{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:06:40Z","timestamp":1760148400604,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,4,26]],"date-time":"2023-04-26T00:00:00Z","timestamp":1682467200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Dispersal among species is an important factor that can govern the prey\u2013predator model\u2019s dynamics and cause a variety of spatial structures on a geographical scale. These structures form when passive diffusion interacts with the reaction part of the reaction\u2013diffusion system in such a way that even if the reaction lacks symmetry-breaking capabilities, diffusion can destabilize the symmetry and allow the system to have them. In this article, we look at how dispersal affects the prey\u2013predator model with a Hassell\u2013Varley-type functional response when predators do not form tight groups. By considering linear stability, the temporal stability of the model and the conditions for Hopf bifurcation at feasible equilibrium are derived. We explored spatial stability in the presence of diffusion and developed the criterion for diffusion-driven instability. Using amplitude equations, we then investigated the selection of Turing patterns around the Turing bifurcation threshold. The examination of the stability of these amplitude equations led to the discovery of numerous Turing patterns. Finally, numerical simulations were performed to validate the outcomes of the analysis. The outcomes of the theoretical study and numerical simulation were accorded. Our findings demonstrate that spatial patterns are sensitive to dispersal and predator death rates.<\/jats:p>","DOI":"10.3390\/sym15050986","type":"journal-article","created":{"date-parts":[[2023,4,27]],"date-time":"2023-04-27T04:30:47Z","timestamp":1682569847000},"page":"986","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Influence of Spatial Dispersal among Species in a Prey\u2013Predator Model with Miniature Predator Groups"],"prefix":"10.3390","volume":"15","author":[{"family":"Shivam","sequence":"first","affiliation":[{"name":"Department of Mathematics, Graphic Era Deemed to be University, Dehradun 248002, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7473-7115","authenticated-orcid":false,"given":"Turki","family":"Aljrees","sequence":"additional","affiliation":[{"name":"Department College of Computer Science and Engineering, University of Hafr Al-Batin, Hafar Al-Batin 39524, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8050-5639","authenticated-orcid":false,"given":"Teekam","family":"Singh","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, Graphic Era Deemed to be University, Dehradun 248002, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6537-7891","authenticated-orcid":false,"given":"Neeraj","family":"Varshney","sequence":"additional","affiliation":[{"name":"Department of Computer Engineering & Applications, GLA University, Mathura 281406, India"}]},{"given":"Mukesh","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Graphic Era Deemed to be University, Dehradun 248002, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7201-6381","authenticated-orcid":false,"given":"Kamred Udham","family":"Singh","sequence":"additional","affiliation":[{"name":"School of Computing, Graphic Era Hill University, Dehradun 248002, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8411-9233","authenticated-orcid":false,"given":"Vrince","family":"Vimal","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, Graphic Era Hill University, Dehradun 248002, India"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1111\/j.0030-1299.2004.12814.x","article-title":"Faced with a choice, sparrowhawks more often attack the more vulnerable prey group","volume":"104","author":"Cresswell","year":"2004","journal-title":"Oikos"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1006\/anbe.1995.0260","article-title":"Predator preferences for attacking particular prey group sizes: Consequences for predator hunting success and prey predation risk","volume":"50","author":"Krause","year":"1995","journal-title":"Anim. 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