Spatiotemporal Dynamics in a Time-Delayed Rhizophagus grandis-Dendroctonus valens Predator-Prey System with Nonlocal Diffusion and Taxis

The objective of this study is to investigate the spatiotemporal dynamics of a Rhizophagus grandis-Dendroctonus valens predator-prey system. Under Neumann boundary conditions, we establish a reaction-diffusion system incorporating Holling-II functional response, nonlocal effect, predator-prey taxis and time delay to elucidate the regulatory mechanisms of chemotactic behaviors and delayed predation pressure on ecosystem stability. In addition, we investigate the existence conditions for Hopf bifurcation and employ the multiple time scales method to derive its normal form. Under appropriate parameter constraints, numerical simulations demonstrate that the system exclusively exhibits Hopf bifurcation in the current parameter configuration, where nonlocal interactions, taxis and time delays synergistically induce the formation of spatially heterogeneous periodic solutions. These results elucidate the intrinsic mechanism underlying periodic pest outbreaks and provide a theoretical foundation for optimizing biological control strategies. Notably, our analysis reveals the critical role of predator taxis capability and delayed predation feedback in stabilizing pest population cycles, offering practical guidance for sustainable forest pest management.

• Predator-Prey
• Hopf Bifurcation
• Time Delay
• Nonlocal
• Taxis

1. Introduction

Dendroctonus valens is a globally distributed forest pest exhibiting significant ecological role divergence across regions. Research has demonstrated that Dendroctonus valens functions as a secondary pest of Pinus spp. across its native range spanning the United States, Mexico and Canada, where it typically coexists with competitively dominant bark beetle species [1] . Stand mortality or large-scale outbreaks attributed solely to Dendroctonus valens are statistically negligible in native ecosystems, with economic impacts confined to timber industry losses. However, since its initial detection in Jincheng, Shanxi Province, China, the invasive population has undergone a behavioral phase transition, exhibiting exponential spread to adjacent regions including Hebei, Henan, Beijing and other neighboring areas, resulting in a cumulative death toll exceeding 10 million host trees [2] . The current climatically suitable habitats for Dendroctonus valens in China are concentrated in southern and northern regions, yet under global warming scenarios, the potential distribution demonstrates a monotonic expansion trend: an outbreak in the Liaoning-Inner Mongolia border region in 2017 confirmed its invasion into northeastern China [3] . The infestation dynamics follow a distinct mechanistic pattern as illustrated in Figure 1 : adults colonize pine trunks through wounds, oviposit between the phloem, cambium and disrupt vascular transport through feeding, inducing host mortality within a two-year time horizon. This delayed mortality mechanism permits multiple larval cohorts to complete development, while adult dispersal to neighboring healthy trees generates a positive feedback loop sustaining infestation cycles.

Traditional control methods, such as aluminum phosphide fumigation and trunk spraying, are partially effective but risk environmental pollution and harm to beneficial insects. In the context of low-carbon environmental protection, biological control has gained prominence. Early studies in Georgia demonstrated the efficacy of Rhizophagus grandis against Dendroctonus micans. Yang et al. [4] proposed using Rhizophagus grandis to control Dendroctonus valens. Zhao et al. [5] explored Braconidae as supplementary agents to control it. However, predatory natural enemies exhibit superior long-term control due to their host-specificity and adaptability.

Mathematical ecology provides a powerful analytical framework for deciphering predator-prey dynamics. From the Malthusian exponential growth model and Verhulst’s resource-limited logistic framework to the Lotka-Volterra system describing interspecific interactions, the development of mathematical models has continued to deepen our understanding of ecological complexity [6] . Modern theoretical advancements incorporating functional response functions, reaction-diffusion systems, and nonlocal effects enable precise characterization of spatial heterogeneity in ecosystems. Holling’s experimental derivation of functional responses addresses the omission of predator handling time and conversion efficiency in classical models [7] . Rabia et al. [8] revealed bifurcation-chaos dynamics and their ecological implications in discrete Holling-II systems through stability analysis. In the realm of reaction-diffusion systems, PDE outperform ODE in resolving spatial heterogeneity. For example, Du et al.’s Lotka-Volterra competition model with nonlocal diffusion and free boundaries, combined with parameter $k$ analysis, elucidated long-term survival-extinction mechanisms of invasive species during boundary expansion [9] . Víctor et al. [10] analyzed the persistence of three species in a three-level food chain model, illustrating that the existence of a zero-Hopf equilibrium occurs exclusively under Holling type III or IV functional responses. Wang’s reaction-diffusion model with spatial memory uncovers Turing-Hopf bifurcation mechanisms and complex spatiotemporal patterns driven by the synergy of perceptual scale and memory delay [11] . Song et al. [12] confirmed the existence of spatially inhomogeneous periodic solutions and heteroclinic orbits through integrating nonlocal interactions with time delays.

It is crucial to emphasize that when analyzing the behavior of complex dynamical systems, normal form analysis serves as a fundamental methodology, with the center manifold deduction [13] - [15] and the multiple scales method [16] [17] being two primary techniques for deriving normal forms. Ding et al. [18] rigorously proved the equivalence between third-order Hopf bifurcation normal forms obtained via MTS and CMR for abstract diffusive equations.

Building upon these theoretical advances, this study investigates Rhizophagus grandis-mediated biocontrol of Dendroctonus valens through the following structure. In Section 2, we establish a spatiotemporal dynamical model integrating nonlocal competition, prey-taxis behavior and time delay effects. In Section 3, we conduct stability analysis of equilibrium points to derive sufficient conditions for Hopf bifurcation occurrence. In Section 4, we employ the multiple scales method to derive the normal form, thereby revealing critical bifurcation characteristics. In Section 5, we perform numerical simulations under biologically feasible parameter configurations to validate theoretical predictions and provide ecological interpretations. A conclusion is given in Section 6.

Figure 1

Figure 1. Schematic diagram of Dendroctonus valens infesting pine trees.

2. Model Construction

Considering the predation mechanism of the natural enemy Rhizophagus grandis on Dendroctonus valens, the classical Lotka-Volterra model is formulated as:

$\{\begin{array}{l}\frac{\text{d}x}{\text{d}t}=x\left(a-by\right)\mathrm{,}\\ \frac{\text{d}y}{\text{d}t}=y\left(kx-l\right)\mathrm{,}\end{array}$(1)

where $a$, $b$, $k$ and $l$ are positive constants, $y$ denotes the predator species and $x$ represents its prey.

To reflect the processing time of predators and other factors, we incorporate Holling-II functional responses [19] . Since Dendroctonus valens has very strong mobility and it can locate hosts by the smell of their prey’s feces and migrate by flying or crawling to find the larvae of Dendroctonus valens [4] . Therefore, inspired by [20] [21] [22] , we introduce the prey-taxis $-\eta \cdot \nabla \left(v\nabla u\right)$ and

$\hat{u}={\displaystyle {\int }_{\Omega }}G\left(x\mathrm{,}y\right)u\left(y\mathrm{,}t\right)\text{d}y$ into our model, where $G\left(x\mathrm{,}y\right)=\frac{1}{\left|\Omega \right|}$ can be justified by averaged density in $\Omega =\left(\mathrm{0,}\pi \right)$.

In this study, we consider the mechanism by which Rhizophagus grandis preys on Dendroctonus valens. Literature indicates that Rhizophagus grandis feeds main on the eggs, larvae and pupae of Dendroctonus valens [4] . In order to incorporate the time-delayed effect of egg predation behavior on the population decline of Dendroctonus valens, we introduce a time delay term into the functional response [23] [24] . Consistent with biocontrol practices, natural enemies are released at pest distribution edges, prompting the use of homogeneous Neumann boundary conditions [5] . The final optimized spatiotemporal model is formulated as:

$\{\begin{array}{l}\frac{\partial u\left(x,t\right)}{\partial t}=d_1\Delta u+ru\left(1-\frac{\hat{u}}{K}\right)-\frac{\alpha u\left(x,t-\tau \right)v\left(x,t-\tau \right)}{1+\alpha u\left(x,t-\tau \right)h},x\in \Omega ,t>0,\\ \frac{\partial v\left(x,t\right)}{\partial t}=d_2\Delta v-d_{3}v+\frac{\beta \alpha uv}{1+\alpha uh}-\eta \cdot \nabla \left(v\nabla u\right),x\in \Omega ,t>0,\\ \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0,x\in \partial \Omega ,t\ge 0,\\ u\left(x,t\right)={\phi }_u\ge 0v\left(x,t\right)={\phi }_v\ge 0,x\in \Omega ,t\in \left[-\tau ,0\right],\end{array}$(2)

where $u\left(x\mathrm{,}t\right)$, $v\left(x\mathrm{,}t\right)$ describe the density of the prey/predator, respectively, at position $x$ and time $t$, $d_1\mathrm{<mo>\; >\; </mo>0}$, $d_2\mathrm{<mo>\; >\; </mo>0}$ are the diffusion coefficients of Dendroctonus valens and Rhizophagus grandis, respectively, $\Delta$ is the Laplacian operator, $d_3$ is the Rhizophagus grandis mortality, $r$ is intrinsic growth rate, $K$ is environmental capacity, $h$ is the time it takes a predator to hunt and process its prey, $\eta$ is the identification diffusion coefficients of Rhizophagus grandis, $\beta$ is the energy conversion efficiency and $n$ is the outer normal vector of the boundary $\partial \Omega$.

3. Stability and Bifurcation Analysis

System (2) admits two boundary equilibria $E_0=\left(\mathrm{0,0}\right)$ and $E_1=\left(K\mathrm{,0}\right)$. If the following assumption holds:

(H₀) $K>\frac{d_3}{\alpha \left(\beta -d_{3}h\right)}>0,$

then system (2) must have a positive constant steady equilibrium $E_{\mathrm{<mo>\; *\; </mo>}}=\left(u_{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{v}_{\mathrm{<mo>\; *\; </mo>}}\right)$, with

$0<u_{*}=\frac{d_3}{\alpha \beta -\alpha d_{3}h}<K$, $v_{*}=\frac{\beta }{d_3}\left[r{u}_{*}\left(1-\frac{u_{*}}{K}\right)\right]$.

The linearized system is given as:

$\begin{array}{c}\frac{\partial }{\partial t}\left(\begin{array}{c}u\left(x,t\right)\\ v\left(x,t\right)\end{array}\right)=J_1\left(\begin{array}{c}\Delta u\left(x,t\right)\\ \Delta v\left(x,t\right)\end{array}\right)+J_2\left(\begin{array}{c}{\left(u{v}_x\right)}_x\\ {\left(v{u}_x\right)}_x\end{array}\right)+J_3\left(\begin{array}{c}u\left(x,t\right)\\ v\left(x,t\right)\end{array}\right)\\ +J_4\left(\begin{array}{c}\hat{u}\left(x,t\right)\\ \hat{v}\left(x,t\right)\end{array}\right)+J_5\left(\begin{array}{c}u\left(x,t-\tau \right)\\ v\left(x,t-\tau \right)\end{array}\right),\end{array}$(3)

where

$\begin{array}{l}{J}_1=\left(\begin{array}{cc}{d}_1& 0\\ 0& d_2\end{array}\right),J_2=\left(\begin{array}{cc}0& 0\\ -\eta v_{*}& 0\end{array}\right),J_3=\left(\begin{array}{cc}{a}_{11}& 0\\ a_{21}& 0\end{array}\right),\\ J_4=\left(\begin{array}{cc}{b}_{11}& 0\\ 0& 0\end{array}\right),J_5=\left(\begin{array}{cc}{c}_{11}& c_{12}\\ 0& 0\end{array}\right),\end{array}$(4)

with

$\begin{array}{l}{a}_{11}=r\left(1-\frac{u_{*}}{K}\right),a_{21}=\frac{\beta \alpha v_{*}}{{\left(1+\alpha u_{*}h\right)}^2},b_{11}=-\frac{r{u}_{*}}{K},\\ c_{11}=-\frac{\alpha v_{*}}{{\left(1+\alpha u_{*}h\right)}^2},c_{12}=-\frac{\alpha u_{*}}{1+\alpha u_{*}h}.\end{array}$(5)

Hence, the characteristic system of (2) at $E=\left(u_{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{v}_{\mathrm{<mo>\; *\; </mo>}}\right)$ is given as follows:

${\lambda }^2+A_n\lambda +B_n+C_n{\text{e}}^{-\lambda \tau }+D_n\lambda {\text{e}}^{-\lambda \tau }=0,$(6)

where

$\{\begin{array}{l}{A}_n=\left(d_1+d_2\right)n^2-a_{11}-b_{11}{\delta }_n,\\ B_n=d_1{d}_2{n}^4-a_{11}{d}_2{n}^2,\\ C_n=-\left(c_{12}\eta v_{*}+c_{11}{d}_2\right)n^2-a_{21}{c}_{12},\\ D_n=-c_{11},\end{array}$

with

$\delta \left(n\right)=\{\begin{array}{ll}1,\hfill & n=0,\hfill \\ 0,\hfill & n\in {\mathbb{Z}}^{+}.\hfill \end{array}$

When $\tau =0$, Equation (6) becomes ${\lambda }^2-\left(A_n+D_n\right)\lambda +\left(B_n+C_n\right)=0$. For the equilibrium $E_{\mathrm{<mo>\; *\; </mo>}}$ to be locally asymptotically stable, two conditions must simultaneously hold for all $n\in \mathbb{N}$: (i) $A_n+D_n>0$, (ii) $B_n+C_n>0$.

For condition (i), we observe that

$A_n+D_n=\left(d_1+d_2\right)n^2-\left(a_{11}+c_{11}\right)>-\left(a_{11}+c_{11}\right)>0$, which directly yields the parameter constraint $K>\frac{d_3{\left(1+\alpha u_{*}h\right)}^2}{\beta {\alpha }^{2}h{u}_{*}\left(1-\frac{u_{*}}{K}\right)}$. Regarding condition (ii), $g\left(n\right)\triangleq B_n+C_n=d_1{d}_2{n}^4-\left(a_{11}{d}_2+c_{11}{d}_2+c_{12}\eta u_{*}\right)n^2-a_{21}{c}_{12}>0$ requires careful analysis. Let $\varsigma =n^2$, we obtain $w\left(\varsigma \right)=d_1{d}_2{\varsigma }^2-\left(a_{11}{d}_2+c_{11}{d}_2+c_{12}\eta u_{\mathrm{<mo>\; *\; </mo>}}\right)\varsigma -a_{21}{c}_{12}$. When the discriminant of $w\left(\varsigma \right)$, doneted as ${\Delta }_1={\left(a_{11}{d}_2+c_{11}{d}_2+c_{12}\eta u_{*}\right)}^2+4d_1{d}_2{a}_{21}{c}_{12}<0$, the inequality automatically holds. When the discriminant ${\Delta }_1\ge 0$, we analyze its symmetry axis given by $\zeta =\frac{a_{11}{d}_2+c_{11}{d}_2+c_{12}\eta u_{\mathrm{<mo>\; *\; </mo>}}}{2d_1{d}_2}$. If $\zeta <0$, that is $\eta >\frac{\alpha d_2{v}_{*}h}{1+\alpha u_{*}h}$, since $\varsigma =n^2>0$, we can obviously find that the inequality holds for all $n\in \mathbb{N}$. If $\zeta >0$, that is $\eta <\frac{\alpha d_2{v}_{*}h}{1+\alpha u_{*}h}$, we need to meet the following conditions: $g\left(\lceil \sqrt{\zeta }\rceil \right)>0$ and $g\left(\lfloor \sqrt{\zeta }\rfloor \right)>0$.

In summary, we establish three mutually exclusive stability hypotheses:

(H₁₁) $\{\begin{array}{l}K>\frac{d_3{\left(1+\alpha u_{*}h\right)}^2}{\beta {\alpha }^{2}h{u}_{*}\left(1-\frac{u_{*}}{K}\right)},\\ {\Delta }_1<0,\end{array}$(7)

(H₁₂) $\{\begin{array}{l}K>\frac{d_3{\left(1+\alpha u_{*}h\right)}^2}{\beta {\alpha }^{2}h{u}_{*}\left(1-\frac{u_{*}}{K}\right)},\\ {\Delta }_1\ge 0,\\ \eta >\frac{\alpha d_2{v}_{*}h}{1+\alpha u_{*}h},\end{array}$(8)

(H₁₃) $\{\begin{array}{l}K>\frac{d_3{\left(1+\alpha u_{*}h\right)}^2}{\beta {\alpha }^{2}h{u}_{*}\left(1-\frac{u_{*}}{K}\right)},\\ {\Delta }_1\ge 0,\\ \eta <\frac{\alpha d_2{v}_{*}h}{1+\alpha u_{*}h},\\ f\left(\lceil \sqrt{\zeta }\rceil \right)>0,\mathrm{<mi>\; </mi>}f\left(\lfloor \sqrt{\zeta }\rfloor \right)>0.\end{array}$(9)

Considering $\tau >0$ and substituting $\lambda =\pm i\omega$ ( $\omega >0$) into Equation (6), we separate the real and imaginary parts to obtain the following results:

$\{\begin{array}{l}\cos \omega \tau =\frac{\left(C_n-A_n{D}_n\right){\omega }^2-B_n{C}_n}{C_n^2+D_n^2{\omega }^2},\\ \sin \omega \tau =\frac{D_n{\omega }^3+\left(A_n{C}_n-B_n{D}_n\right)\omega }{C_n^2+D_n^2{\omega }^2},\end{array}$(10)

with $n=\mathrm{0,1,2}\cdots$.

When both sides of the system are squared and then summed, the following result is obtained:

${\omega }^4+\left(A_n^2-2B_n-D_n^2\right){\omega }^2+B_n^2-C_n^2=0.$(11)

Let $z={\omega }^2$, we obtain

$h\left(z\right)\triangleq z^2+\left(A_n^2-2B_n-D_n^2\right)z+B_n^2-C_n^2=0,$(12)

and the discriminant is $\Delta ={\left(A_n^2-2B_n-D_n^2\right)}^2-4\left(B_n^2-C_n^2\right)$.

Assuming $\Delta \ne 0$, since $h\left(z\right)=0$, we can obtain

$\{\begin{array}{l}{\omega }_{n,1}=\sqrt{z_{n,1}}=\frac{\sqrt{2}}{2}{\left[D_n^2+2B_n-A_n^2+\sqrt{\Delta }\right]}^{\frac{1}{2}},\\ {\omega }_{n,2}=\sqrt{z_{n,2}}=\frac{\sqrt{2}}{2}{\left[D_n^2+2B_n-A_n^2-\sqrt{\Delta }\right]}^{\frac{1}{2}}.\end{array}$(13)

Thereby,

${\tau }_n^{\left(j\right)}=\{\begin{array}{l}\frac{1}{{\omega }_{n,i}}\left(\arccos \frac{\left(C_n-A_n{D}_n\right){\omega }^2-B_n{C}_n}{C_n^2+D_n^2{\omega }^2}+2j\pi \right),\sin \omega \tau >0,\\ \frac{1}{{\omega }_{n,i}}\left(2\pi -\arccos \frac{\left(C_n-A_n{D}_n\right){\omega }^2-B_n{C}_n}{C_n^2+D_n^2{\omega }^2}+2j\pi \right),\sin \omega \tau <0,\end{array}$(14)

with $j\in \mathbb{N}$, $i=\mathrm{1,2}$.

Assume that the finite or infinite sets

$\begin{array}{l}{I}_1=\left\{n\in \mathbb{N}|\Delta >0,B_n-C_n<0\right\},\\ I_2=\left\{n\in \mathbb{N}|\Delta >0,B_n-C_n>0,A_n^2-2B_n-D_n^2<0\right\},\end{array}$(15)

respectively. Denote

${\tau }_c=\min \left\{{\tau }_{*}|{\tau }_{*}={\tau }_{n,i}^{\left(j\right)},n\in I_1\text{or}n\in I_2\right\}.$(16)

Taking $\tau$ as a bifurcation parameter, we have

$\mathrm{Re}\left[{\frac{\text{d}\tau }{\text{d}\lambda }|}_{\lambda =\pm i\omega }\right]=\frac{h^{\prime }\left(z_{n,i}\right)}{D_n^2{\omega }^2+C_n^2}\{\begin{array}{ll}>0,\hfill & h^{\prime }\left(z_{n,1}\right)>0,\hfill \\ <0,\hfill & h^{\prime }\left(z_{n,2}\right)<0.\hfill \end{array}$(17)

In summary, we can get the following conclusions.

Theorem 1. Suppose (H₁₁), (H₁₂) or (H₁₃) holds, for system (2), the following statements hold true.

1) If $I_1=\varnothing$ and $I_2=\varnothing$, then the positive constant steady state $E_{\mathrm{<mo>\; *\; </mo>}}$ is asymptotically stable for all $\tau \ge 0$.

2) If $I_1\ne \varnothing$ and $I_2=\varnothing$, then Equation (12) has only one positive root $z_{n\mathrm{,1}}$ for some $n\in I_1$, which implies that the positive constant steady state $E_{\mathrm{<mo>\; *\; </mo>}}$ is asymptotically stable for $0<\tau <{\tau }_c$, unstable for $\tau >{\tau }_c$ and system (2) undergoes Hopf bifurcation near $E_{\mathrm{<mo>\; *\; </mo>}}$ at $\tau ={\tau }_c$.

3) If $I_1=\varnothing$ and $I_2\ne \varnothing$, then Equation (12) has two positive roots $z_{n\mathrm{,1}}$ and $z_{n\mathrm{,2}}$ for some $n\in I_2$. Denote that ${\tau }_{n\mathrm{,1}}^{\left(0\right)}$ is the smallest time delay corresponding to $n\in I_2$. Then, the positive constant steady state $E_{\mathrm{<mo>\; *\; </mo>}}$ is locally asymptotically stable for $\tau \in \left(\mathrm{0,}{\tau }_{n\mathrm{,1}}^{\left(0\right)}\right)$ and there will occur stability switching for system (2). System (2) undergoes Hopf bifurcation near $E_{\mathrm{<mo>\; *\; </mo>}}$ at $\tau ={\tau }_c$.

4) If $I_1\ne \varnothing$ and $I_2\ne \varnothing$, then Equation (12) has three positive roots: the root $z_{n_1\mathrm{,1}}$ associated with indices $n_1\in I_1$ and two additional roots $z_{n_2\mathrm{,1}}$, $z_{n_2\mathrm{,2}}$ corresponding to indices $n_2\in I_2$, which implies ${\omega }_{n_1\mathrm{,1}}$ ( $n_1\in I_1$), ${\omega }_{n_2\mathrm{,1}}$ and ${\omega }_{n_2\mathrm{,2}}$ ( $n_2\in I_2$). Denote that ${\tau }_{q\mathrm{,1}}^{\left(0\right)}$ is the smallest time delay corresponding to $q$ ( $q\in I_1$ or $q\in I_2$). Then, the positive constant steady state $E_{\mathrm{<mo>\; *\; </mo>}}$ is locally asymptotically stable for $\tau \in \left(\mathrm{0,}{\tau }_{q\mathrm{,1}}^{\left(0\right)}\right)$ and there will occur stability switching for system (2). System (2) undergoes Hopf bifurcation near $E_{\mathrm{<mo>\; *\; </mo>}}$ at $\tau ={\tau }_c$.

4. Normal Form of Hopf Bifurcation

In this section, we use the MTS to derive the normal form of the Hopf bifurcation for system (2). Suppose that the characteristic system (6) has a pair of pure imaginary roots $\lambda =\pm i\omega$ for $\tau ={\tau }_c$, then system (2) undergoes $n_0$-mode Hopf bifurcation near the equilibrium point $E_{\mathrm{<mo>\; *\; </mo>}}$, then we derive the normal form of Hopf bifurcation of system (2) by the MTS. We choose time delay $\tau$ as a bifurcation parameter, where $\tau ={\tau }_c+\epsilon \mu$. The parameter $\epsilon$ is a dimensionless scaling factor and $\mu$ is a perturbation parameter, ${\tau }_c$ is given in Equation (16).

Let $\tilde{u}\left(x\mathrm{,}\tau t\right)=u\left(x\mathrm{,}\tau t\right)-u_{\mathrm{<mo>\; *\; </mo>}}$, $\tilde{v}\left(x\mathrm{,}\tau t\right)=v\left(x\mathrm{,}\tau t\right)-v_{\mathrm{<mo>\; *\; </mo>}}$, we still use $u\left(x\mathrm{,}t\right)$ and $v\left(x\mathrm{,}t\right)$ to represent $\tilde{u}\left(x\mathrm{,}\tau t\right)$ and $\tilde{v}\left(x\mathrm{,}\tau t\right)$, respectively, we have

$\{\begin{array}{l}\frac{\partial u}{\partial t}=\tau \{d_1\Delta u+ru\left(1-\frac{u_{*}}{K}\right)-\frac{r}{K}\left(u+u_{*}\right)\hat{u}-g_{1}u\left(t-1\right)-g_{2}v\left(t-1\right)\\ \stackrel{}{}-g_3{u}^2\left(t-1\right)-g_{4}u\left(t-1\right)v\left(t-1\right)-f_{111}{u}^3\left(t-1\right)-f_{112}{v}^3\left(t-1\right)\},\\ \frac{\partial v}{\partial t}=\tau \{d_2\Delta v-d_{3}v+\beta \left[g_{1}u+g_{2}v+g_3{u}^2+g_{4}uv+f_{111}{u}^3+f_{112}{u}^{2}v\right]\\ +\eta v{u}_{xx}-\eta v_{*}{u}_{xx}-\eta v_x{u}_x\},\end{array}$(18)

with

$\begin{array}{l}{f}_0=\frac{\alpha u_{*}{v}_{*}}{1+\alpha u_{*}h},f_1=\frac{\alpha v_{*}}{{\left(1+\alpha u_{*}h\right)}^2},f_2=\frac{\alpha u_{*}}{1+\alpha u_{*}h},f_{11}=-\frac{{\alpha }^{2}h{v}_{*}}{{\left(1+\alpha u_{*}h\right)}^3},\\ f_{12}=\frac{\alpha }{{\left(1+\alpha u_{*}h\right)}^2},f_{111}=\frac{{\alpha }^3{h}^2{v}_{*}}{{\left(1+\alpha u_{*}h\right)}^4},f_{112}=-\frac{{\alpha }^{2}h}{{\left(1+\alpha u_{*}h\right)}^3},\\ g_1=\frac{\alpha v_{*}\left({\alpha }^2{u}_{*}^2{h}^2+2\right)}{{\left(1+\alpha u_{*}h\right)}^4},g_2=\frac{\alpha u_{*}\left({\alpha }^2{u}_{*}^2{h}^2+2\alpha u_{*}h+2\right)}{{\left(1+\alpha u_{*}h\right)}^3},\\ g_3=\frac{{\alpha }^2{v}_{*}h\left(3\alpha u_{*}h-2\right)}{{\left(1+\alpha u_{*}h\right)}^3},g_4=\frac{\alpha \left(1-\alpha u_{*}h\right)}{{\left(1+\alpha u_{*}h\right)}^3}.\end{array}$

Let $h={\left(h_{11}\mathrm{,}{h}_{12}\right)}^{\text{T}}$ be the eigenvector corresponding to the eigenvalue $\lambda =i\omega {\tau }_c$ for the linear operator of Equation (18) and let $h^{\mathrm{<mo>\; *\; </mo>}}=\rho {\left(h_{21}\mathrm{,}{h}_{22}\right)}^{\text{T}}$ be the normalized eigenvector corresponding to the eigenvalue $\lambda =-i\omega {\tau }_c$ for the adjoint operator of the linear operator in Equation (18). According to the condition $\langle \overline{h^{\mathrm{<mo>\; *\; </mo>}}}\mathrm{,}h\rangle =1$, we can proceed with the following analysis:

$\{\begin{array}{l}h={\left(\mathrm{1,}\frac{\eta v_{\mathrm{<mo>\; *\; </mo>}}{n}^2+a_{21}}{d_2{n}^2+i\omega }\right)}^{\text{T}}\mathrm{,}\\ h^{\mathrm{<mo>\; *\; </mo>}}=\rho {\left(\frac{d_2{n}^2-i\omega }{c_{12}{\text{e}}^{i\omega {\tau }_c}}\mathrm{,1}\right)}^{\text{T}}\mathrm{,}\end{array}$(19)

with $\rho ={\left(\overline{h_{11}}{h}_{21}+h_{22}\overline{h_{12}}\right)}^{-1}$.

We assume that the solution of Equation (18) is:

$Z\left(x,t\right)=Z\left(x,T_0,T_1,T_2,\cdots \right)={\displaystyle \underset{k=1}{\overset{\infty }{\sum }}}{\epsilon }^k{Z}_k\left(x,T_0,T_1,T_2,\cdots \right),$(20)

where $T_i={\epsilon }^{i}t\left(i=0,1,2,\cdots \right)$. The derivation with respect to $t$ is

$\frac{\text{d}}{\text{d}t}=\frac{\partial }{\partial T_0}+\epsilon \frac{\partial }{\partial T_1}+{\epsilon }^2\frac{\partial }{\partial T_2}+\cdots =D_0+\epsilon D_1+{\epsilon }^2{D}_2+\cdots ,$(21)

where the differential operator $D_i=\frac{\partial }{\partial T_i}\mathrm{<mi>\; </mi>}\left(i=0,1,2,\cdots \right)$. We expand $Z\left(x\mathrm{,}t-1\right)$ at $Z\left(x\mathrm{,}{T}_0-\mathrm{1,}{T}_1\mathrm{,}{T}_2\mathrm{,}\cdots \right)$, we obtain

$Z\left(x\mathrm{,}t-1\right)=\epsilon Z_{11}+{\epsilon }^2{Z}_{21}-{\epsilon }^2{D}_1{Z}_{11}+{\epsilon }^3{Z}_{31}-{\epsilon }^3{D}_1{Z}_{21}-{\epsilon }^3{D}_2{Z}_{11}+\cdots \mathrm{,}$(22)

where $Z_{j,1}=Z_j\left(x,T_0-1,T_1,T_2,\cdots \right),j=1,2,3,\cdots$.

Substituting Equations (20)-(22) into Equation (18). For the $\epsilon$, we obtain

$\{\begin{array}{l}{D}_0{u}_1-{\tau }_c\left[d_1\Delta u_1+r{u}_1\left(1-\frac{u_{*}}{K}\right)-\frac{r{u}_{*}}{K\pi }{\displaystyle {\int }_0^{\pi }{u}_1\text{d}x}-g_1{u}_{11}-g_2{v}_{11}\right]=0,\\ D_0{v}_1-{\tau }_c\left[d_2\Delta v_1-d_3{v}_1+\beta \left(g_1{u}_1+g_2{v}_1\right)-\eta v_{*}{u}_{1,xx}\right]=0.\end{array}$(23)

The solution of Equation (19) can be written as follows:

$\{\begin{array}{l}{u}_1=G{\text{e}}^{\text{i}\omega {\tau }_c{T}_0}{h}_{11}\cos \left(nx\right)+c\mathrm{.}c\mathrm{.,}\hfill \\ v_1=G{\text{e}}^{\text{i}\omega {\tau }_c{T}_0}{h}_{12}\cos \left(nx\right)+c\mathrm{.}c\mathrm{.,}\hfill \end{array}$(24)

where $h_{11}$ and $h_{12}$ are given in Equation (19), $c\mathrm{.}c\mathrm{.}$ means the complex conjugate of the preceding terms.

For the ${\epsilon }^2$, we have

$\{\begin{array}{l}{D}_0{u}_2-{\tau }_c\left[d_1\Delta u_2+r{u}_2\left(1-\frac{u_{*}}{K}\right)-\frac{r{u}_{*}}{K\pi }{\displaystyle {\int }_0^{\pi }{u}_2\text{d}x}\right]+g_1{u}_{21}+g_2{v}_{21}\\ =-D_1{u}_1+{\tau }_c\left[-\frac{r{u}_1}{K\pi }{\displaystyle {\int }_0^{\pi }{u}_1\text{d}x}+g_1{D}_1{u}_{11}+g_2{D}_1{v}_{11}-g_3{u}_{11}^2-g_4{u}_{11}{v}_{11}\right]\\ +\mu \left[d_1\Delta u_1+r{u}_1\left(1-\frac{u_{*}}{K}\right)-\frac{r{u}_{*}}{K\pi }{\displaystyle {\int }_0^{\pi }{u}_1\text{d}x}-g_1{u}_{11}-g_2{v}_{11}\right],\\ D_0{v}_2-{\tau }_c\left[d_2\Delta v_2-d_3{v}_2+\beta \left(g_1{u}_2+g_2{v}_2\right)-\eta v_{*}{u}_{2,xx}\right]\\ =-D_1{v}_1+{\tau }_c\left[\beta \left(g_3{u}_1^2+g_4{u}_1{v}_1\right)-\eta v_1{u}_{1,xx}-\eta u_{1,x}{v}_{1,x}\right]\\ +\mu \left[d_2\Delta v_1-d_3{v}_1+\beta \left(g_1{u}_1+g_2{v}_1\right)-\eta v_{*}{u}_{1,xx}\right].\end{array}$(25)

Substituting Equation (24) into Equation (25), by solvability condition, we obtain

$\frac{\partial G}{\partial T_1}=\mu MG\mathrm{,}$(26)

with

$\{\begin{array}{l}M=\frac{\overline{\rho h_{21}}{\ell }_1+\overline{\rho h_{22}}{\ell }_2}{\overline{\rho h_{22}}{h}_{21}-\overline{\rho h_{21}}\left[{\tau }_c{\text{e}}^{-\text{i}\omega {\tau }_c}\left(g_1{h}_{11}+g_2{h}_{12}\right)-h_{11}\right]},\\ {\ell }_1=-d_1{n}_0^2{h}_{11}+r{h}_{11}\left(1-\frac{u_{*}}{K}\right)-g_1{h}_{11}{\text{e}}^{-\text{i}\omega {\tau }_c}-g_2{h}_{12}{\text{e}}^{-\text{i}\omega {\tau }_c},\\ {\ell }_2=-d_2{n}_0^2{h}_{12}-d_3{h}_{12}+\eta v_{*}{n}_0^2{h}_{11}.\end{array}$

Assume the solution of Equation (25) to be as follows:

$\{\begin{array}{l}{u}_2={\displaystyle \underset{k=0}{\overset{+\infty }{\sum }}}\left(j_{0k}G\bar{G}+j_{1k}{\text{e}}^{2\text{i}\omega T_0}{G}^2+{\bar{j}}_{1k}{\text{e}}^{-2\text{i}\omega T_0}{\bar{G}}^2\right)\cos \left(kx\right),\\ v_2={\displaystyle \underset{k=0}{\overset{+\infty }{\sum }}}\left(p_{0k}G\bar{G}+p_{1k}{\text{e}}^{2\text{i}\omega T_0}{G}^2+{\bar{p}}_{1k}{\text{e}}^{-2\text{i}\omega T_0}{\bar{G}}^2\right)\cos \left(kx\right).\end{array}$(27)

Substituting solutions Equation (24) and Equation (27) into Equation (25), we obtain

$\begin{array}{l}{j}_{0k}=\frac{A{y}_2-B{y}_1}{x_1{y}_2-x_2{y}_1},\mathrm{<mi>\; </mi>\; <mi>\; </mi>}{j}_{1k}=\frac{C{y}_4-D{y}_3}{x_3{y}_4-x_4{y}_3},\\ p_{0k}=\frac{A{x}_2-B{x}_1}{y_1{x}_2-y_2{x}_1},\mathrm{<mi>\; </mi>\; <mi>\; </mi>}{p}_{1k}=\frac{C{x}_4-D{x}_3}{y_3{x}_4-y_4{x}_3},\end{array}$(28)

where

Figure 2

with θ K = ∑ k = 0 ∞ ∫ 0 π cos 2 ( n 0 x ) cos ( k x ) d x = { π 2 , k = 0 , π 4 , k = 2 n 0 , 0 , k ≠ 0 , 2 n 0 , ϕ K = ∑ k = 0 ∞ ∫ 0 π sin 2 ( n 0 x ) cos ( k x ) d x = { π 2 , k = 0 , − π 4 , k = 2 n 0 , 0 , k ≠ 0 , 2 n 0 , ψ K = ∑ k = 0 ∞ ∫ 0 π cos ( n 0 x ) cos ( k x ) d x = { π 2 , k = n 0 , 0 , k ≠ n 0 . For the ε 3 , we obtain { D 0 u 3 − τ c [ d 1 Δ u 3 + r u 3 ( 1 − u * K ) − r u * K π ∫ 0 π u 3 d x + g 1 u 31 + g 2 v 31 ] = − D 1 u 2 − D 2 u 1 + τ c [ − r u 1 K π ∫ 0 π u 2 d x − r u 2 K π ∫ 0 π u 1 d x + g 1 D 1 u 21 + g 1 D 2 u 11 + g 2 D 1 v 21 + g 2 D 2 v 11 + 2 g 3 u 11 u 21 − 2 g 3 D 1 u 11 2 + g 4 ( 2 D 1 u 11 v 11 − u 11 v 21 − u 21 v 11 ) − f 111 u 11 3 − f 112 u 11 2 v 11 ] + o ( μ ) , D 0 v 3 − τ c [ d 2 Δ v 3 − d 3 v 3 + β ( g 1 u 3 + g 2 v 3 ) − η v * u 3 , x x ] = − D 1 v 2 − D 2 v 1 + τ c [ β ( 2 g 3 u 1 u 2 + g 4 u 1 v 2 + g 4 u 2 v 1 + f 111 u 1 3 + f 112 u 1 2 v 1 ) − η ( v 2 u 1 , x x + v 1 u 2 , x x + u 1 , x v 2 , x + u 2 , x v 1 , x ) ] + o ( μ ) . (29)Substituting Equation (24) and Equation (27) into Equation (29), by solvability condition, we obtain ∂ G ∂ T 2 = H G 2 G ¯ ρ h 22 ¯ h 21 − ρ h 21 ¯ [ τ c e − i ω τ c ( g 1 h 11 + g 2 h 12 ) − h 11 ] ∫ 0 π cos 2 ( n 0 x ) d x , (30)where H = τ c ρ ¯ ∑ k = 0 + ∞ { h 21 ¯ [ − r h 11 K π j 0 k ∫ 0 π cos 2 ( n 0 x ) d x ∫ 0 π cos ( k x ) d x − r h 11 K π θ K + e − i ω τ c T 0 ( 2 g 3 h 11 j 0 k − g 4 h 11 p 0 k − g 4 h 12 j 0 k ) θ K − e − 3 i ω τ c T 0 ( f 111 h 11 2 h 11 ¯ + f 112 h 11 2 h 12 + 2 f 112 h 11 ¯ h 11 h 12 ) ∫ 0 π cos 4 ( n 0 x ) d x ] + h 22 ¯ [ ( 2 g 3 h 11 j 0 k + g 4 h 11 p 0 k + g 4 h 12 j 0 k ) θ K + ( f 111 h 11 2 h 11 ¯ + f 112 h 11 2 h 12 + 2 f 112 h 11 ¯ h 11 h 12 ) ∫ 0 π cos 4 ( n 0 x ) d x + η ( n 0 2 h 11 p 0 k θ K + k 2 h 12 j 0 k θ K − k n 0 h 11 p 0 k ∫ 0 π sin ( n 0 x ) cos ( n 0 x ) sin ( k x ) d x − k n 0 h 12 j 0 k ∫ 0 π sin ( n 0 x ) cos ( n 0 x ) sin ( k x ) d x ) ] } ,

with

${\displaystyle {\int }_0^{\pi }\sin \left(n_{0}x\right)\cos \left(n_{0}x\right)\sin \left(kx\right)\text{d}x}=\{\begin{array}{ll}\frac{\pi }{4},\hfill & k=n_0+1,\hfill \\ 0,\hfill & k\ne n_0+1.\hfill \end{array}$

Therefore, the third-order truncated normal form near the Hopf bifurcation point for system (2) reduced on the center manifold is given by

$\frac{\partial G}{\partial t}=\mu MG+\frac{H{G}^2\bar{G}}{\overline{\rho h_{22}}{h}_{21}-\overline{\rho h_{21}}\left[{\tau }_c{\text{e}}^{-\text{i}\omega {\tau }_c}\left(g_1{h}_{11}+g_2{h}_{12}\right)-h_{11}\right]{\displaystyle {\int }_0^{\pi }{\cos }^2\left(n_{0}x\right)\text{d}x}},$(31)

where $M$ and $H$ are defined by Equation (26) and Equation (30).

Let $G=r{\text{e}}^{\text{i}\theta }$, the normal form near the Hopf bifurcation of system (2) in polar coordinates is written as:

$\{\begin{array}{l}\stackrel{\dot{}}{r}=\mathrm{Re}\left(M_1\right)\mu r+\mathrm{Re}\left(M_2\right)r^3,\\ \stackrel{\dot{}}{\theta }=\mathrm{Im}\left(M_1\right)\mu +\mathrm{Im}\left(M_2\right)r^2.\end{array}$(32)

Therefore, we have the following theorem.

Theorem 2. For Equation (32), if $\frac{\mathrm{Re}\left(M_1\right)\mu }{\mathrm{Re}\left(M_2\right)}<0$ holds, then Equation (32)

has one equilibrium $r_{\mathrm{<mo>\; *\; </mo>}}$, and there exist bifurcating periodic solutions near the positive constant steady state $E_{\mathrm{<mo>\; *\; </mo>}}$ of system (2). Moreover:

1) if $\mathrm{Re}\left(M_1\right)\mu <0$, the bifurcating periodic solutions are unstable, and the direction of bifurcation is forward (backward) for $\mu >0$ ( $\mu <0$);

2) if $\mathrm{Re}\left(M_1\right)\mu >0$, the bifurcating periodic solutions are stable, and the direction of bifurcation is forward (backward) for $\mu >0$ ( $\mu <0$).

5. Numerical Simulations

In this section, we perform numerical simulations to verify the correctness of the theoretical analysis. Due to the limited availability of actual ecological data, the parameters in this study are established based on the theoretical framework of the system and biological plausibility. Under the constraints that $\alpha$ and $h$ must remain non-zero as established in previous conditions, we implement the following

reformulation of the functional response function: $\frac{\alpha uv}{1+\alpha uh}=\frac{kuv}{D_1+u}$, where $D_1=\frac{1}{\alpha h}$ is the half-saturation constant and $k=\frac{1}{h}$ is a constant.

We provide the following parameters: $d_1=0.2$, $d_2=0.01$, $d_3=0.01$, $D_1=9$, $k=0.2$, $\eta =1$, $\beta =1$, $r=0.2$, $K=5$ and $l=1$. Based on the data above, (H₀) always holds, system (2) has two boundary equilibria $E_0=\left(\mathrm{0,0}\right)$, $E_1=\left(\mathrm{5,0}\right)$ and one nontrivial equilibrium $E_{\mathrm{<mo>\; *\; </mo>}}=\left(u_{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{v}_{\mathrm{<mo>\; *\; </mo>}}\right)=\left(\mathrm{0.4737,8.5762}\right)$. As mentioned earlier, what we ultimately focus on is nothing but the stability of the nontrivial equilibrium $E_{\mathrm{<mo>\; *\; </mo>}}$.

When $\tau =0$ holds, $E_{\mathrm{<mo>\; *\; </mo>}}$ is local asymptotically stable. This means that: although Dendroctonus valens and Rhizophagus grandis can coexist at this time, natural predators are capable of suppressing the reproduction of the pests.

When $\tau >0$ holds, (H₁₁) holds for $n=1$, there does not exist any $n$ such that (H₁₂) and (H₁₃) holds. At this time, we can get the critical delay ${\tau }_1^{\left(0\right)}=2.0075$. From the definition of ${\tau }_c$, we obtain ${\tau }_c=2.0075$, Theorem 1 supports the conclusion that the positive constant steady state $E_{\mathrm{<mo>\; *\; </mo>}}$ achieves local asymptotic stability when $\tau \in \left[\mathrm{0,2.0075}\right)$ and instability when $\tau \in \left[\mathrm{2.0075,}+\infty \right)$. We will select appropriate parameter values within the aforementioned regions (stable and unstable states), provide numerical simulation results and rigorously analyze them to validate the theoretical assertions established above. We choose $\tau =1.75\in \left[\mathrm{0,2.0075}\right)$ and get Figure 2 .

Figure 3

(a) (b)Figure 2. When τ = 1.75 , the simulation of system (2) reveals that the equilibrium E * is locally asymptotically stable.

Figure 3

(a) (b)Figure 2. When τ = 1.75 , the simulation of system (2) reveals that the equilibrium E * is locally asymptotically stable.

Figure 3

(a) (b)Figure 2. When τ = 1.75 , the simulation of system (2) reveals that the equilibrium E * is locally asymptotically stable.

Then, we choose $\tau =2.0095$, from Equation (26) and Equation (30), we obtain $\mathrm{Re}\left(M_1\right)>0$, $\mathrm{Re}\left(M_2\right)<0$. Thus, according to Theorem 2, the Equation (2) will generate inhomogeneous periodic solutions near the positive constant steady state $E_{\mathrm{<mo>\; *\; </mo>}}$ of system (2) and bifurcating periodic solutions are stable and forward, as shown in Figure 3 .

Figure 4

(a) (b)Figure 3. When τ = 2.0095 , system (2) produces stable, inhomogeneous forward periodic solutions near E * .

Figure 4

(a) (b)Figure 3. When τ = 2.0095 , system (2) produces stable, inhomogeneous forward periodic solutions near E * .

Figure 4

(a) (b)Figure 3. When τ = 2.0095 , system (2) produces stable, inhomogeneous forward periodic solutions near E * .

By analyzing Figure 2 and Figure 3 , we can draw the following conclusions.

1) As illustrated in Figure 2 , when the time delay is below the critical threshold, Rhizophagus grandis can rapidly suppress Dendroctonus valens populations to a stable safety level, demonstrating the spatiotemporal controllability of this biocontrol strategy.

2) As illustrated in Figure 3 , when the time delay slightly exceeds the critical threshold, Rhizophagus grandis maintains partial control over Dendroctonus valens populations, but induces periodic outbreak cycles in the pest population while exhibiting synchronous periodic fluctuations in the predator population.

6. Conclusions

In this study, we established a predator-prey system incorporating nonlocal diffusion, prey-taxis and time delay effects to investigate the spatiotemporal dynamics between Dendroctonus valens and its natural enemy Rhizophagus grandis. Our theoretical analysis and numerical simulations yielded the following key findings:

When time delay is absent ( $\tau =0$) or below the critical threshold ( $\tau <{\tau }_c$), the positive equilibrium $E_{\mathrm{<mo>\; *\; </mo>}}$ exhibits local asymptotic stability, indicating effective pest suppression by the natural enemy. However, when $\tau \ge {\tau }_c$, the system undergoes a Hopf bifurcation, generating spatially stable periodic solutions that correspond to cyclical pest outbreaks and predator-prey oscillations. Numerical simulations confirmed the bifurcation behavior at ${\tau }_c=2.0095$, validating our theoretical predictions.

From an application perspective, the critical delay ${\tau }_c$ provides important guidance for optimizing the timing of natural enemy releases to prevent system destabilization caused by excessive delays. Enhancing the prey-taxis capability of natural enemies can improve their pest localization efficiency and help suppress periodic outbreaks.

It should be noted that, our model still has certain limitations. Environmental factors such as temperature variations and terrain heterogeneity were not incorporated, which may affect prediction accuracy. Moreover, field population dynamics data for Dendroctonus valens and Rhizophagus grandis remain scarce. Therefore, all parameters were derived through theoretical analysis.

In our future work, we will collaborate with ecological research teams to systematically enhance both the practical applicability and theoretical rigor of the proposed model. Through comprehensive analysis of the current framework, we intend to identify critical components exerting dominant influences on dynamic properties, subsequently developing simplified yet operationally viable models tailored for practical implementation. These refined models will be rigorously integrated with experimental datasets to establish empirical validation of their real-world predictive capabilities, thereby bridging the gap between theoretical constructs and ecological applications while maintaining methodological coherence across scales.

Acknowledgements

This study was funded by the Heilongjiang Provincial Natural Science Foundation of China (Grant No. LH2024A001), and the College Students Innovations Special Project funded by Northeast Forestry University (No. DCLXY-2025010).

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