In this paper, a three‐species food chain model with Holling type IV and Beddington–DeAngelis functional responses is formulated. Numerical simulations show that this system can generate chaos for some parameter values. But the mechanism behind chaos is still unclear only through numerical simulations. Then, using the topological horseshoe theories and Conley–Moser conditions, we present a computer‐assisted analysis to show the chaoticity of this system in the topological sense, that is, it has positive topological entropy. We prove that the Poincaré map of this model possesses a closed uniformly hyperbolic chaotic invariant set, and it is topologically conjugate to a 2‐shift map. At last, we consider the impact of fear on this three‐species model. It is an important factor in controlling chaos in biological models, which has been validated in other models.
The aim of this paper is to study the dynamical behaviors of a piecewise smooth predator–prey model with predator harvesting. We consider a harvesting strategy that allows constant catches if the population size is above a certain threshold value (to obtain predictable yield) and no catches if the population size is below the threshold (to protect the population). It is shown that boundary equilibrium bifurcation and sliding–grazing bifurcation can happen as the threshold value varies. We provide analytical analysis to prove the existence of sliding limit cycles and sliding homoclinic cycles, the coexistence of them with standard limit cycles. Some numerical simulations are given to demonstrate our results.
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