On the population dynamics of the Australian bushfly

Abstract

Each spring, the Australian bushfly (Musca vetustissima Walker) migrates from its over-wintering areas located in central Australia and repopulates the south-eastern parts of the continent. In this paper, a simple size-structured model, in the form of a reaction–advection equation with delay, is presented for the evolution of the population density of the bushfly. The model takes into account the important role of cattle dung in its life cycle and includes the effects of wind-borne migration. Bifurcation analyses of the model equations in the absence of wind advection reveals the environmental conditions under which the fly population will die out, reach a stable equilibrium or explode. Numerical simulations which include advection reveal the importance of wind-borne migration in building up the population of flies in south-eastern Australia during the summer months.

Appendix

Parameter value determination

In this section, detailed explanation for the selection of various parameter values is provided where this has not been given in the text.

The values of the minimum and maximum physiological head widths were taken to be at the edges of the smallest and largest class sizes in Table I of Sands and Hughes (1977). These have mean 1.2 and 2.7 mm, respectively, and width 0.1 mm, giving w _l  = 1.15 and w _r  = 2.75. The chosen value of w _l is very close to the head width for which the size of the egg complement vanishes according to Eq. 10 of Sands and Hughes (1977), i.e. 34.5/28.7 ≈ 1.2.

Equation 5 is a modification of Eq. 7 of Sands and Hughes (1977), whereby the rate of death of flies is treated as an exponentially decreasing function of the head width, but does not treat the death rate as an increasing function of the physiological age as in that paper (see their Eq. 6). Rather, a constant part to the death rate, β _d , has been introduced and its value chosen so that, as indicated by Sands and Hughes (1977), the expected life span of a fly with a head width of 2.5 mm is 15.5 egg stage periods (esps). Thus, $r_d(2.5)^{-1}=15.5$, i.e. $\beta_d+0.029\exp(-1.22(2.5-1.5))=1/15.5$, giving β _d  = 0.0560 esp^− 1.

Values of σ and l _∞ were determined by fitting Eq. 29 to experimental data from Sands and Hughes (1977) of 11,530 female flies that were caught and assigned to various classes of head widths and physiological ages (see their Table I). Because the model in this paper predicts the abundance of sexually mature females, the data of the newly emerged flies were not included in the fit. To correlate the numbers of flies caught with their actual abundance in the environment, the numbers of flies were divided by the quantities a _p , p = 1...5, which were employed previously by Sands and Hughes (1977) to represent the relative attractiveness of the collector to the fly during the p-th stage of the ovarian cycle. The resulting numbers of flies in each size class was then divided by the total number of flies to obtain the experimental relative abundances. For the given size classes, [w _i ,w _j ], the differences between N(w _i ,w _j ) defined by Eq. 32 and the corresponding experimental relative abundances were minimised over all classes using the MATLAB function lsqcurvefit, to give σ = 0.652 and l _∞ = 1.573.

Tyndale-Biscoe and Hughes (1968) found that the physiological age of a bushfly is related to the number of accumulated day-degrees above 12.5°C. Following Sands and Hughes (1977), the physiological age, φ, is expressed in terms of the number of esps where 1 esp = 7.2 day-degrees above 12.5°C. Thus, the physiological age corresponding to a physical time duration t days is

$$ \phi = \frac{1}{7.2}\int^t_0 (T(\hat{t})-12.5)H(T(\hat{t})-12.5)\, d\hat{t}, \label{eq:espeq} $$

(57)

where T is the temperature in degrees Celsius and H(T) is the Heaviside step function. Following Sands and Hughes (1977), the temperature is taken to be constant at 17.5°C, this being the quoted mean temperature in the study area in that paper. Thus, from Eq. 57, the physical time in days is obtained by multiplying the physiological time by 1.44. (t = 1.44φ). Table 2 gives the duration of the egg, larval and pupal stages in esps, which were taken from Table I of Hughes and Sands (1979), as well as the corresponding duration in days.

To determine the values of the parameters τ _s and γ _s , Eq. 3, for the duration of the immature state, was fitted to the data given in Fig. 2 of Hughes (1974). Note that that figure contains the original data used by Sands and Hughes (1977) to derive a piecewise linear regression (see their Eq. 2).

The value of the exponent p in Eq. 9 was calculated by relating the maximum and minimum head widths to the corresponding maximum and minimum pupal masses, given in Table 2. Thus $m_l/m_r=(w_l/w_r)^p$ implies p = (ln m _l  − ln m _r )/(ln w _l  − ln w _r ) ≈ 2.73.

Equation 7 is a modification of Eq. 5 of Sands and Hughes (1976) to allow the dung quality to be defined on an unbounded range of values, − ∞ < q < ∞. In that paper, the following formula was used:

$$ m=m_r(1-\exp(-q^*(\alpha + \beta q^*))), \label{eq:meqx} $$

(58)

where q ^* is given by Eq. 8 and, for l = 200 L^− 1, the values of the constants α and β were determined so as to make m = 4.5 when q = 0 and m = 16.5 when q = 1. In this study, the values of c ₀ and c ₁ in Eq. 7 were determined using the same approach. It was verified that the curves of Eqs. 7 and 58 are in close agreement.