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We develop a two-stage (four component) model for youths with serious drinking problems and their treatment. The youths with alcohol problems are split into two classes, namely those who admit to having a problem and those who do not. It is shown that the model possesses two steady states, one where people have no alcohol problems and one where there is an endemic state involving those with an alcohol problem. The stability of these states is analyzed and a threshold established such that each state will be stable depending on whether the incidence rate is above or below the threshold. The model is analyzed in the context of actual data.

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A predator᎐prey model with a stage structure for the predator which improves the assumption that each individual predator has the same ability to capture prey is proposed. It is assumed that immature individuals and mature individuals of the predator are divided by a fixed age and that immature predators do not have the ability to attack prey. We obtain conditions that determine the permanence of the populations and the extinction of the populations. Furthermore, we establish necessary and sufficient conditions for the local stability of the positive equilibrium of the model. By exploiting the monotonicity of one equation of the model, we obtain conditions for the global attractivity of the positive equilibrium, which allow for long delay as long as the predator birth rate is large or the death rate of immature predators is small. By constructing Liapunov functionals, we also obtain conditions under which the positive equilibrium is globally stable when the delay is small. ᮊ

No abstract