Stability and Double-Hopf Bifurcations of a Gause-Kolmogorov-Type Predator-Prey System with Indirect Prey-Taxis
Stability and Double-Hopf Bifurcations of a Gause-Kolmogorov-Type Predator-Prey System with Indirect Prey-Taxis
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具有间接猎物趋向性的高斯-柯尔莫哥洛夫型捕食者-被捕食系统的稳定性和双跳分岔
DOI：
10.1007/s10884-020-09878-9
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发表时间：
--
影响因子：
1.3
通讯作者：
Song Yongli
中科院分区：
文献类型：
--
作者：
Zuo Wenjie;Song Yongli
在本文中，我们研究具有间接猎物趋向性的高斯-柯尔莫哥洛夫型捕食者-被捕食者系统，这意味着捕食者的定向运动受到猎物释放的一些化学物质的刺激。研究了正平衡的存在性、间接猎物趋向性对稳定性的影响以及相关的分岔。明确确定了发生Hopf分岔、图灵分岔、图灵-Hopf分岔和双Hopf分岔的临界值。推导了非共振和弱共振双Hopf分岔的范式计算算法。此外，我们将理论结果应用到具有Holling-II型函数响应的系统中，在间接猎物趋向性和自饱和系数的平面内完全确定了稳定区和分岔曲线。双Hopf分岔点附近的动力学分类是明确确定的。在双Hopf分岔附近，存在稳定的空间齐次/非齐次周期解、稳定的空间非齐次四周期解，并且随着间接趋向性和半饱和系数的变化，从一种时空模式到另一种时空模式的转变。结果表明，空间不均匀的 Hopf 分岔是由间接猎物趋向性参数引起的，这对于具有直接猎物趋向性的反应扩散捕食者-猎物模型来说是不可能的。
In this paper, we deal with the Gause–Kolmogorov-type predator–prey system with indirect prey-taxis, which means that directional movement of predators is stimulated by some chemicals emitted by preys. The existence of the positive equilibrium, the effect of the indirect prey-taxis on the stability and the related bifurcations are investigated. The critical values for the occurrence of the Hopf bifurcation, Turing bifurcation, Turing–Hopf bifurcation and double-Hopf bifurcation are explicitly determined. An algorithm for calculating the normal form of the double-Hopf bifurcation for the non-resonance and weak resonance is derived. Moreover, we apply the theoretical results to the system with Holling-II type functional response, the stable region and the bifurcation curves are completely determined in the plane of the indirect prey-taxis and self saturation coefficient. The dynamical classification near the double-Hopf bifurcation point is explicitly determined. In the neighborhood of the double-Hopf bifurcation, there are stable spatially homogeneous/inhomogeneous periodic solutions, stable spatially inhomogeneous quadi-periodic solutions and the pattern transitions from one spatial–temporal patterns to another one with the changes of the indirect taxis and semi saturation coefficients. The results show that spatially inhomogeneous Hopf bifurcations are induced by an indirect prey-taxis parameter, which is impossible for the reaction–diffusion predator–prey model with a direct prey-taxis.