Abstract:
In this paper, we present an extended SI model of Hilker et al. (2009). In the presented model the birth rate and the death rate are both modeled as quadratic polynomials. This approach provides ample opportunity for taking into account the major contributors to an Allee effect and effectively captures species’ differential susceptibility to the Allee effects. It is shown that, the behaviors (persistence or extinction) of the model solutions are characterized by the two essential threshold parameters λ0λ0 and λ1λ1 of the transmissibility λλ and a threshold quantity μ∗μ∗ of the disease pathogenicity μμ. If λ<λ0λ<λ0, the model is bistable and a disease cannot invade from arbitrarily small introductions into the host population at the carrying capacity, while it persists when λ>λ0λ>λ0 and μ<μ∗μ<μ∗. When λ>λ1λ>λ1 and μ>μ∗μ>μ∗, the disease derives the host population to extinction with origin as the only global attractor. For the special cases of the model, verifiable conditions for host population persistence (with or without infected individuals) and host extinction are derived. Interestingly, we show that if the values of the parameters αα and ββ of the extended model are restricted, then the two models are similar. Numerical simulations show how the parameter ββ affects the dynamics of the model with respect to the host population persistence and extinction.
