A　Leslie–Gower　predator–prey　model　with　square　root　response　function　and　generalist　predator　is　considered,　and　the　existence　and　stability　of　equilibria　of　the　system　are　discussed.　It　is　shown　that　the　system　undergoes　a　degenerate　Hopf　bifurcation　of　codimension　exactly　two,　where　there　exist　two　limit　cycles.　In　addition,　we　find　that　the　system　has　a　cusp　of　codimension　two　and　exhibits　a　Bogdanov–Takens　bifurcation　of　codimension　two.　Our　results　reveal　richer　dynamics　than　the　system　with　no　generalist　predator.　©　2024　Elsevier　Ltd