具有 Allee 效應的掠食者-被掠食者模型的數學分析

在這項工作中，我們考慮了一個二維掠食者被掠食者系統，其中被掠食者的成長函數對參數μ具有變化 Allee 效應函數，由無 Allee 效應函數（μ=0，單穩型）和強 Allee 效應（μ=1，雙穩型）的線性組合而成。我們通過 Lyapunov 方法顯示了弱 Allee 效應的正平衡的全局漸近穩定性。此外，當正平衡不穩定時，通過Hopf分岔存在小振幅週期解。我們的數值模擬結果表明振幅相對於參數μ遞增，並且當 Allee 效應強(0≪μ<1)時存在鬆弛振盪。

關鍵字

Allee 效應；兩個物種；掠食者-被掠食者；鬆弛振盪

並列摘要

In this work, we consider a two-dimensional predator prey system where the birth function of the prey has various Allee effect on parameter μ by a linear combination of a no-Allee effects function (μ=0, monostable type) and a strong Allee effect function (μ=1, bistable type). We show the globally asymptotical stability of the positive equilibrium by Lyapunov method for the weak Allee effect. Moreover, it is well known that there is a periodic solution with the small amplitude via the Hopf bifurcation when the positive equilibrium is unstable. Our numerical simulations suggest that the amplitude is increasing with respective to the parameter μ, and there exists the relaxation oscillation when the Allee effect is strong (0≪μ<1).

並列關鍵字

Allee effects ； two species ； Predator-Prey ； relaxation oscillation

參考文獻

References

Google Scholar

[1]E. N. Bodine, Anne E. Yust.Predator–prey dynamics with intraspecific com-petition and an Allee effect in the predator population. (2017). Letters inBiomathematics.4(1):23-38DOI: 10.1080/23737867.2017.1282843.