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The global dynamic behavior of the competition model of three species. (English) Zbl 0974.34047

The competition Lotka-Volterra (LV) model \[ \dot{x}_i=x_i(1-x_i-\alpha_ix_{i+1}-\beta_ix_{i+2}),\quad x_i(0)>0,\;\alpha_i>0,\;\beta_i>0,\;i=1,2,3,\tag{1} \] with \(x_4=x_1,\;x_5=x_2\) is investigated. By a diffeomorphism from \(\mathbb{R}^3-\{O\}\) to \(\mathbb{R}^3-\{O\}\): \(u=\frac{x}{\|x\|}\), the authors decompose the dynamic behavior of the LV model (1) into the dynamic behavior of two-dimensional manifolds, and completely analyse the global asymptotic behavior of (1). They obtain sufficient and necessary conditions for the existence of the positive equilibrium \(E_7\) and the global asymptotical stability of \(E_7\). They also give sufficient and necessary conditions for the LV model (1) having a family of limit cycle solutions and a heteroclinic cycle, both of which are the \(\omega\)-limit set of some other trajectories to LV model (1). In the theorem 2.1, it is said that if \(\alpha_i>0,\;\beta_i>0,\;\alpha_i\neq 1,\;\beta_i\neq 1,\) then model (1) has a positive equilibrium \(E_7=(\frac{D_1}{D},\frac{D_2}{D},\frac{D_3}{D})\) if and only if one of the following 46 conditions holds:
(1) \(\alpha_1,\beta_1,\alpha_2,\beta_2,\alpha_3,\beta_3<1\); (2) \(\alpha_1>1,\beta_1,\alpha_2,\beta_2,\alpha_3,\beta_3<1\);
(3) \(\alpha_2>1,\beta_1,\alpha_1,\beta_2,\alpha_3,\beta_3<1\); (4) \(\alpha_3>1,\beta_1,\alpha_1,\beta_2,\alpha_2,\beta_3<1; \dots\);
(46) \(\alpha_1,\beta_1,\alpha_2,\beta_2,\alpha_3,\beta_3>1\);
with \(D_1=A_2(A_1-B_3)+B_1B_3\), \(D_2=A_3(A_2-B_1)+B_2B_1\), \(D_3=A_1(A_3-B_2)+B_3B_2\), \(D=D_i+\alpha_iD_{i+1}+\beta_iD_{i+2}\), \(A_i=1-\alpha_i\), \(B_i=1-\beta_i\), \(i=1,2,3\).
The reviewer found that if \(A_1>B_3>0\), \(B_1>A_2>0\), \(B_2>0\), \(A_3<0\), \(0<B_3\ll 1\) (\(B_3\) is small enough), then the condition (4) is true. But we have \(D_1>0\), \(D_2>0\), \(D_3<0\). Suppose that \(D>0\), then \(E_7\) is not a positive equilibrium.

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
92D25 Population dynamics (general)
34D23 Global stability of solutions to ordinary differential equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
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References:

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