Department of Mathematics, Yuncheng University, Yuncheng, Shanxi 044000, China.
Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, Canada.
Math Biosci. 2019 Jan;307:33-41. doi: 10.1016/j.mbs.2018.11.004. Epub 2018 Nov 13.
This paper concerns with a stage-structured predator-prey model with cannibalism in the predator, where the predator population is divided into two stages, a juvenile stage and a mature stage. The effects of cannibalism on the model have been partially discussed by Magnússon [12] and Mařík and Přibyloná [13] by means of numerical simulation and simple mathematical analysis. Here we reinvestigate the dynamics of this model. We obtain the global stability of the model, illustrate the effects of the two factors (one describes the cannibalization and the other the benefit from cannibalism) on dynamics of the model, and find that the positive equilibrium can undergo stability switch (from stable to unstable to stable, or from unstable to stable to unstable) with the change of the cannibalization rate. The obtained results indicate that large cannibalization rate can make the positive equilibrium globally stable although its stability would change with the increase of the rate. We also analyze theoretically Hopf bifurcation. It turns out that supercritical/subcritical Hopf bifurcation can occur with the rates of the above mentioned two factors as bifurcation parameters. Finally, we introduce some quantities with clear biological meanings to better understand our results from the point of biology.
本文研究了一个具有捕食者自食行为的阶段结构捕食-被捕食模型，其中捕食者种群分为两个阶段，即幼年阶段和成熟阶段。Magnússon [12] 和 Mařík 和 Přibyloná [13] 通过数值模拟和简单的数学分析部分讨论了自食行为对模型的影响。在这里，我们重新研究了这个模型的动力学。我们得到了模型的全局稳定性，说明了两个因素（一个描述自食行为，另一个描述自食行为的益处）对模型动力学的影响，并发现正平衡点可以随着自食率的变化经历稳定性转变（从稳定到不稳定再到稳定，或从不稳定到稳定再到不稳定）。所得结果表明，尽管自食率的稳定性会随着其增加而变化，但较大的自食率可以使正平衡点全局稳定。我们还从理论上分析了 Hopf 分支。结果表明，超临界/亚临界 Hopf 分支可能会发生，其分支参数为上述两个因素的速率。最后，我们引入了一些具有明确生物学意义的数量，以便从生物学角度更好地理解我们的结果。
