Cost of flight apparatus and optimum body size of aphid migrants

Ecology, July, 1999 by Anthony F. Dixon, Pavel Kindlmann

INTRODUCTION

A central problem of evolutionary biology is to provide a general explanation for the design of organisms (Stearns 1982), and an important feature of design is size. Aerodynamic and physiological features indicate that larger organisms have to devote proportionally more resources to flying than small organisms (Pennycuik 1975, 1989, Casey 1976, Ellington 1991). Thus, it is surprising that Roff (1991, 1992) concludes that insect migrants should be larger than nonmigrants. This conclusion is based on the assumptions that the distance an insect can fly without refuelling is directly proportional to its flight speed and the size of its fuel reserves, and inversely proportional to its rate of metabolism. Roff (1991) offers some empirical evidence in support of his hypothesis.

The definition of migration adopted in this study is that of Taylor and Taylor (1983), who argue that ecological migration is the process by which organisms redistribute themselves in space. In aphids this can be achieved by walking (Harrington and Taylor 1990) or by flying. Although there is a continuum, from short- to long-distance movements, there is a very clear distinction between those individuals that migrate by flying as opposed to walking. This is reflected in their morphology and the distance they are likely to travel, with the unwinged individuals relatively more sedentary than the winged individuals, which in this study are the migrants.

The intraclonal alary dimorphism shown by many species of aphids makes them ideal for determining the cost of producing a flight apparatus. Here the theory of aerodynamics and a model of the optimum partitioning of energy in aphids are used to account for the relative size of winged and unwinged aphids, and the size of the gonads and wing beat frequency of migrant aphids of different sizes. We test the model's predictions against empirical data, and the bearing of the outcome on the optimum size of insect migrants is discussed.

THEORY

Aerodynamics of flight

As aphids are small, their wing muscles are unable to generate sufficient power to overcome wind speed, and as a consequence they have little control over the direction of their flight. Therefore all they require is to be able to take off and remain airborne. The least expensive way of remaining airborne is to fly at the speed that minimizes the sum of the parasite and induced powers (Pennycuik 1989). Here we use Pennycuick's (1989) approach, terminology, and variable names (summarized in Table 1) throughout. In geometrically similar animals the power required for flight, [P.sub.am], at this speed scales with the 7/6 power of the body mass, m (Pennycuik 1975):

[P.sub.am] [approximately equal to] [m.sup.7/6]. (1)

TABLE 1. Definition of the parameters and variables.

Parameter           Explanation

t                   time

[P.sub.am]          power required for flight

m                   body mass

b                   wingspan

V                   flight speed

[P.sub.m]           mass-specific power output of aerobic muscle

[Sigma]             stress

[Lambda]            active strain

f                   wingbeat frequency

[Rho]               density of the muscle

[k.sub.1]           constant

M                   mass of the wing muscles

P(M)                power produced by a muscle of mass M

a                   scaling exponent in the relation between
                    mass of flight apparatus and body mass

s                   somatic mass at time t

g                   gonadal mass at time t

[s.sub.0]           somatic mass at time 0, at birth

[g.sub.0]           gonadal mass at time 0, at birth

w                   mass of the flight apparatus (wing muscles,
                    thorax plus wings) at time t

[Alpha]             assimilation rate

[k.sub.2]           ratio of energy invested in somatic growth
                    to that invested in somatic plus flight
                    apparatus growth

[[Alpha].sub.alt]   [k.sub.2] x [Alpha]

R                   constraint on gonadal growth rate

D                   developmental time

[r.sub.m]           instantaneous growth rate

F(t)                fecundity at time t

[Mu]                mortality

[Pi]                constant

[s.sub.A]           adult mass of soma, [s.sub.A] = s(D)

[g.sub.A]           adult mass of gonads, [g.sub.A] = g(D)

[w.sub.A]           adult mass of the flight apparatus,
                    [w.sub.A] = w(D)

[m.sub.A]           adult mass, [s.sub.A]   [g.sub.A]   [w.sub.A]

[m.sub.opt]         optimum adult mass

[Phi]               fitness

When animals differ in shape, however, the parasite power scales with [V.sup.3] [m.sup.2/3], the induced power with [m.sup.2] [v.sup.-2] [V.sup.-1], where b is wingspan and V flight speed, and the profile power can be assumed to be relatively constant (Pennycuik 1989). The minimum power speed required to remain airborne can be obtained from d[P.sub.am]/dV = 0, which when substituted back in the formula for the power required to fly, gives