Abstract
ABSTRACTIn this paper, a predator–prey system with harvesting prey and disease in prey species is given. In the absence of time delay, the existence and stability of all equilibria are investigated. In the presence of time delay, some sufficient conditions of the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analysing the corresponding characteristic equation, and the properties of Hopf bifurcation are given by using the normal form theory and centre manifold theorem. Furthermore, an optimal harvesting policy is investigated by applying the Pontryagin's Maximum Principle. Numerical simulations are performed to support our analytic results.
Highlights
IntroductionThe pioneer work of Lotka [27] and Volterra [36] opened new door to biological species
In the twentieth century, the pioneer work of Lotka [27] and Volterra [36] opened new door to biological species
In the presence of time delay, some sufficient conditions of the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analysing the corresponding characteristic equation, and the properties of Hopf bifurcation are given by using the normal form theory and centre manifold theorem
Summary
The pioneer work of Lotka [27] and Volterra [36] opened new door to biological species. Sharma and Samanta [34] studied an eco-epidemiology model with two prey population where one prey species is infected by an infectious disease. Khan et al [23] investigated a predator–prey model with density constraints for susceptible prey population, and considered harvesting to both susceptible and infected prey species. Based on the above analysis and work of Johri et al [19], we will propose and analyse an eco-epidemiological model with time delay, harvesting prey population and constant input of susceptible prey population.
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