{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T02:51:01Z","timestamp":1770346261439,"version":"3.49.0"},"reference-count":26,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,10]],"date-time":"2023-03-10T00:00:00Z","timestamp":1678406400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Basque Government","award":["IT1555-22"],"award-info":[{"award-number":["IT1555-22"]}]},{"name":"Basque Government","award":["KK-2022\/00090"],"award-info":[{"award-number":["KK-2022\/00090"]}]},{"name":"Basque Government","award":["PID2021-358 1235430B-C21"],"award-info":[{"award-number":["PID2021-358 1235430B-C21"]}]},{"name":"Basque Government","award":["PID2021-358 1235430B-C22"],"award-info":[{"award-number":["PID2021-358 1235430B-C22"]}]},{"name":"MCIN\/AEI 269.10.13039\/501100011033","award":["IT1555-22"],"award-info":[{"award-number":["IT1555-22"]}]},{"name":"MCIN\/AEI 269.10.13039\/501100011033","award":["KK-2022\/00090"],"award-info":[{"award-number":["KK-2022\/00090"]}]},{"name":"MCIN\/AEI 269.10.13039\/501100011033","award":["PID2021-358 1235430B-C21"],"award-info":[{"award-number":["PID2021-358 1235430B-C21"]}]},{"name":"MCIN\/AEI 269.10.13039\/501100011033","award":["PID2021-358 1235430B-C22"],"award-info":[{"award-number":["PID2021-358 1235430B-C22"]}]},{"name":"M. De la Sen","award":["IT1555-22"],"award-info":[{"award-number":["IT1555-22"]}]},{"name":"M. De la Sen","award":["KK-2022\/00090"],"award-info":[{"award-number":["KK-2022\/00090"]}]},{"name":"M. De la Sen","award":["PID2021-358 1235430B-C21"],"award-info":[{"award-number":["PID2021-358 1235430B-C21"]}]},{"name":"M. De la Sen","award":["PID2021-358 1235430B-C22"],"award-info":[{"award-number":["PID2021-358 1235430B-C22"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The objective of this work was to investigate the dynamics of host\u2013parasitoid model with spatial refuge effect. For this, two discrete host\u2013parasitoid models were considered under spatial refuge effect. Suppose that a constant population of hosts may seek refuge and protection from an attack of parasitoids. We found the parametric factors affecting the existence of the equilibrium points and uniqueness of equilibrium points. A local stability analysis of host\u2013parasitoid models was also carried out. Bifurcation theory was used to observe that the host\u2013parasitoid models undergo Neimark\u2013Sacker bifurcation. The effect of the existence of constant refuge effect on the local stability and bifurcation of models was also explored. Hybrid chaos control methodology was used to control the chaotic behavior of model. In addition, numerical simulations, bifurcation diagrams, and phase portraits of the models are also presented.<\/jats:p>","DOI":"10.3390\/axioms12030290","type":"journal-article","created":{"date-parts":[[2023,3,10]],"date-time":"2023-03-10T03:02:08Z","timestamp":1678417328000},"page":"290","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["The Qualitative Analysis of Host\u2013Parasitoid Model with Inclusion of Spatial Refuge Effect"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7486-2493","authenticated-orcid":false,"given":"Muhammad Sajjad","family":"Shabbir","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Poonch Rawalakot, Rawalakot 12350, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0999-7404","authenticated-orcid":false,"given":"Qamar","family":"Din","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Poonch Rawalakot, Rawalakot 12350, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9701-4809","authenticated-orcid":false,"given":"Wafa F.","family":"Alfwzan","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9320-9433","authenticated-orcid":false,"given":"Manuel","family":"De la Sen","sequence":"additional","affiliation":[{"name":"Department of Electricity and Electronics, Institute of Research and Development of Processes, Faculty of Science and Technology, Campus of Leioa (Bizkaia), University of the Basque Country, 48940 Leioa, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,10]]},"reference":[{"key":"ref_1","first-page":"565","article-title":"Bifurcation analysis of a host-parasitoid model with the beddington-deangelis functional response","volume":"2","author":"Agarwal","year":"2014","journal-title":"Glob. J. Math. 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