J. Theor. Biol. (1975) 49, 179-189
A Population Model for Dermestid Beetle Survival Under Starvation and Canniba!ismt J . P . BRANNmq
Sandia Laboratories, Albuquerque, New Mexico 87115, U.S.A. (Received 18 December 1973) The larvae of dermestid beetles frequently engage in cannibalistic behavior. A mathematical model based on cannibalistic feedback, probability of attack survival, and probability of surviving starvation predicts survivors for larval populations confined without food. 1. Introduction
The cannibalistic tendencies of dermestid beetles provide an interesting parameter in population survival studies. The effect of cannibalism on observed population survival is uncertain. For example, if a fixed population is associated with a given material, does survival significantly beyond that expected from starvation result from utilization of the material as food or is it the consequence of cannibalism ? This is a report on the derivation and application of a mathematical model which quantifies the effects of starvation and cannibalism in dermestid populations. The model is parameterized for two dermestid species, Anthrenus flavipes and Dermestes maculatus. 2. Model Derivation and Parameterization for Anthrenusflavipes Questions which must be considered in model derivation include: (1) What is expected population survival as a function of time when death is the consequence of starvation ? (2) Are members consumed as food the victims of attack by their neighbors or did they die of starvation ? (3) What is the effect through cannibalism on the return of energy t o the biosystem ?
(A) STARVATIONMODEL Before questions (2) and (3) can be properly addressed, it is necessary to first have a model for expected survivors when the population is reduced by starvation alone. t This work supported by the U.S. Atomic Energy Commission. 179
180
J.P.
BRANNEN
Weekly survivor means for four replicate samples of 15 Anthrenusflavipes larvae confined in plastic boxes 2¼ in x 2¼ in x 5 in over a period of nine weeks are indicated by the circles in Fig. I, Brannen, Gennaro, Salb, Aday & Woodfolk (1973). All members of the population were dead after the nine-week period. Ninety-five per cent confidence intervals based on Student's t-distribution are given by arrows passing through the circles. Data which have a lower confidence interval of zero are indicated by a "To zero" legend. 15, I0
";
I
@ •
tO
Q
Q
•
0-5
2 0"1
T I 2
v
I 4
I
I 6
I
I 8
w I I0
w 12
Time (weeks)
FIo. 1. Semilogarithmic plots of survivor means for four replicate samples of Anthrenus flavipes larvae confined in plastic boxes. Means for initial populations of 15 and 1 are indicated by circles and dots respectively. Ninety-five per cent confidence intervals are indicated by arrows.
Similar survivor means for four replicates each containing only one larva are indicated by dots. In this case, all larvae were dead after six weeks. Since there could be no cannibalism or attacks in the boxes containing only one larva, it is assumed that the dots closely resemble a starvation survival curve. Taking a cue from Eyring & Stover (1970), it is assumed that: (1) The process of starvation proceeds at near equilibrium. (2) There are r cells or sites subject to critical change through starvation and that at time t, n of these have undergone the critical change. Let v~ be the rate at which a single unchanged site is changing and vj be the rate at which the change is disappearing. Assumption (1) above is
[3EETLE POPULATION
181
equivalent to saying that v , ( r - n ) = vln.
It follows that the fraction of changed sites
1 nlr = 1 +(vflv~) and that the fraction of unchanged sites is
1 1 - n / r - 1 +(vdvff Using absolute reaction rate theory, Eyring & Stover point out that the fraction of unchanged sites at time t is of the form l-nit
=
1 l+exp [-(a-bt)]"
Assume that the probability of an organism surviving starvation to time t,
p,(t), is the fraction of unchanged sites at time t normalized so thatp~(0) = 1. From this, 1 +exp ( - a )
p,(t) = 1 +exp [ - ( a - b O ] "
(1)
Under these assumptions, a is a measure of the organism's reserves and b is a measure of the rate at which these reserves are being depleted. The derivation carried out by Eyring & Stover was for their "Steady State Theory of Mutation Rates". However, their assumptions seem directly related to the manner in which starvation occurs so that the transfer from mutation rates to starvation is taken to be direct. Under starvation, there were no survivors at the end of six weeks. To parameterize the model, it is assumed that p,(6) = 0.5 and that p,(7) = 0-1. It is also assumed that exp ( - a ) is sufficiently near 0 so that
~03=
l+exp [ - ( a - 6b)]'
Ifp~(6) = 0.5, then
a - 6 b = O, and if ps(7) = 0"1, then ps(7) =
1 +exp [ - ( 6 b - 7 b ) ]
so that 0.1 = [i +exp (b)] - t .
182
J.
P.
BRANNEN
F r o m this it follows that exp (b) = 9 so that b = 2.197. Since a = 6b, a = 13-183. Equation (1), parameterized for Anthrenus flavipes becomes 1 + e x p ( - 13-183) 1 +exp [-(13"183--2.1970]' The dotted curve in Fig. 2 is a graph of equation (2).
ps(t) =
(2)
-'k
0-.-0.) ....... •
¢.0
,-,..,
a
•
II'°
0.5
0-1
'
! 2
)
I1
4
:)
6 0 Time (weeks)
t
1 ) I0 12
FIG. 2. Starvation and attack moded parameterization for Anthrenus flavipes. Survivor means for initial populations of 15 and 1 are indicated by circles and dots respectively. Probability of surviving starvation is given by the dotted curve. Expected starvation or attack survivors for an initial population of 15 are indicated by the solid and dashed curves respectively.
If the initial population is N(O), the expected survivors at time t, E[N(t)], when the population is reduced by starvation alone is given by
e[N(t)] = N(O)plO. This is shown by the solid curve in Fig. 2 for an initial population of N(0) = 15.
BEETLE
POPULATION
183
(B) ATTACK SURVIVAL MODEL
A comparison of the predictions to observations for N(0) = 15 in Fig. 2 strongly suggests that when N(0) > 1, factors other than starvation are important in population reduction. Also, if only the starved dead were eaten, the initial five-week portion of the survival curve would be even higher than the observations. These beetles are known to be cannibalistic. It will then be assumed that when N(0) > 1, population reduction will result from fratricidal behavior within the population. To model a member's probability of surviving attack, the following assumption is made: The probability of an organism's succumbing to attack during a fixed time interval is proportional to the ratio of the ways attacks can occur to the number of encounters that occur during that time interval. If N is the population, attacks can occur (2N) ways where ( ~ ) is the combination of A things taken B at a time. Also, assuming encounter rate proportional to population, during a fixed time interval the number of encounters is proportional to N z. Thus for fixed time intervals the probability of a member being the victim of attack is taken to be proportional to
The probability of surviving attack, po, is taken to be po = 1 - A 0 - l/N),
(3)
where the proportionality constant A includes the 1/2 from ( N ) . The proportionality constant A in equation (3) is dependent on the time interval during which encounters may occur. For Anthrenusflavipes model parameterization, the time interval will be taken to be one week since the data presented in all the figures were taken at weekly intervals. Also the number N will be taken as the population at the beginning of the week. When the initial population, N(0), was 15, N(1) was 12.5. Assuming that starvation was negligible during this period and that equation (3) was completely dominant, it follows that N(O)po = N(I) or
15 [1 - A (1 - 1 ) ] , = From this it follows that A = 0-18.
12"5.
184
J.P.
BRANNEN
Combining equations (2) and (3) gives a single organism's probability for surviving both attack and starvation during a given week to be
P'× P"= l +exp [-(13.183-2.197 )] -_
1-0.18
1-
,
(4)
where N is the population at the beginning of the week. The dashed line in Fig. 2 is from equation (3) with A = 0.18. The dashed curve of Fig. 3 results when equation (4) is used iteratively on weekly intervals to predict survivors from an initial population of 15. The dashed curve and solid curve are indistinguishable during the first four weeks.
\
O.,= l l 1 I 2
4
6
8
I0
12
T~me(weeks)
FIG. 3. Starvation and attack model parameterization for Anthrenusflavipes. The dashed curve represents expected survivors when death is due to either starvation or attack. The solid curve includes cannibalistic feedback. (C) CANNIBALISTIC TEEDBACK
Since the predictions generated by equation (4) are unacceptably low after the sixth week, it is assumed that those organisms which have been eaten increase the eater's probability of surviving starvation. Recall that the a in equation (1) is a measure of the organism's reserves. It is assumed that by eating other members of the population, the eater replenishes to some extent his diminishing reserves. That is, a increases relative to bt. If N~b is the population at the beginning of the week, the expected loss to cannibalism during the week is given by N¢O-~.).
BEETLE P O P U L A T I O N
185
If N1 is the population at week's end, then the replenishment for the following week should be proportional to N¢(1 -po)/N1. Thus after the first week, a of equation (1) will be taken to be a = CN¢(1-po)/N1, (5) where it is understood that N ¢ ( 1 - p , ) is the number of attack victims during the preceding week and N1 is the population at the beginning of the week under consideration. The solid curve of Fig. 3 shows expected survivors from an initial population of 15 with replenishment of reserves. C is taken to be 12. Figure 4 shows predictions determined as above for initial populations of 15, solid curve, and 5, dashed curve. Observations for initial populations
I0
-a
I 0'5
I
\,
0.1
T
I
2
i
T
4
I
!
6
i
I
8
I0
:
12
Time (weeks)
I~G. 4. Comparison of model predictions to observations. The model was parameterized to the solid curve for Anthrenus flaoipes with an initial population of 15. The dashed curve represents the prediction for an initial population of 5. Observations for initial populations of 15 and 5 are indicated by circles and dots respectively. Ninety-five per cent confidence intervals are given by arrows.
of 15 and 5 are denoted by circles and dots respectively. Ninety-five per cent confidence intervals are shown by arrows. (D) COMPUTER FLOWCHART
The iterative features of the model make it ideal for digital computation. Figure 5 gives a flow chart for digital applications.
186
J.
P.
BRANNEN
(
"
)
Input
°,b,A,C'NgJ;S ,vJ
No ~ F'I = ~
[
=~v#.
Yes FI=C (I-Pa)
I T=T+
1
I Ip= I+,~p(-a) I
,+ .p [-c.-~.T~]l
NgJ=NI
T F'2 = F I*N~
i I.. L
r
No
/
I
. ~
Yes
~
./-
\,
Stop
FxG. 5. Flowchart for model computerization.
Input parameters a, b, A and C are as in equations (1), (3) and (5). N~b is the initial population and Tis time in weeks. After the first week, N~k denotes the population for the preceding week. The cannibalistic feedback is carried out through F1, F2, and v. 3. Model Parameterization for Dermestes macnlatus All parameterization discussed above was for Anthrenus flavipes. Sample means and 95yo confidence intervals for Dermestes maculatus are shown in Fig. 6. Only larvae were placed in the sample boxes. Model parameterization was carried out exactly as for Anthrenus flavipes. The solid curve shows model predictions when a = 6.6, b = 2-17, A = 0"5 and C = 8.
BEETLE P O P U L A T I O N
187
15
¢: I0 ~ t
A
O.5
0'1
II
Ii
2
l
I
5 4 Time(weeks)
~l =~
5
6
Fxo. 6. Model parameterization for Dermestes maculatus. The solid curve represents expected survivors. Mean survivors for initial populations of 15 and 1 are indicated by dots and triangles respectively. Ninety-five per cent confidence intervals are indicated by arrows. This suggests that: (1) Dermestes maculatus is a more efficient cannibal than Anthrenus flavipes since the A values are 0.5 and 0.18 respectively. (2) D. maculatus has a much higher metabolic rate than A. flavipes since the a values are 6.6 and 13.2 respectively. In fact, a doubled metabolic rate is indicated.
4. Model Applications Although the model was developed for studying the effects of cannibalism on beetle populations, insight into the utilization of various materials can be gained through application of the model to observations. Suppose, for example, that a material is placed with larvae in identical boxes, survivor means determined, and the model parameterized for these data. Then information relative to beetle-material interaction may be deduced as follows: (1) If the only replenishment of depleted reserves is by cannibalized individuals, then the material is not utilizable as food; (2) If the probability of dying from attack drops significantly, then the material provides shelter.
188
j. p. BRANNEN
(3) If the survivor mean is "very near" that for no food observations, then the material provides neither food nor shelter. (4) If the population stabilizes at above the non-zero level, then the material provides both food and shelter. Figure 7 shows survivor means for four replicate samples of 15 Anthrenus flavipes larvae confined with a low density fibrous material in sample boxes.
1
E
°t
°I 0'1
~
I
2
I
I
4
t
t
6
8
I0
i
IZ
Time (weeks)
FIG. 7. Model application to low density fibrous material. Observations and confidence intervals are indicated by circles and arrows respectively. The dashed curve represents expected survivors when the proportionality constant for attack successes is 0.18, the no material present case. The solid curve gives expected survivors when the proportionality
constant is 0.I. The solid curve shows a model parameterization achieved by changing the proportionality constant, A, for dying due to attack from 0-18 to 0.I. Since the only replenishment of depleted reserves is cannibalized individuals, it is concluded that the material is not a food source. The material does, however, provide shelter from attacks. For comparison, the dashed curve in Fig. 6 shows the model with "no material" parameterization. 5. Concluding Remarks
Under the assumptions used for model derivations, parameterization for bio-observations establishes that for both Anthrenus flavipes and Dermestes maculatus larvae in a closed no food environment:
BEETLE
POPULATION
189
(1) Fratricidal behavior has an effect in populations containing more than one individual. (2) Cannibalism has a non-negligible effect on the observed population curves.
The model was intended for the analysis of cannibalistic effects. However, parameterization for this purpose provides insight into beetle populations under the stress of starvation. For example, when a low density fibrous material was included in the sample boxes, probability of an organism's succumbing to attack was halved. This suggests that the beetles would rather "hide than fight". The evidence is not conclusive since it is also possible that the material simply made encounters less likely. In addition, the model provides means for quantifying the utilization of various test materials by the beetles relative to food and shelter. REFERENCES BRA~-~N, J. P., GE~r~ARO,A. L., SALB,T. J., ADAY,B. J. • WOOLBOLK,C. D. (1973). Studies on Dermestid Beetle-Material Interactions, II. Food Sources. Sandia Laboratories Research Report SLA-73-0823. E~.u~, H. & STOWR,B. I. (1970). Proc. natn. Acad. Sci. U.S.A. 66, 441.
A Population Model for Dermestid Beetle Survival Under Starvation and Canniba!ismt J . P . BRANNmq
Sandia Laboratories, Albuquerque, New Mexico 87115, U.S.A. (Received 18 December 1973) The larvae of dermestid beetles frequently engage in cannibalistic behavior. A mathematical model based on cannibalistic feedback, probability of attack survival, and probability of surviving starvation predicts survivors for larval populations confined without food. 1. Introduction
The cannibalistic tendencies of dermestid beetles provide an interesting parameter in population survival studies. The effect of cannibalism on observed population survival is uncertain. For example, if a fixed population is associated with a given material, does survival significantly beyond that expected from starvation result from utilization of the material as food or is it the consequence of cannibalism ? This is a report on the derivation and application of a mathematical model which quantifies the effects of starvation and cannibalism in dermestid populations. The model is parameterized for two dermestid species, Anthrenus flavipes and Dermestes maculatus. 2. Model Derivation and Parameterization for Anthrenusflavipes Questions which must be considered in model derivation include: (1) What is expected population survival as a function of time when death is the consequence of starvation ? (2) Are members consumed as food the victims of attack by their neighbors or did they die of starvation ? (3) What is the effect through cannibalism on the return of energy t o the biosystem ?
(A) STARVATIONMODEL Before questions (2) and (3) can be properly addressed, it is necessary to first have a model for expected survivors when the population is reduced by starvation alone. t This work supported by the U.S. Atomic Energy Commission. 179
180
J.P.
BRANNEN
Weekly survivor means for four replicate samples of 15 Anthrenusflavipes larvae confined in plastic boxes 2¼ in x 2¼ in x 5 in over a period of nine weeks are indicated by the circles in Fig. I, Brannen, Gennaro, Salb, Aday & Woodfolk (1973). All members of the population were dead after the nine-week period. Ninety-five per cent confidence intervals based on Student's t-distribution are given by arrows passing through the circles. Data which have a lower confidence interval of zero are indicated by a "To zero" legend. 15, I0
";
I
@ •
tO
Q
Q
•
0-5
2 0"1
T I 2
v
I 4
I
I 6
I
I 8
w I I0
w 12
Time (weeks)
FIo. 1. Semilogarithmic plots of survivor means for four replicate samples of Anthrenus flavipes larvae confined in plastic boxes. Means for initial populations of 15 and 1 are indicated by circles and dots respectively. Ninety-five per cent confidence intervals are indicated by arrows.
Similar survivor means for four replicates each containing only one larva are indicated by dots. In this case, all larvae were dead after six weeks. Since there could be no cannibalism or attacks in the boxes containing only one larva, it is assumed that the dots closely resemble a starvation survival curve. Taking a cue from Eyring & Stover (1970), it is assumed that: (1) The process of starvation proceeds at near equilibrium. (2) There are r cells or sites subject to critical change through starvation and that at time t, n of these have undergone the critical change. Let v~ be the rate at which a single unchanged site is changing and vj be the rate at which the change is disappearing. Assumption (1) above is
[3EETLE POPULATION
181
equivalent to saying that v , ( r - n ) = vln.
It follows that the fraction of changed sites
1 nlr = 1 +(vflv~) and that the fraction of unchanged sites is
1 1 - n / r - 1 +(vdvff Using absolute reaction rate theory, Eyring & Stover point out that the fraction of unchanged sites at time t is of the form l-nit
=
1 l+exp [-(a-bt)]"
Assume that the probability of an organism surviving starvation to time t,
p,(t), is the fraction of unchanged sites at time t normalized so thatp~(0) = 1. From this, 1 +exp ( - a )
p,(t) = 1 +exp [ - ( a - b O ] "
(1)
Under these assumptions, a is a measure of the organism's reserves and b is a measure of the rate at which these reserves are being depleted. The derivation carried out by Eyring & Stover was for their "Steady State Theory of Mutation Rates". However, their assumptions seem directly related to the manner in which starvation occurs so that the transfer from mutation rates to starvation is taken to be direct. Under starvation, there were no survivors at the end of six weeks. To parameterize the model, it is assumed that p,(6) = 0.5 and that p,(7) = 0-1. It is also assumed that exp ( - a ) is sufficiently near 0 so that
~03=
l+exp [ - ( a - 6b)]'
Ifp~(6) = 0.5, then
a - 6 b = O, and if ps(7) = 0"1, then ps(7) =
1 +exp [ - ( 6 b - 7 b ) ]
so that 0.1 = [i +exp (b)] - t .
182
J.
P.
BRANNEN
F r o m this it follows that exp (b) = 9 so that b = 2.197. Since a = 6b, a = 13-183. Equation (1), parameterized for Anthrenus flavipes becomes 1 + e x p ( - 13-183) 1 +exp [-(13"183--2.1970]' The dotted curve in Fig. 2 is a graph of equation (2).
ps(t) =
(2)
-'k
0-.-0.) ....... •
¢.0
,-,..,
a
•
II'°
0.5
0-1
'
! 2
)
I1
4
:)
6 0 Time (weeks)
t
1 ) I0 12
FIG. 2. Starvation and attack moded parameterization for Anthrenus flavipes. Survivor means for initial populations of 15 and 1 are indicated by circles and dots respectively. Probability of surviving starvation is given by the dotted curve. Expected starvation or attack survivors for an initial population of 15 are indicated by the solid and dashed curves respectively.
If the initial population is N(O), the expected survivors at time t, E[N(t)], when the population is reduced by starvation alone is given by
e[N(t)] = N(O)plO. This is shown by the solid curve in Fig. 2 for an initial population of N(0) = 15.
BEETLE
POPULATION
183
(B) ATTACK SURVIVAL MODEL
A comparison of the predictions to observations for N(0) = 15 in Fig. 2 strongly suggests that when N(0) > 1, factors other than starvation are important in population reduction. Also, if only the starved dead were eaten, the initial five-week portion of the survival curve would be even higher than the observations. These beetles are known to be cannibalistic. It will then be assumed that when N(0) > 1, population reduction will result from fratricidal behavior within the population. To model a member's probability of surviving attack, the following assumption is made: The probability of an organism's succumbing to attack during a fixed time interval is proportional to the ratio of the ways attacks can occur to the number of encounters that occur during that time interval. If N is the population, attacks can occur (2N) ways where ( ~ ) is the combination of A things taken B at a time. Also, assuming encounter rate proportional to population, during a fixed time interval the number of encounters is proportional to N z. Thus for fixed time intervals the probability of a member being the victim of attack is taken to be proportional to
The probability of surviving attack, po, is taken to be po = 1 - A 0 - l/N),
(3)
where the proportionality constant A includes the 1/2 from ( N ) . The proportionality constant A in equation (3) is dependent on the time interval during which encounters may occur. For Anthrenusflavipes model parameterization, the time interval will be taken to be one week since the data presented in all the figures were taken at weekly intervals. Also the number N will be taken as the population at the beginning of the week. When the initial population, N(0), was 15, N(1) was 12.5. Assuming that starvation was negligible during this period and that equation (3) was completely dominant, it follows that N(O)po = N(I) or
15 [1 - A (1 - 1 ) ] , = From this it follows that A = 0-18.
12"5.
184
J.P.
BRANNEN
Combining equations (2) and (3) gives a single organism's probability for surviving both attack and starvation during a given week to be
P'× P"= l +exp [-(13.183-2.197 )] -_
1-0.18
1-
,
(4)
where N is the population at the beginning of the week. The dashed line in Fig. 2 is from equation (3) with A = 0.18. The dashed curve of Fig. 3 results when equation (4) is used iteratively on weekly intervals to predict survivors from an initial population of 15. The dashed curve and solid curve are indistinguishable during the first four weeks.
\
O.,= l l 1 I 2
4
6
8
I0
12
T~me(weeks)
FIG. 3. Starvation and attack model parameterization for Anthrenusflavipes. The dashed curve represents expected survivors when death is due to either starvation or attack. The solid curve includes cannibalistic feedback. (C) CANNIBALISTIC TEEDBACK
Since the predictions generated by equation (4) are unacceptably low after the sixth week, it is assumed that those organisms which have been eaten increase the eater's probability of surviving starvation. Recall that the a in equation (1) is a measure of the organism's reserves. It is assumed that by eating other members of the population, the eater replenishes to some extent his diminishing reserves. That is, a increases relative to bt. If N~b is the population at the beginning of the week, the expected loss to cannibalism during the week is given by N¢O-~.).
BEETLE P O P U L A T I O N
185
If N1 is the population at week's end, then the replenishment for the following week should be proportional to N¢(1 -po)/N1. Thus after the first week, a of equation (1) will be taken to be a = CN¢(1-po)/N1, (5) where it is understood that N ¢ ( 1 - p , ) is the number of attack victims during the preceding week and N1 is the population at the beginning of the week under consideration. The solid curve of Fig. 3 shows expected survivors from an initial population of 15 with replenishment of reserves. C is taken to be 12. Figure 4 shows predictions determined as above for initial populations of 15, solid curve, and 5, dashed curve. Observations for initial populations
I0
-a
I 0'5
I
\,
0.1
T
I
2
i
T
4
I
!
6
i
I
8
I0
:
12
Time (weeks)
I~G. 4. Comparison of model predictions to observations. The model was parameterized to the solid curve for Anthrenus flaoipes with an initial population of 15. The dashed curve represents the prediction for an initial population of 5. Observations for initial populations of 15 and 5 are indicated by circles and dots respectively. Ninety-five per cent confidence intervals are given by arrows.
of 15 and 5 are denoted by circles and dots respectively. Ninety-five per cent confidence intervals are shown by arrows. (D) COMPUTER FLOWCHART
The iterative features of the model make it ideal for digital computation. Figure 5 gives a flow chart for digital applications.
186
J.
P.
BRANNEN
(
"
)
Input
°,b,A,C'NgJ;S ,vJ
No ~ F'I = ~
[
=~v#.
Yes FI=C (I-Pa)
I T=T+
1
I Ip= I+,~p(-a) I
,+ .p [-c.-~.T~]l
NgJ=NI
T F'2 = F I*N~
i I.. L
r
No
/
I
. ~
Yes
~
./-
\,
Stop
FxG. 5. Flowchart for model computerization.
Input parameters a, b, A and C are as in equations (1), (3) and (5). N~b is the initial population and Tis time in weeks. After the first week, N~k denotes the population for the preceding week. The cannibalistic feedback is carried out through F1, F2, and v. 3. Model Parameterization for Dermestes macnlatus All parameterization discussed above was for Anthrenus flavipes. Sample means and 95yo confidence intervals for Dermestes maculatus are shown in Fig. 6. Only larvae were placed in the sample boxes. Model parameterization was carried out exactly as for Anthrenus flavipes. The solid curve shows model predictions when a = 6.6, b = 2-17, A = 0"5 and C = 8.
BEETLE P O P U L A T I O N
187
15
¢: I0 ~ t
A
O.5
0'1
II
Ii
2
l
I
5 4 Time(weeks)
~l =~
5
6
Fxo. 6. Model parameterization for Dermestes maculatus. The solid curve represents expected survivors. Mean survivors for initial populations of 15 and 1 are indicated by dots and triangles respectively. Ninety-five per cent confidence intervals are indicated by arrows. This suggests that: (1) Dermestes maculatus is a more efficient cannibal than Anthrenus flavipes since the A values are 0.5 and 0.18 respectively. (2) D. maculatus has a much higher metabolic rate than A. flavipes since the a values are 6.6 and 13.2 respectively. In fact, a doubled metabolic rate is indicated.
4. Model Applications Although the model was developed for studying the effects of cannibalism on beetle populations, insight into the utilization of various materials can be gained through application of the model to observations. Suppose, for example, that a material is placed with larvae in identical boxes, survivor means determined, and the model parameterized for these data. Then information relative to beetle-material interaction may be deduced as follows: (1) If the only replenishment of depleted reserves is by cannibalized individuals, then the material is not utilizable as food; (2) If the probability of dying from attack drops significantly, then the material provides shelter.
188
j. p. BRANNEN
(3) If the survivor mean is "very near" that for no food observations, then the material provides neither food nor shelter. (4) If the population stabilizes at above the non-zero level, then the material provides both food and shelter. Figure 7 shows survivor means for four replicate samples of 15 Anthrenus flavipes larvae confined with a low density fibrous material in sample boxes.
1
E
°t
°I 0'1
~
I
2
I
I
4
t
t
6
8
I0
i
IZ
Time (weeks)
FIG. 7. Model application to low density fibrous material. Observations and confidence intervals are indicated by circles and arrows respectively. The dashed curve represents expected survivors when the proportionality constant for attack successes is 0.18, the no material present case. The solid curve gives expected survivors when the proportionality
constant is 0.I. The solid curve shows a model parameterization achieved by changing the proportionality constant, A, for dying due to attack from 0-18 to 0.I. Since the only replenishment of depleted reserves is cannibalized individuals, it is concluded that the material is not a food source. The material does, however, provide shelter from attacks. For comparison, the dashed curve in Fig. 6 shows the model with "no material" parameterization. 5. Concluding Remarks
Under the assumptions used for model derivations, parameterization for bio-observations establishes that for both Anthrenus flavipes and Dermestes maculatus larvae in a closed no food environment:
BEETLE
POPULATION
189
(1) Fratricidal behavior has an effect in populations containing more than one individual. (2) Cannibalism has a non-negligible effect on the observed population curves.
The model was intended for the analysis of cannibalistic effects. However, parameterization for this purpose provides insight into beetle populations under the stress of starvation. For example, when a low density fibrous material was included in the sample boxes, probability of an organism's succumbing to attack was halved. This suggests that the beetles would rather "hide than fight". The evidence is not conclusive since it is also possible that the material simply made encounters less likely. In addition, the model provides means for quantifying the utilization of various test materials by the beetles relative to food and shelter. REFERENCES BRA~-~N, J. P., GE~r~ARO,A. L., SALB,T. J., ADAY,B. J. • WOOLBOLK,C. D. (1973). Studies on Dermestid Beetle-Material Interactions, II. Food Sources. Sandia Laboratories Research Report SLA-73-0823. E~.u~, H. & STOWR,B. I. (1970). Proc. natn. Acad. Sci. U.S.A. 66, 441.
