BULLETIN OF MATHEMATICAL BIOLOGY
VOLUME37, 1975
A THREE STAGE POPULATION MODEL WITH
CANNIBALISM
9 H. D. LANDA~Jand B. D. HANSEN Department of Biochemistry and Biophysics University of California, San Francisco 94143
A simple population model consisting of one adult and two larval stages with cannibalism or competition a m o n g t h e larval stages is presented. The solutions are found to be either periodic or of a steady state n a t u r e depending on the ratios of fertility and cannibalism a m o n g the larvae. Two similar cannibalism pressure functions are compared and the conditions t h a t lead to steady or periodic solutions, or to extinction, are examined.
Studies of ecological communities require various assumptions concerning the number of species that make up the community, their interrelationships, abundances, and spatial distributions. In many studies, the definition of the investigators community is determined by convenience and intuition (Pielou, 1969) which is based, among other things, on the investigators experience with the individual species. In this note we examine the simplest prototype of a single species population (e.g., insects) that consists of adults and several larval stages where a weighted cannibalism exists among the larval stages. Even in this simple system rather complicated solutions arise that would not be intuitively obvious a priori. It is hoped that the system may give some insight into the cyclic variations in certain insect populations. The following model is a representation of a three stage population with age dependent cannibalism. For convenience, the stages are designated the adults (A'), which live for only one season, those larvae (LI') which hatch from eggs 11
12
H.D.
L A N D A H L A N D B. D. H A N S E N
laid during one season and become adults the next season, and those larvae which remain larvae for two seasons before becoming adults. The number of these latter larvae which are in their first season, junior larvae, will be denoted by/2', those in their second season, senior larvae, by L2'. We assume t h a t these senior larvae exert a cannibalistic or competitive pressure on the larvae, L I ' and 12', which are in their first season, and to a lesser extent upon themselves. For simplicity we neglect the competition among young larvae (/2', L2'), and assume t h a t there is no cannibalism on or by the adults. Suppose t h a t at the beginning of a season n, A~ adults emerge and lay CI' eggs per adult t h a t are destined to become adults in one season, and C2' eggs t h a t become adults in two seasons. The adults are assumed to survive only one season. Let P~ be the probability t h a t an egg survives and emerges as a larva, Pr~ the probability of survival for one season in any larval stage but not including competitive effects, and let Pp be the probability of surviving the pupal stage. I f the interactions are independent events, the mortality pressure on 12' will be exp ( - i l L 2 ' ) , the pressure per individual L2' being exp (-fl). We assume the same pressure on L I ' . The pressure on L2' will be exp ( - a L 2 ' ) , the pressure per individual being exp ( - a). Then there would be CI'A~ eggs laid early in season n, of which PECI'A~ soon become larva. A fraction P1,PLexp (-flL2'n) will survive to become adults at the beginning of the next season. A second group of eggs, C2'A'n will require two seasons to mature. A fraction, PEPL exp (ilL2"), of these will become larva and survive the non-competitive and competitive mortality pressures to become L2' larvae, L2~+1, the next season. A fraction, PLexp ( - a L 2 ' ~ + I ) P P, will survive the non-competitive and competitive factors as larvae as well as the hazards of pupation to emerge as adults in the next season, n + 2. Under these assumptions A~+I, L2~+1 and 12~ will be given by A',+~ = PEPLPpCI'A', exp (-flL2~) + P L P p L I ' , exp ( - a L 2 ~ ) ,
(1)
L 2',~+1 = PF~PLC2'A'n exp ( - flL2'n),
(2)
12'~ = PsC2'A'~;
(3)
LI~ = PzC1A'~.
We note t h a t the reduction in survival due to cannibalism is assumed to be dependent on the number of larvae near the beginning of the season. This restriction is introduced in order to make it feasible $o carry out numerical calculations. I f we set C1 = P s P r . P p C I ' and C2 = PEPLC2', and let Pr.Pp = 1, so t h a t non-competitive mortalities in larval and pupal stages are negligible, L2 = ilL2', p = a/fl, A n = flA~ and R = C1[C2, we have the following normalized
A T H R E E STAGE POPULATION MODEL W I T H CANNIBALISM
13
equations, with the parameters R, p and C2, by which to determine A n and L2 n (as well as 12, and L1,) from any initial conditions:* A ~ +1 = C2RA,~ exp ( - L2,) + L2~ exp ( - pL2n) ,
(4)
L2,+t = C2A,~ exp ( - L2,).
(5)
We note that L I , + I = RC2A,+ = C l A n and/2~+1 = C2An, where C1 and C2 are the fertility constants of the one and two generation larvae respectively. Therefore R becomes the fertility ratio of the two larval stages. The parameter p is the cannibalism ratio which is a measure of the relative cannibalism pressure on the L2 (senior) larvae to that on the 12 (junior) and L1 larval stages. Values of p > 1 = 1, or < 1 represent the cases of L2 cannibalism upon themselves to a greater, equal or lesser extent than upon the "younger" larvae where the values of p < 1 are considered to be generally the most biologically meaningful. The limits of R would be defined by the biological system to which this model may apply. The step size or time interval in the recurrence relations is one generation. I f cannibalism is not the result of independent events, the situation is more complex. We may illustrate the possible effects by considering that the pressure varies with number density in much the same way that a first order competitive inhibitor acts, i.e., of the form 1/(1 + x). In this case, the competitive effect per Z2 larva decreases as the number L2 increases, a situation which would occur if there were competition among the L2 larvae reducing the mortality pressure on the young larvae. Thus, if in (1) and (2), exp ( - x ) is replaced by the hyperbolic function 1/(1 + x), where x is L2 or p L 2 , we have a pair of equations which represent the model for hyperbolic interaction. However, it should be noted that cannibalism is likely to increase when food supply is low and hence one might, on this basis, expect the competitive pressure of each L2 larva on the young larvae to increase, rather than decrease with the number L2. The solutions to the normalized equations (4) and (5) are found to be of two types, either constant steady state or periodic depending on the values of R and p. These final population numbers are independent of the initial values used for the zeroth generation with the obvious exception of zero for both A and L2. Also the magnitude of the final population numbers for all stages increases with increasing R and decreasing p. The steady state and periodic solutions are approached in the manner of damped oscillations with convergence being * We note t h a t the same s y s t e m of equations can be obtained if we assume t h a t adults (A*_ 1) lay eggs at the end of the active season (n -- 1) and t h a t these eggs emerge as larvae at the beginning of the n e x t season (n), some (LI~) becoming adults a t the end of the same season, others (12~, L2~) becoming adults a t the end of the n e x t season, provided we replace A.* b y A~ + 1.
14
H. D. L A N D A H L
AND
B. D. H A N S E N
more rapid for the exponential pressure function t h a n for the hyperbolic function. Figure 1 illustrates the solutions of the various population stages as a function of the cannibalism ratio, p, for the case o f the exponential function with C2 = 5.0, and R = 0.4. F o r the periodic solutions, the A and L2 stages are in phase and the L1 values are out of phase with these two stages. Unless C2 is large the solutions have a period of two seasons and converge to s t e a d y state values as p increases. The hyperbolic function gives solutions which have the same form b u t the amplitudes are finite at p = 0.0. LIH I L 2 H
\ \ ,
I0
\ '
\
\ , c
8
g
LI L
:.G o o cL
--
\
\ ',, \
k
"', \ \
" ..................
L2LI o.~
L_ ~-.-~----~ o.z
-- ~"~'" 0.4
........
2 ..........
I
[
0.6
-
I o8
.........
I
I ~.o
p(= a/B) Figure 1. Final population numbers as a function of the cannibalism ratio, p. Fertility ratio R = 0.4, L2 fertility C2 = 5.0 for the exponential function. Curves diverge at p = 0.0. The L1 curves (broken) are out of phase with the A curves (solid) as well as the L2 curves (dotted). The subscripts H and L denote the high and low values respectively. This figure is representative of any particular value of the fertility ratio (R) chosen in parameter space
Figure 2 is a representation of the regions of s t e a d y state and periodic solutions for various values of the L2 larvae fertility as a function of the fertility ratio, R, and the cannibalism ratio, p. B o t h the exponential and hyperbolic functions are illustrated. The solutions are periodic for p a r a m e t e r values which fall to the left of a n y given curve. Convergence to the final stable values is fastest for p a r a m e t e r values f a r t h e s t a w a y from the curves for b o t h periodic and s t e a d y state solutions. The fact t h a t all curves originate f r o m the same
A THREE STAGE POPULATION MODEL WITH CANNIBALISM
15
point at R = 0.0 and p = 1.0 is due to the normalized form of the reeursion equations. The curve for C2 = 1.0 coincides with the coordinate axes for b o t h functions and is not considered physically meaningful. Double roots exist for exponential function whereas none are found for the hyperbolic function. I n the latter case there is an upper bound at C 2 ' = 5.0 and the complexity of curves for values of C2' > 5.0 is illustrated b y the C2' = 10.0 curve. The region of periodic solutions is m u c h smaller for the hyperbolic function t h a n the exponential function and no periodic solutions are found for R values greater t h a n 1.0 for the hyperbolic function. Figure 2 also illustrates t h a t for small values of C2 the exponential curves approach those of the hyperbolic function as would be expected from the c o m m o n n a t u r e of those two functions at small values of their arguments. I
I
I
I I I
I 0"6
1"2
I
/
0"5
/
0"4 Oa
rY
I I /cz~
I I I
0.8 !
_..o '.L
/
I I Ic2.z
1"0
.~C2' = I0
/ / /
I 0"3
/
//
Ic2:,.51 ~C2'=5
/
/
0"6
\ 0-2
0.4
\ \ \
0-1
/
I I
\ \
/ C2 = IO
l \
I
0.6
0'8
0"2
1
T
0"I
0-2
I 0'3
0.4
(p
0"2 =
0"4
1.0
,~/fl)
Figure 2. Regions of periodic and steady state solutions for various values of the L2 larvae fertility (C2) as a function of the fertility ratio (R) and the cannibalism ratio (p). Periodic solutions are to the left of any given curve where primed curves (solid) designate hyperbolic and unpriIned curves (broken) designate exponential representations. On the left is shown an enlarged portion of the figure on the right Finally, we examine the conditions t h a t would force a stable population, either periodic or in a steady state, to extinction. We find t h a t only the severe condition of setting the A and L2 populations to zero lead to extinction. F o r the exponential function, setting only the adult population to zero results in a
16
H.D.
L A N D A H L A N D B. D. H A N S E N
30 per cent decrease in the adult population with no L1 or L2 larvae in the subsequent generation. Further generations show a gradual convergence to the final stable solutions. Setting only the L2 larvae populations to zero result in a 200 per cent increase in the adults, a 300 per cent increase in the L2 larvae, and no change in the L1 larvae in the next generation. The following generation has an 80 per cent decrease in the A and L2 populations (as compared to the original stable values) with a 500 per cent increase in the L1 larvae. The third generation contains small numbers of all members of the population with a slow convergence to the stable values arising with each subsequent generation. Setting L1 equal to zero has no effect on the solutions either singly or in combination with the A or L2 populations. The hyperbolic function shows the same characteristics as above except the percentages are slightly smaller and the convergence is slower. The results shown in Figures 1 and 2 are based on the assumption t h a t competitive interaction depends upon the number of senior larvae (L2) at the beginning of the season. I f one divides the season into 2, 10 or 100 time intervals, the curve for C2 = 5 at R = 1 moves from p = 0.71 to 0.43, 0.34 and 0.33 respectively, for the exponential interaction. In the case of the hyperbolic function, the point R = 0.3, p = 0.1 on the curve for C2 = 5 moves to R = 0.3, p = 0.07 when the season is divided into 100 intervals. These examples indicate t h a t the simplifying assumption, t h a t the season can be treated as an individed unit, is not too restrictive. Furthermore, calculations show t h a t if PLPe < 1, the range in parameter space in which oscillations occur is reduced. A number of simulations were carried out to illustrate the effect of having some larvae which take as m a n y as N seasons before becoming adults. I t was found t h a t oscillations occur very readily even if all the larvae exert the same competitive effect on one another. The period had a value less than the number N, the period depending on the distribution of the fecundity values C1, C 2 , . . . , CN. A number of simulations indicate the effect of eggs being laid throughout the season reduces the likelihood of oscillations, especially in the case of the exponential pressure model where oscillations are eliminated if the value of R is large enough. For small R and p, oscillations occur even if the egg laying rate is constant throughout the season. However, if egg laying is concentrated near the early part of the season, oscillations are more likely. The results of these simulations indicate t h a t periodic variations in populalation numbers are quite likely to occur and draw attention to those factors which enhance and to those which inhibit fluctuations. Thus the factors which favor early egg laying, and thus favor survival of the population, may be counteracted by the increased tendency for the population to fluctuate too
A THREE STAGE POPULATION MODEL WITH CANNIBALISM
17
widely. As a result, one might expect to find parameters occurring in natural populations which tend to reduce the amplitude of oscillations. The authors are indebted to Drs. William hi. Cannon, Jr. and David H. Staley for their valuable comments and suggestions. This investigation was supported by hiIH research grant number GM 17539 from National Institute of General Medical Sciences. LITERATURE
Pielou, E. C. 1969. A n Introduction to Mathematical Ecology. p. 203.
New York: Wiley,
Rv.CEIV~.D 3-7.74
VOLUME37, 1975
A THREE STAGE POPULATION MODEL WITH
CANNIBALISM
9 H. D. LANDA~Jand B. D. HANSEN Department of Biochemistry and Biophysics University of California, San Francisco 94143
A simple population model consisting of one adult and two larval stages with cannibalism or competition a m o n g t h e larval stages is presented. The solutions are found to be either periodic or of a steady state n a t u r e depending on the ratios of fertility and cannibalism a m o n g the larvae. Two similar cannibalism pressure functions are compared and the conditions t h a t lead to steady or periodic solutions, or to extinction, are examined.
Studies of ecological communities require various assumptions concerning the number of species that make up the community, their interrelationships, abundances, and spatial distributions. In many studies, the definition of the investigators community is determined by convenience and intuition (Pielou, 1969) which is based, among other things, on the investigators experience with the individual species. In this note we examine the simplest prototype of a single species population (e.g., insects) that consists of adults and several larval stages where a weighted cannibalism exists among the larval stages. Even in this simple system rather complicated solutions arise that would not be intuitively obvious a priori. It is hoped that the system may give some insight into the cyclic variations in certain insect populations. The following model is a representation of a three stage population with age dependent cannibalism. For convenience, the stages are designated the adults (A'), which live for only one season, those larvae (LI') which hatch from eggs 11
12
H.D.
L A N D A H L A N D B. D. H A N S E N
laid during one season and become adults the next season, and those larvae which remain larvae for two seasons before becoming adults. The number of these latter larvae which are in their first season, junior larvae, will be denoted by/2', those in their second season, senior larvae, by L2'. We assume t h a t these senior larvae exert a cannibalistic or competitive pressure on the larvae, L I ' and 12', which are in their first season, and to a lesser extent upon themselves. For simplicity we neglect the competition among young larvae (/2', L2'), and assume t h a t there is no cannibalism on or by the adults. Suppose t h a t at the beginning of a season n, A~ adults emerge and lay CI' eggs per adult t h a t are destined to become adults in one season, and C2' eggs t h a t become adults in two seasons. The adults are assumed to survive only one season. Let P~ be the probability t h a t an egg survives and emerges as a larva, Pr~ the probability of survival for one season in any larval stage but not including competitive effects, and let Pp be the probability of surviving the pupal stage. I f the interactions are independent events, the mortality pressure on 12' will be exp ( - i l L 2 ' ) , the pressure per individual L2' being exp (-fl). We assume the same pressure on L I ' . The pressure on L2' will be exp ( - a L 2 ' ) , the pressure per individual being exp ( - a). Then there would be CI'A~ eggs laid early in season n, of which PECI'A~ soon become larva. A fraction P1,PLexp (-flL2'n) will survive to become adults at the beginning of the next season. A second group of eggs, C2'A'n will require two seasons to mature. A fraction, PEPL exp (ilL2"), of these will become larva and survive the non-competitive and competitive mortality pressures to become L2' larvae, L2~+1, the next season. A fraction, PLexp ( - a L 2 ' ~ + I ) P P, will survive the non-competitive and competitive factors as larvae as well as the hazards of pupation to emerge as adults in the next season, n + 2. Under these assumptions A~+I, L2~+1 and 12~ will be given by A',+~ = PEPLPpCI'A', exp (-flL2~) + P L P p L I ' , exp ( - a L 2 ~ ) ,
(1)
L 2',~+1 = PF~PLC2'A'n exp ( - flL2'n),
(2)
12'~ = PsC2'A'~;
(3)
LI~ = PzC1A'~.
We note t h a t the reduction in survival due to cannibalism is assumed to be dependent on the number of larvae near the beginning of the season. This restriction is introduced in order to make it feasible $o carry out numerical calculations. I f we set C1 = P s P r . P p C I ' and C2 = PEPLC2', and let Pr.Pp = 1, so t h a t non-competitive mortalities in larval and pupal stages are negligible, L2 = ilL2', p = a/fl, A n = flA~ and R = C1[C2, we have the following normalized
A T H R E E STAGE POPULATION MODEL W I T H CANNIBALISM
13
equations, with the parameters R, p and C2, by which to determine A n and L2 n (as well as 12, and L1,) from any initial conditions:* A ~ +1 = C2RA,~ exp ( - L2,) + L2~ exp ( - pL2n) ,
(4)
L2,+t = C2A,~ exp ( - L2,).
(5)
We note that L I , + I = RC2A,+ = C l A n and/2~+1 = C2An, where C1 and C2 are the fertility constants of the one and two generation larvae respectively. Therefore R becomes the fertility ratio of the two larval stages. The parameter p is the cannibalism ratio which is a measure of the relative cannibalism pressure on the L2 (senior) larvae to that on the 12 (junior) and L1 larval stages. Values of p > 1 = 1, or < 1 represent the cases of L2 cannibalism upon themselves to a greater, equal or lesser extent than upon the "younger" larvae where the values of p < 1 are considered to be generally the most biologically meaningful. The limits of R would be defined by the biological system to which this model may apply. The step size or time interval in the recurrence relations is one generation. I f cannibalism is not the result of independent events, the situation is more complex. We may illustrate the possible effects by considering that the pressure varies with number density in much the same way that a first order competitive inhibitor acts, i.e., of the form 1/(1 + x). In this case, the competitive effect per Z2 larva decreases as the number L2 increases, a situation which would occur if there were competition among the L2 larvae reducing the mortality pressure on the young larvae. Thus, if in (1) and (2), exp ( - x ) is replaced by the hyperbolic function 1/(1 + x), where x is L2 or p L 2 , we have a pair of equations which represent the model for hyperbolic interaction. However, it should be noted that cannibalism is likely to increase when food supply is low and hence one might, on this basis, expect the competitive pressure of each L2 larva on the young larvae to increase, rather than decrease with the number L2. The solutions to the normalized equations (4) and (5) are found to be of two types, either constant steady state or periodic depending on the values of R and p. These final population numbers are independent of the initial values used for the zeroth generation with the obvious exception of zero for both A and L2. Also the magnitude of the final population numbers for all stages increases with increasing R and decreasing p. The steady state and periodic solutions are approached in the manner of damped oscillations with convergence being * We note t h a t the same s y s t e m of equations can be obtained if we assume t h a t adults (A*_ 1) lay eggs at the end of the active season (n -- 1) and t h a t these eggs emerge as larvae at the beginning of the n e x t season (n), some (LI~) becoming adults a t the end of the same season, others (12~, L2~) becoming adults a t the end of the n e x t season, provided we replace A.* b y A~ + 1.
14
H. D. L A N D A H L
AND
B. D. H A N S E N
more rapid for the exponential pressure function t h a n for the hyperbolic function. Figure 1 illustrates the solutions of the various population stages as a function of the cannibalism ratio, p, for the case o f the exponential function with C2 = 5.0, and R = 0.4. F o r the periodic solutions, the A and L2 stages are in phase and the L1 values are out of phase with these two stages. Unless C2 is large the solutions have a period of two seasons and converge to s t e a d y state values as p increases. The hyperbolic function gives solutions which have the same form b u t the amplitudes are finite at p = 0.0. LIH I L 2 H
\ \ ,
I0
\ '
\
\ , c
8
g
LI L
:.G o o cL
--
\
\ ',, \
k
"', \ \
" ..................
L2LI o.~
L_ ~-.-~----~ o.z
-- ~"~'" 0.4
........
2 ..........
I
[
0.6
-
I o8
.........
I
I ~.o
p(= a/B) Figure 1. Final population numbers as a function of the cannibalism ratio, p. Fertility ratio R = 0.4, L2 fertility C2 = 5.0 for the exponential function. Curves diverge at p = 0.0. The L1 curves (broken) are out of phase with the A curves (solid) as well as the L2 curves (dotted). The subscripts H and L denote the high and low values respectively. This figure is representative of any particular value of the fertility ratio (R) chosen in parameter space
Figure 2 is a representation of the regions of s t e a d y state and periodic solutions for various values of the L2 larvae fertility as a function of the fertility ratio, R, and the cannibalism ratio, p. B o t h the exponential and hyperbolic functions are illustrated. The solutions are periodic for p a r a m e t e r values which fall to the left of a n y given curve. Convergence to the final stable values is fastest for p a r a m e t e r values f a r t h e s t a w a y from the curves for b o t h periodic and s t e a d y state solutions. The fact t h a t all curves originate f r o m the same
A THREE STAGE POPULATION MODEL WITH CANNIBALISM
15
point at R = 0.0 and p = 1.0 is due to the normalized form of the reeursion equations. The curve for C2 = 1.0 coincides with the coordinate axes for b o t h functions and is not considered physically meaningful. Double roots exist for exponential function whereas none are found for the hyperbolic function. I n the latter case there is an upper bound at C 2 ' = 5.0 and the complexity of curves for values of C2' > 5.0 is illustrated b y the C2' = 10.0 curve. The region of periodic solutions is m u c h smaller for the hyperbolic function t h a n the exponential function and no periodic solutions are found for R values greater t h a n 1.0 for the hyperbolic function. Figure 2 also illustrates t h a t for small values of C2 the exponential curves approach those of the hyperbolic function as would be expected from the c o m m o n n a t u r e of those two functions at small values of their arguments. I
I
I
I I I
I 0"6
1"2
I
/
0"5
/
0"4 Oa
rY
I I /cz~
I I I
0.8 !
_..o '.L
/
I I Ic2.z
1"0
.~C2' = I0
/ / /
I 0"3
/
//
Ic2:,.51 ~C2'=5
/
/
0"6
\ 0-2
0.4
\ \ \
0-1
/
I I
\ \
/ C2 = IO
l \
I
0.6
0'8
0"2
1
T
0"I
0-2
I 0'3
0.4
(p
0"2 =
0"4
1.0
,~/fl)
Figure 2. Regions of periodic and steady state solutions for various values of the L2 larvae fertility (C2) as a function of the fertility ratio (R) and the cannibalism ratio (p). Periodic solutions are to the left of any given curve where primed curves (solid) designate hyperbolic and unpriIned curves (broken) designate exponential representations. On the left is shown an enlarged portion of the figure on the right Finally, we examine the conditions t h a t would force a stable population, either periodic or in a steady state, to extinction. We find t h a t only the severe condition of setting the A and L2 populations to zero lead to extinction. F o r the exponential function, setting only the adult population to zero results in a
16
H.D.
L A N D A H L A N D B. D. H A N S E N
30 per cent decrease in the adult population with no L1 or L2 larvae in the subsequent generation. Further generations show a gradual convergence to the final stable solutions. Setting only the L2 larvae populations to zero result in a 200 per cent increase in the adults, a 300 per cent increase in the L2 larvae, and no change in the L1 larvae in the next generation. The following generation has an 80 per cent decrease in the A and L2 populations (as compared to the original stable values) with a 500 per cent increase in the L1 larvae. The third generation contains small numbers of all members of the population with a slow convergence to the stable values arising with each subsequent generation. Setting L1 equal to zero has no effect on the solutions either singly or in combination with the A or L2 populations. The hyperbolic function shows the same characteristics as above except the percentages are slightly smaller and the convergence is slower. The results shown in Figures 1 and 2 are based on the assumption t h a t competitive interaction depends upon the number of senior larvae (L2) at the beginning of the season. I f one divides the season into 2, 10 or 100 time intervals, the curve for C2 = 5 at R = 1 moves from p = 0.71 to 0.43, 0.34 and 0.33 respectively, for the exponential interaction. In the case of the hyperbolic function, the point R = 0.3, p = 0.1 on the curve for C2 = 5 moves to R = 0.3, p = 0.07 when the season is divided into 100 intervals. These examples indicate t h a t the simplifying assumption, t h a t the season can be treated as an individed unit, is not too restrictive. Furthermore, calculations show t h a t if PLPe < 1, the range in parameter space in which oscillations occur is reduced. A number of simulations were carried out to illustrate the effect of having some larvae which take as m a n y as N seasons before becoming adults. I t was found t h a t oscillations occur very readily even if all the larvae exert the same competitive effect on one another. The period had a value less than the number N, the period depending on the distribution of the fecundity values C1, C 2 , . . . , CN. A number of simulations indicate the effect of eggs being laid throughout the season reduces the likelihood of oscillations, especially in the case of the exponential pressure model where oscillations are eliminated if the value of R is large enough. For small R and p, oscillations occur even if the egg laying rate is constant throughout the season. However, if egg laying is concentrated near the early part of the season, oscillations are more likely. The results of these simulations indicate t h a t periodic variations in populalation numbers are quite likely to occur and draw attention to those factors which enhance and to those which inhibit fluctuations. Thus the factors which favor early egg laying, and thus favor survival of the population, may be counteracted by the increased tendency for the population to fluctuate too
A THREE STAGE POPULATION MODEL WITH CANNIBALISM
17
widely. As a result, one might expect to find parameters occurring in natural populations which tend to reduce the amplitude of oscillations. The authors are indebted to Drs. William hi. Cannon, Jr. and David H. Staley for their valuable comments and suggestions. This investigation was supported by hiIH research grant number GM 17539 from National Institute of General Medical Sciences. LITERATURE
Pielou, E. C. 1969. A n Introduction to Mathematical Ecology. p. 203.
New York: Wiley,
Rv.CEIV~.D 3-7.74
