On interacting bee/mite populations: a stochastic model with analysis using cumulant truncation

Parasitism by the Varroa mite has had recent drastic impact on both managed and feral bee colonies. This paper proposes a stochastic population dynamics model for interacting African bee colony and Varroa mite populations. Cumulant truncation procedures are used to obtain approximate transient cumulant functions, unconstrained by the usual assumption of bivariate Normality, for an assumed large-scale model. The apparent size of the variance and skewness functions suggest the importance of the proposed truncation procedure which retains some higher-order cumulants, but determining the accuracy of the approximations is problematical. A smaller-scale bee/Varroa mite model is hence proposed and investigated. The accuracy for the means is exceptional, for the second-order cumulants is moderate, and for some third-order cumulants is poor. Notwithstanding the poor accuracy of a skewness approximation, the saddlepoint approximations for the marginal transient population size distributions are excellent. The cumulant truncation methodology is very general, and research is continuing in its application to this new class of host-parasite models.

[1]  Samuel Kotz,et al.  A note on relationships between moments, central moments and cumulants from multivariate distributions , 1998 .

[2]  E. Renshaw,et al.  Applying the saddlepoint approximation to bivariate stochastic processes. , 2000, Mathematical biosciences.

[3]  M. Makela,et al.  Invasive dynamics of africanized honeybees in North America , 1992, Naturwissenschaften.

[4]  T. Wehrly,et al.  Some generalized stochastic compartment models for digesta flow. , 1989, Biometrics.

[5]  John Maynard Smith.  Models in ecology , 1974 .

[6]  Eric Renshaw.  Saddlepoint approximations for stochastic processes with truncated cumulant generating functions , 1998 .

[7]  M. Bartlett,et al.  A comparison of theoretical and empirical results for some stochastic population models , 1960 .

[8]  R. Costantino,et al.  NONLINEAR DEMOGRAPHIC DYNAMICS: MATHEMATICAL MODELS, STATISTICAL METHODS, AND BIOLOGICAL EXPERIMENTS' , 1995 .

[9]  W. Wilson,et al.  Subtropical Apis mellifera (Hymenoptera: Apidae) Swarming Dynamics and Africanization Rates in Northeastern Mexico and Southern Texas , 1996 .

[10]  Peter J. Smith.  A Recursive Formulation of the Old Problem of Obtaining Moments from Cumulants and Vice Versa , 1995 .

[11]  V Isham,et al.  Assessing the variability of stochastic epidemics. , 1991, Mathematical biosciences.

[12]  James H. Matis,et al.  Compartmental models with multiple sources of stochastic variability: The one-compartment models with clustering , 1981 .

[13]  R. May,et al.  Stability and Complexity in Model Ecosystems , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[14]  P. R. Parthasarathy,et al.  On the cumulants of population size for the stochastic power law logistic model. , 1998, Theoretical population biology.

[15]  Stephen Wolfram,et al.  Mathematica: a system for doing mathematics by computer (2nd ed.) , 1991 .

[16]  Eric Renshaw.  Modelling biological populations in space and time , 1990 .

[17]  R. E. Page,et al.  Effect of Varroa jacobsoni (Mesostigmata: Varroidae) on Feral Apis mellifera (Hymenoptera: Apidae) in California , 1995 .

[18]  H. Daniels.  Saddlepoint Approximations in Statistics , 1954 .

[19]  James H. Matis,et al.  ON APPROXIMATING THE MOMENTS OF THE EQUILIBRIUM DISTRIBUTION OF A STOCHASTIC LOGISTIC MODEL , 1996 .

[20]  William Gurney,et al.  Modelling fluctuating populations , 1982 .

[21]  R. Banks.  Growth and diffusion phenomena , 1993 .

[22]  P. Whittle.  On the Use of the Normal Approximation in the Treatment of Stochastic Processes , 1957 .

[23]  J. Matis,et al.  Effects of immigration on some stochastic logistic models: a cumulant truncation analysis. , 1999, Theoretical population biology.

[24]  M. Breed,et al.  The "African" Honey Bee , 1991 .

[25]  J. Matis,et al.  Describing the spread of biological populations using stochastic compartmental models with births. , 1995, Mathematical biosciences.

[26]  B. Mobus,et al.  The New Varroa Handbook , 1993 .

[27]  H. Daniels.  Approximate Solutions of Green's Type for Univariate Stochastic Processes , 1960 .