A mathematical model of the defence mechanism of a...

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ResearchCite this article: James A, Morison K, Todd S.

2013 A mathematical model of the defence

mechanism of a bombardier beetle. J R Soc

Interface 10: 20120801.

http://dx.doi.org/10.1098/rsif.2012.0801

Received: 2 October 2012

Accepted: 1 November 2012

Subject Areas:biomathematics

Keywords:bombardier beetle, dynamical model,

pulse oscillations

Author for correspondence:Alex James

e-mail: [email protected]

& 2012 The Author(s) Published by the Royal Society. All rights reserved.

A mathematical model of the defencemechanism of a bombardier beetle

Alex James1, Ken Morison2 and Simon Todd1

1Biomathematics Research Centre, and 2Chemical and Process Engineering, University of Canterbury,Christchurch, New Zealand

Previous studies of bombardier beetles have shown that some species have a

continuous discharge while others exhibit a pulsed discharge. Here, a math-

ematical model of the defence mechanism of the bombardier beetle is

developed and the hypothesis that almost all bombardiers’ defences have

some sort of cyclic behaviour at frequencies much higher than previously

thought is put forward. The observation of pulses arises from secondary

lower frequency cycles that appear for some parameter values. For realistic

parameter values, the model can exhibit all the characteristics seen in the

various species of bombardier. The possibility that all bombardiers have

the same underlying defence mechanism gives weight to the theory that

all bombardiers’ explosive secretory mechanisms have diversified from a

common ancestral mechanism.

1. BackgroundBombardier beetles are ground beetles (Carabidae) whose defence mechanism

has earned them a special place in entomology. They spray their attackers with

a noxious and often hot spray that is ejected with force from their abdomen, in

some cases with great accuracy [1]. There are many species of bombardier from

the tribes Brachinini, Crepidogastrini, Metriini, Mystropomini, Paussini and

Ozaenini [2], and their collective name derives from the audible pop that

often accompanies the ejection [3] (table 1).

A bombardier beetle stores reactant chemical compounds in a reservoir and,

when attacked, contracts muscle that forces these reactants into a rigid-walled

reaction chamber containing catalytic enzymes. In this chamber, the compound

starts to react exothermically, increasing the chamber pressure and ultimately

forcing the reactants out of glands in the beetle’s abdomen at temperatures of

up to 1008C, in what has been termed an explosive secretory discharge (ESD)

[6]. A system of valves stops these reactants from entering the beetle’s greater

anatomy. The total ejection lasts for only a fraction of a second but is easily

enough to ward off most predators.

The defence mechanism of the bombardier beetle has fascinated entomolo-

gists for close to two centuries since they were first discovered by the naturalist

J. O. Westwood in 1839. Individuals of the Brachinus genus (species unknown)

were amongst the first to be examined [6], and it was found that the reservoir

solution contained 10 per cent hydroquinone and 25 per cent hydrogen per-

oxide with heat of reaction of 48.5 kcal per mol hydroquinone [5]. Although

the range of chemicals found in the secretions of different bombardiers is

remarkably similar, there is a marked difference between species in the

proportions of the different constituent parts [7].

While the outlet temperature of the discharge is often quoted as being 1008C(the maximum outlet temperature calculated by Aneshansley et al. [6]), it has

also been seen to vary over a considerable range depending on species. For

instance, Goniotropis nicaraguesis has an outlet flow with a much lower tempera-

ture (average 658C, maximum observed temperature 818C) [1]. Similarly, the

discharge temperature of some individuals of the tribe Crepidogastrini (Carabi-

dae) is 658C [2], secretions of Metrius contractus have been measured at 558C on

average [8] and Mystropomus regularis has a relatively cold discharge of only

34–478C [9]. The duration of the ESD varies across studies but in general the

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Table 1. Variables and parameters.

independent variables

t time s

dependent variables

Fin, Fout volumetric flow rate of substance in and out m3 s – 1

hO2, hwV specific enthalpy of oxygen and water vapour J mol – 1

HC enthalpy of the chamber wall J

HR enthalpy of the components inside the reaction chamber J

NBQ quantity of benzoquinone in the reaction chamber mol

Nhp quantity of hydrogen peroxide in the reaction chamber mol

NHQ quantity of hydroquinone in the reaction chamber mol

NO2quantity of oxygen in the reaction chamber mol

Nw quantity of water in the reaction chamber mol

NwL quantity of liquid water in the reaction chamber mol

NwV quantity of water vapour in the reaction chamber mol

P chamber pressure Pa

pO2, pwV partial pressures of oxygen and water in the gas phase Pa

Qin, Qout rate of energy flow into and out of the chamber J s – 1

R1, R2 reaction rates 1 and 2 mol s – 1

T, TC temperature of the chamber contents and chamber wall K

Tsat saturation temperature of water at chamber pressure K

vout outlet velocity m s – 1

VBQ volume of benzoquinone in the chamber m3

Vhp volume of hydrogen peroxide in the chamber m3

VHQ volume of hydroquinone in the chamber m3

VG, VL volume of gas and liquid in the chamber m3

VwL volume of liquid water in the chamber m3

�V G, �V in, �V mix molar volume of gas, inlet mixture, mixture m3 mol – 1

WBQ,out molar flow rate of benzoquinone exiting chamber mol s – 1

Whp,in molar flow rate of hydrogen peroxide entering chamber mol s – 1

Whp,out molar flow rate of hydrogen peroxide exiting chamber mol s – 1

WHQ,in molar flow rate of hydroquinone entering chamber mol s – 1

WHQ,out molar flow rate of hydroquinone exiting chamber mol s – 1

WO2,out molar flow rate of oxygen exiting chamber mol s – 1

Ww,in molar flow rate of water entering chamber mol s – 1

WwL,out molar flow rate of liquid water exiting chamber mol s – 1

WwV,out molar flow rate of water vapour exiting chamber mol s – 1

xBQ mole fraction of benzoquinone in the chamber

xhp mole fraction of hydrogen peroxide in the chamber

xHQ mole fraction of hydroquinone in the chamber

xO2mole fraction of oxygen in the chamber

xwL mole fraction of liquid water in the chamber

xwV mole fraction of water vapour in the chamber

rmix mixture density kg m23

j inlet flow constriction factor

adjustable model parameters

A1, A2 Arrhenius factor for reaction 1 and 2 5 � 107 s – 1

AC chamber internal surface area ¼ pDL m2

(Continued.)

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SocInterface10:20120801

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Table 1. (Continued.)

D chamber diameter 0.0006 ma

EA1, EA2 Activation energy for reactions 1 and 2 3.8 � 104 J mol – 1c

h film heat transfer coefficient 10 W m22 K – 1c

L length of the inlet tube 0.0001 ma

LC chamber length 0.003 ma

mc molar mass of chamber wall 1 � 1026 mol

Pres pressure in the reservoir from time zero 109000 Paa

rin radius of inlet tube see main body of report

rout radius of chamber outlet 0.0003 ma

V chamber volume ¼ pðD=2Þ2LC m3a

DH1 heat of reaction 1 298200 J mol – 1b

DH2 heat of reaction 2 (þ177200 2 285500 J mol – 1) 2108300 J mol – 1b

fhp volume fraction of hydrogen peroxide in the inlet 0.2b

fHQ volume fraction of hydroquinone in the inlet 0.02b

Poutopen outlet valve opening pressure 110 000 Paa

Poutclose outlet valve closing pressure 101 200 Paa

Pinclose inlet valve closing pressure see main body of report

invariant parameters

Cd discharge coefficient of the outlet 0.97

Cp,BQ specific heat capacity of liquid benzoquinone 75.3 J mol – 1 K – 1

Cp,hp specific heat capacity of hydrogen peroxide 89.0 J mol – 1 K – 1

Cp,HQ specific heat capacity of liquid hydroquinone 75.3 J mol – 1 K – 1

Cp,O2specific heat capacity of oxygen 30.0 J mol – 1 K – 1

Cp,wL specific heat capacity of liquid water 75.3 J mol – 1 K – 1

MBQ molecular mass of benzoquinone 0.108 kg mol – 1

Mhp molecular mass of hydrogen peroxide 0.034 kg mol – 1

MHQ molecular mass of hydroquinone 0.110 kg mol – 1

MO2molecular mass of oxygen 0.016 kg mol – 1

MW molecular mass of water 0.018 kg mol – 1

P0 ambient pressure 101000 Pa

R gas constant 8.314 J mol – 1 K – 1

Tref reference temperature (ambient in this case) 290 K�VBQ molar volume of benzoquinone 8.24 � 1025 m3 mol – 1

�Vhp molar volume of liquid hydrogen peroxide 2.32 � 1025 m3 mol – 1

�VHQ molar volume of hydroquinone 8.46 � 1025 m3 mol – 1

�VwL molar volume of liquid water 1.8 � 1025 m3 mol – 1

min viscosity of inlet liquid (approx) 0.001 Pa s

rhp density of liquid hydrogen peroxide 1463 kg m23

rwL density of liquid water 998 kg m23

aChamber dimensions and trigger pressures are from Beheshti & McIntosh [4].bChemical parameters, where available, are from Schildknecht [5].cOther parameters were not available and the effect of varying these is tested in the report.



3


discharge has either finished (in the case of jet discharges) or

peaked (for secretory/mist discharges) after 30 ms [2,6,10].

Additionally, the manner in which beetles discharge

varies between species. Many species, e.g. those of the

genus Brachinus and Stenaptus insignus, eject their discharge

as a spray or jet which can often be aimed with very high

accuracy [11]. Other species, e.g. M. contractus, have a more

diffuse, mist-like secretion and in some cases also produce

a froth that can build-up on the individual’s body. Of particu-

lar interest is the outlet flow of S. insignus, which is not

continuous but pulsed [10]. Again, the duration of the

entire discharge is short (2.6–24.1 ms), but each discharge

contains between 2 and 12 pulses (average 7) at a frequency

of between 400 and 800 Hz [10]. The discharge velocities of


R

OH

OH R = H, CH3

reservoir

vestibuleE

H2O2

m

Figure 1. The glands of the bombardier beetle (reproduced with permissionfrom Aneshansley [6]). Pressure on the reservoir forces reactants into thecombustion chamber (vestibule) where they react with the aid of a catalystand are forced out of the chamber. The pressure in the reaction chambercontrols valves at the chamber entrance and exit.



4


this pulsed flow are, in general, an order of magnitude higher

than those of non-pulsed flow in other beetles, averaging

12 ms21 with a range between 3.25 and 19.5 ms21 [10].

Despite the diversity of ESD behaviour, the prevailing litera-

ture [1,2,12]. with only a single exception [13] states that the

internal mechanism is remarkably similar between species,

consisting of a thin and flexible reservoir and separate

rigid-walled reaction chamber. Figure 1 shows schematic of

the bombardier gland system.

Although it is clear that the ESD mechanism of the bom-

bardier beetle can be seen in many forms from pulsed jet

sprays to frothing mists and at many temperatures, many

questions about them remain unanswered. In particular, the

evolutionary pathway and the phyletic relationships between

the different species and tribes is still unknown. Some [14]

believe that bombardiers have a monophyletic lineage (i.e.

the mechanism evolved only once) and that the paussoids

and brachinoids are sister groups. Others [15] believe

the mechanism evolved separately in the two families of

paussoids and brachinoids.

Previous modelling work [4] used methods from compu-

tational fluid dynamics to build a model of the beetle’s

anatomy. This approach neglected the chemistry and injec-

tion phase and instead considered a body of water and

some method of heating. The model gave accurate answers

by including high-resolution spatial variables and allowing

the flow and temperature to vary across the reaction chamber.

Such success enabled the authors to build a working exper-

imental rig in which water is heated electrically in a

combustion chamber approximately 2 cm long and fired up

to 4 m with varying spray temperatures and droplet sizes

[16]. This spray technology, called mMist, is envisaged to

have important applications in fire extinguishers, drug deliv-

ery devices such as nebulizers and inhalers, and fuel-injection

systems in cars and other vehicles, as well as large environ-

mental advantages. By excluding chemical reactions, the

authors were able to show that the discharge velocities seen

in bombardiers can be recreated using only flash evaporation.

The model presented here includes the terms necessary to

exhibit flash evaporation but also includes terms to describe

the chemical reactions within the beetle. This addition

allows us to elucidate the dynamics of beetles with lower

temperature ejections where flash evaporation is not avail-

able. The importance of chemically released oxygen in the

mechanism driving the ESD was originally posited by

Aneshansley in 1969 [6] and, by including both mechanisms,

the model presented here allows comparison between these

two potential driving forces.

In this study, we construct a simple, dynamic mathemat-

ical model of the beetles’ defence mechanism. The model

assumes that the reaction chamber is spatially homogeneous

and the dynamics are only time-dependent; this simplifica-

tion has the advantage of allowing key features such as

reaction chemistry to be included. We show that this simpli-

fied model can exhibit all the modes of behaviour previously

described. On the basis of this model, we put forward the

hypothesis that many more individuals than previously

thought are undergoing a cyclic flow, but at a very high fre-

quency. The model demonstrates that observed pulses are

actually secondary cycles and can also be used to predict

which traits of individuals may lead to the different types

of observed behaviour.

2. Mathematical modelThe defence apparatus is modelled as a non-rigid reservoir

and a rigid-walled cylindrical reaction chamber joined by a

tube with a zero friction non-return valve, the closing of

which is driven by an increase in the chamber pressure.

Further details of the beetles’ valve system can be found in

McIntosh & Beheshti [16]. The reaction chamber also has an

outlet with a sticky valve. An aqueous solution of hydrogen

peroxide and hydroquinone is forced under pressure Pres

from the reservoir into the reaction chamber, on the walls

of which there are catalysts for the decomposition of both.

The hydrogen peroxide decomposes and the hydroquinone

reacts to form benzoquinone (C6H4O2).

H2O2! 1=2O2þH2O with rateR1 and heat of reactionDH1

C6H6O2!H2þC6H4O2 with rateR2

The resultant hydrogen reacts with the oxygen to produce

water.

H2 þ 1=2O2 ! H2O with rate R3 ðwhich is likely to be highÞ

If, as expected, hydrogen peroxide is more abundant than

hydroquinone and reacts at least as fast, then oxygen will be in

excess and the second and third reactions can be combined as

C6H6O2 þ 1=2O2 ! H2Oþ C6H4O2

with rate R2 and heat of reaction DH2:

The combined reaction is

H2O2 þ nC6H6O2 ! ð1þ nÞ H2Oþ 1=2ð1� nÞO2 þ nC6H4O2;

where n is the molar ratio of hydroquinone to hydrogen

peroxide. Schildknecht [5] measured the inlet to be 10 per

cent hydroquinone and 25 per cent hydrogen peroxide.

Assuming this to be by mass, the mole fractions of the

feed would be 2.0 per cent hydroquinone, 17 per cent hydro-

gen peroxide and 81 per cent water. The solubility of

hydroquinone in water is 5.9 g 100 ml21 at 158C (and

7 g 100 ml21 at 258C) [17], which is lower than the




5


concentration given, indicating that the solubility is higher in

a hydrogen peroxide solution [18].

The reaction generates oxygen as a gas, and the heat from

the exothermic reaction can also evaporate some water. Both

of these cause a pressure build-up, which, if higher than some

critical value, Popen, will open a flexible outlet that acts as a

valve and eject a gas/liquid mixture. When the pressure

drops to a critical closing value, Pclose, the valve closes.

The model uses molar balances for the quantities of

hydrogen peroxide (Nhp), water (Nw), comprising both

liquid (NwL) and vapour (NwV) states, oxygen ðNO2Þ, hydro-

quinone (NHQ), and benzoquinone (NBQ) in the reaction

chamber (for full symbol details, see table 1).

dNhp

dt¼Whp;in �Whp;out � R1

dNw

dt¼Ww;in �WwL;out �WwV;out þ R1 þ R2;

where Nw ¼ NwL þNwV:

dNO2

dt¼ �Whp;out þ

ðR1 � R2Þ2

dNHQ

dt¼WHQ;in �WHQ;out � R2

dNBQ

dt¼ �WBQ;out þ R2

Wi denotes the flow rate in moles per second, either in or out

of the chamber, for constituent i; R1 and R2 are the rates of the

two reactions. Enthalpy balances are used for the reaction

chamber contents (HR, including both reactants and pro-

ducts) and for the reaction chamber wall (HC)

dHR

dt¼ �Qout � DHR1

R1 � DHR2R2 � hACðT � TCÞ

dHC

dt¼ hACðT � TCÞ:

The chamber enthalpy is

HR ¼ ðNhpCp;hp þNHQCp;HQ þNBQCp;BQ þNwLCp;wL

þNO2Cp;O2

þNH2Cp;H2

ÞðT � TrefÞ þ hwVNwV

and the wall enthalpy is

HC ¼ mCCP;CðTC � TrefÞ:

This dynamical formulation of mass and enthalpy bal-

ances allows the modelling of the system in various states

and phase transitions, including full and partial (also

known as flash) evaporation and condensation [19].

The volume V occupied by each of the liquid components

in the chamber is expressed in terms of the molar volume, �V.

Vhp ¼ Nhp�Vhp; VHQ ¼ NHQ

�VHQ; VBQ ¼ NBQ�VBQ;

VwL ¼ NwL�VwL

Hence, the total volume of liquid in the chamber is

VL ¼ VwL þ Vhp þ VHQ þ VBQ, and the total volume of gas

is VG ¼ V � VL. Provided there is some liquid water in the

chamber, the partial pressure of the water can be calculated

using Antoine’s equation [20] fitted over the range 20–1008C:

pwV ¼ exp23:423� 3955:6

T � 40:65

� �;

otherwise, the water vapour follows the ideal gas law,

pwVVG ¼ NwVRT:

The partial pressure of the released oxygen is

pO2VG ¼ NO2

RT;

and the total pressure in the chamber is

P ¼ pwV þ pO2:

The reaction rates are modelled with an Arrhenius term and

determined by the quantity of reactant and the temperature.

R1 ¼ NhpA1e�Ea1=RT

R2 ¼ NHQA2e�Ea2=RT :

The overall volumetric inlet flow rate is governed by a

constricted Poiseuille flow

Fin ¼(Pres � P)r4

inp

8Lmin

j:

The constriction is provided by the j parameter, which

varies between 0 (no inlet flow) and 1 (full flow). It is

known that an increase in the chamber pressure will cause

a decrease in flow as the inlet tube is constricted [16] and it

is assumed that this constriction is the primary cause of the

inlet flow decrease (i.e. Pres . Pinclose) and results in a linear

decrease in the flow rate, with full flow at ambient pressure,

P0, and no flow when a critical pressure, Pinclose, is reached.

j ¼ Pinclose � PPinclose � P0

:

The individual inlet components have flow rates

Whp;in ¼fhp;inFin

�Vin

WHQ;in ¼fHQ;inFin

�Vin

Ww;in ¼(1� fhp;in � fHQ;in)Fin

�Vin;

and the molar volume of the inlet flow is

�Vin ¼ �VwLxwL;in þ �Vhpxhp;in þ �VHQxHQ;in:

When the outlet valve is open, the outlet flow is two-phase

liquid and gas; so the volumetric flow rate is determined by

Bernoulli’s equation. Assuming that the outlet acts as an orifice

with a rounded entrance, this outflow rate is then given by

Fout ¼ Cd

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2(P� P0)

rmix

sr2

outp:

The outlet mixture has molar volume

�Vmix ¼ �VwLxwL þ �Vhpxhp þ �VHQxHQ þ �VBQxBQ þ �VG;

and density

rmix ¼P

xiMi�Vmix

;

where xi is the molar fraction of component i,

xi ¼NiP

Ni;

and the molar volume of the gas is

�VG ¼RTP:


1.0

refill andheating

closedchamber

exhaust

0.5

0110

105

100

40

876543time (ms)

outle

t rat

e(m

g s–1

)pr

essu

re(k

Pa)

inle

t siz

e

21

20

0

Figure 2. The three-phase process involved in the beetles explosive secretory discharge (ESD) process. The inlet size is shown as a proportion of the inlet radius.During the first phase of refill and heating (blue), only the inlet valve is open. During the second phase (red), the chamber is closed. Finally, the third phase (markedby an arrow) is the exhaust phase, when only the outlet valve is open. This phase is significantly shorter than the first two phases. This cycle then repeats withphase 1 starting at approximately 5.1 and again at 7.8 ms. (Online version in colour.)



6


The enthalpy of water vapour in the range 20–1008C(when referenced to 208C) can be described by the following

empirical equation [21]:

hwV ¼Mwð�1:568182ðTsat � 273:15Þ2 þ 1916:348ðTsat

� 273:15Þ þ 2416597þ 1904ðT � TsatÞÞ:The inlet flow is at ambient reference temperature and

hence provides no incoming heat energy. The outlet heat

flow is

Qout ¼ ðWwL;outCp;wL þWhp;outChp þWHQ;outCHQ

þWBQ;outCBQÞðT � TrefÞ þWwVhwV þWO2hO2

:

The molar outflow rate of each of the six compo-

nents (liquid water, water vapour, hydrogen peroxide,

hydroquinone, benzoquinone and oxygen) is

Wi;out ¼ xiFout

�Vmix:

The model does not predict the duration of a discharge;

rather, the model is run for 30 ms with the reservoir pressure

kept constant throughout this time. While, during a real dis-

charge, one would expect the reservoir pressure to rise to

some maximum value and then decrease after the conclusion

of the attack, the aim of this study was to show the types of

solution available under a more simple set of assumptions.

Brief explorations of different reservoir pressure time profiles

showed that this addition to the model made very little differ-

ence to the results shown here. During the model

explorations, longer simulations were frequently carried out

and where an extended run would significantly change the

model’s output, this has been described. Details of all

model parameters, values and units can be found in table 1.

The set of stiff, algebraic-differential equations given by

the model was solved using the Matlab ode15s function, a

variable order solver based on the numerical differentiation

formulae and Gear’s method. The system was explored for a

wide range of parameter values and initial conditions, and

the only behaviour types seen were those given here. The stab-

ility of many solutions over a longer time period was also

tested. Unless stated otherwise, all parameters for the solution

figures are as in table 1. Many of the parameters are known

chemical constants and the internal dimensions of the beetle

are available for some species (for references, see table 1).

For the remaining parameters whose values are not widely

known (e.g. reaction rates and activation energies), a range

of values were tested and the results are discussed.

3. ResultsWhen solved, the equations above result in a three-phase

process illustrated in figure 2. Initially, the pressure is low

(P , Pinclose); so the inlet valve is open and the outlet valve is

closed: this is the refilling phase. Reactants enter the chamber,

contact the enzyme catalyst contained there and start to heat

up. This temperature rise has an associated pressure increase

that forces the extremities of the chamber to constrict the inlet

pipe. The pipe is completely closed when the pressure reaches

Pinclose, triggering the second (closed) phase. As the reaction

continues, the pressure increases further owing to the release

of oxygen from the decomposition of the hydrogen peroxide

and to the water vapourizing at the increased temperature. At

some point, the pressure is enough to open the outlet valve

(P � Poutopen) and the third phase, exhaust, is initiated. The

high pressure in the chamber forces the contents out through

the outlet pipe and the pressure decreases. Owing to this

pressure drop, the inlet valve reopens and the chamber is once

more placed in the refilling phase. At the start of the exhaust




7


phase, the system is under pressure and the temperature may be

above 1008C. If this is the case, then the opening of the outlet

valve and resultant pressure drop may cause a significant

increase in the steam quality of the water vapour in the chamber

owing to partial or flash evaporation. The speed of the partial

evaporation event is controlled by the size of the outlet [4].

In all cases, the system was tested for a range of inlet closing

pressures and inlet tube radii. Together, these two parameters

control the flow of reactant entering the chamber. The other

adjustable model parameters were also tested within reason-

able ranges both in turn and, wherever possible, in groups.

Where adjustment of a parameter or a group of parameters

changed the model behaviour significantly, the results have

been detailed below. However, in many cases, the behaviour

of the system was highly robust to parameter changes and,

while it showed some small quantitative changes, there were

no qualitative changes. While a parameter space of such high

dimension can never be fully explored, we are confident that,

qualitatively, the system can show only four different modes

of behaviour, which can be described as follows.

3.1. Inert – no reactionIf either the inlet closing pressure or the inlet pipe radius is

very small, the fluid enters the chamber so slowly that the

heat generated by the reaction is lost to the walls and there

is no subsequent build-up of pressure during the first

30 ms. For higher inlet flows, this mode is not stable, i.e. if

the model were run for a longer time period, the system

would move to one of the other possible modes.

3.2. Continuous cyclesIn this mode, while the inlet valve is open, enough fuel is

entering the chamber to allow the reaction to proceed and

the chamber pressure to increase. The system undergoes the

three phases outlined above and the cycle repeats, putting

the system in a steady oscillating state. While the frequency

of the cycles is usually very high (over 1000 Hz), there is a

small range of parameter space (low inlet closing pressure

and small inlet radius) where these cycles can be observed at

a much lower frequency. As with the inert case, there are

some regions of parameter space where this behaviour is not

stable in the long-term; if the model was run for a longer

time period, the continuous cycles may result in pulsed cycles.

3.3. Pulsed cyclesHere, the system initially follows the behaviour seen in the con-

tinuous cycles case. However, in this case, the inlet flow rate is

low, either due to a narrow inlet tube or small inlet closing

pressure. Without the additional cooling effect caused by the

incoming fluid at ambient temperature in each cycle, the maxi-

mum temperature reached is a little higher and the reaction

starts sooner. This causes the inlet valve to stay open for a

shorter length of time, allowing less fuel to enter the chamber;

so there is less cooling provided by the incoming fluid. At some

point, this positive feedback loop results in a lack of fuel enter-

ing the chamber and causes the reaction to stop, the chamber

temperature and pressure both to drop dramatically, and, con-

sequently, no discharge to be observed for a brief period of

time. However, the drop in pressure causes the inlet valve to

open, and, slowly, the incoming fuel reacts and the sequence

starts again. Ultimately, this results in a series of pulsed

cycles. Pulsed cycles generally have a higher maximum temp-

erature than ordinary cycles and the frequency of the pulsing is

usually around 200 Hz, with the primary cycles occurring at

approximately 2000 Hz. Even in this state, the water in the

chamber remains predominantly in a liquid state, with only a

small fraction of the available water in the chamber turning

to steam. Once the system reaches this state, it appears to be

stable indefinitely, provided there is fuel in the reservoir.

3.4. Steady flowWhen fuel is entering the chamber sufficiently fast, it is possible

for the inlet flow combined with the reaction speed to ‘match’

the outlet flow and thus for the system to reach a steady state

with both valves open and a continuous discharge. This

fourth phase (not included in figure 2) usually occurs after a

small number of cycles when the drop in pressure during the

exhaust phase is not enough to close the outlet valve. In this

mode, the residence time for the reactants within the chamber

is very low and the discharge is predominantly made up of

chemicals contained in the reservoir (hydrogen peroxide and

hydroquinones), rather than the reaction products (water,

oxygen and benzoquinones). This mode is usually much

cooler than the cycling mode due to the fuel having a lower resi-

dence time in the chamber, giving less time for the reaction to

take place and for heat to build-up; typical temperatures

are between 208C and 308C. The final outlet flow rate is also

significantly lower than the maximum seen during cyclic flow.

Figure 3 shows example time series for the four different be-

haviour modes. All system parameters are held constant (at the

values given in table 1, with the inlet closing pressure fixed at

102 kPa), except the inlet radius. One would not expect an indi-

vidual beetle to be able to vary this parameter; so the variations

shown here represent differences between species or genera. As

the inlet radius parameter is varied, the system undergoes the

four modes of behaviour described above.

— Inert (no reaction) (figure 3a). For an inlet radius smaller

than 6 mm, there is not enough inlet flow to sustain a reac-

tion and the chamber temperature does not exceed 308C.

— Continuous cycles (figure 3b). For an inlet radius between 7

and 11 mm, the chamber starts to undergo slow cycles and

the temperature reaches approximately 808C, with the

cycle frequency ranging from 200 to 750 Hz. For many

cases in this region, if the model was run for a longer

time period, the frequency of these cycles would increase

and lead to pulsed cycles.

— Pulsed cycles (figure 3c). For an inlet radius between 12 and

35 mm, the system shows pulsed behaviour, with the cycle

frequency now between 900 and 1300 Hz and the pulses

occurring much slower at approximately 100 Hz. The

temperature is still very high, between 708C and 808C.

— Continuous cycles (figure 3d ). As the inlet radius is increased

further (26–58 mm), the cycles get faster (1400–2000 Hz),

though the temperature drops (70–358C). This behaviour

will continue indefinitely.

— Steady flow (figure 3e). Finally, for the largest inlet radii

(above 58 mm), the cycles are only transient and lead to

cooler (around 308C) steady outlet flow.

Figure 4 shows the different behavioural modes and the

corresponding maximum chamber temperature seen for a

range of inlet radii and inlet closing pressures. For larger


no ejection (rin = 1µm)(a)

mas

s ou

tlet r

ate

(mg

s–1)

pres

sure

(kPa

)te

mpe

ratu

re(°

C)

continuous cycles (rin = 5µm)(b)

(d)

20

40

60

80

100

100

105

110

0

10203040

20

40

60

80

100

100

105

110

0.005 0.010 0.015 0.020 0.025 0.0300

10203040

time (s)

continuous cycles (rin = 30µm)

0 0.005 0.010 0.015 0.020 0.025 0.030

time (s)

pulsed flow (rin = 15µm)(c)

mas

s ou

tlet r

ate

(mg

s–1)

pres

sure

(kPa

)te

mpe

ratu

re(°

C)

20

40

60

80

100

100

105

110

0

10

20

30

40

steady flow (rin = 50µm)

0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020time (s)

(e)

mas

s ou

tlet r

ate

(mg

s–1)

pres

sure

(kPa

)te

mpe

ratu

re(°

C)

Figure 3. (a – e) The different modes of behaviour seen as the flow rate into the reaction chamber is increased. For each behaviour mode, temperature, pressure andmass outlet rate are shown. The inlet closing pressure is 102 kPa; all other values are as given in table 1.



8


pressures, the pulsed flow behaviour is not seen, and for

large inlet radii, the cycles simply speed up without showing

pulsed behaviour during the first 30 ms of the discharge. The

hottest modes are always the pulsed and cyclic modes and

the steady flow mode is the coolest (except the mode with

no reaction). The greatest total discharge during the first

30 ms is, as expected, always seen for the largest inlet radii

and highest inlet closing pressure pressures (up to 80 mg dis-

charge during the first 30 ms of ejection at pressure of 108 kPa

and inlet radius 50 mm). However, the greatest outlet rate (of

40 mg s21) is observed during the cyclic flow (be it pulsed or

not); the outlet flow rate during steady flow is approximately

an order of magnitude smaller. The maximum rate is

governed by the state of the reaction chamber (in particular

the pressure) and is thus not dependent on the inlet closing

pressure or the inlet radius. The total mass discharged is

more strongly dependent on the frequency of the cycles,

which is significantly higher for larger pressures (up to

5000 Hz). The mass flow rates seen here are very similar to

those seen by the detailed model of Beheshti & McIntosh

[4], which used spatial information to convert these rates to

outlet velocities of between 2 and 30 ms21, fitting well with

reported observations. Without detailed spatial information,

the observed outlet velocity of a discharge is best predicted

by the average velocity of the outlet. During cyclic flow,

this is approximately 5 ms21; however, during pulsed flow,


50(a)

(b)

pulses

cycles

steady

inert

40

30

20

10

102 103 104 105 106inlet closing pressure (kPa)

inle

t rad

ius

(µm

)


50 100

90

80

70

60

50

40

30

20

40

30

20

10

inle

t rad

ius

(µm

)

Figure 4. The type of behaviour seen depends on the inlet closing pressureand the dimensions of the inlet tube. (a) The different modes of behaviourfor different inlet radii and inlet closing pressures. (b) The correspondingmaximum temperature (8C) of the ESD.



9


it can increase to higher than 15 ms21 during a pulse. Vari-

ations in inlet radius and inlet closing pressure would only

be expected between species or genera rather than on

different occasions by the same individual. However, an indi-

vidual may be able to alter other parameters in the model

(e.g. reservoir pressure) and hence may be able to exhibit

more than one mode of behaviour.

3.5. Exploring parameter spaceThe name bombardier beetle is used to describe many different

species of beetle that all display the bombardier ESD mechan-

ism. To explore how different species may display the

mechanism, different parameter values were tested. Bombar-

diers come in many sizes: for example, Crepidogaster spp.,

which exhibit a cool (638C) mist-like ejection, have a body

length of less than 5 mm [2], and Stenaptus insignis, which exhi-

bit pulsed discharges, are up to 20 mm long [11]. Similar ranges

have been seen in SEM images of the beetle glands and reaction

chambers: for example, M. contractus, with an external body

length of approximately 10 mm, have a reaction chamber

with length 0.7 mm [12], whereas the smaller Crepidogasterspp. have a reaction chamber with length closer to 0.3 mm

[2]. Figure 5 shows an example of the sensitivity of the different

behavioural modes to the beetles internal measurements: on the

top row, all the beetle’s internal length measurements were

increased by a factor of 2, and on the bottom row, they

were all decreased by a factor of 0.5. This results in a change

in the reaction chamber volume, the outlet radius and the

effective thermal diffusivity in the external heat loss. Larger indi-

viduals are more likely to exhibit cycling or pulsed behaviour

and will also get hotter. By contrast, the smaller individuals

show more steady flow and cooler temperatures.

The chemical make-up of different bombardier species

also varies widely. While there is little information available

on the range of concentrations of the reactants, the products of

the ESD—in particular, 1,4-benzoquinones and n-alkanes—

show large variations [7], implying a potentially wide range

of reactant concentrations. The chemical reaction parameter

values—in particular, the reaction rate and the activation

energy—are not known for any individual bombardier

species, but the activation energy used here is within the

range 30–60 J mol– 1 reported for a range of catalysts [22].

However, the results shown here are generic to a wide

range of parameter values. As would be expected, increasing

the reactivity of the solution (either through the activation

energy or the rate constants) results in individuals that are

more likely to exhibit cycles and pulses rather than steady

flow. Figure 6 shows two examples of beetles with different

reactivity. The top row shows an individual with increased

reactivity (reaction rates and heat of reaction are increased

by 5% and the activation energy is decreased by 5%),

which is more likely to show pulsed oscillations and a

higher temperature discharge. The bottom row shows an

individual with decreased reactivity (reaction rates and heat

of reaction are decreased by 5% and the activation energy is

increased by the same amount). An individual with this

decreased reactivity shows pulsed cycles over a much smaller

range of parameters and has a lower temperature discharge.

Overall, extensive explorations of the system show that

the occurrence of these different modes for different par-

ameter values is very stable. For example, an individual

exerting a lower reservoir pressure or with a differently

sized outlet is most likely to be able to show the range of

behaviours illustrated here.

4. DiscussionIn summary, the model shows four modes of behaviour, which

include two types of oscillations, the individual cycles and the

secondary pulse oscillations. The individual cycles predomi-

nantly have a very high frequency, usually above 1000 Hz;

though it is possible to see slower cycles, these exist only in

very small areas of parameter space. The secondary pulse oscil-

lations are at a much lower frequency, with up to 10 pulses seen

in the 30 ms of the simulation.

We hypothesize that the pulses exhibited by the model

are the same as those seen in S. insignus [10], which were

approximately every 1.4 ms, with roughly seven pulses seen

over 12 ms. Figure 7 (top panel) shows an example of part

of a pulsed solution overlaid on to the results of Dean et al.[10]. This study directed the beetles discharge onto a piezo-

electric crystal and recorded the electric output. In this

data, the pulse oscillations can be seen very clearly, but

there are also faster oscillations within the pulses that are

an excellent match with the fast oscillations seen in the

model. Figure 7 shows a good agreement both qualitatively

in the shape of the solutions and quantitatively in the fre-

quency of the secondary low frequency pulses and primary

high-frequency oscillations within a pulse between the data


50(a) (b)

(c) (d )

40

30

20

10

inle

t rad

ius

(µm

)in

let r

adiu

s (µ

m)

50

40

30

20

10

50 100

90

80

70

60

50

40

30

20

100

90

80

70

60

50

40

30

20

40

30

20

10

pulses

cycles

steady

inert

50

40

30

20

10

pulses

cycles

steady

inert

102 103 104 105 106



102 103 104 105 106

Figure 5. Larger insects are more likely to show pulsed oscillations and high-temperature ejections. (a,b) The results for a larger individual (all linear measurementsin table 1 have been increased by a factor of 2): (a) the region of parameter space exhibiting pulsed behaviour is larger, as is the region showing cycles, and (b) theindividual is also more likely to have high temperature ejections. (c,d) The results for a smaller individual (all linear measurements have been halved) that is morelikely to exhibit behaviour in the low temperature, steady flow regime.



10


and the model. This simulation uses parameter values as

used earlier with a slight increase (4%) in reactivity. We

believe that the primary, very high frequency cycles will be

exhibited by most beetles that have some form of jet spray.

However, these very fast oscillations will usually be undetect-

able and only individuals that undergo the secondary pulses

will be reported as having an oscillatory discharge.

Bombardier beetles are strongly associated with very high

temperature discharges of 1008C. However, reports of these

extreme temperatures all stem from a single study on

Brachinus spp. [6]. Measuring the temperature of such a

small outlet flow is not an insignificant task and usually

involves back-calculations from previous calibrations of

known temperature emissions [6], this could easily lead to

over or under estimations of the true discharge temperature.

Despite this limitation, studies have recorded a wide range of

mean temperatures, all significantly lower than the original

1008C: for instance, M. contractus ejects its secretion at 558C[12], G. nicaraguesis at 558C [1], Crepidogastrine atrata at 658C[2] and M. regularis at 358C and 478C [9]. This variability sup-

ports the model, which shows that only a very small range of

beetle physiologies are able to produce very high temperature

discharges and that lower temperatures are much more

common. Figures 3–6 show a wide range of temperature dis-

charges (the maximum temperature of the pulse in figure 3 is

over 808C). In addition to this, figure 8 shows a pulsed time-

series that reaches the very high temperatures seen in the

original work of Aneshansley. These very high temperatures

can also been in cyclic flow for higher inlet closing pressures.

The model also predicts that the highest temperature dis-

charges are most likely to be seen in individuals with pulsed

flow. This hypothesis is supported by the results of Eisner

et al. [2], which showed pulsed oscillations from a species

thought to be Brachinus elongatulus, the genus that was orig-

inally shown by Aneshansley et al. [6] to have a consistently

high discharge temperature across individuals of different

species. Eisner et al. [2] mention unpublished recordings of

G. nicaraguesis, a beetle with a lower discharge temperature

of 658C [1], which failed to show pulsing, and reports no pul-

sing in the discharge of C. atrata, which also has a lower

discharge temperature of 658C, again supporting the model

predictions. This hypothesis could be tested further by exam-

ining the temperature of the discharge of S. insignus, which

the model predicts would be high.

A final characteristic of the bombardier beetle is the large

throw ratio achieved by the ESD. A high throw ratio (the ratio

of the distance travelled by the discharge and the size of the

combustion chamber) is achieved by having a high mass out-

flow rate through a very small outlet radius. The mass

outflow rates predicted by this model are of the same order

(up to 4 � 1025 kg s21) as those predicted by the more

detailed spatial models of Beheshti & McIntosh [4]. While

the present model does not predict throw ratio, it would be

expected to be of the same magnitude as that obtained by

Beheshti and McIntosh because the mass outlet flow rate

and the outlet radii are similar. The difference between the

two models lies in the mechanism providing the force

behind the two rates. Previous models have not


50(a)

(c) (d )

(b)

40

30

20

10

inle

t rad

ius

(µm

)

pulses

cycles

steady

inert

102 103 104 105 106

inle

t rad

ius

(µm

)

50

40

30

20

10

pulses

cycles

steady

inert


102 103 104 105 106

50

40

30

20

10

100

90

80

70

60

50

40

30

20

50

40

30

20

10


100

90

80

70

60

50

40

30

20

Figure 6. Insects with more reactive chemistry are more likely to show pulsed oscillations. (a,b) The results for an individual with more reactive chemistry (increasedby 5%): (a) the region of parameter space exhibiting pulsed behaviour is larger, as is the region showing cycles, and (b) the individual is also more likely to havehigh temperature ejections. (c,d) The results for an individual with a less reactive mixture (reduced by 5%), which is more likely to exhibit behaviour in the lowtemperature, steady flow regime. The cyclic behaviour is also at a lower temperature.

outle

t mas

s fl

ow r

ate

time (s)

0.018 0.020 0.022 0.024 0.026 0.028 0.030

Figure 7. A comparison of the model results with published observation data. (Top panel) The underlying time series (grey, large dots) is the original data of Deanet al. [10] (reproduced with permission). The black lines are from a model simulation (rin ¼ 15 mm, Pinclose ¼ 102.5 kPa, reactivity increased by 4%, all otherparameters as in table 1). Note that the vertical axis for the observed data is the electric output for a piezoelectric crystal, while the model shows the outlet massflow rate: the units for these quantities are not comparable in absolute terms, but one would expect them to have a similar relative size. The scaling of thehorizontal axis is the same for both time series but the relative position is not fixed.



11


included the beetle’s chemical reactions, so have concluded

that the force could be provided by the chamber temperature

rising above 1008C and the resulting partial evaporation of

steam as the outlet valve opens and the pressure and temp-

erature quickly fall. The present model shows that similar

ejections can be achieved without the chamber temperature

rising above the atmospheric boiling point of water and

that the build-up of pressure needed is due mainly to

released oxygen.

To summarize, the model predicts that the same

underlying mechanism can result in a range of different beha-

viours; all the observations seen in the wide range of

bombardier species can be exhibited by the model for different

parameter values. Both the pulsed and continuous ejection

modes that have been observed can be seen in the model for

a wide range of parameter values that are consistent with

the known information of individual beetles. The observed

temperatures of the different beetle discharges have also


20

40

60

80

100

tem

pera

ture

(°C

)

100

105

110pr

essu

re (

kPa)

0

0.5

1.0

0.005 0.010 0.015 0.020 0.025 0.030

stea

m q

ualit

y

time (s)

Figure 8. The bombardier first became of interest to entomologists owing to its high temperature discharge. The model also shows solutions that reach these high(in this case over 998C) temperatures. In this case (rin ¼ 10 mm, Pinclose ¼ 101.3 kPa, reactivity increased by 4%, all other parameters as in table 1), the water inthe chamber is sometimes (over 60%) vapour rather than liquid.



12


been recreated. We believe the ability of wide-ranging beha-

viours to be described by a single underlying system adds

weight to the theory that different bombardiers’ ESD mechan-

isms could have evolved from a single common mechanism.

References

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http://dx.doi.org/10.1126/science.215.4528.83


http://dx.doi.org/10.1007/PL00001853

http://dx.doi.org/10.1007/PL00001853

http://dx.doi.org/10.1007/PL00001854

http://dx.doi.org/10.1007/PL00001854

http://dx.doi.org/10.1088/1748-3182/2/4/001

http://dx.doi.org/10.1088/1748-3182/2/4/001

http://dx.doi.org/10.1002/anie.196400731

http://dx.doi.org/10.1002/anie.196400731



http://dx.doi.org/10.1016/0022-1910(77)90162-7

http://dx.doi.org/10.1007/BF01963125

http://dx.doi.org/10.1007/BF01963125

http://dx.doi.org/10.1007/BF01240663

http://dx.doi.org/10.1126/science.2349480

http://dx.doi.org/10.1073/pnas.96.17.9705

http://dx.doi.org/10.1073/pnas.96.17.9705

http://dx.doi.org/10.2992/0097-4463-77.1.79

http://www.inchem.org

http://dx.doi.org/10.1021/je0502748

http://dx.doi.org/10.1021/je0502748

http://dx.doi.org/10.1016/0098-1354(91)80026-R

http://dx.doi.org/10.1002/kin.1059


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