Department of Mathematics & Statistics, Texas Tech University, Lubbock, TX, USA.
Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt.
J Biol Dyn. 2023 Dec;17(1):2242372. doi: 10.1080/17513758.2023.2242372.
We develop and investigate a discrete-time predator-prey model with cooperative hunting among predators and a spatial prey refuge. The system can exhibit two positive equilibria if the magnitude of cooperation is large and the maximal reproduction number of predators is less than one. In such a scenario, the predator population may exhibit a strong Allee effect, and therefore the predator may survive if its density is above the threshold. When the positive equilibrium is unique, we prove that hunting cooperation can destabilize the equilibrium through a Neimark-Sacker bifurcation. Numerical findings indicate that a high degree of predator hunting cooperation can help rescue the predator population if the proportion of prey refuge is large, while hunting cooperation becomes destabilizing when the proportion of refuge is small. Despite hunting cooperation's destabilizing effect, it can facilitate predator persistence, particularly with a large proportion of prey refuge. Furthermore, there exists a wide parameter space where the predator-prey interaction may exhibit chaotic behaviour.
我们开发并研究了一个具有捕食者合作狩猎和空间猎物避难所的离散时间捕食者-猎物模型。如果合作的幅度较大且捕食者的最大繁殖数小于 1，则系统可以表现出两个正平衡点。在这种情况下，捕食者种群可能会表现出强烈的阿利效应，因此如果其密度高于阈值，捕食者就可能存活。当正平衡点是唯一的时，我们通过奈马克-萨克分岔证明了捕食者的合作狩猎可以使平衡点失稳。数值研究结果表明，如果猎物避难所的比例较大，那么高程度的捕食者合作狩猎可以帮助拯救捕食者种群，而当避难所的比例较小时，合作狩猎则会变得不稳定。尽管合作狩猎具有不稳定性，但它可以促进捕食者的生存，特别是在猎物避难所比例较大的情况下。此外，还存在一个捕食者-猎物相互作用可能表现出混沌行为的广泛参数空间。
