaystyle \frac{d_2}{(1+w)(e_2-c_1{d}_2)(b_1+m{y}^{*})}},\end{array}$$

(8)

which is positive under the following condition

$$\begin{array}{c}\hfill e_2>c_1{d}_2,\end{array}$$

(9)

and $f\left(0\right):={\displaystyle \frac{d_2}{(1+w)(e_2-c_1{d}_2)b_1}}$. Substituting $u^{*}$ in the first and second equations of model (4), we have $(p^{*},y^{*})$. Substituting $u^{*}$ into the first and second equations yielded the following isoclines:

$$\begin{array}{cc}\hfill {\Delta }_1(p,y)& =r(1-{\displaystyle \frac{p}{k}})-{\displaystyle \frac{a_{1}f\left(y\right)}{(1+w)(1+a_{2}p)}}=0,\hfill \\ \hfill {\Delta }_2(p,y)& ={\displaystyle \frac{e_1{a}_{1}p}{(1+w)(1+a_{2}p)}}-{\displaystyle \frac{(1+w)(b_1+my)y}{1+c_1(1+w)(b_1+my)f\left(y\right)}}-d_1=0.\hfill \end{array}$$

As $y\to 0$, we have

$$\begin{array}{cc}\hfill {\Delta }_1(p,0)& =r(1-{\displaystyle \frac{p}{k}})-{\displaystyle \frac{a_{1}f\left(0\right)}{(1+w)(1+a_{2}p)}}=0,\hfill \\ \hfill {\Delta }_2(p,0)& ={\displaystyle \frac{e_1{a}_{1}p}{(1+w)(1+a_{2}p)}}-d_1=0.\hfill \end{array}$$

Note that from ${\Delta }_1(p,0)$, one can get that $p_{{\Delta }_1}$ is a root of second degree polynomial

$$\begin{array}{c}\hfill {\displaystyle \frac{a_2}{k}}{p}^2+\left({\displaystyle \frac{1}{k}}-a_2\right)p+{\displaystyle \frac{a_{1}f\left(0\right)}{r(1+w)}}-1=0,\end{array}$$

which has a unique root provided that

$$\begin{array}{c}\hfill r(1+w)>a_{1}f\left(0\right),\end{array}$$

(10)

and

$$\begin{array}{c}\hfill {\Delta }_2(p,0)={\displaystyle \frac{d_1(1+w)}{e_1{a}_1-d_1(1+w)a_2}}.\end{array}$$

Now, we assume that

$$\begin{array}{c}\hfill 0<p_{{\Delta }_2}<p_{{\Delta }_1}.\end{array}$$

(11)

Also, we take for granted that the next two requirements are met:

$$\begin{array}{cc}\hfill {\displaystyle \frac{dy}{dp}}& =-{\displaystyle \frac{\partial {\Delta }_1}{\partial p}}/{\displaystyle \frac{\partial {\Delta }_1}{\partial y}}<0,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\displaystyle \frac{dy}{dp}}& =-{\displaystyle \frac{\partial {\Delta }_2}{\partial p}}/{\displaystyle \frac{\partial {\Delta }_2}{\partial y}}>0.\hfill \end{array}$$

(13)

Therefore, the slope of ${\Delta }_1(p,y)$ is negative. The isoclines ${\Delta }_1(p,y)$ and ${\Delta }_2(p,y)$ intersect at a point $(p^{*},b^{*})$ in the positive $py$-plane. As a result, the interior equilibrium point is real and defined as $S_4=(p^{*},u^{*},y^{*})$. The following result is obtained:

Theorem 3.

Model (4) has a unique interior equilibrium point $S_4=(p^{*},u^{*},y^{*})$ provided that the conditions (9)–(13) hold.

4.2. Stability of Equilibria

1.

The Jacobian matrix at $S_1=(0,0,0)$, is given by:

$$J\left(S_1\right)=\left[\begin{array}{ccc}r& 0& 0\\ 0& -d_1& 0\\ 0& 0& -d_2\end{array}\right].$$

(14)

Then, the characteristic equation of matrix (14) is given by:

$$(r-\lambda )(-d_1-\lambda )(-d_2-\lambda )=0.$$

(15)

The eigenvalues of the preceding Equation (15) can be represented as follows:

${\lambda }_{11}=r$, ${\lambda }_{12}=-d_1$, ${\lambda }_{13}=-d_2$. Therefore, $S_1$ is a saddle point.

2.

The Jacobian matrix at $S_2=(k,0,0)$ is given by:

$$J\left(S_2\right)=\left[\begin{array}{ccc}-r& {\displaystyle \frac{a_{1}k}{(1+w)(1+a_{2}k)}}& 0\\ 0& -d_1& 0\\ 0& 0& -d_2\end{array}\right].$$

(16)

Then, the characteristic equation of matrix (16) is given by:

$$(-r-\lambda )(-d_1-\lambda )(-d_2-\lambda )=0.$$

(17)

The eigenvalues of the preceding Equation (17) can be represented as follows:

${\lambda }_{21}=-r$, ${\lambda }_{22}=-d_1$, ${\lambda }_{23}=-d_2$. Thus, the equilibrium point $S_2$ is (LSA) locally asymptotically stable.

3.

The Jacobian matrix for $S_3=(\overline{p},\overline{u},0)$ is presented:

$$J\left(S_3\right)=\left[\begin{array}{ccc}{c}_{11}^{\left[2\right]}& c_{12}^{\left[2\right]}& 0\\ c_{21}^{\left[2\right]}& c_{22}^{\left[2\right]}& 0\\ 0& 0& c_{33}^{\left[2\right]}\end{array}\right],$$

(18)

where

$$\begin{array}{cc}\hfill c_{11}^{\left[2\right]}& ={\displaystyle \frac{r(k-2\overline{p})}{k}}-{\displaystyle \frac{a_1\overline{u}}{(1+w)(1+a_2\overline{p}){)}^2}},\hfill \\ \hfill c_{12}^{\left[2\right]}& =-{\displaystyle \frac{a_1\overline{u}}{(1+w)(1+a_2\overline{p})}},\hfill \\ \hfill c_{21}^{\left[2\right]}& ={\displaystyle \frac{e_1{a}_1\overline{u}}{(1+w){(1+a_2\overline{p})}^2}},\hfill \\ \hfill c_{22}^{\left[2\right]}& ={\displaystyle \frac{e_1{a}_{1}p}{(1+w)(1+a_{2}p)}}-d_1,\hfill \\ \hfill c_{33}^{\left[2\right]}& ={\displaystyle \frac{e_2{b}_1(1+w)(1+c_1{b}_1(1+w)\overline{u})\overline{u}}{1+c_1{b}_1(1+w)\overline{u}}}-d_2.\hfill \end{array}$$

Then, the characteristic equation of $J_{s_3}$ is given by:

$$(c_{33}^{\left[2\right]}-\lambda )[{\lambda }^2-Tr\left(J\left(S_3\right)\right)\lambda +Det\left(J\left(S_3\right)\right)]=0.$$

(19)

The eigenvalues of the preceding equation can be represented as follows:

$$\begin{array}{c}\hfill {\lambda }_{31}=-{\displaystyle \frac{e_2{b}_1(1+w)(1+c_1{b}_1(1+w)\overline{u})\overline{u}}{1+c_1{b}_1(1+w)\overline{u}}}+d_2,\end{array}$$

$$\begin{array}{cc}\hfill Tr\left(J\left(S_3\right)\right)& ={\displaystyle \frac{(1+w){(1+a_2\overline{u})}^2{(1+a_2\overline{p})}^2\left[r(k-2\overline{p})+d_{1}k\right]-a_{1}k\left[\overline{u}{(1+a_2\overline{p})}^2+e_1\overline{p}{(1+a_2\overline{u})}^2\right]}{k(1+w){(1+a_2\overline{u})}^2{(1+a_2\overline{p})}^2}}\hfill \\ \hfill Det\left(J\left(S_3\right)\right)& ={\displaystyle \frac{e_1{a}_1^2{\overline{p}}^2\overline{u}}{{(1+w)}^2{(1+a_2\overline{p})}^3}}+{\displaystyle \frac{\left[r(k-2\overline{p})(1+w){(1+a_2\overline{u})}^2-k{a}_1\overline{u}\right]e_1{a}_1\overline{p}}{k{(1+w)}^2{(1+a_2\overline{u})}^2{(1+a_2\overline{p})}^2}}\hfill \\ & -{\displaystyle \frac{d_1\left[r(k-2\overline{p})(1+w){(1+a_2\overline{u})}^2-k{a}_1\overline{u}\right]}{(1+w){(1+a_2\overline{u})}^2}}.\hfill \end{array}$$

Clearly, $S_3$ demonstrates local asymptotic stability if and only if the following requirements are met:

$$\begin{array}{cc}\hfill {\displaystyle \frac{e_2{b}_1(1+w)(1+c_1{b}_1(1+w)\overline{u})\overline{u}}{1+c_1{b}_1(1+w)\overline{u}}}& >d_2,\hfill \end{array}$$

(20a)

$$\begin{array}{cc}\hfill a_{1}k\overline{u}{(1+a_2\overline{p})}^2+k{e}_1{a}_1\overline{p}{(1+a_2\overline{u})}^2& >(1+w){(1+a_2\overline{u})}^2{(1+a_2\overline{p})}^2\left[r(k-2\overline{u})+d_{1}k\right],\hfill \end{array}$$

(20b)

$$\begin{array}{cc}\hfill Det\left(J\left(S_3\right)\right)& >0.\hfill \end{array}$$

(20c)

4.

The Jacobian matrix at the $S_4=(p^{*},u^{*},y^{*})$ is given:

$$J\left(S_4\right)=\left[\begin{array}{ccc}{c}_{11}^{\left[3\right]}& c_{12}^{\left[3\right]}& 0\\ c_{21}^{\left[3\right]}& c_{22}^{\left[3\right]}& c_{32}^{\left[3\right]}\\ 0& c_{32}^{\left[3\right]}& c_{33}^{\left[3\right]}\end{array}\right],$$

(21)

where

$$\begin{array}{cc}\hfill c_{11}^{\left[3\right]}& ={\displaystyle \frac{r(k-2p^{*})}{k}}-{\displaystyle \frac{a_1{u}^{*}}{(1+w){(1+a_2{p}^{*})}^2}};\hfill \\ \hfill c_{12}^{\left[3\right]}& =-{\displaystyle \frac{a_1{p}^{*}}{(1+w)(1+a_2{p}^{*})}};\hfill \\ \hfill c_{21}^{\left[3\right]}& ={\displaystyle \frac{e_1{a}_1{u}^{*}}{(1+w){(1+a_2{p}^{*})}^2}};\hfill \\ \hfill c_{22}^{\left[3\right]}& ={\displaystyle \frac{e_1{a}_1{p}^{*}}{(1+w){(1+a_2{p}^{*})}^2}}-{\displaystyle \frac{(1+w)(b_1+m{y}^{*})y^{*}}{{(1+c_1(1+w)(b_1+m{y}^{*})u^{*})}^2}}-d_1;\hfill \\ \hfill c_{23}^{\left[3\right]}& ={\displaystyle \frac{(1+w)\left((b_1+2m{y}^{*})(1+c_1)(1+w)b_1+c_1(1+w)m^2{y}^{*2}{u}^{*}\right)u^{*}}{{(1+c_1(1+w)(b_1+m{y}^{*})u^{*})}^2}};\hfill \\ \hfill c_{32}^{\left[3\right]}& ={\displaystyle \frac{(1+w)(b_1+m{y}^{*})e_2{y}^{*}}{{(1+c_1(1+w)(b_1+m{y}^{*})u^{*})}^2}};\hfill \\ \hfill c_{33}^{\left[3\right]}& ={\displaystyle \frac{(1+w)(b_1+2m{y}^{*}+c_1{b}_1(1+w)(b_1+2my+b_1{m}^2{y}^{*2})u^{*})e_2{u}^{*}}{{(1+c_1(1+w)(b_1+m{y}^{*})u^{*})}^2}}.\hfill \end{array}$$

Then, the characteristic equation of $J_{S_4}$ is given by:

$${\lambda }^3+D_1{\lambda }^2+D_2\lambda +D_3=0,$$

(22)

where

$$\begin{array}{cc}\hfill D_1& =-\left(c_{33}^{\left[3\right]}+c_{11}^{\left[3\right]}+c_{22}^{\left[3\right]}\right),\hfill \\ \hfill D_2& =c_{11}^{\left[3\right]}{c}_{33}^{\left[3\right]}+c_{22}^{\left[3\right]}{c}_{33}^{\left[3\right]}+c_{11}^{\left[3\right]}{c}_{22}^{\left[3\right]}-c_{12}^{\left[3\right]}{c}_{21}^{\left[3\right]}-c_{32}^{\left[3\right]}{c}_{23}^{\left[3\right]},\hfill \\ \hfill D_3& =c_{12}^{\left[3\right]}{c}_{21}^{\left[3\right]}{c}_{33}^{\left[3\right]}+c_{32}^{\left[3\right]}{c}_{23}^{\left[3\right]}{c}_{11}^{\left[3\right]}-c_{11}^{\left[3\right]}{c}_{22}^{\left[3\right]}{c}_{33}^{\left[3\right]},\hfill \end{array}$$

$$\begin{array}{cc}\hfill \Delta =D_1{D}_2-D_3=& -\left(c_{33}^{\left[3\right]}+c_{11}^{\left[3\right]}+c_{22}^{\left[3\right]}\right)\left(c_{11}^{\left[3\right]}{c}_{33}^{\left[3\right]}+c_{22}^{\left[3\right]}{c}_{33}^{\left[3\right]}+c_{11}^{\left[3\right]}{c}_{22}^{\left[3\right]}-c_{12}^{\left[3\right]}{c}_{21}^{\left[3\right]}-c_{32}^{\left[3\right]}{c}_{23}^{\left[3\right]}\right)\hfill \\ & -c_{12}^{\left[3\right]}{c}_{21}^{\left[3\right]}{c}_{33}^{\left[3\right]}-c_{11}^{\left[3\right]}\left(c_{32}^{\left[3\right]}{c}_{23}^{\left[3\right]}-c_{22}^{\left[3\right]}{c}_{33}^{\left[3\right]}\right).\hfill \end{array}$$

Now, according to the Routh–Hurwitz criteria, $S_4$ is a LAS point, provided that

$$\begin{array}{c}\hfill D_1>0,D_3>0,\Delta >0.\end{array}$$

(23)

5. Local Bifurcation

In bifurcation theory, small changes in system parameters can cause major changes in behavior. In ecological models, this means that slight variations in biological factors can shift population dynamics, for example, from stable coexistence to oscillations or extinction. In mathematics, it is most frequently employed to investigate systems that undergo variations over time. Local bifurcations occur when parameters exceed critical thresholds, resulting in modifications to the local stability of equilibria [25]. This section evaluates the likelihood of local bifurcation. To achieve this, we revise the PUY model as follows:

$${\displaystyle \frac{dS}{dt}}=G\left(S\right),withS=\left(\begin{array}{c}p\\ u\\ y\end{array}\right),andG=\left(\begin{array}{c}{g}_1(p,u,y)\\ g_2(p,u,y)\\ g_3(p,u,y).\end{array}\right)$$

For any non-zero vector $q={(q_1,q_2,q_3)}^T$, we set

$$D^{2}G(n,n)=\left[\begin{array}{c}{b}_{11}^{*}\\ b_{21}^{*}\\ b_{31}^{*}\end{array}\right],$$

(24)

where

$$\begin{array}{cc}\hfill b_{11}^{*}& =\left[-{\displaystyle \frac{-2r}{k}}+{\displaystyle \frac{2a_1{a}_{2}u}{(1+w){(1+a_{2}p)}^3}}\right]q_1^2-{\displaystyle \frac{a_1(1-a_2^2{p}^2)q_1{q}_2}{(1+w){(1+a_{2}p)}^4}}-{\displaystyle \frac{a_1{q}_1{q}_2}{(1+w){(1+a_{2}p)}^2}},\hfill \\ \hfill b_{21}^{*}& ={\displaystyle \frac{2a_2{e}_1{a}_{1}u{q}_1{q}_2}{(1+w){(1+a_{2}p)}^3}}+{\displaystyle \frac{e_1{a}_1{q}_2^2}{(1+w){(1+a_{2}p)}^2}}+{\displaystyle \frac{e_1{a}_1{q}_1{q}_2}{(1+w){(1+a_{2}p)}^2}}+{\displaystyle \frac{2c_1(1+w){\overline{x}}_1^{2}y{q}_2^2}{{(1+{\overline{x}}_1{\overline{x}}_2)}^3}}\hfill \\ \hfill & -{\displaystyle \frac{(1+w)((b_1+2my)(1-{\overline{x}}_2{\overline{x}}_1^2)+2{\overline{x}}_2{\overline{x}}_1^2(1+{\overline{x}}_1{\overline{x}}_2))q_3{q}_2}{{(1+{\overline{x}}_1{\overline{x}}_2)}^4}}\hfill \\ \hfill & -{\displaystyle \frac{(1+w)(b_1(1+b_1{\overline{x}}_2(2+b_1{\overline{x}}_2(b_1-m^2{y}^2))+2(1+{\overline{x}}_2{\overline{x}}_3+b_1{\overline{x}}_2^2{\overline{x}}_1)my)q_2}{{(1+{\overline{x}}_1{\overline{x}}_2)}^4}}\hfill \\ & -{\displaystyle \frac{2(1+w)mu((2+b_{1}m{\overline{x}}_2){\overline{x}}_8+{\overline{x}}_2(my(1+{\overline{x}}_2{\overline{x}}_3+b_1{\overline{x}}_2^2{\overline{x}}_1-b_1{\overline{x}}_6(1+{\overline{x}}_2{\overline{x}}_1))q_3{q}_2}{{(1+{\overline{x}}_1{\overline{x}}_2)}^4}},\hfill \\ \hfill b_{31}^{*}& ={\displaystyle \frac{c_1(1+w)((1+w)\left((2\left({\overline{x}}_1(2+my)\right)+(b_1^2+my(3(b_1+m)+m^2{y}^2))u)b_1{e}_{2}y\right)q_2{q}_3}{{(1+{\overline{x}}_1{\overline{x}}_2)}^4}}\hfill \\ \hfill & +{\displaystyle \frac{e_2(1+w)((b_1+2my)(1-(b_1^2+my{\overline{x}}_3{\overline{x}}_2^2)+2{\overline{x}}_2(b_1^2+my{\overline{x}}_3)(1+b_1{\overline{x}}_1{\overline{x}}_2))q_3^2}{{(1+{\overline{x}}_1{\overline{x}}_2)}^4}}\hfill \\ \hfill & +{\displaystyle \frac{e_2(1+w)(b_1{\overline{x}}_8+2my(1+{\overline{x}}_2({\overline{x}}_3+b_1{\overline{x}}_1{\overline{x}}_2))q_2{q}_3}{{(1+{\overline{x}}_1{\overline{x}}_2)}^4}}\hfill \\ \hfill & +{\displaystyle \frac{2e_{2}mu(1+w){\overline{x}}_6({\overline{x}}_8+{\overline{x}}_2{m}^{2}y(1+\left({\overline{x}}_2({\overline{x}}_3+{\overline{x}}_1{\overline{x}}_2{b}_1)\right))-(1+w)b_{1}u{\overline{x}}_7(c_{1}m{\overline{x}}_9{q}_3^2}{{(1+{\overline{x}}_1{\overline{x}}_2)}^4}},\hfill \end{array}$$

where

${\overline{x}}_1=b_1+my$,

${\overline{x}}_2=c_1(1+w)u$,

${\overline{x}}_3=2b_1+my$,

${\overline{x}}_4=b_1^2-m^2{y}^2$,

${\overline{x}}_5=2b_1+{\overline{x}}_2{\overline{x}}_4$,

${\overline{x}}_6=1+b_1{\overline{x}}_2$,

${\overline{x}}_7=1+e_2{b}_1{\overline{x}}_2$,

${\overline{x}}_8=1+{\overline{x}}_2{\overline{x}}_5$,

${\overline{x}}_9=1+m{\overline{x}}_1{\overline{x}}_2$.

Theorem 4.

For $a_2=a_2^{*}$, the system (4) at the equilibrium point $S_3$ has

(1) 

saddle-node bifurcation if

$${\left(K^{\left[2\right]}\right)}^T{G}_{a_2}(S_3,a_2^{*})\ne 0,$$

(25)

$${\left(Q^{\left[2\right]}\right)}^T\left[D^2{G}_{a_2}(S_3,a_2^{*})(K^{\left[2\right]},K^{\left[2\right]})\right]\ne 0,$$

(26)

(2) 

a transcritical bifurcation if condition (25) is violated and condition (26) is satisfied along with

$${\left(Q^{\left[2\right]}\right)}^{T}D{G}_{a_2}(S_2,a_2^{*})K^{\left[2\right]}\ne 0.$$

(27)

Proof. 

At $a_2=a_2^{*}$, where the critical value of $a_2^{*}$ could be evaluated from the following equation:

$$\begin{array}{c}\hfill Det\left(J\left(s_3\right)\right){|}_{a_2=a_2^{*}}=0,\end{array}$$

$J\left(S_3\right)$ has a zero eigenvalue, say ${\lambda }_{31}=0$. Therefore, $J\left(S_3\right)$ at $a_2^{*}$ becomes

$$J\left(S_3\right)=\left[\begin{array}{ccc}{c}_{11}^{*\left[2\right]}& c_{12}^{*\left[2\right]}& 0\\ c_{21}^{*\left[2\right]}& c_{22}^{*\left[2\right]}& 0\\ 0& 0& c_{33}^{*\left[2\right]}\end{array}\right],$$

(28)

where

$$\begin{array}{cc}\hfill c_{11}^{*\left[2\right]}& ={\displaystyle \frac{r(k-2\overline{p})}{k}}-{\displaystyle \frac{a_1\overline{u}}{(1+w)(1+a_2^{*}\overline{p}){)}^2}};\hfill \\ \hfill c_{12}^{*\left[2\right]}& =-{\displaystyle \frac{a_1\overline{u}}{(1+w)(1+a_2^{*}\overline{p})}};\hfill \\ \hfill c_{21}^{*\left[2\right]}& ={\displaystyle \frac{e_1{a}_1\overline{u}}{(1+w){(1+a_2^{*}\overline{p})}^2}};\hfill \\ \hfill c_{22}^{*\left[2\right]}& ={\displaystyle \frac{e_1{a}_{1}p}{(1+w){(1+a_2^{*}p)}^2}}-d_1;\hfill \\ \hfill c_{33}^{*\left[2\right]}& ={\displaystyle \frac{e_2{b}_1(1+w)(1+c_1{b}_1(1+w)\overline{u})\overline{u}}{1+c_1{b}_1(1+w)\overline{u}}}-d_2.\hfill \end{array}$$

Now, let $K^{\left[2\right]}={\left(k_1^{\left[2\right]},k_2^{\left[2\right]},k_3^{\left[2\right]}\right)}^T$ be an eigenvector corresponding to ${\lambda }_{31}=0$. Thus, $\left(J^{*}\left(S_3\right)-{\lambda }_{31}I\right)k^{\left[2\right]}=0$, which gives $K^{\left[2\right]}={\left({\displaystyle \frac{e_1{a}_1{\overline{p}}^{*}-d_1(1+w){(1+a_2^{*}{\overline{p}}^{*})}^2}{e_1{a}_1{\overline{p}}^{*}{\overline{u}}^{*}}},1,1\right)}^T$.

Let $Q^{\left[2\right]}={\left(q_1^{\left[2\right]},q_2^{\left[2\right]},q_3^{\left[2\right]}\right)}^T$ be an eigenvector corresponding to ${\lambda }_{31}=0$. Thus, $\left(J^{*T}\left(S_3\right)-{\lambda }_{31}I\right)q^{\left[2\right]}=0$, which gives $Q^{\left[2\right]}={\left(-{\displaystyle \frac{e_1{a}_1{\overline{p}}^{*}-d_1(1+w)(1+a_2^{*}{\overline{p}}^{*})}{a{\overline{p}}^{*}(1+a_2^{*}{\overline{p}}^{*})}},1,1\right)}^T$.

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial G}{\partial a_2}}=G_{a_2}(S,a_2)={\left({\displaystyle \frac{\partial g_1}{\partial a_2}},{\displaystyle \frac{\partial g_2}{\partial a_2}},{\displaystyle \frac{\partial g_3}{\partial a_2}}\right)}^T={\left({\displaystyle \frac{a_1(1+w)u{p}^2}{{\left(a_2(1+w)p\right)}^2}},-{\displaystyle \frac{e_1{a}_1(1+w)u{p}^2}{{\left(a_2(1+w)p\right)}^2}},0\right)}^T.\end{array}$$

So,

$$\begin{array}{c}\hfill G_{a_2}(S_3,a_2^{*})={\left({\displaystyle \frac{a_1(1+w)\overline{u}{\overline{p}}^2}{{\left(a_2^{*}(1+w)\overline{p}\right)}^2}},-{\displaystyle \frac{e_1{a}_1(1+w)\overline{u}{\overline{p}}^2}{{\left(a_2^{*}(1+w)\overline{p}\right)}^2}},0\right)}^T,\end{array}$$

under condition (25). Therefore, the first condition of saddle-node bifurcation is met. Subsequently,

$$\begin{array}{c}\hfill {\left(K^{\left[2\right]}\right)}^T{G}_{a_2}(S_3,a_2^{*})={\displaystyle \frac{a_1(1+w)\overline{u}{\overline{p}}^2\left[e_1{a}_1{\overline{p}}^{*}-d_1(1+w){(1+a_2^{*}{\overline{p}}^{*})}^2\right]}{e_1{a}_1{\overline{p}}^{*}{\overline{u}}^{*}{\left(a_2^{*}(1+w)\overline{p}\right)}^2}}-{\displaystyle \frac{e_1{a}_1(1+w)u{p}^2}{{\left(a_2(1+w)p\right)}^2}}.\end{array}$$

Next, suppose that condition (26) applies, i.e.,

$$\begin{array}{c}\hfill {\left(Q^{\left[2\right]}\right)}^T\left[D^2{G}_{a_2}(S_3,a_2^{*})(K^{\left[2\right]},K^{\left[2\right]})\right]\ne 0.\end{array}$$

Therefore, condition (25) ensures that the second requirement of saddle-node bifurcation is met. The PUY model exhibits saddle-node bifurcation with parameter $a_2^{*}$.

On the other side, if condition (25) is not satisfied, then the following is obtained.

As a result, the first requirement for transcritical bifurcation is met, whereas saddle-node bifurcation cannot occur. Subsequently,

$$\begin{array}{c}\hfill DG(S_3,a_2^{*})=\left[\begin{array}{ccc}0& {\displaystyle \frac{(1+w)a_1{\overline{p}}^2}{{\left((1+w)a_2^{*}\overline{p}\right)}^2}}& 0\\ 0& -{\displaystyle \frac{e_1(1+w)a_1{\overline{p}}^2}{{\left((1+w)a_2^{*}{\overline{p}}^2\right)}^2}}& 0\\ 0& 0& 0\end{array}\right],\end{array}$$

where $D{S}_{a_2}(S,a_2^{*})$ represents the derivative of $S_{a_2}(S,a_2)$ with respect to $S={(p,u,y)}^T$.

Furthermore,

$$\begin{array}{c}\hfill DG(S_3,a_2^{*})K^{\left[2\right]}=\left[\begin{array}{ccc}0& {\displaystyle \frac{(1+w)a_1{\overline{p}}^2}{{\left((1+w)a_2^{*}\overline{p}\right)}^2}}& 0\\ 0& -{\displaystyle \frac{e_1(1+w)a_1{\overline{p}}^2}{{\left((1+w)a_2^{*}{\overline{p}}^2\right)}^2}}& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{k}_1^{\left[2\right]}\\ k_2^{\left[2\right]}\\ k_3^{\left[2\right]}\end{array}\right].\end{array}$$

$$\begin{array}{c}\hfill {\left(Q^{\left[2\right]}\right)}^{T}D{G}_{a_2}(S_3,a_2^{*})K^{\left[2\right]}={\displaystyle \frac{(1+w)a_1{\overline{p}}^2}{{\left((1+w)a_2^{*}\overline{p}\right)}^2}}-{\displaystyle \frac{e_1(1+w)a_1{\overline{p}}^2}{{\left((1+w)a_2^{*}{\overline{p}}^2\right)}^2}}.\end{array}$$

That means the second condition occurs if condition (27) is satisfied. Then, the system (4) presents a transcritical bifurcation near $S_3$ at $a_2$. □

Theorem 5.

For $w=w^{*}$, where $w^{*}$ is given the proof of this theorem, the system (4) at the equilibrium point $S_4$ has

(1) a saddle-node bifurcation if

$${\left(Q^{\left[3\right]}\right)}^T{G}_w(S_4,w^{*})\ne 0,$$

(29)

$${\left(Q^{\left[3\right]}\right)}^T\left[D^2{G}_w(S_4,w^{*})(K^{\left[3\right]},K^{\left[3\right]})\right]\ne 0;$$

(30)

(2) a transcritical bifurcation if condition (30) is violated and condition (29) is satisfied along with

$${\left(Q^{\left[3\right]}\right)}^T\left[D{G}_w(S_4,w^{*})(K^{\left[3\right]},K^{\left[3\right]})\right]\ne 0.$$

(31)

Proof. 

From the Jacobian matrix $J\left(S_4\right)$ given by (21) of the model (4), $S_4$ has a zero eigenvalue, say ${\lambda }_{41}=0$, at

$$\begin{array}{c}\hfill w^{*}={\displaystyle \frac{a_1{a}_{21}{a}_{33}{p}^{*}-(1+a_2{p}^{*})(a_{33}+a_{11}+a_{22})(a_{11}(a_{33}+a_{22})+a_{22}{a}_{33}-a_{12}{a}_{21}-a_{32}{a}_{23})+a_{11}(a_{32}{a}_{23}-a_{22}{a}_{33})}{(1+a_2{p}^{*})(a_{33}+a_{11}+a_{22})(a_{11}(a_{33}+a_{22})+a_{22}{a}_{33}-a_{12}{a}_{21}-a_{32}{a}_{23})+a_{11}(a_{32}{a}_{23}-a_{22}{a}_{33})}},\end{array}$$

where

$$\begin{array}{c}\hfill (a_{33}+a_{11}+a_{22})(a_{11}(a_{33}+a_{22})+a_{22}{a}_{33}-a_{12}{a}_{21}-a_{32}{a}_{23})+a_{11}(a_{32}{a}_{23}-a_{22}{a}_{33})\ne 0,\end{array}$$

and the Jacobian matrix $J^{*}\left(S_4\right)=J^{*}(S_4,w^{*})$, becomes:

$$J^{*}\left(S_4\right)=\left[\begin{array}{ccc}{c}_{11}^{[*3]}& c_{12}^{[*3]}& 0\\ c_{21}^{[*3]}& c_{22}^{[*3]}& c_{23}^{[*3]}\\ 0& c_{32}^{[*3]}& c_{33}^{[*3]}\end{array}\right],$$

(32)

where

$$\begin{array}{cc}\hfill c_{11}^{[\ast 3]}& ={\displaystyle \frac{r(k-2p^{*})}{k}}-{\displaystyle \frac{a_1{u}^{*}}{(1+w^{*}){(1+a_2{p}^{*})}^2}};\hfill \\ \hfill c_{12}^{[\ast 3]}& =-{\displaystyle \frac{a_1{p}^{*}}{(1+w^{*})(1+a_2{p}^{*})}};\hfill \\ \hfill c_{21}^{[\ast 3]}& ={\displaystyle \frac{e_1{a}_1{u}^{*}}{(1+w^{*}){(1+a_2{p}^{*})}^2}};\hfill \\ \hfill c_{22}^{[\ast 3]}& ={\displaystyle \frac{e_1{a}_1{p}^{*}}{(1+w^{*}){(1+a_2{p}^{*})}^2}}-{\displaystyle \frac{(1+w^{*})(b_1+m{y}^{*})y^{*}}{{(1+c_1(1+w^{*})(b_1+m{y}^{*})u^{*})}^2}}-d_1;\hfill \\ \hfill c_{23}^{[\ast 3]}& ={\displaystyle \frac{(1+w^{*})\left((b_1+2m{y}^{*})(1+c_1)(1+w^{*})b_1+c_1(1+w^{*})m^2{y}^{*2}{u}^{*}\right)u^{*}}{{(1+c_1(1+w^{*})(b_1+m{y}^{*})u^{*})}^2}};\hfill \\ \hfill c_{32}^{[\ast 3]}& ={\displaystyle \frac{(1+w^{*})(b_1+m{y}^{*})e_2{y}^{*}}{{(1+c_1(1+w^{*})(b_1+m{y}^{*})u^{*})}^2}};\hfill \\ \hfill c_{33}^{[\ast 3]}& ={\displaystyle \frac{(1+w^{*})(b_1+2m{y}^{*}+c_1{b}_1(1+w^{*})(b_1+2my+b_1{m}^2{y}^{*2})u^{*})e_2{u}^{*}}{{(1+c_1(1+w^{*})(b_1+m{y}^{*})u^{*})}^2}}.\hfill \end{array}$$

Let $K^{\left[3\right]}={\left(k_1^{\left[3\right]},k_2^{\left[3\right]},k_3^{\left[3\right]}\right)}^T$ be an eigenvector corresponding to ${\lambda }_{41}=0$. Thus, $\left(J^{*}\left(S_4\right)-{\lambda }_{41}I\right)k^{\left[3\right]}=0$, which gives $K^{\left[3\right]}={\left({\displaystyle \frac{-c_{12}^{[*3]}}{c_{11}^{[*3]}}},1,{\displaystyle \frac{-c_{32}^{[*3]}}{c_{33}^{[*3]}}}\right)}^T$.

Let $Q^{\left[3\right]}={\left(q_1^{\left[3\right]},q_2^{\left[3\right]},q_3^{\left[3\right]}\right)}^T$ be an eigenvector corresponding to ${\lambda }_{41}=0$. Thus, $\left(J^{*T}\left(S_4\right)-{\lambda }_{41}I\right)q^{\left[3\right]}=0$, which gives $Q^{\left[3\right]}={\left({\displaystyle \frac{-c_{21}^{[*3]}}{c_{11}^{[*3]}}},1,{\displaystyle \frac{-c_{23}^{[*3]}}{c_{33}^{[*3]}}}\right)}^T$.

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial G}{\partial w}}=G_w(S,w)={\left({\displaystyle \frac{\partial g_1}{\partial w}},{\displaystyle \frac{\partial g_2}{\partial w}},{\displaystyle \frac{\partial g_3}{\partial w}}\right)}^T={\left({\displaystyle \frac{a_1(1+a_{2}p)pu}{{\left((1+w)(1+a_{2}p)\right)}^2}},-{\displaystyle \frac{e_1{a}_1(1+a_{2}p)pu}{{\left((1+w)(1+a_{2}p)\right)}^2}}+{\displaystyle \frac{-(b_1+my)uy}{{(1+c_1(1+w)(b_1+my)u)}^2}},{\displaystyle \frac{-e_2(b_1+my)uy}{{(1+c_1(1+w)(b_1+my)u)}^2}}\right)}^T.\end{array}$$

So,

$$\begin{array}{c}\hfill G_w(S_3,w)={\left({\displaystyle \frac{a_1(1+a_2{p}^{*})p^{*}{u}^{*}}{{\left((1+w^{*})(1+a_2{p}^{*})\right)}^2}},-{\displaystyle \frac{e_1{a}_1(1+a_2{p}^{*})p^{*}{u}^{*}}{{\left((1+w^{*})(1+a_2{p}^{*})\right)}^2}}+{\displaystyle \frac{-(b_1+m{y}^{*})u^{*}{y}^{*}}{{(1+c_1(1+w^{*})(b_1+m{y}^{*})u^{*})}^2}},{\displaystyle \frac{-e_2(b_1+m{y}^{*})u^{*}{y}^{*}}{{(1+c_1(1+w^{*})(b_1+m{y}^{*})u^{*})}^2}}\right)}^T,\end{array}$$

and hence

$$\begin{array}{cc}\hfill {\left(Q^{\left[3\right]}\right)}^T{G}_w(S_3,w^{*})& ={\displaystyle \frac{-a_1(1+a_2{p}^{*})p^{*}{u}^{*}}{{\left((1+w^{*})(1+a_2{p}^{*})\right)}^2}}{\displaystyle \frac{c_{21}^{[*3]}{q}_2^{\left[3\right]}}{c_{11}^{[*3]}}}-{\displaystyle \frac{e_1{a}_1(1+a_2{p}^{*})p^{*}{u}^{*}}{{\left((1+w^{*})(1+a_2{p}^{*})\right)}^2}}+\hfill \\ & {\displaystyle \frac{-(b_1+m{y}^{*})u^{*}{y}^{*}}{{(1+c_1(1+w^{*})(b_1+m{y}^{*})u^{*})}^2}}{q}_2^{\left[3\right]}+{\displaystyle \frac{e_2(b_1+m{y}^{*})u^{*}{y}^{*}}{{(1+c_1(1+w^{*})(b_1+m{y}^{*})u^{*})}^2}}{\displaystyle \frac{c_{23}^{[*3]}{q}_2^{\left[3\right]}}{c_{33}^{[*3]}}},\hfill \end{array}$$

Under condition (29). Therefore, the first condition of saddle-node bifurcation is met. Next, suppose that condition (30) satisfied, i.e.,

$$\begin{array}{c}\hfill {\left(Q^{\left[3\right]}\right)}^T\left[D^2{G}_w(S_3,w^{*})(K^{\left[3\right]},K^{\left[3\right]})\right]\ne 0.\end{array}$$

Therefore, condition (29) ensures that the second requirement of saddle-node bifurcation is met. Thus, the model (4) exhibits saddle-node bifurcation with the parameter $w^{*}$.

On the other side, if condition (29) is not satisfied, then the following is obtained.

$$\begin{array}{c}\hfill {\left(Q^{\left[3\right]}\right)}^T{G}_w(S_3,w^{*})=0.\end{array}$$

So, according to Sotomayor’s theorem, the first condition of transcritical bifurcation is satisfied. Now,

$$\begin{array}{c}\hfill D{G}_w(S_3,w^{*})=\left[\begin{array}{ccc}{t}_{11}^{[\ast 3]}& t_{12}^{[\ast 3]}& t_{13}^{[\ast 3]}\\ t_{21}^{[\ast 3]}& t_{22}^{[\ast 3]}& t_{23}^{[\ast 3]}\\ t_{31}^{[\ast 3]}& t_{32}^{[\ast 3]}& t_{33}^{[\ast 3]}\end{array}\right],\end{array}$$

where

$$\begin{array}{cc}\hfill t_{11}^{[\ast 3]}& ={\displaystyle \frac{a_1{u}^{*}(1+a_2{p}^{*}{u}^{*}(2+a_2))}{{(1+w^{*})}^2{(1+a_2{p}^{*})}^4}},\hfill \\ \hfill t_{12}^{[\ast 3]}& ={\displaystyle \frac{a_1(1+a_2{p}^{*})p^{*}}{{\left((1+w^{*})(1+a_2{p}^{*})\right)}^2}},\hfill \\ \hfill t_{13}^{[\ast 3]}& =0.\hfill \\ \hfill t_{21}^{[\ast 3]}& =-{\displaystyle \frac{e_1{a}_1{u}^{*}(1+a_2{p}^{*}{u}^{*}(2+a_2))}{{(1+w^{*})}^2{(1+a_{2}p)}^4}},\hfill \\ \hfill t_{22}^{[\ast 3]}& =-{\displaystyle \frac{e_1{a}_1(1+a_2{p}^{*})p^{*}}{{\left((1+w^{*})(1+a_2{p}^{*})\right)}^2}}-{\displaystyle \frac{(b_1+m{y}^{*})y^{*}(1-{\left[m_1(b_1+m{y}^{*})\right]}^2{u}^{*2})}{(1+m_1(b_1+m{y}^{*}))u^{*}}},\hfill \\ \hfill t_{23}^{[\ast 3]}& ={\displaystyle \frac{b_1{u}^{*}(1+m_1(b_1{u}^{*}(2+m_1{b}_1{u}^{*})+m{u}^{*}{y}^{*}(1+m_{1}m{u}^{*}{y}^{*})))+m_2+m{y}^{*}\left)\right)}{{(1+m_1(b_1+m{y}^{*})u^{*})}^4}},\hfill \\ \hfill t_{31}^{[\ast 3]}& =0,\hfill \\ \hfill t_{32}^{[\ast 3]}& =-{\displaystyle \frac{e_2(b_1+m{y}^{*})y^{*}(1-{\left[m_1(b_1+m{y}^{*})\right]}^2{u}^{*2})}{(1+m_1)(b_1+m{y}^{*}))u^{*}}},\hfill \\ \hfill t_{33}^{[\ast 3]}& ={\displaystyle \frac{e_2{b}_1{u}^{*}(1+m_1(b_1{u}^{*}(2+m_1{b}_1{u}^{*})+m{u}^{*}{y}^{*}(1+m_{1}m{u}^{*}{y}^{*})))+m_2+m{y}^{*}\left)\right)}{{(1+m_1(b_1+m{y}^{*})u^{*})}^4}},\hfill \end{array}$$

where $m_1=c_1(1+w^{*})$, $m_2=2m{u}^{*}{y}^{*}(1+2m_1{u}^{*}(b_1(1+b_1{m}_1{u}^{*})$, and $D{S}_w(S,w^{*})$ represent the derivatives of $G_w(S,w^{*})$ with respect to $S={(p,u,y)}^T$. Furthermore,

$$\begin{array}{c}\hfill D{G}_w(S_3,w^{*})k^{\left[3\right]}=\left[\begin{array}{ccc}{t}_{11}^{[\ast 3]}& t_{12}^{[\ast 3]}& t_{13}^{[\ast 3]}\\ t_{21}^{[\ast 3]}& t_{22}^{[\ast 3]}& t_{23}^{[\ast 3]}\\ t_{31}^{[\ast 3]}& t_{32}^{[\ast 3]}& t_{33}^{[\ast 3]}\end{array}\right]\left[\begin{array}{c}{k}_1^{\left[3\right]}\\ k_2^{\left[3\right]}\\ k_3^{\left[3\right]}\end{array}\right].\end{array}$$

$$\begin{array}{cc}\hfill & {\left(Q^{\left[3\right]}\right)}^{T}D{G}_w(S_3,w^{*})(K^{\left[3\right]},K^{\left[3\right]})=-{\displaystyle \frac{a_1{u}^{*}(1+a_2{p}^{*}{u}^{*}(2+a_2))c_{12}^{*2}{q}_1}{{(1+w^{*})}^2{(1+a_2{p}^{*})}^4}}+{\displaystyle \frac{a_1(1+a_2{p}^{*})p^{*}{q}_2}{{\left((1+w^{*})(1+a_2{p}^{*})\right)}^2}}+\hfill \\ & {\displaystyle \frac{e_1{a}_1{u}^{*}(1+a_2{p}^{*}{u}^{*}(2+a_2))c_{12}^{*2}{q}_1}{{(1+w^{*})}^2{(1+a_{2}p)}^4}}-{\displaystyle \frac{e_1{a}_1(1+a_2{p}^{*})p^{*}}{{\left((1+w^{*})(1+a_2{p}^{*})\right)}^2}}-{\displaystyle \frac{(b_1+m{y}^{*})y^{*}(1-{\left[m_1(b_1+m{y}^{*})\right]}^2{u}^{*2})}{(1+m_1(b_1+m{y}^{*}))u^{*}}}\hfill \\ & -{\displaystyle \frac{b_1{u}^{*}(1+c_1(1+w^{*})(b_1{u}^{*}(2+c_1(1+w^{*})b_1{u}^{*})+m{u}^{*}{y}^{*}(1+c_1(1+w^{*})m{u}^{*}{y}^{*})))}{{(1+c_1(1+w^{*})(b_1+m{y}^{*})u^{*})}^4{c}_{33}^{*2}}}\hfill \\ & +{\displaystyle \frac{2m{u}^{*}{y}^{*}(1+2c_1(1+w^{*})u^{*}(b_1(1+b_1{c}_1(1+w^{*})u^{*})+m{y}^{*}))c_{32}^{*2}{q}_3}{{(1+c_1(1+w^{*})(b_1+m{y}^{*})u^{*})}^4{c}_{33}^{*2}}}-\hfill \\ & {\displaystyle \frac{e_2(b_1+m{y}^{*})y^{*}(1-{\left[c_1(1+w^{*})(b_1+m{y}^{*})\right]}^2{u}^{*2})q_2}{(1+c_1(1+w^{*})(b_1+m{y}^{*}))u^{*}}}\hfill \\ & -{\displaystyle \frac{e_2{b}_1{u}^{*}(1+c_1(1+w^{*})(b_1{u}^{*}(2+c_1(1+w^{*})b_1{u}^{*})+m{u}^{*}{y}^{*}(1+c_1(1+w^{*})m{u}^{*}{y}^{*})))}{{(1+c_1(1+w^{*})(b_1+m{y}^{*})u^{*})}^4{c}_{33}^{*2}}}\hfill \\ & +{\displaystyle \frac{2m{u}^{*}{y}^{*}(1+2c_1(1+w^{*})u^{*}(b_1(1+b_1{c}_1(1+w^{*})u^{*})+m{y}^{*}))c_{32}^{*2}{q}_3}{{(1+c_1(1+w^{*})(b_1+m{y}^{*})u^{*})}^4{c}_{33}^{*2}}}.\hfill \end{array}$$

Hence, condition (31) guarantees that the third condition of transcritical bifurcation is satisfied. Therefore, the conditions for transcritical bifurcation hold. □

Hopf Bifurcation

Theorem 6.

Assume that condition (20c) is satisfied along with the following condition

$$k(r+d_1)>2r\overline{p}.$$

(33)

The theorem’s proof defines $\overline{a_1}$. The PUY model shows a Hopf bifurcation at $a_1=\overline{a_1}$.

Proof. 

From Equation (19) we have;

$${\lambda }^2-Tr\left(J\left(S_3\right)\right)+DetJ\left(S_3\right)=0.$$

(34)

The criteria for the Hopf bifurcation are listed below.

(a)

$\left[TrJ\left(S_3\right)\right]{|}_{a_1=\overline{a_1}}=0$,

(b)

$\left[DetJ\left(S_3\right)\right]{|}_{a_1=\overline{a_1}}>0$,

(c)

${\displaystyle \frac{d}{d{a}_1}}\left[Re\left({\lambda }_{1,2}\right)\right]{|}_{a_1=\overline{a_1}}\ne 0$.

Condition (a) has been satisfied at $\overline{a_1}={\displaystyle \frac{(1+w){(1+a_2\overline{u})}^2{(1+a_2\overline{p})}^2\left[r(k-2\overline{p})+d_{1}k\right]}{k\left[\overline{u}{(1+a_2\overline{p})}^2+e_1\overline{p}{(1+a_2\overline{u})}^2\right]}}$.

Clearly, $\overline{a_1}>0$ if condition (33) holds. At $a_1=\overline{a_1}$, the characteristic equation given by (34) is rewritten as

$${\lambda }^2+DetJ\left(S_3\right)=0,$$

(35)

which has two roots ${\lambda }_{1,2}=\mp i\sqrt{DetJ\left(S_3\right)}$. Under condition (20c), it is evident that $a_1=\overline{a_1}$ has two entirely imaginary eigenvalues, ${\lambda }_1$ and ${\lambda }_2$, which are complex conjugates. Additionally, the generic roots of Equation (34) in the vicinity of $\overline{a_1}$ are written as

$$\begin{array}{c}\hfill {\lambda }_{1,2}={\displaystyle \frac{Tr\left(J\left(S_3\right)\right)\mp i\sqrt{DetJ{(}_3)}}{2}}.\end{array}$$

Then,

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{d{a}_1}}{\left[Re\left({\lambda }_{1,2}\right)\right]}_{a_1=\overline{a_1}}={\displaystyle \frac{d}{d{a}_1}}{\left[{\displaystyle \frac{Tr\left(J\left(S_3\right)\right)}{2}}\right]}_{a_1=\overline{a_1}}={\displaystyle \frac{-\left[\overline{u}{(1+a_2\overline{p})}^2+e_1\overline{p}{(1+a_2\overline{u})}^2\right]}{(1+w){(1+a_2\overline{u})}^2{(1+a_2\overline{p})}^2}}\ne 0.\end{array}$$

This indicates that the third condition (c) has been confirmed, guaranteeing that for $a_1=\overline{a_1}$, the PUY model presents a Hopf bifurcation. □

Theorem 7.

Assuming the following

$$D_i>0,i=1,2,$$

(36)

$$\eta \left(\overline{w}\right)\mu \left(\overline{w}\right)+\tau \left(\overline{w}\right)\varphi \left(\overline{w}\right)\ne 0,$$

(37)

$$\overline{w}>0.$$

(38)

Then, the PUY model around $S_4$ presents a Hopf bifurcation at $w=\overline{w}$.

Proof. 

The bifurcation parameter value can be determined by setting $D_1\left(\overline{w}\right)D_2\left(\overline{w}\right)-D_3\left(\overline{w}\right)=0$ instead in Formula (22). This gives:

$$\overline{w}={\displaystyle \frac{a_1{a}_{21}{a}_{33}{p}^{*}-(1+a_2{p}^{*})(a_{33}+a_{11}+a_{22})(a_{11}(a_{33}+a_{22})+a_{22}{a}_{33}-a_{12}{a}_{21}-a_{32}{a}_{23})+a_{11}(a_{32}{a}_{23}-a_{22}{a}_{33})}{(1+a_2{p}^{*})(a_{33}+a_{11}+a_{22})(a_{11}(a_{33}+a_{22})+a_{22}{a}_{33}-a_{12}{a}_{21}-a_{32}{a}_{23})+a_{11}(a_{32}{a}_{23}-a_{22}{a}_{33})}}.$$

Clearly, $\overline{w}>0$ if condition (38) holds. Now, at $w=\overline{w}$, Equation (22) can be written as

$$\begin{array}{c}\hfill \left(\lambda +D_1\right)\left({\lambda }^2+D_2\right)=0.\end{array}$$

The aforementioned equation has three roots, according to condition (36): two fully imaginary roots ${\lambda }_{2,3}=\mp i\sqrt{D_2}$ and a negative root ${\lambda }_1=-D_1$. In a neighborhood of $\overline{w}$, the roots have the following forms:

$$\begin{array}{c}\hfill {\lambda }_1=-D_1,{\lambda }_{2,3}={\chi }_1\left(w\right)\mp i{\chi }_2\left(w\right).\end{array}$$

Clearly, $Re\left({\lambda }_{2,3}\right){|}_{w=\overline{w}}={\chi }_1\left(\overline{w}\right)=0$ indicates that the first condition for Hopf bifurcation has been met at $w=\overline{w}$. Now, to confirm the transversality condition, we substitute ${\chi }_1\left(w\right)\mp i{\chi }_2\left(w\right)$ into Equation (22) and then compute its derivative with respect to $\overline{w}$, $\eta \left(\overline{w}\right)\mu \left(\overline{w}\right)+\tau \left(\overline{w}\right)\varphi \left(\overline{w}\right)\ne 0$, where the form of $\eta \left(\overline{w}\right)$,$\mu \left(\overline{w}\right)$,$\tau \left(\overline{w}\right)$, and $\varphi \left(\overline{w}\right)$, are

$$\begin{array}{c}\hfill \mu \left(\overline{w}\right)=3{\chi }_1^2\left(w\right)+2F_1\left(w\right){\chi }_1\left(w\right)+F_2\left(w\right)-3{\chi }_2^2\left(w\right),\\ \hfill \varphi \left(\overline{w}\right)=6{\chi }_1\left(w\right){\chi }_2\left(w\right)+2F_1\left(e\right){\chi }_2\left(w\right),\\ \hfill \eta \left(\overline{w}\right)={\chi }_1^2\left(w\right)F_1^{\prime }\left(w\right)+F_2^{\prime }\left(w\right){\chi }_1\left(w\right)+F_3^{\prime }\left(w\right)-F_1\left(w\right){\chi }_2^2\left(w\right),\\ \hfill \overline{w}=2{\chi }_1\left(w\right){\chi }_2\left(w\right)F_1^{\prime }\left(w\right)+F_2^{\prime }\left(e\right){\chi }_2\left(w\right).\end{array}$$

Now, at $w=\overline{w}$, substituting ${\chi }_1$ = 0 and ${\chi }_2=\sqrt{D_2}$ into Equation (22), the following is obtained:

$$\mu \left(\overline{w}\right)=-2D_2\left(\overline{w}\right),$$

$$\varphi \left(\overline{w}\right)=2D_1\left(\overline{w}\right)\sqrt{D_2}\left(\overline{w}\right),$$

$$\eta \left(\overline{w}\right)=D_3^{\prime }\left(\overline{w}\right)-D_1^{\prime }\left(\overline{w}\right)D_2\left(\overline{w}\right),$$

$$\tau \left(\overline{w}\right)=D_2^{\prime }\left(\overline{w}\right)\sqrt{D_2\left(\overline{w}\right)},$$

where

$$\begin{array}{cc}\hfill D_1^{\prime }\left(\overline{w}\right)& =-\left(c_{33}^{*\left[3\right]}+c_{11}^{*\left[3\right]}+c_{22}^{*\left[3\right]}\right),\hfill \\ \hfill D_2^{\prime }\left(\overline{w}\right)& =c_{33}^{*\left[3\right]}\left(c_{11}^{*\left[3\right]}+c_{22}^{*\left[3\right]}\right)+c_{11}^{*\left[3\right]}{c}_{22}^{*\left[3\right]}-c_{12}^{*\left[3\right]}{c}_{21}^{*\left[3\right]}-c_{32}^{*\left[3\right]}{c}_{23}^{*\left[3\right]},\hfill \\ \hfill D_3^{\prime }\left(\overline{w}\right)& =c_{12}^{\ast \left[3\right]}{c}_{21}^{\ast \left[3\right]}{c}_{33}^{\ast \left[3\right]}+c_{11}^{\ast \left[3\right]}\left(c_{32}^{\ast \left[3\right]}{c}_{23}^{\ast \left[3\right]}-c_{22}^{\ast \left[3\right]}{c}_{33}^{\ast \left[3\right]}\right).\hfill \end{array}$$

Hence, condition (37) gives

$$\begin{array}{cc}\hfill \eta \left(\overline{w}\right)\mu \left(\overline{w}\right)+\tau \left(\overline{w}\right)\varphi \left(\overline{w}\right)& =2\left(c_{33}^{\ast \left[3\right]}\left(c_{11}^{\ast \left[3\right]}+c_{22}^{\ast \left[3\right]}\right)+c_{11}^{\ast \left[3\right]}{c}_{22}^{\ast \left[3\right]}-c_{12}^{\ast \left[3\right]}{c}_{21}^{\ast \left[3\right]}-c_{32}^{\ast \left[3\right]}{c}_{23}^{\ast \left[3\right]}\right),\hfill \\ \hfill & \left[D_3^{\prime }\left(\overline{w}\right)-D_2(D_1^{\prime }\left(\overline{w}\right)-D_1\left(\overline{w}\right))\right]\ne 0.\hfill \end{array}$$

That means the Hopf bifurcation has occurred at $\overline{w}$. □

Remark 1.

The sizes of plant populations, herbivores, and predators fluctuate in a recurring, regular pattern over time due to periodic oscillations. Each population experiences cycles—booms followed by declines—at roughly predetermined periods rather than reaching steady values. This tendency is specifically described mathematically by a limit cycle. It describes a stable, closed loop in the system’s dynamics: the trajectories of the populations gradually approach and follow the same repeating cycle, regardless of where they begin (within a tolerable range). From an ecological perspective, this indicates that the ecosystem has a natural rhythm that endures even in the face of minor disruptions.

Remark 2.

Table 2. Transcritical and Hopf bifurcations in population models: an ecological interpretation.

Bifurcation Type	Population Outcome	Ecological Interpretation
Transcritical	Extinction ↔ coexistence	Species invasion or exclusion
Hopf	Equilibrium ↔ limit cycle	Sustained oscillatory dynamics’ onset
Transcritical + Hopf	Threshold + cycles	Following invasion, oscillatory coexistence

describes the environmental effects of Hopf dendritices and transcribed dendritices (e.g., coexistence, exclusion, oscillatory dynamics).

6. Numerical Simulation

In this section, we discuss the dynamical behavior of model (4) numerically. For the numerical simulation of model (4), we have to set a fixed set of parameters for the whole simulation. The fixed set of parameters is as follows:

$$\begin{array}{cc}\hfill & r=1;k=5;a_1=0.8;w=0.7;a_2=0.1;e_1=0.8;e_2=0.4;b_1=0.02;‘m=0.01;c_1=0.7;d_1=0.05.d_2=0.01;\hfill \end{array}$$

(39)

The parameter units are described in Table 1. For the numerical simulation, we employed the Runge–Kutta 4th-order (RK-4) technique to generate the phase portrait and time series plot using MATLAB version R2018b. Additionally, we utilized the Matcont toolbox to generate the bifurcation diagram.

6.1. Effect of Wind Flow (w) on the Model (4)

This subsection examined the effect of wind flow on the model (4), revealing that the model becomes stable, unstable, or oscillatory as the wind flow effect is increased. The bifurcation diagram about the parameter wind flow (w) shown in Figure 2 shows that Hopf bifurcation arises at point $w=1.356194$; before that point, the interior equilibrium is stable, and after that, periodic oscillation behavior is obtained. The stable and periodic oscillation for both regions is described through phase portraits and time series plots. The phase portrait and time series for the stable interior equilibrium point are shown in Figure 3. The effect of wild flow is illustrated through the phase portrait and time series plot, where we observe that the wind flow effect induces oscillatory behavior in the model’s interior equilibrium, as shown in Figure 4. As wind flow increases, model dynamics change from stable to oscillatory.

Figure 3a shows that the interior equilibrium is a stable spiral, and the corresponding time series in Figure 3b shows that there is no oscillation in the population or converges near the interior equilibrium, which means it is stable over time.

6.2. Effect of Parameter $a_2$ on Model (4)

For the fixed set of parameters as mentioned in Equation (39), we have demonstrated the effect of parameter $a_2$ on model (4). In this section, we find that the stable equilibrium point becomes unstable and shows periodic, periodic doubling, and chaotic behavior about the interior equilibrium. Figure 5 shows that increasing the handling-time parameter $a_2$, which represents grazing saturation in ungulates, destabilizes the coexistence equilibrium through a Hopf bifurcation. Beyond this critical value, the plant–ungulate–wolf system exhibits stable population oscillations. As $a_2$ increases further, the system undergoes a sequence of period-doubling bifurcations and eventually enters a chaotic regime, leading to increasingly irregular and unpredictable population fluctuations. For example, instead of repeating regular annual cycles, plant biomass may peak every two or four seasons, while ungulate and wolf populations respond with noticeable delays and uneven amplitudes. In the chaotic regime, even small changes in initial plant density or grazing pressure can result in very different population trajectories, making long-term prediction of plant, ungulate, and wolf abundances difficult.

Phase Portraits and Time Series Corresponding to Bifurcation Diagram Figure 5

The periodic, period-doubling oscillations, and chaotic behavior of model (4) with respect to the parameter $a_2$ occur as obtained in the bifurcation Figure 5 region. To further illustrate these dynamics, we have plotted the corresponding phase portraits and time series for specific values of $a_2$ within the bifurcation. Figure 6a shows a stable spiral at the interior equilibrium, Figure 6b exhibits limit cycle behavior around the unstable equilibrium, Figure 6c displays a periodic double limit cycle, and Figure 6d illustrates a chaotic attractor.

The time series corresponding to the Figure 6 have been plotted in Figure 7 which clearly described the stable, periodic and chaotic behaviour about the parameter $a_1$.

6.3. Bifurcation Diagram and Corresponding for the Parameter $a_1$

The bifurcation diagram (Figure 8) was plotted for a fixed set of parameters with $d_2=0.1$. Figure 8 shows that a Hopf bifurcation occurs at the interior equilibrium points $a_1=0.135291$ and $a_1=0.034220$. A transcritical bifurcation is observed at $a_1=0.135862$, denoted by a black line. Branch points are located at $a_1=0.031388$ and $a_1=0.135862$. Additionally, model (4) exhibits a saddle-node bifurcation at $a_1=0.272189$, where the number of equilibrium points changes from two to one to zero (see Figure 8). In Figure 8, the red curve indicates that the equilibrium is unstable, while the green curve shows the stable equilibrium, as also demonstrated in Figure 9 and Figure 10. The two interior equilibria exist after the TC point shown in Figure 8, where one is a stable equilibrium, and the other is an unstable equilibrium. The phase portraits are plotted for the bifurcation diagram (Figure 8) for all possible regions of $a_1$. Figure 9 shows the phase portrait and time series for the unstable region before Hopf points $a_1=0.135291$. Figure 9a shows the stable limit cycle around the unstable equilibrium point, with trajectories converging to the limit cycle. Figure 9b–d are time series plots that demonstrate all species persist and fluctuate periodically over a long time.

Figure 10a indicates that the interior equilibrium point is a stable spiral, and the corresponding time series for the stable spiral is shown in Figure 10b. Figure 10c shows the existence of two interior equilibrium points: one is a stable spiral equilibrium denoted as $E_1$, while the other is an unstable equilibrium denoted as $E_2$. Figure 10c clearly shows that the two trajectories taken around $E_1$ pass through it and converge to the equilibrium point $E_2$. Figure 11 shows a phase portrait, a stable spiral, and a time series for the saddle node bifurcation point to ensure the validation of the bifurcation plot Figure 7. Also, we have plotted the phase portrait and time series in Figure 12 for the boundary equilibrium point to show the dynamics of the system.

6.4. Numerical Result for the Boundary Equilibrium

Advantages and Limitations of the Model

The proposed model incorporates several ecologically realistic mechanisms, including wind flow effects, grazing saturation, and cooperative hunting, which allow for a more comprehensive understanding of plant–ungulate–wolf interactions. The analytical and numerical investigation of saddle-node, transcritical, and Hopf bifurcations provides valuable insight into stability transitions, oscillatory behavior, and potential ecological collapse. These features make the model useful for exploring complex population dynamics under varying environmental and biological conditions. However, the model also has certain limitations. It assumes homogeneous mixing of populations and constant parameter values, which may not fully capture spatial heterogeneity, seasonal variations, or stochastic environmental effects present in real ecosystems. Additionally, factors such as age structure, migration, and adaptive behavior are not taken into account. Despite these limitations, the model provides a meaningful framework for understanding the key mechanisms driving prey–predator dynamics and can be extended in future work to incorporate spatial structure and environmental variability.

7. Results and Discussions

This article investigates the impacts of wind flow on the prey–predator model. The wind flow causes the destabilization of the model, as discussed through analytical methods and validated using numerical methods with the aid of a bifurcation diagram, phase portrait, and time series. The theoretical part’s outcome can be summed up as follows:

• While the plant equilibrium point $S_2$ is always stable, the trivial equilibrium point $S_1$ is always saddle. It is evident that $S_3$ demonstrates local asymptotic stability provided that requirements ((20a)–(20c)) are met. Additionally, the condition (4) ensures the local stability of the interior equilibrium point.
• The system (4) exhibits saddle-node bifurcation at $S_3$ with $a_2=a_2^{*}$ as long as (25) and (26) are met. If condition (25) is broken and condition (26) is met together with (27), the system (4) has a transcritical bifurcation. However, provided that (29) and (30) are satisfied, the system (4) displays saddle-node bifurcation at $S_4$ with $w=w^{*}$. The system (4) exhibits a transcritical bifurcation if condition (29) is broken and condition (30) is met along with (31).
• The Hopf bifurcation phenomena may occur around $S_4$ at $w=w^{*}$ provided that ((36)–(38)).

This article numerically examines the saddle-node bifurcation and the Hopf bifurcation. Figure 2 is a bifurcation diagram regarding the parameter wind flow (w), which shows that the stability of model (4) changes from a stable interior to an unstable state, and a limit cycle emerges after the Hopf point. Phase portraits and time series corresponding to the bifurcation diagram (2) are shown in Figure 3 and Figure 4. Figure 3b shows that all populations, including plants, unregulated species, and wolves, coexist and remain stable over a long period. Figure 4a shows the behavior of the limit cycle, and the corresponding time series are shown in Figure 4b–d, which illustrates that all species exist periodically over time.

7.1. Impact of Parameter Hindrance Rate in Plant Capture for Ungulate Species $a_2$

Figure 5 shows the impact of parameter hindrance rate in plant $a_2$ as a stable system about the interior equilibrium destabilizes and causes the periodic doubling and chaos as $a_2$ increases. The time series and phase portrait ensure the stability, periodic doubling, and chaotic behavior of the system, as seen in Figure 6 and Figure 7.

7.2. Impact of Plant Consumption Rate $\left(a_1\right)$ by Ungulate Species

In this model, the plant consumption rate $a_1$ plays a key role in shaping the system dynamics. Changes in $a_1$ give rise to different types of bifurcations with clear ecological interpretations.

A Hopf bifurcation occurs when the stable interior equilibrium loses stability and periodic oscillations emerge. Ecologically, this represents cyclic fluctuations in plant and ungulate populations, caused by a feedback loop between grazing pressure and plant regrowth. Overgrazing reduces plant biomass, followed by recovery phases.

A transcritical bifurcation corresponds to an exchange of stability between a coexistence equilibrium and a boundary equilibrium. In ecological terms, it marks a threshold between species coexistence and exclusion, indicating that either plants or ungulates can no longer persist together when grazing pressure becomes too strong.

A saddle-node bifurcation occurs when two equilibria (one stable and one unstable) collide and disappear. Ecologically, this implies a sudden loss of viable population states, meaning that small increases in grazing rate can abruptly eliminate sustainable plant–ungulate coexistence. Together, these bifurcations demonstrate that moderate grazing supports stable coexistence, whereas excessive plant consumption destabilizes the system, leading to oscillations, loss of equilibria, and potential ecological collapse, as illustrated in Figure 7, Figure 8, Figure 9 and Figure 10.

8. Conclusions

This study examined the impact of wind flow and grazing-related parameters on the dynamics of a prey–predator ecological model using both analytical and numerical methods. The stability analysis revealed that the trivial equilibrium is always a saddle point, while the plant-only equilibrium remains stable. The interior coexistence equilibrium is locally asymptotically stable under suitable parameter conditions, indicating balanced coexistence among plants, ungulates, and predators.

The influence of wind flow was shown to play a destabilizing role in the system. As the wind intensity parameter w increases, the stable interior equilibrium loses stability through a Hopf bifurcation, leading to the emergence of stable limit cycles. This transition is characterized by periodic oscillations in plant, ungulate, and predator populations, as confirmed by bifurcation diagrams, phase portraits, and time series. These results highlight that environmental forces, such as wind, can significantly alter predator–prey interactions and long-term population behavior.

The hindrance rate in plant capture by ungulates ($a_2$) further enriches the system dynamics. Increasing $a_2$ destabilizes the interior equilibrium, giving rise to period-doubling bifurcations and chaotic dynamics, which indicate increasingly irregular and unpredictable population fluctuations. This suggests that grazing inefficiency can strongly amplify nonlinear feedback mechanisms within the ecosystem.

Moreover, the plant consumption rate $a_1$ was found to be a key driver of ecological transitions. Variations in $a_1$ lead to Hopf, transcritical, and saddle-node bifurcations, each with clear ecological implications. Moderate grazing promotes stable coexistence, whereas excessive plant consumption destabilizes the system, causing oscillations, loss of coexistence equilibria, and potential ecological collapse.

Overall, the results demonstrate that both environmental factors (wind flow) and biological interactions (grazing intensity and capture hindrance) critically shape ecosystem stability. The combined analytical and numerical findings provide valuable insights into how external disturbances and grazing pressure can drive complex dynamics, including oscillations and chaos, in plant–ungulate–predator systems.