Moment closure equations for the stochastic logistic model

This web-page contains the Maple worksheets that have been used to derive the results reported in the paper "An extension of the moment closure method", appearing in Theoretical Population Biology, Volume 64, pp 233--239, 2003. The paper uses a slight variation of Whittle's moment closure method to derive approximations of the first few cumulants of the quasi-stationary distribution (QSD) for the stochastic logistic model. This is done by determining asymptotic approximations of the critical points of the moment closure equations. Also, the stabilities of these critical points are determined by deriving asymptotic approximations of the eigenvalues of the jacobian matrices of the right-hand sides of the systems of differential equations at each critical point.

Maple worksheets for studying the critical points of the first TWO moment closure equation

• Derivation based on the normal assumption.
• Derivation based on the Poisson assumption.
• Derivation based on the lognormal assumption.
• Derivation based on the binomial assumption.
• Derivation based on assuming that the third cumulant is of order N.

Maple worksheets for studying the critical points of the first THREE moment closure equation

• Derivation based on the normal assumption.
• Derivation based on the Poisson assumption.
• Derivation based on the lognormal assumption.
• Derivation based on the binomial assumption.
• Derivation based on assuming that the forth cumulant is of order N.

Maple worksheets for plotting phase portraits of the first two moment closure equations

• Equations derived from the normal assumption.
• Equations derived from the Poisson assumption.
• Equations derived from the lognormal assumption.
• Equations derived from the binomial assumption.

A Maple worksheet for expressing the third and forth cumulants in terms of the first two ones for three elementary distributions

• Cumulants for Poisson, binomial, and lognormal distributions.