Supercritical Hopf bifurcation in baleen whale populations

This paper investigates a continuous time model for the baleen whale population, which is a diverse and widely distributed parvorder of carnivorous marine mammals. We use theoretical and schematic designs to explore stability charts, rightmost characteristic roots, and supercritical Hopf bifurcation of the positive equilibrium. Our research on the Hopf bifurcation and stability of the bifurcating periodic solutions is based on the center manifold reduction and Poincaré normal form theory. Interestingly, the characteristic equation has pure imaginary roots at the second, third, and subsequent critical values. However, Hopf bifurcation theorem is not satisfied because all other characteristic roots of the characteristic equation at these critical values do not have strictly negative real parts, except the pure imaginary roots. We also use the parameter values reported in the previous studies to simulate the unstable periodic solutions at the second and third critical values through bifurcation diagrams. The numerical results obtained under specific parameter values align closely with our theoretical derivations.