Evaluation of Markov Models with Discontinuities

Background. Several methods, such as the half-cycle correction and the life-table method, were developed to attenuate the error introduced in Markov models by the discretization of time. Elbasha and Chhatwal have proposed alternative “corrections” based on numerical integration techniques. They present an example whose results suggest that the trapezoidal rule, which is equivalent to the half-cycle correction, is not as accurate as Simpson’s 1/3 and 3/8 rules. However, they did not take into consideration the impact of discontinuities. Objective. To propose a method for evaluating Markov models with discontinuities. Design. Applying the trapezoidal rule, we derive a method that consists of adjusting the model by setting the cost at each point of discontinuity to the mean of the left and right limits of the cost function. We then take from the literature a model with a cycle length of 1 year and a discontinuity on the cost function and compare our method with other “corrections” using as the gold standard an equivalent model with a cycle length of 1 day. Results. As expected, for this model, the life-table method is more accurate than assuming that transitions occur at the beginning or the end of cycles. The application of numerical integration techniques without taking into account the discontinuity causes large errors. The model with averaged cost values yields very small errors, especially for the trapezoidal and the 1/3 Simpson rules. Conclusion. In the case of discontinuities, we recommend applying the trapezoidal rule on an averaged model because this method has a mathematical justification, and in our empirical evaluation, it was more accurate than the sophisticated 3/8 Simpson rule.

Bisexual and oedipal reproduction of Macrocheles muscaedomesticae (Acari, Macrochelidae) feeding on Musca domestica (Diptera, Muscidae) eggs

Macrocheles muscaedomesticae (Scopoli) is a predatory mesostigmatic mite that inhabits different manure microhabitats and preys mostly on housefly (Musca domestica L.) eggs. When a virgin female colonizes a new manure substrate, it produces male offspring through parthenogenesis (arrhenotoky); when her sons reach maturity, oedipal mating takes place and the female begins to produce bisexual offspring. In order to examine the consequence of oedipal reproduction on population development, we designed two separate experiments to compare life history traits and life table parameters of oedipal versus bisexual cohorts of M. muscaedomesticae, using the age-stage, two sex life table method. Experiments were conducted at 28 +/- 1 °C, using a photoperiod of 14:10 (L: D) h, and 65 +/- 5% relative humidity, with housefly eggs used to feed mites. Mean adult female longevity was 38.63 days, and fecundity 128.51 offspring under bisexual reproduction, and 37.48 days and 68.23 offspring under oedipal reproduction. In the bisexual cohort, the intrinsic rate of increase (rm), the finite rate of increase (λ), the net reproduction rate (R0), the gross reproductive rate (GRR) and the mean generation time (T) of M. muscaedomesticae were 0.2938 d-1, 1.3415 d-1, 54.216 offspring/individual, 77.7 offspring/individual and 13.5885 days, respectively. Because only male eggs were produced during the first 5.62 days (on average) of the oviposition period in the oedipal cohort, it was theoretically incorrect to compute the population parameters using the survival and fecundity values for this group, even though bisexual reproduction did occur after this period. Our findings determined that the effect of oedipal reproduction could be correctly defined and analyzed by using the age-stage, two-sex life table method. Our results demonstrated that virgin females are able to produce and copulate with their sons (oedipal mating), which then allows those females to produce both sexes. This reproductive system can enable this valuable natural enemy to considerably extend its distribution potential.