在這項研究中，我們考慮一掠食者-兩被掠食者模型依據Holling type II功能性反應且兩被掠食者之間不存在競爭關係。首先建立有界且有正解的初始條件，然後證明所有的邊界平衡點的存在性和穩定性在三維中。此外，我們利用掠食者的死亡率푑作為參數，分類我們的動態系統(1.2)，以及正平衡點存在性和穩定性的分析，最後利用我們的分類，進行一些數值模擬。

In this work, we consider the one-predator-two-prey models with Holling type II functional response without competition between the two renewable base resource. We first establish the boundedness and positivity of solution with positive initial conditions. Then the existence and local stability of all boundary equilibria are clarified in ℝ3. Moreover, we use the death rate of predator 푑 as a parameter to classify the dynamics of system (1.2) as well as the existence and local stability of positive equilibrium. Finally, some numerical simulations are performed for each region of our classification.

Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Boundedness of Solutions. . . . . . . . . . . . . . . . . . . . . 3
2.2 Subsystems and its Equilibria . . . . . . . . . . . . . . . . . . 4
3 Existence of Boundary Equilibria and its Stability in R3 . . . . . . 7
3.1 Su cient Condition for Extinction of the Predator in R3 . . . . . 7
3.2 Stability Analysis in of Boundary Equilibria R3 . . . . . . . . . 8
4 Existence of Coexistence States and Stability of quilibria . . . . . 12
4.1 Existence of Positive Equilibrium . . . . . . . . . . . . . . . . 12
4.2 Stability Analysis of Positive Equilibria . . . . . . . . . . . . 19
5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 20
6 Conclusions, Discussions and Future Works . . . . . . . . . . . . . 30
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

List of Figures
3.1 The region of parameters pace with various a1, a2 and  xed m1, m2,
r, m, d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.1 Typical pictures of parameter space with varied d. . . . . . . . . . . 13
4.2 Typical pictures of parameter space with parameter 0 < d < m2 < m1. 14
4.3 Typical pictures of P1 and P2 with parameter m2 < d <m1. . . . . 17
4.4 Typical pictures of P1 and P2 with parameter m1 < d < m1/(1+a1)+ m2/(1+a2). 18
5.1 Solutions with parameters in the class (a)-(2)-(i) of Fig. 4.1 are periodic on the x1-y plane and x2 dies out. . . . . . . . . . . . . . . 21
5.2 Solutions with parameters in the class (a)-(2)-(ii) of Fig. 4.1 converge to E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.3 Solutions with parameters in the class (a)-(3) of Fig. 4.1 converge to E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.4 Solutions with parameters in the class (a)-(4) of Fig. 4.1 converge to E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.5 Solutions with parameters in the class (a)-(1) of Fig. 4.1 converge to Ex1x2 and y dies out. . . . . . . . . . . . . . . . . . . . . . . . . 23
5.6 Solutions with parameters in the class (b)-(2)-(i) of Fig. 4.1 are periodic on the x1-y plane and x2 dies out. . . . . . . . . . . . . . . 24
5.7 Solutions with parameters in the class (b)-(2)-(ii) of Fig. 4.1 converge to Ex1y and x2 dies out. . . . . . . . . . . . . . . . . . . . . . 24
5.8 Solutions with parameters in the class (b)-(3) of Fig. 4.1 converge to E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.9 Solutions with parameters in the class (b)-(1) of Fig. 4.1 converge to Ex1x2 and y dies out. . . . . . . . . . . . . . . . . . . . . . . . . 25
5.10 Solutions with parameters in the class (a)-(2)-(i) of Fig. 4.1 converge to Ex1y and x2 dies out. . . . . . . . . . . . . . . . . . . . . . 27
5.11 Solutions with parameters in the class (a)-(2)-(ii) of Fig. 4.1 are periodic in R3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.12 Solutions with parameters in the class (a)-(3) of Fig. 4.1 are periodic in R3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.13 Solutions with parameters in the class (b)-(2)-(i) of Fig. 4.1 converge to Ex1y and x2 dies out. . . . . . . . . . . . . . . . . . . . . . 28
5.14 Solutions with parameters in the class (b)-(2)-(ii) of Fig. 4.1 converge to E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.15 Solutions with parameters in the class (b)-(3) of Fig. 4.1 converge to Ex1y and x2 dies out. . . . . . . . . . . . . . . . . . . . . . . . . 29

List of Tables
2.1 Subsystems, equilibria and its stability for system (1.2) . . . . . . . 6
5.1 Figure 4.1 (a)m1 < m1/(1+a1)+ m2/(1+a2). Fixed r = m = 1:0, a1 = 0:6,a2 = 0:5, m1 = 2:0 and m2 = 1:5 . . . . . . . . . . . . . . . . . . . . 20
5.2 Figure 4.1 (b)m1>=m1/(1+a1)+ m2/(1+a2). Fix r = m = 1:0, a1 = 0:6,a2 = 0:5, m1 = 2:5 and m2 = 1:2 . . . . . . . . . . . . . . . . . . . . 21
5.3 Compare Figure 3.1 with Figure 4.1. Fixed r = 1:0, m = 0:5 . . . . 26

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