{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T22:22:12Z","timestamp":1778710932045,"version":"3.51.4"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,2,1]],"date-time":"2025-02-01T00:00:00Z","timestamp":1738368000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,2,1]],"date-time":"2025-02-01T00:00:00Z","timestamp":1738368000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2025,6]]},"DOI":"10.1007\/s12190-025-02394-3","type":"journal-article","created":{"date-parts":[[2025,2,1]],"date-time":"2025-02-01T02:47:36Z","timestamp":1738378056000},"page":"4227-4256","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Impact of nonlocal fear effect and mutual interference between predators in a diffusive predator\u2013prey model"],"prefix":"10.1007","volume":"71","author":[{"given":"Chenyu","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3131-9498","authenticated-orcid":false,"given":"Wensheng","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,2,1]]},"reference":[{"issue":"5","key":"2394_CR1","doi-asserted-by":"crossref","first-page":"1530","DOI":"10.2307\/1940005","volume":"73","author":"AA Berryman","year":"1992","unstructured":"Berryman, A.A.: The orgins and evolution of predator\u2013prey theory. Ecology 73(5), 1530\u20131535 (1992)","journal-title":"Ecology"},{"key":"2394_CR2","unstructured":"Lotka, A.J.: Elements of Mathematical Biology, Dover Publications (1956)"},{"issue":"4","key":"2394_CR3","doi-asserted-by":"crossref","first-page":"3775","DOI":"10.1007\/s11071-021-06780-x","volume":"105","author":"X Zhang","year":"2021","unstructured":"Zhang, X., An, Q., Wang, L.: Spatiotemporal dynamics of a delayed diffusive ratio-dependent predator\u2013prey model with fear effect. Nonlinear Dyn. 105(4), 3775\u20133790 (2021)","journal-title":"Nonlinear Dyn."},{"issue":"1","key":"2394_CR4","doi-asserted-by":"crossref","DOI":"10.1016\/j.jmaa.2019.123471","volume":"481","author":"SK Sasmal","year":"2020","unstructured":"Sasmal, S.K., Takeuchi, Y.: Dynamics of a predator\u2013prey system with fear and group defense. J. Math. Anal. Appl. 481(1), 123471 (2020)","journal-title":"J. Math. Anal. Appl."},{"key":"2394_CR5","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/j.matcom.2020.04.025","volume":"177","author":"D Mukherjee","year":"2020","unstructured":"Mukherjee, D.: Role of fear in predator\u2013prey system with intraspecific competition. Math. Comput. Simul. 177, 263\u2013275 (2020)","journal-title":"Math. Comput. Simul."},{"issue":"1857","key":"2394_CR6","first-page":"20170878","volume":"284","author":"KH Elliott","year":"2017","unstructured":"Elliott, K.H., Betini, G.S., Norris, D.R.: Fear creates an Allee effect: experimental evidence from seasonal populations. Proc. Biol. Sci. 284(1857), 20170878 (2017)","journal-title":"Proc. Biol. Sci."},{"issue":"6","key":"2394_CR7","doi-asserted-by":"crossref","first-page":"977","DOI":"10.1007\/s00265-015-1910-4","volume":"69","author":"A Basso","year":"2015","unstructured":"Basso, A., Richner, H.: Effects of nest predation risk on female incubation behavior and offspring growth in great tits. Behav. Ecol. Sociobiol. 69(6), 977\u2013989 (2015)","journal-title":"Behav. Ecol. Sociobiol."},{"issue":"6061","key":"2394_CR8","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1126\/science.1210908","volume":"334","author":"LY Zanette","year":"2011","unstructured":"Zanette, L.Y., White, A.F., Allen, M.C., Clinchy, M.: Perceived predation risk reduces the number of offspring songbirds produce per year. Science 334(6061), 398\u2013401 (2011)","journal-title":"Science"},{"issue":"1","key":"2394_CR9","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1656\/045.017.0102","volume":"17","author":"GB Fanson","year":"2010","unstructured":"Fanson, G.B.: Effect of direct and indirect cues of predation risk on the foraging behavior of the white-footed mouse (Peromyscus leucopus). Northeast. Nat. 17(1), 19\u201328 (2010)","journal-title":"Northeast. Nat."},{"issue":"3","key":"2394_CR10","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1016\/S0006-3207(00)00014-8","volume":"95","author":"WJ Ripple","year":"2000","unstructured":"Ripple, W.J., Larsen, E.J.: Historic aspen recruitment, elk, and wolves in northern Yellowstone National Park, USA. Biol. Conserv. 95(3), 361\u2013370 (2000)","journal-title":"Biol. Conserv."},{"issue":"5","key":"2394_CR11","doi-asserted-by":"crossref","first-page":"1181","DOI":"10.1016\/j.anbehav.2004.07.022","volume":"69","author":"S Creel","year":"2005","unstructured":"Creel, S., Winnie, J.A.: Responses of elk herd size to fine-scale spatial and temporal variation in the risk of predation by wolves. Anim. Behav. 69(5), 1181\u20131189 (2005)","journal-title":"Anim. Behav."},{"issue":"5","key":"2394_CR12","doi-asserted-by":"crossref","first-page":"1179","DOI":"10.1007\/s00285-016-0989-1","volume":"73","author":"XY Wang","year":"2016","unstructured":"Wang, X.Y., Zanette, L.Y., Zou, X.F.: Modelling the fear effect in predator\u2013prey interactions. J. Math. Biol. 73(5), 1179\u20131204 (2016)","journal-title":"J. Math. Biol."},{"key":"2394_CR13","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1016\/j.matcom.2024.07.010","volume":"226","author":"HP Rashi","year":"2024","unstructured":"Rashi, H.P., Singh, S.: Singh, Effect of fear with saturated fear cost and harvesting on aquatic food chain model (plankton-fish model) in the presence of nanoparticles. Math. Comput. Simul. 226, 283\u2013305 (2024)","journal-title":"Math. Comput. Simul."},{"key":"2394_CR14","doi-asserted-by":"crossref","DOI":"10.1016\/j.ecocom.2020.100826","volume":"42","author":"K Sarkar","year":"2020","unstructured":"Sarkar, K., Khajanchi, S.: Impact of fear effect on the growth of prey in a predator\u2013prey interaction model. Ecol Complex. 42, 100826 (2020)","journal-title":"Ecol Complex."},{"key":"2394_CR15","volume":"476","author":"HK Qi","year":"2024","unstructured":"Qi, H.K., Liu, B., Li, S.: Stability, bifurcation, and chaos of a stage-structured predator\u2013prey model under fear-induced and delay. Appl. Math. Comput. 476, 128780 (2024)","journal-title":"Appl. Math. Comput."},{"key":"2394_CR16","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1016\/j.matcom.2024.01.003","volume":"220","author":"D Pal","year":"2024","unstructured":"Pal, D., Kesh, D., Mukherjee, D.: Cross-diffusion mediated Spatiotemporal patterns in a predator\u2013prey system with hunting cooperation and fear effect. Math. Comput. Simul. 220, 128\u2013147 (2024)","journal-title":"Math. Comput. Simul."},{"key":"2394_CR17","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1016\/j.matcom.2024.09.022","volume":"228","author":"XB Jiao","year":"2025","unstructured":"Jiao, X.B., Liu, L., Yu, X.: Rich dynamics of a reaction\u2013diffusion Filippov Leslie\u2013Gower predator\u2013prey model with time delay and discontinuous harvesting. Math. Comput. Simul. 228, 339\u2013361 (2025)","journal-title":"Math. Comput. Simul."},{"issue":"1","key":"2394_CR18","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1007\/s44198-024-00238-5","volume":"31","author":"MW Yasin","year":"2024","unstructured":"Yasin, M.W., Ahmed, N., Saeed, J., et al.: numerical study of the reaction diffusion prey\u2013predator model having Holling II Increasing Function in the predator under noisy environment. J. Nonlinear Math. Phys. 31(1), 68 (2024)","journal-title":"J. Nonlinear Math. Phys."},{"issue":"3","key":"2394_CR19","doi-asserted-by":"crossref","first-page":"387","DOI":"10.15388\/NA.2019.3.5","volume":"24","author":"L Liu","year":"2019","unstructured":"Liu, L.: Stability and Hopf bifurcation of a delayed reaction\u2013diffusion predator\u2013prey model with anti-predator behaviour. Nonlinear Anal. Model. Control 24(3), 387\u2013406 (2019)","journal-title":"Nonlinear Anal. Model. Control"},{"issue":"2","key":"2394_CR20","first-page":"97","volume":"26","author":"K Owolabi","year":"2018","unstructured":"Owolabi, K., Pindza, E.: Mathematical and computational studies of fractional reaction\u2013diffusion system modelling predator\u2013prey interactions. J. Numer. Math. 26(2), 97\u2013110 (2018)","journal-title":"J. Numer. Math."},{"issue":"10","key":"2394_CR21","doi-asserted-by":"crossref","first-page":"6217","DOI":"10.1002\/mma.8167","volume":"45","author":"HK Qi","year":"2022","unstructured":"Qi, H.K., Meng, X.Z., Tasawar, H., Aatef, H.: Bifurcation dynamics of a reaction\u2013diffusion predator\u2013prey model with fear effect in a predator-poisoned environment. Math. Methods Appl. Sci. 45(10), 6217\u20136254 (2022)","journal-title":"Math. Methods Appl. Sci."},{"issue":"1","key":"2394_CR22","first-page":"53","volume":"52","author":"AM Turing","year":"1990","unstructured":"Turing, A.M.: The chemical basis of morphogenesis. Bull. Math. Biol. 52(1), 53\u201397 (1990)","journal-title":"Bull. Math. Biol."},{"key":"2394_CR23","doi-asserted-by":"crossref","first-page":"775","DOI":"10.3934\/mbe.2018035","volume":"15","author":"X Wang","year":"2018","unstructured":"Wang, X., Zou, X.: Pattern formation of a predator\u2013prey model with the cost of anti-predator behaviors. Math. Biosci. Eng. 15, 775\u2013805 (2018)","journal-title":"Math. Biosci. Eng."},{"issue":"1","key":"2394_CR24","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/BF00276081","volume":"27","author":"J Furter","year":"1989","unstructured":"Furter, J., Grinfeld, M.: Local vs non-local interactions in population-dynamics. J. Math. Biol. 27(1), 65\u201380 (1989)","journal-title":"J. Math. Biol."},{"key":"2394_CR25","first-page":"115497","volume":"188","author":"YF Du","year":"2024","unstructured":"Du, Y.F., Sui, M.T.: Stability and spatiotemporal dynamics in a diffusive predator\u2013prey model with nonlocal prey competition and nonlocal fear effect. Chaos Solitons Fract. Interdiscip. J. Nonlinear Sci. Nonequilib. Complex Phenom. 188, 115497\u2013115497 (2024)","journal-title":"Chaos Solitons Fract. Interdiscip. J. Nonlinear Sci. Nonequilib. Complex Phenom."},{"key":"2394_CR26","first-page":"114","volume":"177","author":"XL Sun","year":"2023","unstructured":"Sun, X.L.: Dynamics of a diffusive predator\u2013prey model with nonlocal fear effect. Chaos Solitons Fract. 177, 114\u2013221 (2023)","journal-title":"Chaos Solitons Fract."},{"key":"2394_CR27","volume":"132","author":"XS Dong","year":"2022","unstructured":"Dong, X.S., Niu, B.: On a diffusive predator\u2013prey model with nonlocal fear effect. Appl. Math. Lett. 132, 108156 (2022)","journal-title":"Appl. Math. Lett."},{"issue":"4","key":"2394_CR28","first-page":"877","volume":"69","author":"S Dash","year":"2023","unstructured":"Dash, S., Khajanchi, S.: Dynamics of intraguild predation with intraspecies competition. J. Appl. Math. Comput. 69(4), 877\u2013906 (2023)","journal-title":"J. Appl. Math. Comput."},{"issue":"1","key":"2394_CR29","first-page":"1","volume":"137","author":"B Mondal","year":"2022","unstructured":"Mondal, B., Rahman, M.S., Sarkar, S., Ghosh, U.: Studies of dynamical behaviours of an imprecise predator\u2013prey model with Holling type II functional response under interval uncertainty. Eur. Phys. J. Plus 137(1), 1\u201320 (2022)","journal-title":"Eur. Phys. J. Plus"},{"issue":"8","key":"2394_CR30","first-page":"66","volume":"33","author":"JL Xiao","year":"2023","unstructured":"Xiao, J.L., Xia, Y.H.: Global dynamics of a predator\u2013prey model with simplified Holling type IV functional response and fear effect. Int. J. Bifurc. Chaos 33(8), 66 (2023)","journal-title":"Int. J. Bifurc. Chaos"},{"key":"2394_CR31","doi-asserted-by":"crossref","unstructured":"Xie, B.F.: Impact of the fear and Allee effect on a Holling type II prey\u2013predator model. Adv. Differ. Equ. 2021(1), 66 (2021)","DOI":"10.1186\/s13662-021-03592-6"},{"issue":"5","key":"2394_CR32","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1007\/s00285-024-02070-3","volume":"88","author":"S Wu","year":"2024","unstructured":"Wu, S., Wang, Z., Ruan, S.: Hopf bifurcation in an age-structured predator\u2013prey system with Beddington\u2013DeAngelis functional response and constant harvesting. J. Math. Biol. 88(5), 56 (2024)","journal-title":"J. Math. Biol."},{"issue":"1","key":"2394_CR33","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1007\/s11071-020-06140-1","volume":"103","author":"T Saha","year":"2021","unstructured":"Saha, T., Pal, P.J., Banerjee, M.: Relaxation oscillation and canard explosion in a slow-fast predator\u2013prey model with Beddington\u2013DeAngelis functional response. Nonlinear Dyn. 103(1), 195\u2013217 (2021)","journal-title":"Nonlinear Dyn."},{"issue":"05","key":"2394_CR34","first-page":"28","volume":"30","author":"J Gao","year":"2020","unstructured":"Gao, J., Guo, S.: Patterns in a modified Leslie\u2013Gower model with Beddington\u2013DeAngelis functional response and nonlocal prey competition. Int. J. Bifurc. Chaos 30(05), 28 (2020)","journal-title":"Int. J. Bifurc. Chaos"},{"issue":"3","key":"2394_CR35","doi-asserted-by":"crossref","first-page":"211","DOI":"10.2307\/1467324","volume":"8","author":"PH Crowley","year":"1989","unstructured":"Crowley, P.H., Martin, E.K.: Functional responses and interference within and between year classes of a dragonfly population. J. N. Am. Benthol. Soc. 8(3), 211\u2013221 (1989)","journal-title":"J. N. Am. Benthol. Soc."},{"issue":"4","key":"2394_CR36","doi-asserted-by":"crossref","first-page":"2463","DOI":"10.3934\/dcdsb.2022177","volume":"28","author":"N Li","year":"2023","unstructured":"Li, N., Sun, W., Liu, S.: A stage-structured predator\u2013prey model with Crowley\u2013Martin functional response. Discrete Continu. Dyn. Syst. Ser. B 28(4), 2463\u20132489 (2023)","journal-title":"Discrete Continu. Dyn. Syst. Ser. B"},{"issue":"1","key":"2394_CR37","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1111\/1365-2656.12280","volume":"84","author":"B Zimmermann","year":"2015","unstructured":"Zimmermann, B., Sand, H., Wabakken, P., Liberg, O., Andreassen, H.P.: Predator-dependent functional response in wolves: from food limitation to surplus killing. J. Anim. Ecol. 84(1), 102\u2013112 (2015)","journal-title":"J. Anim. Ecol."},{"issue":"15","key":"2394_CR38","first-page":"66","volume":"30","author":"SZ Rida","year":"2020","unstructured":"Rida, S.Z., Yousef, A.M., Arafat, S.: Stability, analytic bifurcation structure and chaos control in a mutual interference host-parasitoid model. Int. J. Bifurc. Chaos 30(15), 66 (2020)","journal-title":"Int. J. Bifurc. Chaos"},{"key":"2394_CR39","volume-title":"Nonlinear Parabolic and Elliptic Equations","author":"CV Pao","year":"1992","unstructured":"Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992)"},{"key":"2394_CR40","volume-title":"Introduction to Reaction\u2013Diffusion Equations","author":"Q Ye","year":"2011","unstructured":"Ye, Q., Li, Z., Wang, M., Wu, Y.: Introduction to Reaction\u2013Diffusion Equations. Science Press, Beijing (2011)"},{"issue":"4","key":"2394_CR41","first-page":"48","volume":"42","author":"SJ Guo","year":"2018","unstructured":"Guo, S.J.: Bifurcation and spatio-temporal patterns in a diffusive predator\u2013prey system. Nonlinear Anal. Real World Appl. 42(4), 48\u201377 (2018)","journal-title":"Nonlinear Anal. Real World Appl."},{"issue":"5","key":"2394_CR42","doi-asserted-by":"crossref","first-page":"995","DOI":"10.1016\/j.chaos.2003.09.048","volume":"20","author":"SL Yuan","year":"2004","unstructured":"Yuan, S.L., Han, M.A.: Bifurcation analysis of a chemostat model with two distributed delays. Chaos Solitons Fract. 20(5), 995\u20131004 (2004)","journal-title":"Chaos Solitons Fract."},{"issue":"1","key":"2394_CR43","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s002850100109","volume":"44","author":"SA Gourley","year":"2002","unstructured":"Gourley, S.A., So, J.W.H.: Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain. J. Math. Biol. 44(1), 49\u201378 (2002)","journal-title":"J. Math. Biol."},{"key":"2394_CR44","doi-asserted-by":"crossref","DOI":"10.1016\/j.nonrwa.2024.104182","volume":"81","author":"M Khachatryan","year":"2025","unstructured":"Khachatryan, M., Onyido, M.A., Salako, R.B.: Persistence and positive steady states of a two-stage structured population model with mixed dispersals. Nonlinear Anal. Real World Appl. 81, 104182 (2025)","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"2394_CR45","first-page":"10100710","volume":"66","author":"D Bhattacharjee","year":"2024","unstructured":"Bhattacharjee, D., Ray, N., Das, D.: A discrete-time dynamical model of prey and stage-structured predator with juvenile hunting incorporating negative effects of prey refuge. Part. Differ. Equ. Appl. Math. 66, 10100710 (2024)","journal-title":"Part. Differ. Equ. Appl. Math."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02394-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-025-02394-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02394-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,23]],"date-time":"2025-05-23T16:45:58Z","timestamp":1748018758000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-025-02394-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,1]]},"references-count":45,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["2394"],"URL":"https:\/\/doi.org\/10.1007\/s12190-025-02394-3","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,1]]},"assertion":[{"value":"19 November 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 January 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 January 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 February 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there is no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}