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。

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).

Contents
1.Introduction . . . . . . . .. . . . . . . 2
2.Preliminary . . . . . . . .. . . . . . .  4
  2.1 Existence of Equilibria . . . . . . . . 4
  2.2 Local Stability of Equilibria . . . . . 6
3.Main Results . . . . . . . .. . . . . . . 8
4.Numerical Simulations and Discussion . . .10
References . . . . . . . .. . . . . . . .. .12
Appendix . . . . . . . .. . . . . . . .. . .14

List of Figures
1.strong, weak and no Allee effect . . . . . . . .. . .. . 2
2.The local stability of the positive equilibrium E. is dependent on the positive of the intersection of the vertical isocline (the blue dashed line) and another isocline (the black line). . . . . . . . . . . . . . . . . . .. . .. . .. 8
3.μ = 0, a = 0.2, m = 0.35, d = 0.23, θ = 0.1 . . . . . .  10
4.μ = 0.6, a = 0.2, m = 0.35, d = 0.23, θ = 0.1 . . . . . .11
5.μ = 0.8, a = 0.2, m = 0.35, d = 0.23, θ = 0.1 . . . . . .11

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