Towards achieving maximum sustainable yield

Journal of Biological Dynamics

ISSN: 1751-3758 (Print) 1751-3766 (Online) Journal homepage: http://www.tandfonline.com/loi/tjbd20

Towards achieving maximum sustainable yield management of the bearded pig (Sus barbatus) in a logged-over planted forest C. S. Lee , C. H. Diong , B. S. Goh , R. B. Stuebing & D. J. Junau To cite this article: C. S. Lee , C. H. Diong , B. S. Goh , R. B. Stuebing & D. J. Junau (2011) Towards achieving maximum sustainable yield management of the bearded pig (Sus barbatus) in a logged-over planted forest, Journal of Biological Dynamics, 5:3, 268-276, DOI: 10.1080/17513758.2010.535912 To link to this article: http://dx.doi.org/10.1080/17513758.2010.535912

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Date: 29 April 2016, At: 05:19

Journal of Biological Dynamics Vol. 5, No. 3, May 2011, 268–276

Towards achieving maximum sustainable yield management of the bearded pig (Sus barbatus) in a logged-over planted forest C.S. Leea *, C.H. Diongb , B.S. Gohc† , R.B. Stuebingd and D.J. Junaud

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a c/o

Mathematics & Mathematics Education Academic Group, 1 Nanyang Walk, Nanyang Technological University, Singapore 637616, Singapore; b Natural Sciences and Science Education Academic Group, 1 Nanyang Walk, Nanyang Technological University, Singapore 637616, Singapore; c Mathematics Department, Nanjing University, Nanjing, Jiangsu 210093, China; d Conservation Department, Grand Perfect Sdn Bhd, No. 144, Lot 3430, Parkcity Commerce Square, 97000 Bintulu, Sarawak, Malaysia (Received 1 September 2010; final version received 23 October 2010 ) In this paper, we use the Verhulst logistic equation to model the population dynamics of a bearded pig population in a logged-over planted forest in Sarawak, Borneo Island. Using available information on the population, we generate a set of potential growth data for a pair of adult bearded pigs. Based on the data, we obtain two parameters of the logistic model, namely, the intrinsic growth rate r of the managed population and the carrying capacity K of the environment. Two harvesting strategies, constant effort versus constant quota strategy for maximum sustainable yield of the natural resource, are then considered; for system stability, the former harvesting strategy is preferred over the latter. However, as it may take some time for the benefits of implementation of an alternative harvesting policy to be realized, the cooperation of all stakeholders is paramount and necessary for the management strategy to succeed. Keywords: planted forest; bearded pig Sus barbatus; intrinsic growth rate r; carrying capacity K; harvest strategy

1.

Introduction

Harvesting strategies of natural renewable resources have attracted the interest of many researchers in the past few decades. Among others, Clark [6] and Gulland [16] reported the demise of a few large fisheries in the 1960s and 1970s due to overfishing. In view of this, it is of great concern to many that natural renewable resources should be utilized with extreme care and not over exploited or driven to extinction. In many wildlife populations, conservation through sustainable use that is premised on the thesis that wildlife populations are dynamic entities and capable of adapting to harvest is now a mainstream conservation strategy [28]. Following this awareness of and concern over suitable sustainable management strategies of renewable natural resources to *Corresponding author. Email: [email protected] Author Emails: [email protected]; [email protected]; [email protected]; [email protected] † B.S. Goh is currently affiliated with the Institute of Mathematical Sciences, University of Malaysia 50603, Kuala Lumpur, Malaysia.

ISSN 1751-3758 print/ISSN 1751-3766 online © 2011 Taylor & Francis DOI: 10.1080/17513758.2010.535912 http://www.informaworld.com



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service human needs, large amounts of research has been carried out in the past few decades (see [6,11,14,16,18,26,27] and references cited therein). In this paper, we are concerned with the management of a population of bearded pigs (Sus barbatus) in a logged-over planted forest in the east Malaysian state of Sarawak in Borneo Island. The planted forest zone (PFZ) is a planted forest project of the Sarawak Forest Department comprising an area of approximately 4900 km2 . Inside PFZ, 1960 km2 (40%) is planted (and to be planted) with monoculture of Acacia mangium for industrial wood production, 1470 km2 (30%) of patches of unlogged forests, steep lands, river buffers and wetlands protected as conservation zone, whereas the rest of the land area of secondary forests and traditional agricultural lands are set aside for use by indigenous people [25]. About 5000 families live in 254 longhouses in PFZ; 90% of the inhabitants are ethnically Iban and the remaining 10% Punan and Baketan people. Humans have hunted bearded pigs in Borneo Island for some 40,000 years [7,21]. Human predation of pigs continues in the east Malaysian Borneo state of Sarawak where the bearded pig is the principal source of animal protein and economic product in the livelihoods of indigenous people in the forests. Bearded pig meat constitutes between 67 and 93% of the total hunted wild meat biomass in Sarawak [5] and between 50 and 95% of the total weight of animals hunted elsewhere in Borneo [1,22]. Maintenance of bearded pig populations and sustainable harvest of this forest resource are important for the socio-economic well being and livelihoods of forest indwellers.

2. Analysis of harvest data collected in PFZ A three-age class model had been used in an earlier study [8] to estimate the population growth rate λ using the Lotka equation [23]. The purpose of the study is to determine whether current levels of pig harvest and subsistence hunting with a quota system by forest indwellers are sustainable. The objective of our paper is to use the data presently available to estimate the intrinsic growth rate (natural birth rate-natural death rate) r of the bearded pig population and the carrying capacity K in the whole of PFZ. With estimates of r and K, we hope to review existing harvest management strategy towards achieving maximum sustainable yield (MSY) of the population. From January 2005 to December 2006, a biodiversity inventory that incorporated field studies on spatial distribution and hunting activity of bearded pigs was conducted in PFZ [25]. Data on trap night per survey month in a camera project gave a density estimate of 2.5 pigs/km2 [12]; this gave an initial density estimate of 12,000 pigs. Eighty-seven hunters in 48 of the 254 longhouses in PFZ were interviewed about their hunting methods and hunting season preference. Based on this field sample, the total number of pig hunters in PFZ was estimated at 460. A total of 646 jaw specimens were collected from 32 longhouses. We used the data from 243 voucher specimens of bearded pig jaws that were aged by tooth eruption and attrition [8] in this paper. This sample constitutes approximately 40% of all jaw samples collected; therefore, we estimate the total annual bearded pig harvest in PFZ to approximately 6000. The proportion bearded pigs killed by hunters in our study sample in each of the three age classes are shown in Table 1. Table 1. Fraction of total bearded pig harvested by age class (n = 243). Age class

Number of pigs harvested

Fraction of total harvested

0–1 year 1–2 years >2 years

76 38 129

0.3127 0.1564 0.5309


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Bearded pigs have not been investigated scientifically. Hunters have been interviewed by several workers to gather local knowledge of the pig, and several articles on feeding habits, behaviours, migratory movements, and subsistence hunting of bearded pigs have been published on the basis of hunters’ interviews [1–3,17]. In captive populations in zoos, S. barbatus sows mature within 18 months; litter size averages six piglets per birth [4], with a range of 2–11 piglets annually [4]. In the wild, bearded pigs are known to produce two litters a year and undergo periodic irruptions and migratory movements when food is seasonally abundant, particularly during mast fruiting of the Dipterocarpaceae and Fagaceae [24]. No biological data are available on reproductive output and mortality within wild populations. Natural sources of mortality are poorly known, but natural predators such as pythons (Python reticulatu) and clouded leopards (Neofelis nebulosa) have either been documented or reported by workers from several sections of the landscape within PFZ [25]. We use life-history parameters of the Eurasian wild pig Sus scrofa in our model development for the bearded pig population [8]. Age-specific mortality rates in the juvenile age class (0–1 year), the sub-adult age class (1–2 years) and the adult age class (>2 years) were assumed to be 0.5, 0.1, and 0.3, respectively.

3.

Model for an unexploited population

We observe that the unexploited population dynamics of many species of fish and wildlife in nature may be described by the Verhulst logistic equation: dN r = N (K − N ) dt K

(1)

where N(t) denotes the population at time t of a species of animal or fish, and the parameters r and K are the intrinsic rate of increase and the carrying capacity, respectively. The solution of Equation (1) satisfying the initial condition N (0) = No is given by N (t) =

KNo No + (K − No )e−rt

(2)

In order to employ Equation (1) in the development of our harvesting models for bearded pigs in PFZ, we first estimate the parameters r and K in the following section. Based on the above review, we assume that a female bearded pig matures in 18 months, and reproduces annually with a litter size of six piglets. We classify the age structure of a population of bearded pigs in our model into four age groups, namely, juveniles (individuals 21 year), yearlings (individuals between 21 year and 1 year), sub-adult (individuals between 1 year and 1 21 years), and adults (individuals >1 21 years). Table 2 is obtained by tracking the reproduction rate of the adults and the mortality rates of the four age groups at half yearly intervals (see Appendix for the model used in the calculations). We display, in Table 2, the reproductive potential and recruitment in the bearded pig population that began with a one male and one female adult pig. Using Equation (2), N (0) = 2, and the three sets of values of (t, N ) in Table 2, namely (1, 6.3942), (3, 30.3236), and (5, 118.8701), we obtain three equations in r and K. By means of a transformation x = e−r , we may express Kin terms of x as: K =2−

1.4898 x 3 (x 2 − 0.2551)

After elimination of K, we obtain: x 5 − 0.2551x 3 − 1.0839x + 0.3390 = 0


271

Table 2. Projected growth trajectory of a bearded pig population starting with one pair of adult male and female pigs with age-specific and mortality. See text for details. Number of bearded pigs after time interval (years) Age groups

1 2

1

Adults Sub-adults Yearlings Juveniles Total

2 0 0 6 8

1.72 0 4.67 0 6.39

1

1 2

1.48 3.64 0 4.44 9.57

2

2

4.74 0 3.46 10.39 18.58

1 2

4.08 2.70 8.09 3.29 18.15

3 6.07 6.30 2.56 15.39 30.32

3

1 2

11.22 2.00 11.98 20.41 45.61

which has roots:

√ x = 1, 0.3084, −1.1442, and − 0.0821 ±


4 11.56 9.33 15.90 17.10 53.89

4

1 2

18.82 12.38 13.32 41.76 86.28

5 27.98 10.37 32.52 48.00 118.87

−3.8150 2

Among all the five roots, only x = 0.3084 is applicable, and it corresponds to r = 1.1764, and K = 319.4643. Here, it is important to emphasize that the unit of area for K is yet to be fixed. Based on Stuebing’s [24] density estimate of 4–11 bearded pigs/km2 at the ITTO Model Forest Management Area in Central Sarawak, it is reasonable to assume a density of 7.5 pigs/km2 in the conservation zone in PFZ. We further assume that the area unit for K is 25 km2 , so that the carrying capacity per km2 in the conservation zone is 12.78 animals. In the management of an exploited population, two common harvesting policies are generally considered, namely, the constant quota policy and the constant effort harvesting policy. In the constant quota policy, a fixed number of animals in a population are harvested per unit time, whereas in the constant harvesting policy, the effort applied per unit time to harvest the animals in a population is fixed. Qualitatively speaking, the two harvesting policies are quite different. Goh [13,15] pointed out that constant quota MSY policies are usually unstable, whereas the constant effort harvesting policies are usually stable. In what follows, we will obtain the harvest rates of the two different types of harvesting policies that correspond to the MSY of the exploited population in PFZ.

4. Two models for an exploited bearded pig population Harvesting an ecological system modelled by Equation (1) with a constant quota harvest rate U is described by dN r = N (K − N ) − U (3) dt K For sustainable yield of the population, r N (K − N ) − U = 0 K

(4)

Thus, r N (K − N ) (5) K U attains the maximum value when N = 1/2K, hence when the exploited animal population is at 1/2K, harvesting it with the constant rate U=

U∗ =

rK 4

(6)

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will result in MSY of the exploited population. On the other hand, if we assume that the harvest rate is proportional to the product of the amount of effort made and the population density of the animal, then harvesting the ecosystem modelled by Equation (1) with a constant effort E may be described by dN r = N (K − N ) − αEN dt K

(7)

where α, 0 < α < 1, is a constant commonly referred to as the catchability coefficient. E may be regarded as the number of licensed hunters (or hunting licenses issued). Again, for sustainable yield of the population,


r N (K − N ) − αEN = 0 K

(8)

Thus αEN =

r N (K − N ) K

(9)

We have seen that the right-hand side of Equation (9) attains the maximum value when N = 1/2K. Thus αE∗ =

1 r 2

(10)

As has been mentioned earlier, the maximum sustainable constant quota harvest rate U ∗ = 1/4rK in model (3) gives an unstable equilibrium at N ∗ = 1/2K, whereas the maximum sustainable constant effort harvest rate αEN∗ = 1/4rK in model (7) yields a globally stable equilibrium at N ∗ = 1/2K. However, Goh [14] further pointed out that if the control variables U and E are used in a feedback manner as prescribed by optimal control theory, the qualitative difference between the two models (3) and (7) disappears. In other words, the qualitative difference between the two harvest rates disappears if reliable data are collected regularly at strategic locations frequently visited by the pigs in PFZ, such as salt licks, so that an accurate estimation of r and K can be made. Consequently, for any variation of r and K, the harvest rates can be adjusted accordingly in a feedback manner. As mentioned earlier, the bearded pig population in PFZ is estimated at 12,000 if Gilman’s [12] estimate of 2–3 pigs/km2 is applied. Clearly, since bearded pigs are highly mobile, their density will vary in different sections of the land mosaic in PFZ. Stuebing’s [24] estimate of 4– 11 animals/km2 in a logged forest adjoining PFZ is a good reference to apply to density estimates of pigs in the conservation zone of PFZ. If the density estimate of 7.5 animals/km2 is used in the conservation zone and a density of 2.5 animals/km2 is applied to the PFZ, then the population density is 15,925. The impacts of logging and A. mangium plantation forestry on pig metapopulation dynamics, movements, and recruitment in this mosaic landscape are not presently known. However, bearded pigs have been found to range in the mosaic landscape, including the planted acacia forest [19]. If it is assumed that planted forests are no longer capable of providing food, shelter, or serve as breeding sanctuaries to function as ‘sources’for dispersal, then the planted forest area has little or no contribution to the estimation of K. If this assumption holds true, then the bearded pig population in PFZ is 11,025. Applying the estimate of 12.78 animals/km2 for K in Table 3, the corresponding K for PFZ is 18,787 animals. In Table 3, we display the MSY numbers for the whole of PFZ. In scenario 2, suppose the area unit for K in Table 4 is increased to 36 km2 , then the carrying capacity per km2 in the conservation zone becomes 8.87 animals. The corresponding K for PFZ is 13,039, and the MSY numbers for PFZ are shown in Table 4.


273

Table 3. MSY hunting effort, annual harvest rate, and steady-state bearded pig population number that correspond to the carrying capacity K of 12.78 animals per km2 . Hunting effort αE ∗

Annual harvest U ∗

Steady-state population at N ∗

Carrying capacity K

5525

9394

18,787

0.5882

Table 4. MSY hunting effort, annual harvest rate, and steady-state bearded pig population size that correspond to the carrying capacity K of 8.87 animals per km2 . Hunting effort αE ∗

Annual harvest U ∗

Steady-state population at N ∗

Carrying capacity K

3835

6519

13,039


0.5882

5.

Review of harvest strategies and concluding remarks

As mentioned at the outset, humans have hunted bearded pigs for food for millennia, yet populations of bearded pigs persist, are still widespread in Borneo [10], and are relatively stable over time despite intense human predation. It still is the most commonly hunted wild animal in Borneo; its reproductive potential and ability to adapt to its environment indicate that the bearded pig is resilient to human exploitation. Compared to other ungulates in the weight range of 50–80 kg, the fecundity of wild pigs, measured as average litter size, far exceeds that typically attained by other ungulate species [9,20]. Elsewhere in its range, habitat loss and forest fragmentation pose a greater threat to bearded pig populations than subsistence hunting. The logistic model with a constant effort harvesting term modelled by Equation (7), which has only one positive globally stable equilibrium (steady state), is the most suitable model for sustainable yield management of this biological resource to meet the aspirations of forest dwelling communities in PFZ as well as to improve their livelihoods. The usefulness of the model hinges on how close the estimated r and K are to the population; the estimation of bearded pig density in PFZ using standardized census protocols is important in improving model development. In Tables 3 and 4, which display the MSY of the bearded pig population described by the logistic growth model in PFZ that corresponds to K = 12.78 and 8.87 animals/km2 in the conservation zone, respectively, we would like to investigate whether the population is over- or under-exploited under different scenarios. Towards this objective, we depict in Table 5 three different estimates of the bearded pig population in PFZ. Row A corresponds to the density of 2.5 animals/km2 in both the planted forest area and the conservation zone and row B corresponds to the density of 2.5 animals/km2 in the planted forest area and 7.5 animals/km2 in the conservation zone, Table 5. Three scenarios for density estimates of bearded pig population in PFZ. Estimated population size Scenario A B C

Planted forest area (1960 km2 )

Conservation zone (1470 km2 )

Whole of PFZ (4900 km2 )

4900 4900 0

3675 11,025 11,025

8575–12,250a 15,925 11,025

a The upper density estimate includes the remaining 30% of PFZ landscape comprising human settlements, traditional agricultural lands, and secondary forests.


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whereas row C corresponds to zero density in the planted forest area and 7.5 animals/km2 in the conservation zone. Scenario C gives the most conservative density estimate in the landscape. Although the population size 11,025 is larger than the MSY steady-state population sizes of 9394 and 6519 in Tables 3 and 4, respectively, it is smaller than both the carrying capacity values, as shown in Tables 3 and 4. If the current harvest level of 6000 pigs by 460 hunters annually is accurate, then the current harvest level is larger than the MSY annual harvest rates of 5525 and 3835 in Tables 3 and 4, respectively. Our model suggests that the bearded pig population is currently over-exploited. If this harvest rate continues without adjustments, then the yield may not be sustainable. As pig population declines, the probability of an encounter between hunters and pigs decreases progressively with time, resulting in an increasingly higher effort to harvest animals or a switch to hunt other protected and less fecund wildlife for wild meat. Viewed in another way, when pig population size N is smaller than the MSY steady-state N ∗ , then with the harvest effort (E) applied remaining the same, the amount of annual harvest will always be less than the MSY of the population. Therefore, an over-exploited renewable resource must be allowed to grow back to its MSY steady-state level in the shortest possible time. All natural dynamical systems are complex, and the PFZ bearded pig population is no exception. The two parameters in the logistic model are important in studying wildlife demography and population dynamics, but the model and its predicted outcome have limitations. Our model did not consider dispersal in this open population, seasonal migration in the population, irruptions that are coupled with cyclical abundance of food resources, and the plasticity of the species life-history traits. It is hoped that the dynamics of the bearded pig population will be better understood with further model development over time when life-history data of the species become available from research. Life history data from long-term systematic study of bearded pigs in the logged-over forest should enable us to either construct more dynamic models or improve estimates of r and Kof the logistic model, and the corresponding MSY values of the population at steady state N ∗ , harvesting effort αE ∗ , annual harvest rate U ∗ , and ascertain the gap N ∗ − N . To achieve MSY of the population, this gap must be closed. Implementation of an alternative harvest management policy may take some time for the benefits to be realized. A shared community-based sustainable co-management of the bearded pig population involving stake holders will ensure that this dietary protein and common wealth of the indigenous people are accessible to and used by the communities.

References [1] J.O. Caldecott, Hunting and wildlife management in Sarawak, IUCN Tropical Forest Programme Monograph No. 7, pp. 1–168. IUCN, Gland, Switzerland, 1988. [2] J. Caldecott, Ecology of the Bearded Pig in Sarawak, in Biology of Suidae, R.H. Barrett and F. Spitz, eds., INRA, France, 1991, pp. 117–129. [3] J.O. Caldecott, R.A. Blouch, and A.A. MacDonald, The bearded pig (Sus barbatus), in Pigs, Peccaries, and Hippos: Status Survey and Conservation Plan, W.L.R. Oliver, ed., IUCN/SSC Pigs and Peccaries Specialist Group and IUCN/SSC Hippo Specialist Group, IUCN, Gland, Switzerland, 1993, pp. 136–145. [4] C. Chandran, H.C. Sim, and C.H. Diong, Captive management and reproduction in the Borden bearded pig Sus barbatus barbatus (Artiodactyla: Suidae) at the Singapore Zoological Gardens, Proc. 6th Conf. Southeast Asian Zoological Parks Association, 1996, pp. 44–47. [5] C. Chin, Pig in the pot: comments on Sus barbatus in the hunting lifestyle of the Penan in Sarawak, Asian Wild Pig News 1(1) (2003), pp. 10–11. [6] C.W. Clark, Bioeconomic Modelling and Fisheries Management, John Wiley & Sons, New York, 1985. [7] E. Cranbrook, Late quaternary turnover of mammals in Borneo: the zooarchaeological record, Biodivers. Conserv. 19(2) (2009), pp. 373–371. [8] C.H. Diong, D.J. Junau, and R.B. Stuebing, Harvest management strategy for the bearded pig Sus barbatus in a planted forest, Sarawak, Malaysia, inA. Náhlik and T. Tari, eds., Proceedings of the Seventh International Symposium on Wild Boar (Sus scrofa) and on Sub-order Suiformes, Sopron, Hungary, 2008, pp. 30–32.



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[9] S. Focardi, J.M. Gaillard, F. Ronchi, and S. Rossi, Survival of wild boars in a variable environment: unexpected life-history variation in an unusual ungulate, J. Mammal. 89(5) (2008), pp. 1113–1123. [10] C.M. Francis, A Field Guide of the South-east Asia Mammals, New Holland Publishers, London, 2008. [11] W.M. Getz and R.G. Haight, Population Harvesting; Demographic Models of Fish, Forest, and Animal Resources, Princeton University Press, Princeton, NJ, 1989. [12] B.R. Gilman, R.B. Stuebing, and N. Megum, A camera trapping inventory for mammals in a mixed use planted forest in Sarawak, Raff. Bull. Zool. 55(1) (2007), pp. 209–215. [13] B.S. Goh, Stability in a stock-recruitment model of an exploited fishery, Math. Biosci. 33(3–4) (1977), pp. 359–372. [14] B.S. Goh, Management and Analysis of Biological Populations, Elsevier, Amsterdam, 1980. [15] B.S. Goh, Optimal control of a fish resource, Malay. Sci. 5(1) (1969), pp. 65–70. [16] J.A. Gulland, The Management of Marine Fisheries, Scientechnica, Bristol, 1974. [17] P.A. Hancock, E.J. Milner-Gulland, and M.J. Keeling, An individual based model of bearded pig abundance, Eco. Model. 181(2–3) (2005), pp. 123–137. [18] S. Ling and E.J. Milner-Gulland, When does spatial structure matter in models of wildlife harvesting?, J. Appl. Ecol. 45(1) (2008), pp. 63–71. [19] W.J. McShea, C. Stewart, L. Peterson, P. Erb, R. Stuebing, and B. Gilman, The Acacia/secondary forest matrix in Sarawak Malaysia, Biol. Conserv. 142(12) (2009), pp. 3108–3119. [20] E. Morgan, Life history characteristics of placental non-volant mammals, Ecology 84(12) (2003), p. 3402, Ecological Archives E084-093. [21] W.L.R. Oliver and B. Fruzinski, People and Pigs, in Biology of Suidae, R.H. Barrett and F. Spitz, eds., Institute National de Recherche Agronomique, Castanet, France, 1991, pp. 93–116. [22] R.K. Puri, Hunting knowledge of the Penan Benalui of East Kalaimantan, Indonesia, Ph.D. diss., University of Hawaii, 1997. [23] J.R. Skalski, K.E. Ryding, and J.H. Millspaugh, Wildlife Demography: An Analysis of Sex, Age, and Count Data, Elsevier Academic Press, CA, USA, 2005. [24] R.S. Stuebing, Wildlife diversity, abundance and management in the ITTO Model Forest Management Area, Tech. Rep. submitted to the ITTO Unit, Sarawak Forest Department, Kuching, Sarawak, Malaysia, 1995, p. 96 [25] R.S. Stuebing and D.J. Junau, Managing the bearded pig (Sus barbatus) for sustainable use in the planted forest zone, Planted Forest Project, Forest Department Sarawak and Conservation Department, Grand Perfect Sdn Bhd, Sarawak, Malaysia, 2007, p. 27 [26] T.L. Vincent and J.M. Skowronski (eds.) Renewable Resource Management, Proceedings, Christchurch, New Zealand, 1980. Springer-Verlag, Berlin Heidelberg, 1981. [27] T.L. Vincent, Y. Cohen, W.J. Grantham, G.P. Kirkwood, and J.M. Skowronski (eds.) Modeling and Management of Resources Under Uncertainty, Proceedings, Honolulu, 1985. Springer-Verlag, Berlin Heidelberg, New York, 1987. [28] G.J.W. Webb, Conservation and sustainable use of wildlife – an evolving concept, Pac. Conserv. Biol. 8(1) (2002), pp. 12–26.

Appendix Let x1 (t), x2 (t), x3 (t), and x4 (t) denote the adults, sub-adults, yearlings, and juveniles of the bearded pig population at time t, respectively. Assume that the annual reproductive rate of an adult female is six piglets. Also, assume that the mortality rates of x1 , x2 , x3 , and x4 are 0.3, 0.1, 0.5, and 0.5, respectively. From dxi = −mi xi , dt

i = 1, . . . 4,

we have

xi (t) xi (t0 )

dxi = −mi xi

t

dt. t0

Hence, xi (t) = xi (t0 )e−mi (t−t0 ) . Let x1 t + 21 = x11 t + 21 + x12 t + 21 where x11 (t + 21 ) represents those adults that reproduce during (t, t + 21 ] and x12 (t + 21 ) represents those adults that do not reproduce in the same period. Then, the projected population growth as displayed in Table 2 is predicted by the

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following model: x1 t x11 t x12 t x2 t x3 t x4 t

+

1 2

+

1 2

+

1 2

+

1 2

+

1 2

+

2

1


The set of initial values is (x1 ( 21 ), x2 ( 21 ), x3 ( 21 ), x4 ( 21 )) = (2, 0, 0, 6), x11 (0) = x11 ( 21 ), x12 (0) = x12 ( 21 ) = 0.

= x11 t + 21 + x12 t + 21 = x12 (t)e−m1 ×0.5 + x2 (t)e−m2 ×0.5 . = x11 (t)e−m1 ×0.5 = x3 (t)e−m3 ×0.5 , = x4 (t)e−m4 ×0.5 , = 21 × 6 × x11 t + 21