Homoclinic bifurcation in a predator-prey model
The authors deal with predator-prey models \[ N' = N \Biggl[ \frac{\varepsilon}{K}(K - N) - \frac{aP}{\beta + N}\Biggr], \quad P' = P \Biggl[-M(P) + \frac{bN}{\beta + N}\Biggr], \tag{1} \] where \(N(t)\) and \(P(t)\) are the quantities of prey and predator, respectively, the function \(M(P) = (\gamma + \delta P) / (1 + P)\) describes the specific mortality of predators in absence of prey. They consider codimension-2 bifurcations which occur in some two-dimensional parameter region. There is a continuous curve in the parameter region for which system (1) has a homoclinic orbit. The homoclinic orbit bifurcates generating a locally unique periodic orbit.
scientific article; zbMATH DE number 1261149