The Approach to Equilibrium and the Steady-State Probability Distribution of Adult Numbers in Tribolium brevicornis

The rate of population growth in adult numbers, A, for the flour beetle Tribolium was characterized by the mathematical model dA/dt = X A exp(-CA) - A D with the biological entities pupal productivity, X, adult inhibition of the immature life stages, C, and the death rate among the adults, D. A local stability analysis of the equilibrium A* = log(X/D)/C revealed that the eigenvalue λ = D log(D/X) and A* was stable if X > D. The time it takes for a perturbation to decay was evaluated using the time constant τ = 1/|λ|. The changes in adult numbers were then viewed as a stochastic birth-death process. The numbers of adults were found to asymptotically assume a constant mean value of Ā(t) = A* = log(X/D)/C and a constant variance of V(t) = V* = 1/C. Equations were established for the approach of Ā(t) and V(t) to their respective equilibrium values together with the steady-state probability distribution of adult numbers. Formulas to estimate A*, X, D, C, and λ were obtained based on the adult population size data. Experimental observations on T. brevicornis showed a good correspondence to the theoretical construct.