Hopf bifurcation and hybrid control of a delayed diffusive semi-ratio-dependent predator-prey model

1.   Introduction

The rich dynamics of nonlinear systems have significant implications for modern technology, advancing developments in fields such as natural sciences and engineering applications. Non-linear dynamical systems typically serve as mathematical formalizations of conventional scientific concepts and complex entities, with widespread applications in biology, ecology, and chemistry^[1,2,3]. Zhang and Lu ^[4] studied a type of predator-prey system with a Holling-IV type functional response function

where $x_{1}(t), x_{2}(t)$ represent the densities of the prey and the predator at time $t$, respectively, $m\neq 0, n\geq 0$ are all constant, and $r_{1}, r_{2}$ represent the intrinsic growth rates of the prey and the predator respectively, $b_{1}(t)$ is the intra-specific competition rate of the prey. $a_{1}(t)$ is the capturing rate of the predator, $a_{2}(t)$ is a measure of the food quality that the prey provided for conversion into predator birth^[4]. In ^[4], Zhang and Lu demonstrated the global stability of the positive equilibrium point using the Lyapunov function. In ^[5,6,7], many scholars have studied the properties of periodic solutions for the semi-ratio-dependent prey-predator model. Zhao^[8] investigated the stability and bifurcation of model (1.1) with two delays.

With the continuous deepening and expansion of research, scholars have found that in the natural world, the emergence of many phenomena is not only influenced by the current state but also closely related to the state of a certain moment or period in the past. This phenomenon is called time delay. Many scholars have investigated predator-prey models with various types of time delays, such as infinite delay, time-varying delays, and multiple delays ^[9,10,11]. Under specific conditions, delays induce variations in the stability of equilibrium states, leading to the emergence of bifurcations or spiral wave patterns in the considered systems^{[12,13,14,15,16,17]}. Therefore, the application of time delay in predator-prey models is important for studying the dynamic behavior of ecological systems and predicting their future development trends.

Reaction-diffusion models are commonly employed to characterize the movement and evolution. In both macroscopic and microscopic worlds, every particle, such as bacteria, cells, and organisms, moves in an apparently random manner, commonly referred to as diffusion processes^[18]. The diffusion process may lead to environmental changes. Turing's initial studies demonstrated that stable homogeneous states can become unstable under certain conditions in reaction-diffusion systems, leading to the formation of patterns^[19]. The corresponding instability is referred to as diffusion-driven instability^[20]. Li et al.^[21,22] studied the relevant issues of Turing patterns. Dynamics studies on the stability and bifurcation issues of reaction-diffusion systems have also been quite extensive^{[23,24,25,26,27]}. Hence, the factor of diffusion is considered.

Stability and bifurcation are crucial considerations in the study of predator-prey models. Hopf bifurcation is a significant mathematical tool for describing periodic behavior, and studying Hopf bifurcation contributes to the understanding and modeling of important phenomena in biological systems such as periodic behavior, stability transitions, and dynamic oscillations ^{[28,29,30,31]}. Considering the system instability caused by Hopf bifurcation, there has been an increasing depth of research on bifurcation control by scholars in recent years ^[32,33,34]. Control strategies can effectively enhance dynamic behavior in model research. Jiang et al.^[35] proposed a new control scheme that effectively controls bifurcations. Control strategies such as PD control, state feedback control, time-delay feedback control, and hybrid control have been continuously evolving ^{[36,37,38,39]}. The concept of a hybrid control strategy was initially introduced by Luo et al.^[40], research has shown that compared to pure feedback control, the control parameters in hybrid control can be adjusted over a wider range, making it more convenient and feasible for practical implementation. Peng ^[41] explored the role of hybrid controllers in predator-prey models within the field of biological systems. From a biological perspective, adjusting the stable range of bifurcation period solutions enables the coexistence of two species under oscillatory patterns. These control strategies have predominantly been formulated and examined based on established principles of ordinary differential equations (ODEs). However, there has been limited exploration into control in predator-prey systems described by PDEs. Ghosh ^[42] investigated a practical approach of utilizing time-delayed feedback for bifurcation control in reaction-diffusion systems, thus highlighting the substantial theoretical research significance of the work presented in our paper.

A delayed diffusive system established in this paper is as follows:

here $u(x, t)$ and $v(x, t)$ represent the population densities of the prey and predator, respectively. Parameters of the system (1.2) are regarded as constants. $d_{1}$ and $d_{2}$ represent the constant diffusion coefficient. $\tau$ represents that the consumption of prey by predators needs some time to be converted into effective energy. $\Delta$ represents the Laplacian operator, the system subject to the Neumann boundary condition, and we assume that area $[0, \Omega]$ is closed, where $\Omega=l\pi(l>0)$; the population cannot go through the boundary of the region.

The main contributions of this paper include: (1) To better approximate real ecological systems, the semi-ratio-dependent prey-predator model with Holling-IV functional response in ordinary differential equations is being extended. Time delays and diffusion terms are being introduced into the model. By analyzing the system's corresponding characteristic equations, the conditions for Turing instability induced by diffusion in the absence of time delays are being deduced. (2) The impact of time delays on the stability of positive equilibrium points in reaction-diffusion systems is being investigated. Sufficient conditions for the existence of spatially homogeneous and inhomogeneous Hopf bifurcation are being provided. Utilizing partial differential equation normalization theory and the center manifold theorem, explicit methodologies for determining the bifurcation direction and the stability of periodic solutions at positive equilibrium points are being derived. (3) The application of control strategies for Hopf bifurcations in reaction-diffusion systems formed by partial differential equations is currently limited. A hybrid controller is being integrated to regulate the bifurcation behavior in reaction-diffusion systems with a Holling-IV functional response. Results demonstrate that by adjusting control parameters, the spatial stability range is expanded, leading to modifications in the positions of periodic solutions of the Hopf bifurcation and enhancing the system's performance and controllability. (4) Currently, there are few researchers conducting sensitivity analysis on the reaction-diffusion prey-predator system, at least from our current perspective. Thus sensitivity analysis of the system is being carried out using Latin hypercube sampling/partial rank correlation coefficient (LHS/PRCC), exploring the parameter space of the model. This analysis offers valuable insights for comprehending the uncertainty and intricacy of the system.

The paper is arranged as follows: In Section 2, we analyze the stability of the model with the inclusion of reaction-diffusion terms and the conditions that lead to Turing instability; We also study the existence of spatially homogeneous and inhomogeneous Hopf bifurcations. In Section 3, We present the properties of the Hopf bifurcation. In Section 4, the dynamical behavior and control strategy of a controlled diffusion system are investigated by incorporating a hybrid controller into the prey-predator model. In Section 5, numerical simulations are accomplished to substantiate conclusions. At last, it is a conclusion of this paper.

2.   Stability of equilibrium points and Hopf bifurcation analysis

It is calculated that the system (1.2) has two possible equilibrium points.

(1) The boundary equilibrium $E_{1} = (\frac{r_{1}}{b_{1}}, 0)$.

(2) The coexisting equilibrium point $E_{*} = (u_{*}, v_{*})$, which $v_{*} = \dfrac{r_{2}u_{*}}{a_{2}}$ and the following equation have at least one positive root $u_{*}$.

Let $u(t) = u(x, t), v(t) = v(x, t), u(t-\tau) = u(x, t-\tau)$, and $v(t-\tau) = v(x, t-\tau)$. The system (1.2) is linearized at $(u_{*}, v_{*})$:

where

and

The characteristic equation of (3.1) is

where $I = \left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array} \right),$ and $M_{k} = -\frac{k^{2}}{l^{2}}D^{2}$ for $k\in \{0, 1, 2, ...\}: = N_{0}$. Then we have

where

2.1.   Stability of the equilibria as $\tau =0$

(1) The characteristic equation to $E_{1} = (\frac{r_{1}}{b_{1}}, 0)$ is

Obviously,

There is at least one positive characteristic root of (2.5), hence $E_{1} = (\frac{r_{1}}{b_{1}}, 0)$ is unstable.

(2) When $\tau = 0$, the characteristic equation to $E_{*} = (u_{*}, v_{*})$ is

Define

Assume that

When $d_{1} = d_{2} = 0$ and $\tau = 0$, all roots of (2.6) have negative real parts under hypothesis $\boldsymbol{(H1)}$, $E_{*} = (u_{*}, v_{*})$ is locally asymptotically stable.

Here are the three cases for dividing the parameters,

Denote

Theorem 1. Suppose $\boldsymbol{(H1)}$ holds and $\tau = 0$.

(1) In $\boldsymbol{Case 1}$(or $\boldsymbol{Case 2})$, $E_{*} = (u_{*}, v_{*})$ of system (1.2) is locally asymptotically stable.

(2) In $\boldsymbol{Case 3}$, when $k\in\mathbb{S}$, then $E_{*} = (u_{*}, v_{*})$ of system (1.2) is Turing unstable.

Proof. If $\boldsymbol{Case 1}$(or $\boldsymbol{Case 2})$ hold, we have $\sigma_{k} < 0$ and $\varphi_{k} > 0$ for $k\in N_{0}$, the above (1) holds. If $\boldsymbol{Case 3}$ hold, we have $\varphi_{k} < 0$ for $k\in\mathbb{S}$. This ensures Eq (2.6) has a positive real part root, so $E_{*} = (u_{*}, v_{*})$ of system (1.2) undergoes Turing bifurcation. □

Remark 1. The system (1.2) exhibits an unstable predator-free equilibrium point, which may not occur in natural systems. At the positive equilibrium point $E_{*} = (u_{*}, v_{*})$, there is a richer dynamic behavior. According to the conditions of Turing, spatial diffusion can lead to the destabilization of the stable equilibrium of the ordinary differential equations corresponding to reaction-diffusion systems, forming regular pattern structures in space, known as diffusion-induced instability. Example 1 of numerical simulations can illustrate the occurrence of Turing instability.

2.2.   Hopf bifurcation ($\tau > 0$)

We will derive the conditions for the Hopf bifurcation. Suppose $i\omega(\omega > 0)$ be a solution of Eq (2.4), then

Then we have

which lead to

Let $z = \omega^{2}$, then (2.10) becomes

where

We have $B_{0}^{2}-C_{0}^{2} < 0$, there must exist some $k_{0}\in \{1, 2, \cdots\}$ satisfied $B_{k}^{2}-C_{k}^{2} < 0$ for $k\in \{0, 1, 2, \cdots, k_{0}-1\}$, and $B_{k}^{2}-C_{k}^{2}\geq 0$ for $k\in \{k_{0}, k_{0}+1, \cdots\}$.

When $a_{11}\in (-\infty, -a_{22})$, then we have $-A_{k}^{2}+2B_{k}+a_{22}^{2}\leq a_{22}^{2}-a_{11}^{2} < 0$ for any $k\in N_{0}$.

When $a_{11}\in (-a_{22}, a_{22})$, we make the following assumption:

If $\boldsymbol{(H2)}$ holds, then we have

for any $k\geq k_{0}$. These imply that there are no purely imaginary roots in Eq (2.4).

Since $B_{k}^{2}-C_{k}^{2} < 0$, when $k\in \{0, 1, 2, \cdots, k_{0}-1\}$, the equation (2.11) has one positive root $z_{k}$, namely

Equation (2.4) has a pair of roots with purely imaginary $\pm i\omega_{k}$ at $\tau_{k}^{j}$$(k\in \{0, 1, 2, \cdots, k_{0}-1\}, j\in N_{0})$, where

and

Obviously $\tau _{k}^{0} = \mathop{min}\limits_{j\in N_{0}}\{\tau_{k}^{j}\}$. Denote

Let $\lambda _{k}(\tau) = \alpha _{k}(\tau)\pm i\omega _{k}(\tau)$ be the root of (2.4) near $\tau = \tau_{k}^{j}, k\in \{0, 1, 2, \cdots, k_{0}-1\}$ satisfying $\alpha _{k}(\tau_{k}^{j}) = 0$ for $\omega _{k}(\tau_{k}^{j}) = \omega _{k}$.

Lemma 1. The following traversal condition holds: $Re(\dfrac{d\lambda}{d \tau})|_{\tau = \tau _{k}^{j}} > 0$.

Proof. Take the derivative of both sides of (2.4) with respect to $\tau$, and we have

then

□ These critical values $\tau_{k}^{j}$ for $j\in\{0, 1, 2, \cdots\}; k\in \{0, 1, 2, \cdots, k_{0}-1\}$ are possible Hopf bifurcations. Suppose $\tau_{k_{1}}^{i}\neq \tau_{k_{2}}^{j}$ for any $i, j\in\{0, 1, 2, \cdots\}; {k_{1}}, {k_{2}}\in \{0, 1, 2, \cdots, k_{0}-1\}$. Based on the above, we derive the main result.

Theorem 2. When $a_{11}\in (-\infty, -a_{22})$, if $\boldsymbol{(H1)}$ holds; When $a_{11}\in (-a_{22}, a_{22})$, if $\boldsymbol{(H1)(H2)}$ hold, there are the following conclusions.

(1) When $\tau \in [0, \widetilde{\tau})$, the coexisting equilibrium $E_{*} = (u_{*}, v_{*})$ of system (1.2) is locally asymptotically stable, where $\widetilde{\tau}$ is defined in (2.14).

(2) System (1.2) undergoes a Hopf bifurcation at the equilibrium $E_{*} = (u_{*}, v_{*})$ when $\tau = \tau_{k}^{j}, j = 0, 1, 2, \cdots; k = 0, 1, 2, \cdots, k_{0}-1$. Furthermore, when $k = 0$, bifurcating periodic solutions are spatially homogeneous; otherwise, they are spatially inhomogeneous.

3.   Stability and direction of Hopf bifurcation

We will apply Wu ^[23] and Hassard's ^[28] method to compute properties of periodic solutions for system (1.2). For fixed $k\in \{0, 1, 2, \cdots, k_{0}-1\}, j \in N_{0}$, we denote $\tau^{*} = \tau_{k}^{j}$. We perform a transformation $\overline{u}(x, t) = u(x, \tau t)-u_{*}$ and $\overline{v}(x, t) = v(x, \tau t)-v_{*}$, but we still use $u(x, t), v(x, t)$ instead of $\overline{u}(x, t), \overline{v}(x, t)$. Then (1.2) can be transformed into the following system:

for $x \in(0, l \pi)$, and $t > 0$. Let

We use $\mathcal{C}: = C([-1, 0], X)$ to represent the phase space, then (3.1) can be written as an abstract differential equation.

where $L(\tau)(\phi):\mathcal{C}\rightarrow X$ and $F(\phi, \mu):\mathcal{C}\times R \rightarrow X$ are defined by

with

where

with $\phi (\theta) = (\phi_{1} (\theta), \phi_{2}(\theta))^{T}\in \mathcal{C}$.

Consider the linear equation

Clearly, $(0, 0)$ is an equilibrium point of the system (3.1). Consider the functional differential equation

By the Riesz representation, there exists a bounded variation function $\eta(\theta, \tau^{*}) (-1\leq \theta \leq 0)$, such that

We choose

For $\phi \in C^{1}\left([-1, 0], \mathbb{R}^{2}\right), \psi \in C^{1}\left([0, 1], \mathbb{R}^{2}\right)$, we define

Define the following bilinear pairing

Through the above analysis, $\pm i \omega_{k} \tau^{*}$ are the eigenvalues of $A\left(\tau^{*}\right)$ and $A^{*}$. Let $P$ and $P^{*}$ be the two-dimensional center spaces of $A\left(\tau^{*}\right)$ and $A^{*}$ associated with $\pm i\omega_{k} \tau^{*}$, then $P^{*}$ is the adjoint space of $P$.

Let $p_{1}(\theta) = (1, \xi)^{T} e^{i \omega_{k}{\tau^{*}} \theta}$ and $p_{2}(\theta) = \overline{p}_{1}(\theta)(\theta \in[-1, 0])$ be the bases of $A\left(\tau^{*}\right)$ and $A^{*}$ corresponding to $i\omega_{k}\tau^{*}, -i\omega_{k}\tau^{*}$ respectively. By calculations, we have

Let $\Phi = \left(\Phi_{1}, \Phi_{2}\right),$ and $\Psi^{*} = \left(\Psi_{1}^{*}, \Psi_{2}^{*}\right)^{T}$ with

for $\theta \in[-1, 0]$, and

for $t \in[0, 1]$.

Now we define $\left(\Psi^{*}, \Phi\right) = \left(\begin{array}{ll}{\left(\Psi_{1}^{*}, \Phi_{1}\right)} & {\left(\Psi_{1}^{*}, \Phi_{2}\right)} \\ {\left(\Psi_{2}^{*}, \Phi_{1}\right)} & {\left(\Psi_{2}^{*}, \Phi_{2}\right)}\end{array}\right)$, and it can be computed by (3.11), then we construct a new basis $P^{*}$ by

Then $(\Psi, \Phi) = I_{2}$. In addition, define $f_{k} : = \left(\alpha_{k}^{1}, \alpha_{k}^{2}\right)$, where

Define

We have $P_{C N} \mathcal{C}$ to represent the center space of (3.7), where

And $\mathcal{C} = P_{C N} \mathcal{C} \oplus P_{S} \mathcal{C}$, here $P_{S} \mathcal{C}$ and $P_{C N} \mathcal{C}$ are complementary in $\mathcal{C}$, where

for $u = \left(u_{1}, u_{2}\right)^{T}, v = \left(v_{1}, v_{2}\right)^{T}, u, v \in X$ and $\langle\phi, f_{k}\rangle = \left(\langle\phi, \alpha_{k}^{1}\rangle, \langle\phi, \alpha_{k}^{2}\rangle\right)^{T}$.

Let $A_{\mathrm{\tau}^{*}}$ be the infinitesimal generator generated by the solution of the linear Eq (3.7), and rewrite (3.1) as

where

Induced by $\mathcal{C} = P_{C N} \mathcal{C} \oplus P_{S} \mathcal{C}$, the solution can be obtained as

where $\left(\begin{array}{l}{x_{1}} \\ {x_{2}}\end{array}\right) = \left(\Psi, \langle U_{t}, f_{k}\rangle\right)$, $h\left(x_{1}, x_{2}, \mu\right) \in P_{S} \mathcal{C}, \quad h(0, 0, 0) = 0, \quad D h(0, 0, 0) = 0$. The solution of (3.2) can be obtained as

Let $z = x_{1}-i x_{2}$, and notice that $p_{1} = \Phi_{1}+i \Phi_{2}$, then

and (3.16) can be transformed into

where

From^[23], $z$ satisfies

where

from Eqs (3.18) and (3.19), we have

Hence

with

$\begin{aligned} \chi & = \frac{1}{l \pi} \int_{0}^{l \pi} \cos ^{3}\left(\frac{k x}{l}\right) d x, \\ \varsigma_{1} & = \frac{ c_{13}}{l \pi} \int_{0}^{l \pi} \cos ^{2}\left(\frac{k x}{l}\right) d x+\frac{c_{14}}{l \pi} \int_{0}^{l \pi} \cos ^{4}\left(\frac{k x}{l}\right) d x, \\ \varsigma_{2} & = \frac{c_{23}}{l \pi} \int_{0}^{l \pi} \cos ^{2}\left(\frac{k x}{l}\right) d x+\frac{c_{24}}{l \pi} \int_{0}^{l \pi} \cos ^{4}\left(\frac{k x}{l}\right) d x, \\ c_{11} & = \frac{1}{2}\left(a_{13}+\xi a_{14}\right), \quad c_{12} = \frac{1}{4}\left(2 a_{13}+(\overline{\xi}+\xi) a_{14}\right), \\ c_{13} & = W_{11}^{(1)}(0)\left(2 a_{13}+\xi a_{14}\right)+W_{20}^{(1)}(0)\left(a_{13}+\frac{1}{2} \overline{\xi} a_{14}\right)+W_{11}^{(2)}(0) a_{14}+\frac{1}{2} W_{20}^{(2)}(0) a_{14}, \\ c_{14} & = \frac{1}{4}\left(3 a_{15}+(\overline{\xi}+2 \xi) a_{16}\right), \\ c_{21} & = \frac{1}{2}(a_{23}+a_{26}\xi)e^{-2i\omega_{k}\tau^{*}}+\frac{1}{2}(a_{24}\xi+ a_{25}\xi^{2})e^{-i\omega_{k}\tau^{*}}, \\ c_{22} & = \frac{1}{2}a_{23}+\frac{1}{4}(a_{24}\xi+a_{25}\xi\overline{\xi})e^{i\omega_{k}\tau^{*}}+ \frac{1}{4}(a_{24}\overline{\xi}+a_{25}\xi\overline{\xi})e^{-i\omega_{k}\tau^{*}}+ \frac{1}{4}a_{26}(\xi+\overline{\xi}), \\ c_{23} & = W_{20}^{(1)}(-1)(a_{23}e^{i\omega_{k}\tau^{*}}+\frac{1}{2}a_{26}\overline{\xi}+ \frac{1}{2}a_{24}\overline{\xi})+W_{11}^{(1)}(-1)(2a_{23}e^{-i\omega_{k}\tau^{*}}+a_{24}+a_{26}\xi e^{-i\omega_{k}\tau^{*}})\\ & \quad +\frac{1}{2}W_{20}^{(2)}(0)(a_{24}e^{i\omega_{k}\tau^{*}}+a_{25}\overline{\xi}+a_{25}\overline{\xi} e^{i\omega_{k}\tau^{*}})+W_{11}^{(2)}(0)e^{-i\omega_{k}\tau^{*}}(a_{24}+a_{25}\xi)\\ & \quad +W_{11}^{(2)}(-1)(a_{25}\xi+a_{26}e^{-i\omega_{k}\tau^{*}})+\frac{1}{2}a_{26}W_{20}^{(2)}(-1) e^{i\omega_{k}\tau^{*}}, \\ c_{24} & = \frac{1}{2}a_{27}(e^{-i\omega_{k}\tau^{*}}+\frac{1}{2}e^{i\omega_{k}\tau^{*}})+\frac{1}{2}a_{28}(\xi+\frac{1}{2} \overline{\xi}e^{-2i\omega_{k}\tau^{*}})+\frac{1}{2}a_{29}e^{-i\omega_{k}\tau^{*}}(\xi +\frac{1}{2}\overline{\xi})\\ & \quad +\frac{1}{4}a_{30}(\xi^{2}+\overline{\xi}\xi e^{-2i\omega_{k}\tau^{*}}+\xi \overline{\xi}). \end{aligned}$

Let $\left(\nu_{1}, \nu_{2}\right) = \Psi_{1}(0)-i \Psi_{2}(0)$, and notice that

Then, by the (3.19) and (3.21), we can obtain the following quantities:

To calculate $g_{21}$, we need to find $W_{20}(\theta), W_{11}(\theta)$ for $\theta \in [-1, 0]$. From (3.19) we have

and by ^[23], $\dot{W}(z, \overline{z})$ satisfy

where

and $X_{0} :[-1, 0] \rightarrow B(X, X)$ is given by $X_{0}(\theta) = \left\{\begin{array}{ll}{0, } & {-1 \leq \theta < 0}, \\ {I, } & {\theta = 0}.\end{array}\right.$

Hence, we have

that is

From (3.26), we have that for $\theta \in[-1, 0]$,

Therefore, by (3.26), for $\theta \in[-1, 0]$

and

where

By the definition of $A_{\tau^{*}}$ and (3.27), we have

That is

where

Using the definition of $A_{\tau^{*}}$ and (3.27), we have that for $-1\leq \theta < 0$,

Notice that

We have

Therefore

Similarly, we have

Calculate $W_{20}$ using the same method, we have

Thus, we can evaluate the following values:

Theorem 3. For any critical value $\tau_{k}^{j}$, we have the Hopf bifurcation is forward $(\mu_{2} > 0)$ or backward $(\mu_{2} < 0)$. The bifurcating periodic solutions are orbitally asymptotically stable $(\beta_{2} < 0)$ or unstable $(\beta_{2} > 0)$. The period increases $(T_{2} > 0)$ or decreases $(T_{2} < 0)$.

4.   Hybrid control of bifurcation

In this section, we design a hybrid controller to control Hopf bifurcations and expand the stability range of equilibrium points. The system (1.2) with hybrid controller control can be described as follows:

where the parameter $K$ is treated as a feedback parameter, and $v_{*}$ represents the value of $v(t, x)$ at the equilibrium point.

Linearize the system affected by the hybrid controller at the equilibrium point, and obtain the linearized controlled model as follows:

where

and

The characteristic equation of (4.2) is

then we have

where

When $\tau > 0$, let $i\widetilde{\omega}(\widetilde{\omega} > 0)$ be a solution of Eq (4.4), then decompose into the real and imaginary parts, we have

which lead to

where

We propose the following hypotheses:

Suppose $\boldsymbol{(H3)}$ holds, $Q_{k} > 0, \forall k\geq 1$; Suppose $\boldsymbol{(H4)}$ or $\boldsymbol{(H5)}$ holds, $P_{k} < 0, \forall k\geq 1$. These imply that there are no purely imaginary roots in Eq (4.6) for $k\geq 1$. On the other hand, when $k = 0$, we have $Q_{0} < 0$, then the Eq (4.6) has one positive root :

On account of (4.5), it is obvious that

Therefore

Denote $\tau _{0}^{0} = \mathop{min}\limits_{j\in N_{0}}\{\tau_{0}^{j}\}$.

Let $\lambda(\tau) = \alpha(\tau)\pm i\omega (\tau)$ be the root of (4.4) near $\tau = \tau_{0}^{j}, j = 0, 1, 2, \cdots,$ satisfying $\alpha (\tau_{0}^{j}) = 0$ for $\omega (\tau_{0}^{j}) = \omega _{0}$. Take the derivative of both sides of (4.4) with respect to $\tau$, and we have

Then

If $Re\big(\frac{d\lambda}{d \tau}\big)^{-1}_{\tau = \tau _{0}^{j}}\neq 0$, the traversal condition will hold. Based on the above, we derive the main result.

Theorem 4. Assume that $\boldsymbol{(H1)(H3)(H4)}$ or $\boldsymbol{(H1)(H3)(H5)}$ hold. There are the following conclusions.

(1) The coexisting equilibrium $E_{*} = (u_{*}, v_{*})$ is locally asymptotically stable for $\tau \in[0, \tau _{0}^{0})$.

(2) The coexisting equilibrium $E_{*} = (u_{*}, v_{*})$ is unstable for $\tau \in(\tau _{0}^{0}, +\infty)$. And system (4.1) undergoes a Hopf bifurcation at the equilibrium $E_{*} = (u_{*}, v_{*})$ when $\tau = \tau_{0}^{j}, j = 0, 1, 2, \cdots$.

Remark 2. The above analysis demonstrates that by adjusting the feedback control gains, it is possible to change the values of the Hopf bifurcation without altering the original equilibrium point. This allows the system to transition from its original unstable state back to a stable state, effectively expanding the stability region and maintaining the predator-prey dynamic equilibrium (see Section 5).

5.   Numerical simulations

Example 1. Consider system $(1.2)$ with the following parameters: $d_{1} = 0,$ $d_{2} = 0,$ $r_{1} = 0.35,$ $r_{2} = 0.22, b_{1} = 0.27, a_{1} = 0.57, a_{2} = 0.11, m = 0.36, n = 0.30$, and $\tau = 0$. Hypothesis $\boldsymbol{(H1)}$ is satisfied, then $(u_{*}, v_{*}) = (0.043, 0.086)$ is locally asymptotically stable (Figure 1). We take $d_{1} = 5, d_{2} = 12$, by Theorem 1, $(u_{*}, v_{*}) = (0.043, 0.086)$ is still locally asymptotically stable (Figure 2). We choose $d_{1} = 0.01, d_{2} = 5$, and we have that $(u_{*}, v_{*}) = (0.043, 0.086)$ is Turing unstable, and Turing patterns appear (Figures 3 and 4).

Example 2. Consider system $(1.2)$ with the following parameters: $d_{1} = 0.49,$ $d_{2} = 0.8,$ $r_{1} = 0.97,$ $r_{2} = 0.96, b_{1} = 0.14, a_{1} = 0.42, a_{2} = 0.92, m = 0.79$ and $n = 0.96$. The system $(1.2)$ has a unique coexisting equilibrium $(u_{*}, v_{*}) = (6.5, 6.5)$, we compute that the critical value is $\widetilde{\tau}\approx 1.5551$. By Theorem 2, system $(1.2)$ is locally asymptotically stable for $\tau = 1.4\in [0, \widetilde{\tau}]$ (Figure 5). As $\tau = 1.8 > \widetilde{\tau}$, the system (1.2) undergoes oscillations (Figure 6). In addition, we calculate $Re(C_{1}(0)) = -12.687$, then we have $\mu_{2} > 0$, $\beta_{2} < 0$.

In the process of selecting parameter values for numerical simulations of the model, there exists a certain degree of uncertainty in parameter selection. In order to identify parameters that significantly impact the densities of prey and predator populations, those parameters that have a substantial effect on model outputs, precise values should be assigned, while parameters with minor impacts on model outputs can be assigned rough estimates^[43,44]. This allocation of values is crucial for assessing the sensitivity of the model relative to parameter manipulations. We employed Latin hypercube sampling/partial rank correlation coefficient (LHS/PRCC) sensitivity analysis ^[45] to explore the entire parameter space of the model. In this study, parameter values were obtained from Example 2 with a reference deviation of $\pm {25 \%}$, and a uniform spread was assigned to each model parameter. Each LHS run consisted of 200 simulations, and the sampling was conducted autonomously. The PRCC values range between -1 and 1, where positive and negative PRCC values respectively reveal the positive or negative correlation between model parameters and model outputs, while the magnitude indicates the strength of the linear relationship.

From Figure 7, it can be observed that the growth rate of the prey, food conversion rate, and digestion delay have a positive impact on the prey population density. That is, an increase in these parameters will lead to an increase in the output of the prey population density. On the other hand, the internal competition rate and predation rate have a negative impact on the prey population density. A decrease in these parameters will result in a decrease in the output of the prey population density. Additionally, we find that the intrinsic growth rate of both the prey and predator has a significant positive effect on the predator population density. The competition rate within the prey population, digestion delay, and the diffusion coefficient of the prey have a significant negative impact on the predator population density, and the predator population is more sensitive to theses parameters.

Example 3. We consider the influence of the hybrid bifurcation control strategy of system (4.1). We choose $K = 0.83,$ other parameters remain the same as in Example 2; the bifurcation critical point of the controlled system (4.1) is $\tau_{0}^{0}\approx 2.3$. By Theorem 4 that when $\tau = 1.8 < \tau_{0}^{0}$, the controlled system (4.1) returns to locally stability at the equilibrium point (Figure 8), when $\tau = 2.5 > \tau_{0}^{0}$, the system (4.1) undergoes oscillations (Figure 9). Therefore, adjusting the controller coefficient can effectively expand the stable region and change the position of the bifurcation point. It is evident that reducing the feedback gain leads to a faster convergence of the system towards a stable state; in other words, a smaller feedback gain results in better control of the controller's impact on Hopf bifurcation (Figure 10).

6.   Conclusions

This paper delves into a delayed diffusive semi-ratio-dependent predator-prey model. We first analyzed how diffusion can lead to Turing instability for the system without time delay. Second, we considered the time delay $\tau$ as a parameter for bifurcation and provided conditions for the occurrence of Hopf bifurcation. Our results indicated that time delay can induce complex dynamical phenomena; the model can bifurcate from the normal equilibrium solution to spatially homogeneous and inhomogeneous periodic solutions. The time delay effect causes the system to transition from a stable state to periodic oscillations, reflecting a dynamic imbalance between predator and prey populations. Over time, the populations of predators and prey alternate between increase and decrease, forming a periodic fluctuation pattern. Furthermore, through calculations, it was determined that the Hopf bifurcation is forward and the system possesses stable branch periodic solutions. This implies that the interaction between prey and predators is regulated through stable periodic oscillations, maintaining ecological balance and species diversity. Third, a hybrid controller has been incorporated into the system (4.1) to optimize the dynamic characteristics of the predator-prey model. With the addition of a hybrid controller and reasonable parameter adjustment, the previously oscillating waveforms regain stability. This signifies that the introduction of the controller has effectively controlled the range of model stability. Loading the controller onto the predator and adjusting its control parameters can achieve objectives such as population control, population dynamics regulation, and ecosystem management, playing a significant role in the fields of biology and ecology. Furthermore, through PRCC analysis, we have obtained the sensitivity relationships between the two population densities and the parameters. It is evident that the inclusion of diffusion and time delay has a more significant impact on the predator population density. Finally, numerical examples are introduced to validate the theoretical results.

The bifurcation studied in this paper is limited to one-dimensional space. To be more realistic, future research will continue to explore higher codimension branching problems, such as Turing-Hopf bifurcations of codimension two or even three, and their more complex dynamical phenomena. }We also will explore the analysis and comparison of alternative control strategies and their corresponding simulation scenarios, thereby enriching the development of the current stage of research.

Author contributions

Hairong Li: Conceptualization, Methodology, Software, Validation, Writing-original and editing; Yanling Tian: Conceptualization, Methodology, Validation, Writing-review, Supervision; Ting Huang: Software; Pinghua Yang: Supervision, Resources. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

The research was supported by Basic and Applied Basic Projects of Guangzhou, P. R. China (No. 202002030228); The Youth Innovation Talent Program of Education Department of Guangdong Province (No.2019KQNCX211); Scientific research project of Guangzhou City University of Technology (No. 56-K0223016; No.56-K0223006).

Conflict of interest

The authors declare no conflict of interest.