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On the convexity of the carrying simplex of planar Lotka-Volterra competitive systems. (English) Zbl 1036.34057

Consider the competitive Lotka-Volterra system \[ dx/dt=x(a-bx-cy), \qquad dy/dt=y(d-ex-fy), \] with positive parameters \(a,\dots,f.\) Besides the equilibria \(A=(0,0)\), \(B=(0,d/f)\), \(C=(a/b,0)\) on the boundary of the positive quadrant, there exists at most one interior equilibrium \(D\) which is a stable node or a saddle point.
The author investigates the separatrices connecting \(B\) and \(D\) (briefly: \(BD\)), \(DC\) and \(BC\). He shows that if \(D\) exists as a stable node or saddle, then the union of \(BD\) and \(DC\) is strictly concave or convex, respectively. Moreover, in case, \(D\) does not exist, \(BC\) is concave or convex or linear if \(q:=b(af-cd)+f(bd-ae)\) is positive or negative or zero, respectively.
The paper also contains results on the existence and properties of the second derivatives of the three separatrices at their endpoints. The proofs are elementary but lengthy. The analysis is comprehensive with exhaustive results. They correct earlier statements by Zeeman and Zeeman.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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