Ecological Modelling 34 (1986) 115-132 115 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

A G E N E R A L M O D E L O F P O P U L A T I O N G R O W T H IN T H E H Y R A X P R O C A VIA C A P E N S I S

N. FAIRALL, P.J. VERMEULEN * and M. VAN DER MERWE

Mammal Research Institute, University of Pretoria, Pretoria 0002 (South Africa) • Department of Applied Mathematics, University of Pretoria, Pretoria 0002 (South Africa)

(Accepted 23 September 1985)

ABSTRACT

Fairall, N., Vermeulen, P.J. and Van der Merwe, M., 1986. A general model of population growth in the hyrax Procavia capensis. Ecol. Modelling, 34: 115-132.

A study of an expanding hyrax population when the parameters of such a sub-population provided demographic parameters which were used to construct a model of population growth. The model simulates the observed growth of the population and its stabilization once an increasing death-rate factor, dependent on population size, was incorporated. The hyrax is dependent on shelter for protection from predation, and a biological evaluation of the model suggests that shelter is the limiting factor in population growth and control of predation effects. The study population is not limited by food, but comparisons with populations from semi-arid environments, where food shortage necessitates movement away from shelter thereby increasing predator susceptibility, suggest that the model has general application to population growth in this species.

INTRODUCTION

Studies of popu l a t i on growth or decl ine and the dynamics of these processes are f undamen ta l to the unde r s t and ing of the survival of species t h rough t ime and cur ren t ly to the m a n a g e m e n t of species popu la t ions to ensure their long- te rm survival.

Each popu l a t i on within a species range is subjected to specific environ- men ta l factors tha t inf luence their age-specific fecundi ty and mor ta l i ty schedules. I f enough da ta are available models for these popu la t ions can be cons t ruc ted which descr ibe the dynamics of the specific group. Unde r ly ing all these sub-popula t ions , however , there should be a general model tha t descr ibes the total popula t ion .

Gene ra l l y speaking this model should represent the ideal s i tuat ion and should be capable of descr ibing any sub-popu la t ion when the pa rame te r s of such a sub-popu la t ion are entered.

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Modelling the dynamics of population growth has many problems (Caughley, 1977; Elseth and Baumgartner, 1981) and describing biology in mathematical terms inevitably leads to either a simplification of the system being studied, or such a mass of detail that the main concepts become hidden in the complexity of the model. Finding a balance between these extremes without making invalid assumptions is the real art of modelling biological populations (Elseth and Baumgartner, 1981).

A study of the hyrax Procavia capensis as an agricultural problem animal in the semi-arid, small-stock farming areas of South Africa necessitates the formulation of a model of population processes for management purposes. An expanding natural population in the Tsitsikama National Park has provided, suitable parameters to model the ideal population. The results from a study of this population form the basis of this paper.

BIOLOGICAL DATA

An expanding population of hyrax in the Tsitsikama National Park has been studied since 1979. Population estimates for the juvenile ( < 1 year) and mature ( > 1 year) segments of the population are made four times a year by means of the Robson-Whit lock technique (Fairall and Crawford, 1983). Counts are made during November/December, February/March, June and September. The hyrax is a strictly seasonal breeder, the young being born late October/early November. The first counts thus provide an estimate of the annual births while the September counts provide an estimate of adult mortality for the year as they are conducted shortly before the next birth pulse.

The age of hyrax can be accurately determined up to 4 years by using tooth replacement and wear criteria developed by Fairall (1980). It was

• decided, therefore, to derive the model from specific population parameters calculated for yearly groups of 0-1 year, 1-2 years, 2-3 years, 3-4 years and > 4 years.

Road casualties have until now been the largest single mortality factor and all animals (n = 80) found dead had their reproductive status and age determined. These data were supplemented with information on age-specific fecundity from ten pregnant females shot at the beginning of the study. Combining this data it was thus possible to calculate age-specific fecundity for the female segment of the population.

At the start of the study a group of 22 animals were captured for translocation and a further 26 were captured for a mark/recapture estimate of the population. The ages of these animals were determined and this information together with the annual estimates of population size, were used to calculate the population age structure.

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TABLE 1

Chemical analysis of hyrax dung from Tsitsikama illustrating the consistently high quality of the diet

Date Crude Soluble Lignin (%) protein (%) cell wall (%)

Dec. 1981 22.8 58.0 43.3 Febr. 1982 21.0 59.5 33.5 June 1982 18.6 56.5 43.0 Sept. 1982 20.6 45.7 33.1

Food is not considered to be a limiting factor in the study population. The area is covered by extensive lawns and the natural vegetation in this area is abundant and extremely lush. The quality of the diet was evaluated by faecal analysis for protein and fibre fractions (Erasmus, et al., 1978) and the results in Table 1 indicate a high quality diet throughout the year.

The study population exists within a discrete isolated area and migration is considered to have minimal, if any, effect on population growth (Fairall and Crawford, 1983). The occupied area is a tourist rest camp which had been extended just prior to the present study. The population expanded during the study until all suitable shelter was occupied. Hyrax are mainly diurnal and the human activity within the rest camp area during the day would tend to deter major predators, such as the caracal Felis caracal and large eagles, from entering the area at this time. At night the hyrax are in or very near their shelters in the rocks out of reach of predators.

The population size appears to have been in the process of stabilising since 1982 (Table 2) and it is postulated that this is due to shelter becoming limiting. Some hyrax would hence be forced to disperse into sub-optimal shelter and hence probably become more prone to predation. The increasing population would also attract predators to the area and increased sighting of caracals in the rest camp during 1983 by park staff seems to confirm this assumption.

The limiting factor for this population is thus assumed to be shelter, whilst control is effected through predation. These interacting factors cannot be quantified individually at present and therefore cannot be used in the proposed simulation model. However, the mean of the last three September counts (147.3) can be taken to represent the carrying-capacity of the area and this would incorporate these two factors. A value of 150 hyrax was taken to be the carrying capacity limit.

Another factor that should be taken into consideration in a general model is the incidence of catastrophic mortality affecting the population. This is known to occur within the distribution range of the hyrax during exceptional

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TABLE 2

Robson-Whitlock estimates of the Tsitsikama hyrax population with 9570 confidence limits and the calculated adult mortality

Date Population estimate Adult

Mature Immature Total 4- limits of mortality estimate

Sept. 1979 53 39 92 75-154 Jan. 1980 64 65 129 105-200 Oct. 1980 60 61 121 110-300 34.8% Jan. 1981 89 94 183 175-327 Sept. 1981 65 73 132 125-239 46.3% Dec./Jan. 1982 101 120 221 204-275 Sept. 1982 60 75 135 114-152 50.5% Dec./Jan. 1983 115 105 220 212-288 Sept. 1983 91 84 175 123-177 33.6% Dec. 1983 126 119 245 231-326

d rough t s and p r o b a b l y also a result of highly infec t ious diseases. In the s tudy a rea it could take the f o r m of heavy seas dur ing a spr ingt ide. The shel ter occup ied b y a pa r t of this p o p u l a t i o n is s i tua ted in areas tha t have been tota l ly covered b y wa te r in the past . W a t e r levels of this m a g n i t u d e are t hough t to occur at a f r equency of one in 20 years and would r e m o v e 50% of

the popu la t ion , a f fec t ing all ages and sexes equally.

The resul ts of the p o p u l a t i o n es t imates f r o m S e p t e m b e r 1979 to D e c e m - be r 1983 are shown in Tab l e 2. A rough es t ima te of the to ta l p o p u l a t i o n in 1975 is a b o u t 30 animals . Th is gives an obse rved ra te of increase f r o m 1975 to 1979 of 29% and f r o m 1979 to 1983 of 16% ind ica t ing tha t the p o p u l a t i o n m a y be stabil ising. Age-speci f ic f ecund i ty is shown in T a b l e 3 and it can be seen to increase wi th age up to 4 years.

TABLE 3

Mean age-specific fecundity of the Tsitsikama hyrax with standard deviations (SD)

Age Sample size Mean number of young n per female + SD

1 year 10 0.0 + 0.0 2 years 8 2.0 + 0.0 3 years 12 2.4 + 0.5 4 years 6 3.2 + 0.7 4 + years 2 4.0 + 0.0

119

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AGE GROUPS

Fig. 1. Age structure of the population: (a) initial; (b) during exponential growth; (c) after population stabilisation.

Mortality in the mature segment of the population fluctuates around a mean of 41.5% (see Table 2). In the juvenile segment the mortality is unknown and it is anticipated that the model will provide an estimate for this parameter.

The age structure of the population is detailed in Fig. l(a). The increase in the last age group is due to the lumping of all the age classes above 4 years.

TABLE IV

Age-specific sex ratio of the Tsitsikama hyrax

Age groups n Female Male

0-1 14 85% 15% 1-2 15 80% 20% 2-3 15 70% 30% 3-4 14 58% 42% > 4 0 58% 42%

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Age-specific sex ratios are indicated in Table 4. The low values for males in the lower age groups are ascribed to mortality caused by the highly agressive dominant males or to these young males being driven to peripheral areas where they are more prone to predation.

MODEL ASSUMPTIONS

The structure of the model describing the dynamics of the population is based on information gathered during the field study. This information will be reflected in an aggregate way through parameter estimates. The model permits parameter adjustments as more information becomes available.

The following assumptions were made in deriving the model: (1) The population is devided in five age groups as in Fig. l(a). (2) Five constant parameters, B1, B2, etc., are introduced to describe the

birth rate of each age group using the fecundity values in Table 3. (3) The death rate of each age group is assumed to be a combined effect

of natural death, predation and road mortality. (4) As indicated previously, censuses were conducted four times a year. As

there is no information on the mortality of the population--especially the immature segment of the population--during the time period September to December, it was decided to use a "lumped" value, evenly distributed over 1 year. For this calculation the immature population in September formed part of the total mature population. The mortality was then taken as the decline in the mature population during the following year. An average mortality for the mature population of 40% was calculated from the data in Table 2 and was used for all the mature age groups except for the last age group where mortality of 100% was assumed.

(5) The death rate of the immature hyrax in the 2 months following the birth pulse during October/November is not known and is probably inde- terminable due to the breeding season occurring over approximately 3 weeks. Determining the average annual mortality for the young is one of the objectives of the modelling exercise. This mortality value, assigned to the parameter MIB, can be estimated by fitting the simulated population figures to the actual data given in Table 2.

(6) Biological data (see Table 4) indicate that there is a 5.6 : 1 female t o male ratio in the 0-1 age group. This ratio changes to about 2.3:1 in the 2-3 year age group, and to 1.4 : 1 in the group > 4 years. This information is represented by the parameters R1, R2, etc.

(7) Maturing from one age group to another takes place at the end of each annual cycle and is calculated as the difference between the present popula- tion and the deaths that occurred in an age group during that year.

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THE SIMULATION MODEL

The method used in the construction of this model is known as System Dynamics (Forrester, 1968). The method describes the real system in terms of levels, in this case numbers in each age group. The levels are influenced by rates which add to or withdraw from a level. The rates are determined by the birth and death rates in each group. As stated in the assumptions, the updating of the levels takes place at the end of a time interval, presently taken as 1 year over a simulation period of 35 years.

Figure 2 illustrates the flows into and out of the levels describing each age group.

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P3 1

Fig. 2. Flow diagram of the model. The levels are depicted by rectangles (v-n) and the death and aging rates of each age group by valves (txz3). The births in each age group are described by auxiliary variables, represented in Fig. 2 by circles ( 0 ) which contribute to a single birth rate.

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The model can also be described mathematically by a system of ordinary differential equations as follows:

dY : 'x , h-7 =J(- T) (1)

where Y is a vector describing the population in the five age groups; f is a vector function, dependent on the different populations of the age groups and parameter values describing birth rates, death rates and maturation rates; ~ is the parameter vector describing birth rate, death rate and maturation rate; T is a non-linear table function vector; and Y(0) denotes the initial values of ~.

As an example the first scalar equation of equation (1) will be discussed. The other equations can be explained in a similar manner. The complete set of equations is given in Appendix A.

The first component of the vector Y describes the population of the first age group. The function on the right-hand side, corresponding to the rate of change of the population in the first age group is then given by the right-hand side of equation (2).

dP1 - B - D 1 B - MAT1 (2)

dt

The rate of change of the population in the first age group is, in the first instance, dependent on the inflow of young from the other groups and is indicated by B (see Fig. 2). The depletion due to mortality should be subtracted from this inflow. This is assumed to be M I B % of the population less than 1 year old. The total mortality in the first age group (DIB) is thus given by P1 * M I B / I O 0 . The remainder of the population P1 which survives the first year matures to the second age group and should also be subtracted from the births. The numbers transferred to the second group is then given by MAT1 which is equal to (P1 - D1B).

The constituent equations of the model are extremely versatile due to the large number of parameters that may be altered when more field data becomes available.

SIMULATION RESULTS

Table 5 summarises the initial values of the population in 1975 and the constant parameter values which were used in the model. A standard projection was obtained for the hyrax population by numerically solving the system of differential equations (1). A simple forward difference scheme (the Euler method) was used to approximate the derivatives on the left-hand side

T A B L E 5

A s s u m e d initial p o p u l a t i o n and p a r a m e t e r values for the s imula t ion p resen ted

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Age Init ial Fecund i ty Mor ta l i ty F rac t ion

p o p u l a t i o n p a r a m e t e r s p a r a m e t e r s female in

p o p u l a t i o n

0 - 1 P I I = 1 5 B1 = 0.0 M 1 B = 0.4 R1 = 0.85

1 - 2 P 2 I = 5 B2 = 2.0 M 2 B = 0.4 R 2 = 0.80

2 - 3 P 3 I = 5 B3 = 2.4 M 3 B = 0.4 R3 = 0.70

3 - 4 P 4 I = 2 B4 = 3.2 M 4 B = 0.4 R4 = 0.58

> 4 P 5 I = 3 B5 = 4.0 M 5 B = 1 . 0 R5 = 0.58

of the equation. In the standard projection no limiting carrying capacity for the population was assumed.

Figure 3 shows the projection of the total hyrax population for a 12-year period starting with an initial population of 30 in 1975. It can be seen from the figure that an exponential growth is forecast as expected with an average annual growth rate of 28%. The age distribution of the population after the 35-year simulation period is shown in Fig. lb. As expected it approximates the observed distribution of Fig. l a which is the stable age distribution during the exponential growth phase.

C A R R Y I N G C A P A C I T Y L I M I T A T I O N

Figure 3 indicates that unrestricted population growth occurs when no limiting factors are introduced in the model. The biological data as given in Table 2 seem to indicate that the population has been stabilising since 1982. Taking the habitat, as well as the observed population trend, into account, it is estimated that the carrying capacity for the hyrax in the park is approxi- mately 150.

This restriction was taken into account by including a non-linear table function viz. Death-Rate-from-Crowding-Multiplier (DRCM) in the model. The assumption is made that a higher motality will result in all age groups when the total population exceeds 150. The DRCM is an increasing function with rising population (Fig. 4). A distinction was made between the values of the table function for the first age group and the other groups. Figure 4 depicts the table functions that were used to introduce the population limiting factor.

It can be argued that the young are more vulnerable to predators than the mature population when the assumed carrying capacity is exceeded. No additional mortality effect is therefore taken into account when the popula- tion is less than the assumed carrying capacity of 150. The solid line in Fig.

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Fig. 5. Predicted populat ion growth with increased death rate stablizing numbers after 1982.

4 represents the mortality effects when the limiting capacity of the park is exceeded. Figure 5 shows the trajectory of the hydrax population when the suggested limitation is introduced.

Figure 5 also indicates that the hyrax population will stabilise at about 240 animals just after the birth pulse but will decline to reach the approxi- mate carrying capacity just before the next birth pulse, shown by the dotted line. The decline is ascribed to mortality generated by the DRCM function.

At the equilibrium level, the model indicates that juvenile mortality will be about 62% per year and the predicted stable age distribution is shown in Fig. lc. This is different to that which was obtained during the exponential growth phase (Fig. lb) due to the increase in overall mortality when the carrying capacity is exceeded.

C A T A S T R O P H I C M O R T A L I T Y

The model is able to simulate the consequences of highly infectious diseases, exceptional droughts or floods on the population. Catastrophies are not predictable and a Monte-Carlo technique may be used to randomly evoke a catastrophy during the simulation period. It was felt, however, that the reaction of the model to a 50% reduction in the population at a specified time would be sufficient to study the consequences of such a disaster. Figure 6 simulates the population trend assuming a reduction of 50% in 1990. The assumption of equal mortality in all age classes was maintained for this simulation run.

It is clear from Fig. 6 that the population again enters an exponential growth period after the simulated population collapse. This is to be expected as the event does not change any of the factors causing population growth in

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Fig. 6. Predicted population growth with catastrophic mortality during 1990.

the unlimited environment. It is important to note that a relatively short period is then needed to regain the equilibrium position.

SENSITIVITY ANALYSIS

It is necessary to perform sensitivity tests on the values of the parameters to determine the effect of parameter changes on the results of the model. The identification of sensitive parameters will indicate the areas where more detailed information and data are needed to estimate reliable and undis- puted parameter values.

A sensitivity analysis of the present model was performed by using a well-known approach to parameter sensitivity in the engineering literature (Tomovic and Vukobratovic, 1972). The algorithm was employed to calcu- late the time histories of changes in the state variables due to small but constant changes in the values of the 15 parameters and two non-linear table functions.

The five age groups are taken as the state variables of the dynamic system. The 15 parameters are the five fecundity parameters, B1 to B5, the five age-specific sex ratios, R1 to R5, and the five mortality parameters, M I B to M5B. The nominal values for these parameters are given in Table 5. The two table functions are DRCM1 and DRCM2 as depicted in Fig. 4.

The parameter sensitivity functions are obtained by partial differentiation of equation (1) with respect to the 15 components of the parameter vector ~. This procedure results in the following system of 75 simultaneous linear differential equations:

(ax, I 1 § = = - - - J r - - -

~ j dt ] ~-~ ()qJ ] m=l O-'~-'-m aqj aqj

i = 1 , 2 . . . . ,5 , j ----I ,2, . . . , 1 5 (3)

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The first equality follows from the fact that ~ (or f-) is differentiable and continuous.

Denot ing the partial derivatives Oxi/Oqj (the parameter sensitivity func- tions) by U(I, J), the functions OfJOx m by D(I, M) and the non-homoge- neous term Of JOqj by E(I, J), the following set of linear differential equations with time-varying coefficients describing the rate of change of the sensitivity functions are obtained:

15

0 ( I , J ) = Y'. D(I, M)U(M, J) + E(I, J) M = I

1 = 1 , 2 , . . . , 5, J = l , 2 , . . . , 15 (4)

The initial condit ions are U(I, J) (at t = 1975)= 0; V I, J. Normalized sensitivity functions can be defined as follows:

N(I, J)= U(I, J) / (xJqj) V I, J (5)

Similarly in the case of the table functions, the differential equations for the sensitivity functions are:

15

l ) ( I , J ) = Y'~ D(I, M)U(M, J) + G(I, J) M = I

I = 1 , 2 . . . . . 5, J = 1 , 2 , (6)

where U(I, J) now denotes the partial derivative ~x,/OTj and G(I, J) the derivative 0f,/0Tj. The initial condit ions are again U(I, J) (at t = 1975) = 0; VI, J.

The normalizat ion procedure is chosen as:

N(I, J ) = V(I, J) / (xJTj) VI, J (7)

The expressions for the partial derivatives D(I, M), E(I, J) and G(I, J ) were analytically determined from the functional representation (1) of the original model. When solving the derived differential equations (4) and (6), the values of the coefficients of D, E and G are determined by the unper turbed solution of equation (1).

The normalized parameter sensitivity functions N(I, J) in equation (5) provide an approximat ion of the percentage change in the state variable x i for a 1% increase in the constant parameter q j, as a function of time. This approximat ion improves as the per turbat ion of the parameter is reduced. The normalized sensitivity functions describe the limiting case as the per- turbat ion tends to zero.

In the case of the table function sensitivity the function N(I, J), as defined in equat ion (7), provides an approximat ion to the percentage change in the state variable x, for a constant 1% increase of every output value in

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TABLE 6

Maxima and minima of sensitivity functions for parameters and table functions (times when extremes occurred are given in parentheses)

Parameters State variables

P1 P2 P3 P4 P5

B1 0.0 0.0 0.0 ,0.0 0.0 B2 1.382(1980) * 0.975(1979) 0.719(1979) -0.792(1983) -1.187(1984) * B3 0.724(1980) 0.491(1979) 0.372(1979) 0.382(1983) - 0.537(1983) B4 0.353(1980) 0.249(1979) 0.221(1979) - 0.189(1983) - 0.270(1983) B5 0.282(1980) 0.264(1979) 0.150(1979) 0.233(1979) - 0.192(1983) R1 0.0 0.0 0.0 0.0 0.0 R2 1.382(1980) * 0.975(1979) 0.719(1979) -0.792(1983) - 1.187(1984) * R 3 0.724(1980) 0.491(1979) 0.372(1979) - 0.382(1983) - 0.537(1983) R4 0.363(1980) 0.249(1979) 0.221(1979) -0.189(1983) -0.270(1983) R5 0.283(1980) 0.264(1979) 0.150(1979) 0.233(1979) -0.192(1983) M1B - 1.398(1980) * - 1.474(1978) * - 1.290(1979) * - 0.983(1979) 1.362(1990) * M2B -0.633(1980) -0.328(1979) -0.936(1979) -0.797(1979) -0.618(1980) M3B -0.302(1980) -0.179(1979) -0.158(1979) -0.800(1995) -0.693(1979) M4B -0.123(1980) -0.076(1979) -0.067(1979) 0.099(1983) -0.934(2003) M5B -0.220(1980) -0.103(1979) -0.193(1979) 0.141(1982) -0.967(1983) Table functions DRCM1 -0.328(1981) -0.458(1984) -0.129(1981) 0.358(1984) 0.755(1991) DRCM2 -0.215(1982) -0.071(2003) -0.268(1999) -0.607(1990) -0.944(2002)

the table function ~ . Again the approximation improves as this perturba- tion becomes smaller.

Table 6 shows the maximum (or minimum) of the most sensitive normal- ized parameter and table function sensitivity functions as measured between the years 1975 and 2010. A value greater than unity indicates a change in the state variable which is larger than the change in the parameter or table function value that caused it. The times when the extremes occurred are indicated in parentheses.

It is evident from Table 6 that the model is relatively insensitive to small changes in the values of the constant parameters and the table functions. The largest change occurred in the population of the second age group, P2, when the mortality of the previous age group, M1B, was increased by a small amount. Only eight sensitivity values were found to be marginally greater than unity. These values are indicated by a superscript star in Table 6. None of the table function sensitivity functions reached values of more than one, which is an indication of their insensitivity in the present model structure.

The values of all the sensitivity functions which were obtained by means of the described mathematical treatment are in close agreement with experi-

129

mental sensitivity calculations; that is calculations whereby the percentage change in the value of a state variable is computed by a physical alteration of 1% in the parameter and table function values.

DISCUSSION

As can be seen from the results presented, it was possible to simulate the data obtained in the field study with an accuracy well within the limits of experimental error expected in obtaining these data.

The accuracy of the results on which the model is based will affect both the generality of the model and more specific interpretation with respect to the population studied. The accuracy of the census method used has been discussed and depends on the number of counts conducted as well as how closely the two largest counts correspond (Fairall and Crawford, 1983) but it must be emphasised that no census gives absolute results. Fecundity, as determined here, is considered to be as accurate as can be obtained. Unpublished data from other areas gives essentially the same result. Pub- lished information available only gives general values with no age-specific detail (Millar, 1972; Van der Merwe and Skinner, 1982). Age-specific sex ratios are only available from one study on a different species (Hoeck, 1982) but indicates the same trend as was found here. The population age structure can affect demographic analysis in numerous ways and is likely to be the major difference between different study populations. While the data used here might affect the specific results, it is unlikely to affect the general pattern significantly. The mortality factors determined for the mature popu- lation would reflect the accuracy of the census method, while the results describing the immature segment remain speculative and an accurate field determination is considered an intractable problem. This is confirmed by Hoeck (1982), whose Serengeti study is the only other of comparable intensity.

The model derived here fits the classic pattern of a population growing initially in an unlimiting environment, then reaching a restraint as it approaches the carrying capacity after which the growth rate declines and the population stabilises (Elseth and Baumgartner, 1981). In the general model of population growth this restraint is typically a shortage of nutrients that affects both birth and death processes. In the present case, we propose that the limiting factor is suitable shelter and that this affects only mortality through an increased susceptibility to predation.

Caughley (1977) defines a general model as one that concentrates on the dominant features of the system dispensing with the details of accurate parameters. Although the present model is based on reasonably accurate parameter values, we believe that the results are of general importance to the study and management of hyrax populations.

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The results indicate an increase in the young segment of the population that contributes the major input to growth under the present ideal situation. Any factor causing a change in fecundity under these circumstances would tend to have dramatic results on population growth if predation remains the same.

Climate, through its influences on the food supply, is known to affect fecundity in the hyrax (Millar, 1972). The hyrax is, however, very conserva- tive in its use of energy (Fairall and McNairn, 1981) and, consequently, this factor would not be sensitive in population control. It would rather act in a catastrophic fashion during severe droughts, causing drastic declines in population level such as that simulated in Fig. 6. By the use of appropriate parameter values changes in fecundity can be accommodated in the model.

Climate and its effect on the food supply does have an effect on the interaction between predat ion and shelter that is proposed as the main regulating factor here. In the semi-arid areas of the Karoo in South Africa the vegetation is not as dense or of as high a quality as in the present study. Here a build-up of numbers would create an over-use of vegetation close to shelter causing the animals to move further from shelter to feed, thus becoming more prone to predation. This effect is implicit in the general form. of the present model but would justify more detailed definition; the modelling technique used can accommodate this.

One reason for attempting the present simulation was to define areas where research can validly be intensified to provide a better understanding of the processes underlying the population dynamics of the hyrax. The sensitivity analysis of the model points to mortality as such an area. This concurs with the present biological understanding of the hyrax numbers problems (Fairall, 1981).

The model is, at present, deterministic and predictions based on it should be conservative and not extend too far beyond the period of data acquisi- tion. However, we believe that the validity of the model can be evaluated in three ways. In the first instance, continued monitoring of the population will provide data that can be compared with the projected stability that is indicated by the model. Secondly, the present age structure of the popula- tion can be assessed and should conform to that predicted and thirdly an experimental 50% cull of the population can be instituted and the reaction of the population compared to that predicted.

ACKNOWLEDGEMENTS

The biological part of this study has been financially supported by the Department of Agriculture, the National Parks Board and the Mammal Research Institute of the University of Pretoria. This support is gratefully

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acknowledged, for without it the work could not be done. Various people have assisted in counting and collecting material, Drs. R. Crawford and N. Hanekom having made a major contribution in this regard.

REFERENCES

Caughley, G., 1977. Analysis of Vertebrate Populations. Wiley, Chichester. Elseth, G.D. and Baumgartner, K.D., 1981. Population Biology. Van Nostrand, New York,

232 pp. Erasmus, T., Penzhorn, B.L. and Fairall, N., 1978. Chemical composition of faeces as an

index of veld quality. S. Afr. J. Wildl. Res., 8: 19-24. Falrall, N., 1980. Growth and age determination in the hyrax. S. Afr. J. Zool., 15: 16-21. Fairall, N., 1981. An integrated scheme for the control of the hyrax P r o c a v i a capens i s in

South Africa. In: J.A. Chapman and D. Pursely (Editors), Proc. World Furbearer Conf. University Maryland, College Park, MD, pp. 1875-1884.

Falrall, N. and Crawford, R.J.M., 1983. Application of the Robson-Whitlock technique to estimate hyrax numbers. S. Afr. J. Wildl. Res., 13: 25-26.

Fairall, N. and McNalrn, I.S., 1981. A metabolic energy conserving mechanism in the hyrax P r o c a v i a capens i s . In: J.A. Chapman and D. Pursely (Editors), Proc. World Furbearer Conf. University Maryland, College Park, MD, pp. 343-354.

Forrester, J.W., 1968. Principles of Systems. Wright-Allen Press, Cambridge, MA, 210 pp. Hoeck, H.N., 1982. Population dynamics, dispersal and genetic isolation in two species of

hyrax on habitat islands in the Serengeti. Z. Tierpsychol., 59: 177-210. Millar, R.P., 1972. Reproduction in the rock hyrax P r o c a v i a capens i s Zool. Afr., 6: 243-261. Tomovic, R. and Vukobratovic, R., 1972. General Sensitivity Theory. Elsevier, New York,

258 pp. Van Der Merwe, M. and Skinner, J.D., 1982. Annual reproductive pattern in the dassie

P r o c a v i a capens i s . S. Afr. J. Zool., 1,7: 130-135.

APPENDIX A

List of model equations and factors used in the DR CM functions

dP1 - - = B - D I B - MAT1

dt B = BI* PI* R I + B2* P2* R 2 + B3* P3* R 3 + B4* P4* R 4 + B5* P5* R5 D I B = P1 * ( M 1 B + DRCM1)

MAT1 = P1 - D1 B d P 2

= MAT1 - D 2 B - MAT2 dt

D 2 B = P 2 . ( M 2 B + DRCM2)

M A T 2 = P 2 - D 2 B

dP3 = MAT2 - D 3 B - MAT3

dt

D 3 B = P 3 * ( M 3 B + DRCM2)

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MAT3 = P 3 - D 3 B

d P 4 = MAT3 - D 4 B - MAT4

dt

D 4 B = P 4 * ( M 4 B + D R C M 2 )

MAT4 = P 4 - D 4 B

d P 5 = M A T 4 - D 5 B

dt

D 5 B = P 5 * M 5 B

F U N C T I O N D R C M I ( P O P ) = (0, 0), (150, 0), (200, 0.12), (250, 0.25), (300, 0.4)

F U N C T I O N DRCM2(POP) = (0, 0), (150, 0), (200, 0.10), (250, 0.25), (300, 0.5)

POP = P I + P 2 + P 3 + P 4 + P5