Global dynamics of a predator-prey model with prey-taxis and hunting cooperation

1.   Introduction

In natural habitats, there exist diverse interspecific interactions, such as competition, cooperation, predation, and so on. Predation serves as a fundamental interspecific relationship, which plays a pivotal role in maintaining ecological balance by regulating the populations of predators and prey. To capture the pattern of the quality of the species and the dynamics of predation, Lotka and Volterra presented a classical ordinary differential equation (ODE) model, called the predator-prey model, in the 1920s. Subsequently, many scholars began to study various predator-prey models with different predatory mechanisms, such as direct predation ^[1], selective predation ^[2,3], cooperative predation ^[4,5], and so on.

The random motion of species in space is a natural movement behavior, and scholars introduce random diffusion of predators and prey into the classical predator-prey model ^[6,7,8] to depict the spatiotemporal distribution of species. In addition, predators seek to improve their survival prospects by making directional movements toward areas with higher density of prey. To describe this phenomenon, the following partial differential equation model, called the prey-taxis model:

was proposed by Kareiva and Odell ^[9]. The functions $u\left(x, t\right)$ and $v\left(x, t\right)$ stand for the density of prey and predators, respectively. The parameters $d_i(i = 1, 2) > 0$ mean the random diffusion coefficients. The term $-\chi\nabla\cdot\left(v\nabla u\right)$ signifies the directional movements of the predator toward the prey with prey-taxis sensitivity coefficient $\chi > 0$. $f\left(u\right)$ measures the growth rate of prey, while $g\left(v\right)$ is the mortality rate of the predator. $F\left(u, v\right)$ denotes the interspecific interaction, also called the functional response function, and $b > 0$ indicates the conversion efficiency. In recent years, for model (1.1) with various functional response functions (for example, Holling type ^[10], ratio-dependent ^[11] and Beddington-DeAngelis type ^[12,13]), many scholars have achieved numerous results, involving topics such as the global existence, boundedness, stability, traveling waves, global bifurcation, and so on. Specifically, Jin and Wang ^[14] provided that the solution of (1.1) with some certain assumptions on the functions $F(u, v)$, $f(u)$, and $g(v)$ is globally bounded in two dimensions. In addition, they also investigated that the prey-only steady state of (1.1) with weak predation is globally asymptotically stable, while the coexistence steady state of (1.1) with strong predation and weak prey-taxis is globally asymptotically stable.

In nature, cooperative behavior of populations is a widespread and important phenomenon in ecosystems. To ensure their survival, reproduction, and development, predators often engage in cooperative hunting, such as wolves, African wild dogs, Harris' hawks, etc. In order to investigate the impact of hunting cooperation on the density of the predator and the dynamics of the ecological community, Alves and Hilker ^[4] discussed an ODE model, which reads

with various functional response functions $F\left(u, v\right)$, such as Lotka-Volterra type, Holling type II, etc. Subsequently, many scholars have conducted extensive research on the properties for the solutions of (1.2) with various functional response functions, including local existence, global boundedness, stability, global bifurcation, pattern formation. Moreover, scholars also have studied a wide range of mechanisms on (1.2), such as the Allee effect ^[15,16], spatial diffusion ^[17], time delay ^[18], and so on. Specifically, inspired by (1.1) and (1.2), Zhang et al. ^[19] studied a prey-taxis model with Holling type II hunting cooperative functional response function $F(u, v) = \frac{(1+av)v}{1+h(1+av)u}$, where $a > 0$ represents the intensity of cooperative hunting among predators and $h > 0$ is the average handing time of the predator for the prey, which is formulated as follows:

They demonstrated the uniform boundedness and global existence of time-varying solutions for (1.3). Concurrently, they also analyzed the stability and prey-taxis-driven instability of positive equilibrium through linearization analysis. When $\chi = 0$, Zhang ^[20] incorporated predator-taxis into (1.3) and established the global existence of the classical solution for (1.3) in any spatial dimension. Additionally, he analyzed the instability induced by predator-taxis.

Hunting cooperation not only affects the population size of predators, but also has a significant impact on their spatiotemporal distribution. Therefore, considering (1.1) with the Lotka-Volterra-type hunting cooperative functional response function $F(u, v) = (1+av)u$ (e.g., see ^[4]), we also introduce the random movement of species, the directional movement of predators toward prey, and the intra-specific competition within predator populations (e.g., see ^[14,21]) into (1.2). This derived

in a bounded smooth domain $\Omega\subset\mathbb{R}^2$. The constant $\sigma > 0$ is the intrinsic growth rate on prey, $\kappa > 0$ stands for the carrying capacity of prey, $\beta > 0$ means the mortality rate of predators, and $\gamma > 0$ denotes the mortality rate caused by intra-specific competition. The initial date satisfies

In what follows, without confusion, we shall abbreviate $\int_\Omega f dx$ as $\int_\Omega f$ for simplicity.

Main ideas and results: It is straightforward to achieve that $\|u\|_{L^\infty\left(\Omega\right)}$ is bounded by utilizing the comparison principle on parabolic equations. We remark that from a mathematical perspective, the analysis of the global boundedness of solutions for the Lotka-Volterra-type functional response function is more difficult than that for the Holling type II functional response function derived from (1.3). After all, the holling type II functional response function allows for the estimates $\left|\frac{(1+av)u}{1+h(1+av)u}\right|\leq \frac{1}{h}$ due to a priori estimates for $u$ and $v$. However, the Lotka-Volterra-type functional response function does not possess such a directly useful property for establishing the global boundedness of solutions. Therefore, to derive the $L^\infty(\Omega)$ estimate for $\nabla u$ in two-dimensional space, we need an a priori $\|v\|_{L^6(\Omega)}$ estimate. First, we construct an energy function $\int_\Omega v^2+\int_\Omega |\nabla u|^2$, which can be utilized to prove the boundness of $\|\nabla u\|_{L^2(\Omega)}$ and $\| v \|_{L^2(\Omega)}$ when $a$ is suitably small and also establish the boundedness of $\int_t^{t+\tau}\int_\Omega |\Delta u|^2$ and $\int_t^{t+\tau}\int_\Omega v^3$ for some appropriately small $\tau\in(0, 1]$. Based on the estimate of $\|v\|_{L^3(\Omega)}$, we construct the energy function $\int_\Omega |\nabla u|^4+\int_\Omega v^4$ and demonstrate that $v$ and $\nabla u$ are bounded in $L^4\left(\Omega\right)$, which can help to get the a priori estimate of $\|v\|_{L^6\left(\Omega\right)}$. So far, the estimates of $\|\nabla u\|_{L^{\infty}\left(\Omega\right)}$ and $\|v\|_{L^{\infty}\left(\Omega\right)}$ are proved by Neumann heat semigroups. Zhang et al. primarily discussed the stability of the coexistence equilibrium for (1.3) in ^[19]. In this paper, by constructing suitable Lyapunov functionals, we demonstrate the long-time behavior of the prey-only and coexistence steady state of (1.4) when $a$ falls within a specific range. These results mean that the weak hunting cooperation of predators can avoid population overcrowding, enrich the diversity of biological populations, and enhance ecological balance.

The first main result is as follows.

Theorem 1.1. Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain. Then $\left(u_0, v_0\right)$ satisfies (1.5), if

where $K_1: = \max\{\kappa, \|u_0\|_{L^\infty\left(\Omega\right)}\}$, and then (1.4) possesses a positive global classical solution

Furthermore, the solution satisfies

where constant $K > 0$ does not depend on $t$. In particular, we also have $0 < u\leq K_1$.

Our next result aims to present the impact of the cooperative hunting on the predator-prey model for its dynamic behavior. By a simple calculation, the constant equilibrium point $\left(u_s, v_s\right)$ of (1.4) satisfies

which admits three possible homogeneous equilibria:

$\bullet$ Extinction steady state $\left(0, 0\right)$.

$\bullet$ Prey-only steady state $\left(\kappa, 0\right)$.

$\bullet$ Coexistence steady state $\left(u^*, v^*\right)$.

Set

with $x \in (0, +\infty)$, and its derivative functions are

It is easy to verify that $h''\left(x\right) > 0$, which implies that $h'\left(x\right)$ is strictly monotonically increasing on $\left(0, \infty\right)$.

On the other hand, from the equations presented in (1.7), we can calculate that the equilibrium point $u^*$ satisfies

where $v^*$ is the positive root of the equation $h(x) = 0$. In order to compute the value of $v^*$, we undertake an analysis encompassing the following three cases:

Case 1: $\kappa > \beta$. It follows from $h\left(0\right) < 0$ and the Descartes' rule of signs^[22] that $h\left(x\right) = 0$ has a unique root $v_*\in(0, \sigma)$ and then $u_* = \frac{\beta+\gamma v_*}{1+av_*}\in(\beta, \beta+\gamma\sigma)$.

Case 2: $\kappa = \beta$. It holds that $h\left(0\right) = 0$. When $a\leq \frac{\sigma\gamma+\kappa}{\sigma\kappa}$, $h'\left(x\right) > 0$. Thus $h\left(x\right) = 0$ does not have a positive root. If $a > \frac{\sigma\gamma+\kappa}{\sigma\kappa}$, this deduces that $h'\left(x\right) < 0$ on $x\in\left(0, v_{1}\right)$ and $h'\left(x\right) > 0$ on $x\in\left(v_{1}, +\infty\right)$, where

Therefore, $h\left(x\right) = 0$ has a unique root $v_2 = \frac{\sqrt{\sigma\kappa\left(a\kappa-\gamma\right)}-\kappa}{a\kappa} > 0$.

Case 3: $\kappa < \beta$. Note that $h\left(0\right) > 0$. If $a\leq\frac{\sigma\gamma+\kappa}{\sigma\kappa}$, $h\left(x\right) = 0$ has no positive root. While $a > \frac{\sigma\gamma+\kappa}{\sigma\kappa}$, $h\left(x\right) = 0$ admits three statuses: no positive root as $h\left(v_{1}\right) > 0$, one positive root $v_1$ as $h\left(v_{1}\right) = 0$, and two positive roots $v_3, v_4$ as $h\left(v_{1}\right) < 0$, where $v_3\in\left(0, v_{1}\right)$ and $v_4\in\left(v_{1}, \sigma\right)$.

All in all, the coexistence steady state of (1.4) satisfies

Theorem 1.2. Suppose that the assumptions of Theorem 1.1 hold. Then:

1) Let $\kappa\leq\beta$. If the model parameters satisfy

then for all $t > 0$, the classical solution $\left(u, v\right)$ in (1.4) converges to $\left(\kappa, 0\right)$ in an exponential manner, as described below:

where constants $C > 0$ and $\lambda > 0$ are independent of $t$.

2) Let $\kappa > \beta$. If the model parameters satisfy

and

where $u_*$ and $v_*$ are independent of $\chi$, then for all $t > 0$, the classical solution $\left(u, v\right)$ converges to $\left(u_*, v_*\right)$ in an exponential manner, as described below:

where constants $C > 0$ and $\lambda > 0$ are independent of $t$.

Remark 1.1. Without the hunting cooperation (i.e., $a = 0$), our results of Theorem 1.2 are consistent with the results of Proposition 1.6 in ^[14]. In fact, Theorem 1.2 shows that the hunting cooperation mechanism does not change the stability of (1.4) solutions when a is appropriately small.

Remark 1.2. When $\kappa\leq\beta$, we are unable to verify the long-time behavior of the coexistence steady state $\left(u_*, v_*\right)$ on account of the range of the intensity of cooperative hunting among predators conflicts with the condition of Theorem 1.1. However, for one-dimensional cases, the long-time behavior of the coexistence steady state is an open problem when $\kappa\leq\beta$ and $a$ is appropriately large.

2.   Preliminaries

First, we provide the local existence of solutions for (1.4).

Lemma 2.1. Provided that $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain, and $\left(u_0, v_0\right)$ satisfies (1.5), if the condition (1.6) holds, there exists $T_{max}\in\left(0, \infty\right]$ ensuring that (1.4) possesses a classical solution

satisfying $u, v > 0$. Furthermore, if $T_{max} < \infty$, then

The conclusions of Lemma 2.1 are established from Amann's theorem ^[23,24].

Lemma 2.2. Provided that the assumptions of Lemma 2.1 hold, then for all $t\in(0, T_{max})$, we have

where $K_1: = \max\{\kappa, \|u_0\|_{L^\infty(\Omega)}\}$.

Proof. This lemma has been proven by using an approach similar to ^[25]. □

The following lemma gives a fundamental inequality.

Lemma 2.3. ^[26] Assume that $\Omega$ is a smooth bounded domain, and let $g\in C^2\left(\bar{\Omega}\right)$ satisfy $\frac{\partial g}{\partial\nu} = 0$ on $\partial\Omega$. Then there exists an upper bound $l = l(\Omega)$ of the curvatures of $\partial\Omega$ guaranteeing that

In order to prove Lemma 3.3, we need a lemma.

Lemma 2.4. ^[27] Suppose that $T > 0, \tau\in\left(0, T\right), m_1 > 0,$ and $m_2 > 0$. Provided that $\varphi:[0, T)\rightarrow [0, \infty)$ is absolutely continuous, and satisfies

where the constant $\theta > 0$, the functions $\phi\left(t\right), \psi\left(t\right)\in L_{loc}^1\left([0, T)\right)$ are nonnegative and

Then we can obtain

and

where

3.   Global boundedness

This section is devoted to establishing Theorem 1.1.

Lemma 3.1. Provided that the assumptions of Lemma 2.1 are fulfilled, if the condition (1.6) holds, then for all $t\in(0, T_{max})$, the classical solution $(u, v)$ of (1.4) satisfies

where constant $K_2 > 0$ does not depend on $t$.

Proof. (1.4) implies

Then, using Young's inequality yields

Combining (3.3) and (3.2), we deduce

which implies (3.1) by ODE comparison. □

Then we prove that $\|\nabla u\|_{L^2(\Omega)}$ and $\|v\|_{L^2(\Omega)}$ are bounded.

Lemma 3.2. Provided that the assumptions of Lemma 2.1 are fulfilled, if the condition (1.6) holds, then for all $t\in(0, T_{max})$, the classical solution $(u, v)$ of (1.4) satisfies

and for all $t\in\left(0, T_{max}-\tau\right)$, it holds that

where constants $K_3 > 0$ and $K_4 > 0$ do not depend on $t$.

Proof. Intergating the sum of $-\frac{\Delta u}{u}$ times the first equation of (1.4) by parts, we have

Note that

which substituted into (3.6) gives us

Using $\nabla u\cdot \nabla\Delta u = \frac{1}{2}\Delta|\nabla u|^2-|D^2u|^2$, we conclude

Note that

Combining (3.8) and (3.9) yields

Bringing (3.10) into (3.7), it holds that

Multiplying both sides of the second equation of (1.4) by $\frac{2av}{\chi}$, we derive

Utilizing the condition $a < \frac{\gamma}{K_1}$, (3.11), and (3.12), one has

In the light of the inequality $\int_\Omega u|D^2\ln u|^2\geq C_1 \left(\int_\Omega \frac{|D^2 u|^2}{u}+\int_\Omega \frac{|\nabla u|^4}{u^3}\right)$ with the constant $C_1 > 0$, plugged into (3.13) gives

Combining the trace inequality ^[28], the Cauchy-Schwarz inequality, and Lemma 2.3, we have

where constant $C_2 > 0$. According to (2.1) and Hölder's inequality, then

where constant $C_3 > 0$. Ustilizing $|\Delta u|\leq\sqrt{2}|D^2 u|$, (2.1), and Young's inequality, we achieve

We infer from the Gagliardo-Nirenberg inequality and (3.1) that

where constants $C_i \; (i = 4, 5, 6) > 0$. Substituting (3.14)–(3.18) and $|\Delta u|\leq \sqrt{2}|D^2u|$ yields

which gives (3.4) by the fact $0 < u\leq K_1$ and ODE comparison. Then integrating (3.19) over $(t, t+\tau)$ gives (3.5). □

Next, we get the a priori estimate of $\|v\|_{L^3\left(\Omega\right)}$.

Lemma 3.3. Provided that the assumptions of Lemma 2.1 are fulfilled, if the condition (1.6) holds, then, the classical solution $(u, v)$ of (1.4) satisfies

where constant $K_5 > 0$ does not depend on $t$.

Proof. By direct computations, we have

Applying the Gagliardo-Nirenberg inequality as $n = 2$ yields

and

where constants $C_1 > 0$ and $C_2 > 0$. Thanks to (3.22), (3.23), and Young's inequality, one gets

where constants $C_3 > 0$ and $C_4 > 0$. Substituting (3.24) into (3.21) results in

which gives (3.20) by (3.5) and Lemma 2.4. □

Subsequently, we achieve the bounds $\|v\|_{L^4\left(\Omega\right)}$ and $\|\nabla u\|_{L^4\left(\Omega\right)}$.

Lemma 3.4. Provided that the assumptions of Lemma 2.1 are fulfilled, if the condition (1.6) holds, then the classical solution $(u, v)$ of (1.4) satisfies

where constant $K_6 > 0$ does not depend on $t$.

Proof. Multiplying both side of the second equation of (1.4) by $v^3$, we deduce

Integrating the first equation of (1.4), while taking into account the fact that $0 < u\leq K_1$, it follows that

Thanks to the trace inequality and Lemma 2.3, we know

Utilizing $|\Delta u|\leq \sqrt{2}|D^2 u|$ and $\nabla|\nabla u|^2 = 2D^2u\cdot\nabla u$ gives rise to

and

The Gagliardo-Nirenberg inequality implies

where constants $C_2 > 0$ and $C_3 > 0$. Adding (3.27)–(3.31) and (3.26), we conclude

where constant $C_4 > 0$. Applying (3.4) and the Gagliardo-Nirenberg inequality yields

where constants $C_5 > 0$ and $C_6 > 0$. In addition,

where constant $C_7 > 0$. Applying Young's inequality along with (3.34), one has

where constants $C_8 > 0$ and $C_9 > 0$. Utilizing the Gagliardo-Nirenberg inequality and Lemma 3.3 yields

where constants $C_{10} > 0$ and $C_{11} > 0$. Plugging (3.33), (3.35), and (3.36) into (3.32), there exists a constant $C_{12} = C_3+C_6+C_9+C_{11}$ such that

which in conjunction with ODE comparison results in (3.25). □

Lemma 3.5. Provided that the assumptions of Lemma 2.1 are fulfilled, if the condition (1.6) holds, then for all $t\in(0, T_{max})$, we can get

where constant $K_7 > 0$ does not depend on $t$.

Proof. We can directly compute

Utilizing the Gagliardo-Nirenberg inequality, it follows that

where constants $C_1 > 0$ and $C_2 > 0$. Then

where constants $C_3 > 0$ and $C_4 > 0$. Plugging (3.39) and (3.40) into (3.38), we conclude

which results in (3.37) with ODE comparison. □

We are currently capable of deducing the bounded property of $\|\nabla u\|_{L^{\infty}\left(\Omega\right)}$ and $\|v\|_{L^{\infty}\left(\Omega\right)}$ in the case where the dimension $n = 2$.

Lemma 3.6. Provided that the assumptions of Lemma 2.1 are fulfilled, if the condition (1.6) holds, then for all $t\in(0, T_{max})$, we have

and

where constants $K_8 > 0$ and $K_9 > 0$ do not depend on $t$.

Proof. The variation-of-constants formula implies

and hence

Then by (3.1), (3.20), and (3.37), it holds that

Applying the Neumann heat semigroup ^[29] and (3.43), there exist constants $\gamma_1 > 0$ and $\lambda_1 > 0$ ensuring that

which gives (3.41) and gamma function $\Gamma\left(z\right) = \int_0^\infty t^{z-1}e^{-t}dt$. We use (3.20) and (3.41) to obtain

and combining with (3.1), (3.20), and (3.37), we conclude

Applying the Neumann heat semigroup ^[29], (3.44), and(3.45), there exist constants $\gamma_2 > 0$, $\gamma_3 > 0$, and $\lambda_1 > 0$ ensuring that

which gives (3.42). □

Proof of Theorem 1.1. Combining Lemma 2.2 and Lemma 3.6, then, there exists a positive constant $K$ which guarantees that

which proves Theorem 1.1 with the extension criterion outlined in Lemma 2.1.

4.   Long time behavior

Based on Theorem 1.1, this part aims to establish the convergence property of the solution. Before this, we introduce a lemma as follows.

Lemma 4.1. For all given $\theta\in \left(0, 1\right)$, we have

with a constant $C > 0$.

Proof. The derived conclusion stems directly from the regularity properties of parabolic equations as outlined in ^[30]. □

4.1.   The prey-only steady state: $\left(\kappa, 0\right)$

In this part, we will demonstrate that the solution $(u, v)$ converges to $\left(\kappa, 0\right)$ under certain conditions. To achieve this aim, we introduce the Lyapunov functional, denoted as

Lemma 4.2. Under the assumed conditions of Theorem 1.1, if the parameter satisfies $\kappa\leq \beta$ and $a < \frac{\gamma}{K_1}$, this ensures that

where constants $K_{10} > 0$ and $\delta_1 > 0$.

Proof. From (1.4), we have

and

Adding (4.1) and (4.2) results in

Then, due to $\kappa\leq\beta$ and $a < \frac{\gamma}{K_1}$, one can choose a constant $c_1 > 0$ such that

which provides

where constant $c_2 > 0$. Using (4.4) and the uniform continuity of $u$ and $v$ due to Lemma 4.1 yields

Utilizing the Gagliardo-Nirenberg inequality, for all $t > 1$, we derive

and

where constants $c_3 > 0$ and $c_4 > 0$, which proves the claim with Lemma 4.1 and (4.4). Furthermore, applying L'Hoptial's rule, it holds that

There exists a constant $\varepsilon > 0$ ensuring that

By (4.5) and (4.6), there exists $t_0 > 0$ ensuring that

Therefore, by (4.7), we get

which gives

where constant $c_5 > 0$. Plugging (4.8) into (4.3), it follows that

Therefore, we obtain

with some constants $c_6 > 0$ and $\delta_1 > 0$. □

4.2.   The coexistence steady state: $\left(u_*, v_*\right)$

In this part, we will demonstrate that the solution $(u, v)$ converges to $\left(u_*, v_*\right)$ under certain conditions. To achieve this aim, we define the Lyapunov functional

where $u_*\in\left(\beta, \beta+\gamma\sigma\right)$ and $v_*\in\left(0, \sigma\right)$.

Lemma 4.3. Under the assumed conditions of Theorem 1.1, if the parameters satisfy $\kappa > \beta$ and

as well as

this ensures that

where constants $K_{11} > 0$ and $\delta_2 > 0$.

Proof. Using $\sigma\left(1-\frac{u_*}{\kappa}\right)-\left(1+av_*\right)v_* = 0$, it can be directly calculated that

Utilizing $\left(1+av_*\right)u_*-\beta-\gamma v_* = 0$, we have

Combining (4.11) and (4.12), it follows that

where $X = \left(\frac{\nabla u}{u}, \frac{\nabla v}{v}\right)$, $Y = \left(u-u_*, v-v_*\right)$, and the matrices $Q$ and $M$ stand for

and

If (4.9) and (4.10) hold, we check

and

which means that the matrices $Q$ and $M$ are positive definite with $K_1: = \max\{\kappa, \|u_0\|_{L^{\infty}\left(\Omega\right)}\}$. Then for all $u, v$, it implies that

where constants $c_1 > 0$ and $c_2 > 0$. Subsequently, the rest is analogous to the reasoning employed in Lemma 4.2, and we readily demonstrate that the solution $(u, v)$ exponentially converges to $(u_*, v_*)$ as $t\rightarrow \infty$ in $L^\infty(\Omega)$. □

Proof of Theorem 1.2. Combining Lemmas 4.2 and 4.3, we directly obtain Theorem 1.2.

5.   Conclusions

In this paper, we have proposed a prey-taxis model with hunting cooperation mechanism and explored the effect of predator hunting cooperation mechanism on predator and prey populations. Through mathematical analysis, we have demonstrated that weak cooperative hunting can prevent the blow-up of the classical solutions for model (1.4) in two-dimensional space. On the one hand, we have received that the long time behavior of the prey-only steady state is established under the weak hunting cooperation. On the other hand, we have only obtained the long time behavior for the coexistence steady states under weaker hunting cooperation. However, for the strong hunting cooperation mechanism, whether the long time behavior of the coexistence steady state can be established remains an open problem that necessitates further research.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by the Science and Technology Project of YiLi Prefecture (Grant No. YZ2022B038), the Special Project of Yili Normal University to Improve Comprehensive Strength of Disciplines (Grant No. 22XKZY14), and the Natural Science Foundation of Xinjiang Autonomous Region (Grant No: 2022D01C335).

Conflict of interest

The authors declare that there is no conflict of interest.