Effects of whaling and krill fishing on the whale–krill predation dynamics: bifurcations in a harvested predator–prey model with Holling type I functional response

In the Antarctic, the whale population had been reduced dramatically due to the unregulated whaling. It was expected that Antarctic krill, the main prey of whales, would grow significantly as a consequence and exploratory krill fishing was practiced in some areas. However, it was found that there has been a substantial decline in abundance of krill since the end of whaling, which is the phenomenon of krill paradox. In this paper, to study the krill–whale interaction we revisit a harvested predator–prey model with Holling I functional response. We find that the model admits at most two positive equilibria. When the two positive equilibria are located in the region \(\big \{(N,P)|0\le N< 2N_c,\ P\ge 0\big \}\) $\{(N,P)|0\le N<2N_c,P\ge 0\}$ , the model exhibits degenerate Bogdanov–Takens bifurcation with codimension up to 3 and Hopf bifurcation with codimension up to 2 by rigorous bifurcation analysis. When the two positive equilibria are located in the region \(\big \{(N,P)|N>2N_c,\ P\ge 0\big \}\) $\{(N,P)|N>2N_c,P\ge 0\}$ , the model has no complex bifurcation phenomenon. When there is one positive equilibrium on each side of \(N=2N_c\) $N=2N_c$ , the model undergoes Hopf bifurcation with codimension up to 2. Moreover, numerical simulation reveals that the model not only can exhibit the krill paradox phenomenon but also has three limit cycles, with the outmost one crosses the line \(N=2N_c\) $N=2N_c$ under some specific parameter conditions.