Stable Demographic Limit Cycles in Laboratory Populations of Tribolium castaneum

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Abstract
(1) We present a general population matrix model in which the age-specific vital rates depend upon the age structure of the population. The fecundity and survivorship of each age-class are assumed to decrease exponentially at rates which depend on the densities of each age-class. We specialize this model to describe the physiological and behavioural interactions among eggs, larvae, pupae and adults in laboratory populations of Tribolium flour beetles. (2) A non-trivial equilibrium age structure exists provided the population can grow without density dependence. If such as equilibrium exists, it is unique. We linearize the model in the neighbourhood of its equilibrium and state the necessary and sufficient conditions for local asymptotic stability. (3) Using several simplifying assumptions, we estimate the parameters of the model using data from our own work and from the literature. With these estimates we predict the existence of an unstable equilibrium age structure. (4) Computer simulations are used to compare the behaviour of the model with census data from experimental populations of Tribolium castaneum. After 70 days of culture, the experimental populations were subjected to demographic perturbations. Both the simulations and the experimental populations exhibit stable oscillations. In general, there is good agreement between the model and the data. (5) we simulated the model using a variety of parameter values. We show how each parameter affects the equilibrium and stability of the model. Increases in the rates of mortality or rates of egg and pupal cannibalism by adults are stabilizing, while high rates of fecundity or egg cannibalism by larvae lead to demographic oscillations. For each parameter, we obtain numerical estimates of the threshold between a stable and unstable point equilibrium. (6) Considering the variation in the rates of survivorship, reproduction, and cannibalism reported in the literature for different species and genetic strains of Tribolium under different environmental conditions, we conclude that laboratory populations of Tribolium can exhibit dynamic behaviours ranging from stable equilibria to demographic limit cycles.