An iterative continuous-event model of the population outbreak of a phytophagous Hemipteran

Abstract

Based on the analysis of changes in the local density of the insect within the family Psillidae in Australia we have developed a model for a scenario of a dramatic increase in the number of jumping plant lice at the expense of primary and secondary Encyrtidae parasitoid microwasps. A phenomenological model describes on a case by case basis the efficiency of reproduction in several ranges of population conditions. We have proposed a continuous-event structure, where the rate of a decrease of the number of psyllid generations is uneven at different stages of ontogenesis of the insect with an incomplete metamorphosis. The moments when the rate is changing are determined by the state of internal variables of the auxiliary equation of a continuous system. Spontaneous time-limited local outbreak occurs after overcoming the threshold balancing in iterative dynamic system that reduces the effect of normal regulatory mechanisms of psyllid reproduction and the speed of a decrease of the number of generations changes. The method with the addition of the right side of the first equation by special functionality with limited range of values simulates a sharp decrease in survival with the exhaustion of resources. The limited availability of leaves causes a backward tangent bifurcation. After a few iterations from the tangent bifurcation population transfers to the mode of ordinary fluctuations without explicit of cyclical component at a low average psyllids population.