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On the convexity of carrying simplices in competitive Lotka-Volterra systems. (English) Zbl 0799.92016

Elworthy, K.D. (ed.) et al., Differential equations, dynamical systems, and control science. A Festschrift in Honor of Lawrence Markus. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 152, 353-364 (1994).
Consider a competitive Lotka-Volterra system: \[ dx_ i/dt= x_ i* \biggl( b_ i- \sum_{j=1}^ n a_{ij} x_ j\biggr), \qquad i=1,\dots, n. \tag{1} \] M. W. Hirsch [Nonlinearity 1, No. 1, 51–71 (1988; Zbl 0658.34024)] proved that there is a hypersurface \(\Sigma\) being the carrying simplex for the system (1). Some results:
i) The carrying simplex \(\Sigma\) of the system (1), when \(n=2\), is convex;
ii) The system (1) with strictly convex carrying simplex has no non- trivial recurrence in \(R_ +^ n\);
iii) If \(P\) repels on \(\Sigma\), the boundary \(\partial\Sigma\) attracts on \(\Sigma\), and \(Q\) is definite, then the basin of repulsion of \(P\) is the interior of \(\Sigma\) (that means that there can be no coexistence of all \(n\) species, at least one of the species will be driven to extinction);
iv) There are examples of system (1), with \(n=3\), which have non-convex carrying simplex, but only trivial recurrence.
Open questions: What are the global dynamics on the carrying simplex and how many closed orbits can there be in system (1), when \(n=3\).
For the entire collection see [Zbl 0780.00045].

MSC:

92D25 Population dynamics (general)
37N99 Applications of dynamical systems
37G99 Local and nonlocal bifurcation theory for dynamical systems

Citations:

Zbl 0658.34024
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