density-dependent effects: if predator density cannot increase in proportion to that of prey, destabilising periodicities may not develop. Prey toxicity: if there is a significant cost to the predator of consuming the (now very dense) prey species, predator numbers may not increase sufficiently to give periodicity. Link with Hopf bifurcation The paradox of enrichment can be accounted for by the bifurcation theory. As the carrying capacity increases, the equilibrium of the dynamical system becomes unstable. The bifurcation can be obtained by modifying the Lotka–Volterra equation. First, one assumes that the growth of the prey population is determined by the logistic equation. Then, one assumes that predators have a nonlinear functional response, typically of type II. The saturation in consumption may be caused by the time to handle the prey or satiety effects. Thus, one can write the following (normalized) equations: x is the prey density; y is the predator density; K is the prey population's carrying capacity; γ and δ are predator population's parameters (rate of decay and benefits of consumption, respectively). The term represents the prey's logistic growth, and the predator's functional response. The prey isoclines (points at which the prey population does not change, i.e. dx/dt = 0) are easily obtained as and . Likewise, the predator isoclines are obtained as and , where . The intersections of the isoclines yields three steady-states: The first steady-state corresponds to the extinction of both predator and prey, the second one to the predator-free steady-state and the third to co-existence, which only exists when is sufficiently small. The predator-free steady-state is locally linearly unstable if and only if the coexistence-steady-state exists. By the Hartman–Grobman theorem, one can determine the stability of the steady states by approximating the nonlinear system by a linear system. After differentiating each and with respect to and in a neighborhood of , we get: It is possible to find the exact solution of this linear system, but here, the only interest is in the qualitative behavior. If both eigenvalues of the community matrix have negative real part, then by the stable manifold theorem the system converges to a limit point. Since the determinant is equal to the product of the eigenvalues and is positive, both eigenvalues have the same sign. Since the trace is equal to the sum of the eigenvalues, the co-existence steady-state is locally linearly stable if At that critical value of the parameter K,