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June 1, 2026 14:14 WSPC/S0218-1274 IJBC 2650157International Journal of Bifurcation and Chaos (2026) 2650157 (21 pages)© World Scientif i c Publishing CompanyDOI: 10.1142/S0218127426501579Spatial Pattern Formation in Host–ParasitoidModel of Massicus raddei Infestation with Fear Ef f ectand Nonlocal CompetitionMin Wang and Yuting Ding∗Department of Mathematics, Northeast Forestry University,Harbin 150040, P. R. Chinayuting840810@163.comReceived February 6, 2026; Accepted March 30, 2026; Published June 2, 2026In this paper, we construct a delayed host–parasitoid model with homogeneous Neumann bound-ary condition, incorporating fear ef f ect and nonlocal competition, to investigate the biologicalcontrol of Massicus raddei. Through linear stability and eigenvalue analysis, we derive the con-ditions for Turing, Hopf, and Turing–Hopf bifurcations. Theoretically, the multiple time scalesmethod is employed to derive the normal form of this codimension-2 bifurcation. By conduct-ing systematic analysis and numerical simulations with biologically meaningful parameters, wediscover an interesting phenomenon that the dynamics of the system are closely related to thedomain size. For large domains, the system can undergo Hopf–pitchfork bifurcation, exhibitingspatially homogeneous steady states, spatially inhomogeneous steady states, and spatially inho-mogeneous periodic solutions. Numerical simulations further demonstrate that increasing thedif f usion coef f i cient of the parasitoid can signif i cantly enhance the biocontrol ef f i cacy againstM. raddei. However, when the domain size is reduced under the same parameter set, the sys-tem may undergo degenerate Hopf–transcritical bifurcation. Moreover, further reducing thepregnancy delay and the dif f usion coef f i cient of the parasitoid can induce stable quasi-periodicsolutions, a phenomenon not observed in large domains. Accordingly, these f i ndings provide animportant theoretical basis for designing biological control strategies against M. raddei tailoredto dif f erent spatial scales.Keywords: Host–parasitoid; nonlocal competition; fear ef f ect; Turing–Hopf bifurcation; normalform.1. IntroductionThe beetle Massicus raddei, a major forest pestwidely distributed across East Asia, primarilyinfests oak trees and poses a serious threat to foresthealth. The larvae of M. raddei have a long devel-opmental period and inhabit concealed gallerieswithin the host trees, which largely protects themfrom adverse environmental conditions and chem-ical control measures. Furthermore, this speciesexhibits strong stress resistance, with tolerance toextreme weather and other unfavorable conditions,and demonstrates remarkable vitality. In addition,due to its strong mobility and rapid populationspread, M. raddei can quickly expand into adja-cent healthy forest areas, exacerbating the threatto regional forest ecosystems [Wu et al., 2023].Notably, this pest has caused regional outbreaksin multiple countries including China and SouthKorea, resulting in substantial ecological and eco-nomic losses with a growing trend of deterioration[Lee et al., 2024].∗ Author for correspondence2650157-1Int. J. Bifurcation Chaos Downloaded from www.worldscientific.comby Lin sino on 06/12/26. Re-use and distribution is strictly not permitted, except for Open Access articles.