Abstract
Abstract This paper generalizes the existing minimal mathematical model of a given generic touristic site by including a distributed time-delay to reflect the whole past history of the number of tourists in their influence on the environment and capital flow. A stability and bifurcation analysis is carried out on the coexisting equilibria of the model, with special emphasis on the positive equilibrium. Considering general delay kernels and choosing the average time-delay as bifurcation parameter, a Hopf bifurcation analysis is undertaken in the neighborhood of the positive equilibrium. This leads to the theoretical characterization of the critical values of the average time delay which are responsible for the occurrence of oscillatory behavior in the system. Extensive numerical simulations are also presented, where the influence of the investment rate and competition parameter on the qualitative behavior of the system in a neighborhood of the positive equilibrium is also discussed.
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