Mathematical analysis demonstrates conditions for stationary bifurcations in multi-parameter vector fields, indicating broader applicability in ecological predator-prey models.
Given an [Formula: see text]-parameterized family of [Formula: see text]-dimensional vector fields, such that (i) for some parameter values, the family has one equilibrium point, and (ii) its linearization has one zero eigenvalue and no others in the imaginary axis, we establish sufficient conditions for the existence of the cusp and the stationary bifurcations: saddle-node, transcritical and pitchfork. The center manifold theory is used to simplify the analysis to one dimension. This result generalizes the Sotomayor’s theorem for stationary bifurcations. Finally, we apply the obtained result to a predator–prey model.
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Baca-Carrasco et al. (2026) studied this question.
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