Modeling and analysis of a predator-prey model with disease in the prey.

*(English)* Zbl 0978.92031
Summary: A system of retarded functional differential equations is proposed as a predator-prey model with disease in the prey. Mathematical analyses of the model equations with regard to invariance of non-negativity, boundedness of solutions, nature of equilibria, permanence and global stability are analyzed. If the coefficient in conversing prey into predator $k={k}_{0}$ is constant (independent of delay $\overline{\tau}$, gestation period), we show that positive equilibrium is locally asymptotically stable when time delay $\overline{\tau}$ is suitably small, while a loss of stability by a Hopf bifurcation can occur as the delay increases. If $k={k}_{0}{e}^{-d\overline{\tau}}$ $(d$ is the death rate of predator), numerical simulation suggests that time delay has both destabilizing and stabilizing effects, that is, positive equilibrium, if it exists, will become stable again for large time delays. A concluding discussion is then presented.

##### MSC:

92D40 | Ecology |

92D30 | Epidemiology |

34K20 | Stability theory of functional-differential equations |

34K60 | Qualitative investigation and simulation of models |