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On a geometrical study of population ecosystems. (English) Zbl 0907.92024

We survey some contribution to the geometrical study of mathematical models of ecosystems with emphasis upon some classes of models of interacting species. This class of ecosystems plays an important role in mathematical ecology. The models considered in this paper are of the form \[ \dot x_i=x_iF_i(x_1,x_2,\dots,x_n),\quad i=1,2,\dots,n,\tag{1} \] where \(x_i\) is the density of the \(i\)th species in the community at time \(t\), and \(\dot x_i\) denotes \({dx_i/dt}\). Each \(F_i\) is a continuous function from \(R^n_+\), the nonnegative cone in \(R^n\), to \(R\) and is sufficiently smooth to guarantee that initial value problems associated with (1) have unique solutions in the population orthant, \(R^n_+\).
As a rule, the stability of such a system is studied by using the eigenvalues of its linear approximation. This method gives answer only concerning the stability relative to infinitesimal perturbations of the initial state. In real cases these systems are subjected to large perturbations. So, the study of stability relative to finite perturbations is useful. This requires an extension from a local property to a global concept.
We study a class of Lyapunov functions to be applied for the study of global asymptotic stability of systems of the form (1). In a two-dimensional case, a locally geometrical study of (1) is given. We prove that, in general, total extinction cannot appear in (1) and introduce a Riemannian structure to study it.

MSC:

92D40 Ecology
34D20 Stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
37-XX Dynamical systems and ergodic theory
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