In the study of dynamics and bifurcations of delay differential systems, computing the normal form associated with a singular point is both unavoidable and often technically challenging. Developing efficient computational methods or algorithms for such systems is therefore of great importance for the analysis of real-world problems. The multiple time scales (MTS) method is frequently applied in physics and engineering, but it lacks mathematical rigorousness. In contrast, the center manifold reduction (CMR) method is founded on rigorous mathematical theory, yet it is difficult to apply to practical problems. In this paper, we establish the third-order equivalence of the MTS and CMR methods in computing the normal form of Turing-Hopf bifurcation in general delayed reaction-diffusion equations with advection-diffusion and nonlocal effects. This result provides a rigorous mathematical foundation for applying the simpler MTS method to bifurcation problems in delayed reaction-diffusion systems. An illustrative example is presented, comparing the normal forms obtained by the two approaches and verifying the theoretical results.

Hyphantria cunea (H. cunea) is a serious forest pest, while its parasitic natural enemy, Chouioia cunea Yang (C. cunea), along with sex pheromone traps, can effectively control the population of H. cunea. H. cunea exhibits intraspecific competition for food, space, and other resources, and C. cunea preferentially targets areas with higher H. cunea density. Both situations are non-local and can be characterized by top-hat kernels. Therefore, we formulate a H. cunea-C. cunea (H-C) model that incorporates both nonlo cal perception and nonlo cal competition with top-hat perception kernels, and investigate how these factors affect the stability and spatiotemp oral patterns. We use perception coefficient and perceptual range of C. cunea as bifurcation parameters to derive conditions for the existence of several bifurcations. We simulate spatiotemporal patterns near Turing bifurcation points with higher release rates of C. cunea, and near Hopf and Turing-Hopf bifurcation points with lower release rates. We find that the combination of the two nonlo cal effects can generate boundary-enriched spatiotemp oral patterns when bifurcation parameters belong to specific intervals. Finally, the results indicate that if boundary-enriched spatiotemp oral patterns occur, the artificial release rate of C. cunea should be immediately increased. Additionally, if C. cunea has strong perception, it can effectively control the pest population. This study provides an important theoretical basis for comprehensive pest management.

The host-taxis mechanism with time delay and nonlocal competition are innovatively introduced into a host-generalist parasitoid system for studying pine wilt disease control. First, the wellposedness of the model is established. The existence and stability of the equilibrium points are demonstrated. The existence of Turing, Hopf, and Turing-Hopf bifurcations is analyzed. Subsequently, the normal form of the Turing-Hopf bifurcation is derived using the multiple time scales method. Furthermore, the normal form is linearly transformed into a computationally tractable equivalent form. Second, this study performs numerical simulations from two aspects. On the one hand, with time delay serving as the Hopf bifurcation parameter, it is found that when the delay exceeds its critical value, the system transitions from a homogeneous steady state to an inhomogeneous periodic solution. Research on delays shows that shorter search delays aid epidemic control. Moreover, with fixed delay, parasitoids with weak host-taxis are better for prevention. On the other hand, to investigate the combined effects of host-taxis and time delay, the system undergoing a saddle-node-Hopf bifurcation is simulated. Rich dynamical phenomena, such as various inhomogeneous steady states and inhomogeneous periodic solutions, are obtained. Research on the combined effects of host-taxis and time delays shows that purposefully adjusting these two parameters can steer the population toward a stable state, facilitating the control of pine wilt disease.

In this paper, we construct a delayed host-parasitoid model with homogeneous Neumann boundary condition, incorporating fear effect and nonlocal competition, to investigate the biological control of Massicus raddei. Through linear stability and eigenvalue analysis, we derive the conditions for Turing, Hopf, and Turing-Hopf bifurcations. Theoretically, the multiple time scales method is employed to derive the normal form of this codimension-2 bifurcation. By conducting systematic analysis and numerical simulations with biologically meaningful parameters, we discover an interesting phenomenon that the dynamics of the system are closely related to the domain size. For large domains, the system can undergo Hopf-pitchfork bifurcation, exhibiting spatially homogeneous steady states, spatially inhomogeneous steady states, and spatially inhomogeneous periodic solutions. Numerical simulations further demonstrate that increasing the diffusion coefficient of the parasitoid can significantly enhance the biocontrol efficacy against M. raddei. However, when the domain size is reduced under the same parameter set, the system may undergo degenerate Hopf-transcritical bifurcation. Moreover, further reducing the pregnancy delay and the diffusion coefficient of the parasitoid can induce stable quasi-periodic solutions, a phenomenon not observed in large domains. Accordingly, these findings provide an important theoretical basis for designing biological control strategies against M. raddei tailored to different spatial scales.

Sclerodermus guani is a facultative natural enemy with dual functional responses of predation and parasitism, which has been widely used in the biological control of Monochamus saltuarius, the vector insect of Pine Wilt Disease (PWD). However, most existing studies treat its predation and parasitism behaviors in isolation and ignore the synergistic effect of the dual functions. Based on this, we innovatively construct a facultative predation-parasitism reaction-diffusion model with top-hat nonlocal perception, dual predation-parasitism functional responses, and parasitic time delay, and we systematically analyze the spatiotemporal dynamic behaviors of the model. Theoretical analysis shows that, in the absence of time delay, the nonlocal perception intensity eta and perception range R can induce local over-predation of the host by natural enemies and population collapse, forming stable spatially nonhomogeneous steady-state solutions. When the parasitic time delay exceeds the critical value, the delay effect leads to periodic population oscillation. The synergistic effect of nonlocal perception and time delay induces Turing-Hopf bifurcation, presenting complex spatiotemporally coupled patterns. Numerical simulations based on parameters with realistic biological backgrounds verify the correctness of the theoretical analysis, and intuitively demonstrate the dynamic modes corresponding to different parameters, including local asymptotic stability, stable spatially nonhomogeneous steady-state solutions, and spatially nonhomogeneous periodic solutions. The results reveal the regulation mechanism of top-hat nonlocal perception and parasitic time delay on the spatiotemporal dynamics of the Monochamus saltuarius-Sclerodermus guani system. They not only enrich the spatiotemporal dynamics theory of facultative predation-parasitism systems, but also provide a quantitative mathematical basis for optimizing the release timing and release strategy of Sclerodermus guani in prevention and control as well as lay a theoretical foundation for the biological control of PWD.

Pine wilt disease (PWD) is characterized by its rapid transmission, wide geographic distribution, and high mortality rate, earning it the epithet "the cancer of pine trees". The longhorn beetle is the primary transmission vector, and therefore, controlling its population is key to mitigating the spread of pine wilt disease. In our system, in addition to the memory capacity of Dryocopus martius and the nonlocal intraspecific competition of the longhorn beetles, we also incorporate nonlocal interspecific competition between Dryo copus martius and the longhorn beetles as a habitat constraint. We investigate the existence conditions for Turing, Hopf, and Turing-Hopf bifurcations, and derive the normal form of Turing-Hopf bifurcation for the system by using the method of multiple time scales. We select a set of biologically realistic parameters to perform numerical simulations and our findings show rich dynamics, such as the four-stable phenomenon. Specifically, for a fixed set of parameters, the system possesses four distinct stable solutions depending on the initial functions, namely, a pair of stable spatially nonhomogeneous steady states and two spatially nonhomogeneous periodic solutions with large amplitude. Based on the simulation results, we provide several suggestions and biological interpretations for the control of PWD.

Since spatial memory and nonlocal effect have significant impact on animal movement modeling, we construct a pine wilt disease model with memory-based diffusion and nonlocal effect incorporating Holling-II functional response function to study the control of pine wilt disease. We establish the existence conditions for Turing, Hopf and Turing-Hopf bifurcations, and find that the joint effect of memory-based diffusion coefficient and memory delay can give rise to Turing-Hopf bifurcation. Moreover, the multiple time scales method is extended, showing that it is applicable to the reaction-diffusion system with both nonlocal effect and memory-based diffusion, and the methods for eliminating the secular terms and solving high-order terms for such systems are novelly provided as well in the process of derivation. By using this method, the degenerate Hopf-transcritical bifurcation is derived. Via concisely comparing it with the classical Hopf-zero bifurcation, some interesting results are presented, which suggest that the proposed model exhibits various complex spatiotemporal patterns, such as the six-stable phenomenon, that is, a pair of stable spatially inhomogeneous steady states and periodic solutions induced by Turing-Hopf bifurcation and a pair of stable spatially inhomogeneous large amplitude periodic solutions found numerically, which may provide an alternative approach in explaining the periodic outbreaks of pine wilt disease. Numerical simulations are also performed by adopting scientifically obtained parameters with actual biological meaning. Based on this, the corresponding ecological significance and disease control suggestions have more practical reference value.

Different from the existing studies on the influence of self-diffusion or cross-diffusion on Turing instability, this paper originally focuses on the effect of nonlocal competition and host-taxis on Turing instability in a more realistic two-dimensional space, and novelly applies it to study the control of pine wilt disease. It turns out that the incorporation of host-taxis is not conducive to the generation of Turing instability, whereas nonlocal competition can promote the formation of pattern structure by facilitating the occurrence of it. The results not only reveal the new mechanism for the emergence of spatial heterogeneity patterns, but also provide an alternative theoretical explanation for the actually observed multi-point aggregation and multiple outbreaks of pine wilt disease. The various spatial patterns induced by nonlocal competition and host-taxis are numerically illustrated. We find that the Turing patterns can preserve the symmetry of the initial distribution, and contrary to the taxis diffusion, the self-diffusion of D. helophoroides promotes the pattern formation. Furthermore, the high consistency between the simulated and actual distribution patterns of pine wilt disease strongly validates the practical reference value of the paper. The most interesting finding is that we obtain the circular aggregation distribution pattern from simulations, which is consistent with the actual spread trend of pine wilt disease, and our study theoretically reveals the intrinsic evolution mechanism behind its occurrence.

This paper proposes a reaction-diffusion system with time delay and nonlocal perception via a top-hat kernel for controlling Pine Wilt Disease. We analyze the existence conditions of Turing, Hopf, and Turing-Hopf bifurcations. By selecting appropriate parameters, we conduct numerical simulations. The simulation results indicate that the nonlocal perception can induce stable spatially inhomogeneous solutions. The time delay can induce stable spatially inhomogeneous periodic solutions. The nonlocal perception and time delay can jointly induce Turing-Hopf bifurcation, exhibiting rich dynamical phenomena near the bifurcation points. Small perturbations can lead to transitions between different spatiotemporal patterns, such as locally asymptotically stable solutions, stable spatially inhomogeneous solutions, and stable spatially inhomogeneous periodic solutions. Finally, we interpret the simulated phenomena and propose practical recommendations for Pine Wilt Disease control, thereby offering practical guidance for real-world prevention and management efforts.

Predator gestation delay and nonlocal competition play key roles in controlling population density and maintaining ecosystem stability. In order to control the Dendrolimus superans that cause serious damage to forests, we propose a predator-prey reaction-diffusion equation with Holling type-II functional response function, gestation delay, and nonlocal competition. We investigated the existence conditions of the Hopf bifurcation and obtained its normal form of Hopf bifurcation by employing the multiple time scales method. We selected the appropriate parameters for numerical simulation and found that the gestation delay is helpful to maintain the stability of the population density of Dendrolimus superans.

Nonlocal perception plays a crucial role in studying animal cognitive movement modeling. In this paper, the impact of nonlocal perception on pattern formation is analyzed, and it is applied to study the control of pine wilt disease. It turns out that perceptual movement can provide a theoretical scientific basis for the multi-point outbreaks and spatiotemporal aggregation of pine wilt disease. For the top-hat kernel, we concentrate on the joint effect of perception scale and delay on the stability, and find that Turing-Hopf bifurcation occurs due to their interaction. Besides, the patterns near the bifurcation points are simulated in detail by adopting parameters with actual biological meaning, which are selected by analyzing real data, and diverse complicated spatiotemporal patterns are obtained, such as peak alternating periodic patterns and spatiotemporal aggregation patterns. Finally, we demonstrate that the artificial release of the parasitic natural enemy of the pest can drive the populations to reach stability in the form of the steady state or periodic solutions. The obtained results not only well explain the transmission mechanism of pine wilt disease, but also contribute to the study of biological phenomena such as the formations of flocks and swarms.

This study focuses on poplar root's dynamic salt stress responses, finding that Hsp20s may play an important role and screening its upstream regulators. Populus davidiana × P. alba, an excellent tree species, which is widely planted in China, has been seriously affected by salt stress. In this study, the response of poplar roots to salt stress was deeply studied by time-course transcriptome, and a large number of differentially expressed genes (DEGs) were identified at different times. Through weighted gene co-expression network analysis (WGCNA), it was discovered that oxidative and osmotic stress regulation played a crucial role in resisting salt stress in the early salt stress response (3–6 h), and nitrogen metabolize and transport genes were identified as hub genes. At the middle stage of salt stress (12–24 h), the plants initiated extensive reprogramming to adapt to stress, and the transcription factors (TFs), WRKY53, MYB13 and NFXL1, were identified as hub genes. After 48 h of salt stress, seven PdaHsp20 genes were identified as hub genes, which may alleviate the damage of salt stress. The genome-wide analysis of Hsp20s showed that the Hsp20 proteins were divided into 11 groups. A three-layer gene regulatory network with PdaHsp20s as the underlying gene was constructed and the unique PdaERF72 was found by association analysis with the co-expression network, which may have important functions in regulating PdaHsp20s under salt stress. The expression level analysis of PdaERF72 and PdaHsp20s, which have a direct connection with it, also indicated that some of them may have a negative regulation relationship after salt stress. In a word, poplar dynamically responds to salt stress, and different hub genes play a role in different stress stages, which provided a new perspective to reveal the response mechanism of poplar to salt stress.

Pine wilt disease (PWD) is mainly spread by Monochamus alternatus (in short, M. alternatus). Woodpecker, as the natural predator of M. alternatus, is considered for biological prevention and controlling the PWD. In this paper, we propose a new M. alternatus-woodpecker model with nonlocal competition and memory-based diffusion, which makes the model more realistic for the PWD control. We focus on the dynamics and bifurcations of the model with various combinations of the memory diffusion and nonlocal competition. It is shown that the nonlocal competition can only cause the stable constant steady state to lose stability, while the memory-based diffusion can induce unstable spatially inhomogeneous periodic solutions due to Hopf bifurcation. Consequently, we can explain the spatiotemporal heterogeneity problem in ecology by innovatively using mathematical modelling. Normal form theory with the multiple time scales method is applied to particularly consider Hopf bifurcation, showing complex dynamical behaviours involving various oscillating motions. Finally, numerical simulations are presented with the parameter values chosen from the real forest data of Yuan'an County, Hubei Province, China, confirming the theoretical results of the spatiotemporal heterogeneity of forest diseases and pests, as well as the PWD control.

Prey-taxis is a biological phenomenon, which plays a key role in biological control and ecological balance. For the controlling of pine wilt disease, we establish a reaction-diffusion equation with prey-taxis and nonlocal intraspecific competition of prey in this paper. We investigate the spatial formation of the spread system of pine wilt disease. It is concluded that the spatial pattern formation induced by the prey-taxis and time delay is characterized by Turing, Hopf, and Turing-Hopf bifurcation. Moreover, we extend the multiple time scales method to derive the normal form of the co-dimension-2 TuringHopf bifurcation for the system with prey-taxis and nonlocal effect. Through analyzing the normal form near the critical point of Turing-Hopf bifurcation, we obtain that there exist a pair of spatially nonhomogeneous non-constant steady states. Especially, Turing instability does not occur when prey-taxis coefficient is greater than the critical value. The time delay influences spatiotemporal patterns arising from Hopf bifurcation and Turing-Hopf bifurcation. It is noticed that there exist spatially nonhomogeneous periodic solutions with large amplitude when parameters are selected to some specific values. Furthermore, we also reveal some biological explanations and provide some theoretical support for controlling of pine wilt disease.

Pine wilt disease is a destructive forest disease with strong infectivity, a wide spread range and high difficulty in prevention and control. Since controlling Monochamus alternatus, the vector of pine wood nematode (Bursaphelenchus xylophilus) can reduce the occurrence of pine wilt disease efficiently, the parasitic natural enemy of M. alternatus, Dastarcus helophoroides, is introduced in this paper. Considering the influence of parasitic time of D. helophoroides on the control effect, based on the mutualistic symbiosis and parasitic relationship among pine wood nematode, M. alternatus and D. heloporoides, this paper establishes a pine wood nematode prevention and control model with delay. Then, the stability of positive equilibrium and the existence of Hopf bifurcation are discussed. Besides, we obtain the normal form of Hopf bifurcation by applying the multiple time scales method. Finally, numerical simulations with two sets of meaningful parameters selected by means of statistical analysis are carried out to support the theoretical findings. Through the comparative analysis of numerical simulations, the factors affecting the control effect of pine wilt disease are obtained, and some suggestions are put forward for practical control in the forest.

Pine wilt disease is one of the most serious forest diseases and pests in China, which seriously influences the realization of the goal of ``carbon peak and carbon neutrality."" In our article, we divide longhorns into susceptible ones and infected ones since pine wilt disease is spread by longhorns. Considering the saturation incidence of pine wilt disease, we establish a delayed reaction- diffusion model with nonlo cal effect for susceptible and infected longhorns. First, we consider the well-posedness of solutions and the type of equilibria for the nonspatial system. Next, we discuss the dynamics of the spatial system with nonlo cal effect. According to the multiple time scales method, we derive the normal form of Hopf bifurcation for a system associated with nonlo cal effect, and the stability and direction of bifurcating periodic solutions are analyzed. Finally, using real data for China to perform data analysis, we select suitable values of parameters. Numerical simulations are presented to illustrate the ecological significance. Combined with the current situation, we provide some theoretical support for the prevention and control of pine wilt disease in China. Especially, we find that the nonlo cal term can induce spatially stable inhomogeneous bifurcating periodic solutions.

The protection of forests and the mitigation of pest damage to trees play a crucial role in mitigating the greenhouse effect. In this paper, we first establish a delayed differential equation model for Ips subelongatus Motschulsky-Larix spp., where the delay parameter represents the time required for trees to undergo curing. Second, we analyze the stability of the equilibrium of the model and derive the normal form of Hopf bifurcation using a multiple-time-scales method. Then, we analyze the stability and direction of Hopf bifurcating periodic solutions. Finally, we conduct simulations to analyze the changing trends in pest and tree populations. Additionally, we investigate the impact of altering the rate of artificial planting on the system and provide corresponding biological explanations.

Pine wilt disease is one of the most serious forest pest and disease in China, which seriously influences the realization of the dual carbon goal. In this paper, considering susceptible longhorns and infected longhorns, we study a delayed reaction–diffusion pine wilt disease model both with memory diffusion and nonlocal effect. We analyze the dynamic properties for with and without memory diffusion and nonlocal effect, respectively. Especially, for this model, we find that memory diffusion plays a leading role in spatial dynamics. Memory diffusion can induce spatially inhomogeneous periodic solutions and steady-state solutions. Nonlocal effect mainly affects the amplitude and period of spatially inhomogeneous periodic solutions of the system. In addition, we give some biological explanations for the different phenomena in this paper.

Tamarix hispida is highly tolerant to salt, drought and heavy metal stress and is a potential material for the remediation of cadmium (Cd)-contaminated soil under harsh conditions. In this study, T. hispida growth and chlorophyll content decreased, whereas flavonoid and carotenoid contents increased under long-term Cd stress (25 d). The aboveground components of T. hispida were collected for RNA-seq to investigate the mechanism of Cd accumulation. GO and KEGG enrichment analyses revealed that the differentially expressed genes (DEGs) were significantly enriched in plant hormone-related pathways. Exogenous hormone treatment and determination of Cd2+ levels showed that ethylene (ETH) and abscisic acid (ABA) antagonists regulate Cd accumulation in T. hispida. Twenty-five transcription factors were identified as upstream regulators of hormone-related pathways. ThDRE1A, which was previously identified as an important regulatory factor, was selected for further analysis. The results indicated that ThABAH2.5 and ThACCO3.1 were direct target genes of ThDRE1A. The determination of Cd2+, ABA, and ETH levels indicated that ThDRE1A plays an important role in Cd accumulation through the antagonistic regulation of ABA and ETH. In conclusion, these results reveal the molecular mechanism underlying Cd accumulation in plants and identify candidate genes for further research.

Forest pests and diseases can diminish forest biodiversity, damage forest ecosystem functions, and have an impact on water conservation. Therefore, it is necessary to analyze the interaction mechanism between plants and pests. In this paper, the prevention and control of a specific pest—namely the larva of Paranthrene tabaniformis (Rott.) (hereinafter referred to as larva)—are studied. Based on the invasion mechanism of the larva in poplar, we establish a delayed differential equation and analyze the existence and stability of equilibria. Next, we assess the existence of a Hopf bifurcation to determine the range of parameters that ensures that the equilibria are stable. Then, we select a set of parameters to verify the results of the stability analysis. Finally, we provide biological explanations and effective theoretical control methods for poplar pests and diseases.