e c o l o g i c a l m o d e l l i n g 2 1 1 ( 2 0 0 8 ) 142–152
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A three species competition model as a decision support tool Temple H. Fay, Johanna C. Greeff ∗ Department of Mathematical Technology, Tshwane University of Technology, P/Bag X680, Pretoria 0001, South Africa
a r t i c l e
i n f o
a b s t r a c t
Article history:
An overcrowding problem of nyala, and lately also of impala in the Ndumo Game Reserve,
Received 26 March 2007
South Africa, has been detrimental to other species and vegetation structures over a period
Received in revised form
of two decades. In the present study a deterministic model for three competing species
14 August 2007
(where two species tend to be overpopulated while the third faces probable localized extinc-
Accepted 30 August 2007
tion) is constructed, while future trends coupled with their coexistence are projected.
Published on line 15 October 2007
On a mathematical basis, we seek reasons for the failure of the cropping strategies implemented by management over the last two decades, and suggest alternative,
Keywords:
scientifical-based approaches to the calculation of cropping quotas to ensure the future
Logistic growth
coexistence of all three species. A system of three first-order nonlinear differential equa-
Inter- and intraspecies competition
tions is used, with parameter values based on field data and opinions of specialist ecologists.
Cropping
The effect of various cropping strategies, and the introduction of a fourth species (man as a
Eigenvalue analysis
predator) to the system, is investigated mathematically. This model was implemented as a
Overpopulation
harvesting strategy in 2002, and is being continuously tested. Final assessment can only be
Localized extinction
done over a 10–15-year period, but so far indications are promising. © 2007 Elsevier B.V. All rights reserved.
Stability
1.
Introduction
Over the past two decades, cropping of nyala (Tragelaphus angasii) in the Ndumo Game Reserve was necessitated on a continuous basis: overcrowding of the species proved to be detrimental to the species itself and other competing species. Vegetation structure also changed drastically, aggravating the problem. Over-utilization of vegetation has increased selection pressure on indigenous species, allowing invasive alien plants to gain a foothold in the reserve and further reducing carrying capacity for the three species in question. Cropping quotas were not necessarily scientifically based, and were mostly determined intuitively by managers over the years. Furthermore, the three existing Park Boards in Kwa-Zulu Natal were merged in 1997, resulting in a clash of opinions on the cropping strategies and game count methods, interrupted actions and management indecision. The qual-
∗
ity of the data collected at Ndumo over the past two decades is thus questionable. Inaccuracies can also be attributed to extremely dense vegetation in large parts of the reserve. Population estimates for the impala in particular seem to be contradictory, reporting on extreme fluctuations between consecutive years. Fitting a model to questionable data has its problems: if the data is not correct, inevitably the model will be questionable, yet some solution to the problem has to be found. For this reason the model proposed in the next sections is rather a description of the population trends observed, and not a time-dependent fit. No data on the age structure of the species under discussion is available, and the fecundity rates for each age group cannot be deduced from the data. The discrete model approach to modelling population interaction, using difference equations and Leslie matrices, is not suitable in this case. We use continuous models consisting of differential equations, since population sizes are
Corresponding author. Tel.: +27 12 3186326; fax: +27 12 3186114. E-mail addresses:
[email protected] (T.H. Fay),
[email protected] (J.C. Greeff). 0304-3800/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2007.08.023
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large enough (generally taken as larger than 1000) to justify a continuous approach. To our knowledge there is no recognized model that specifically describes interaction between three competing species similar to nyala, impala (Aepyceros melampus) and red duiker (Cephalophus natalensis), and their responses to cropping, to which the results of our model could be compared. The challenge therefore is to adapt general theoretical models for multiple species interaction to describe what we observe in the Ndumo Game Reserve. We postulate and investigate mathematically the effect of introducing terms that represent various phenomena, such as inter- and intraspecies competition and human interference, and apply our newly obtained knowledge to anticipate possible future trends in order to address the present problem. To compare our model with other multiple competing species models, interested readers may profitably use the search engine Science Direct on the journal Ecological Modelling web page.
Table 1 – Abbreviated table of nyala, impala and red duiker census numbers, and numbers cropped in the Ndumo Game Reserve since 1983 Year
Census numbers Nyala
1983 1988 1993 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006
Impala
5015 502 5700 680 4009 1180 3268 2048 Did not count 3885 1336 4963 2731 3635 2116 4142 1660 3512 1832 2327 1491 2460 2099 2472 1366
Duiker 860 620 466 237 350 357 141 196 179 284 Unknown Unknown
Removals Nyala
Impala
900 600 1424 1210 650 1869 942 177 1098 1646 1206 1235 1260
69 40 0 191 0 638 371 54 45 476 820 885 623
Note that the proposed model was implemented in 2002.
2.
Biological background
The Ndumo Game Reserve, situated on the southern border of Mozambique and northern Kwa Zulu Natal in South Africa and, has a tropical climate and is well suited to various species of grazers and browsers. The park is relatively small (10,000 ha) with only small predator species present (both leopard and caracal are present), which appear not to have significant impact on any of the herbivore species. Since 1964 an overpopulation of nyala has been impacting heavily on vegetation structures and smaller competing species. The bushbuck (Tragelaphus scriptus), suni (Neotragus moschatus) and red duiker are examples of such species that may be facing localized extinction. In this paper the red duiker is used as an example to represent these smaller threatened species. Simultaneously, larger species such as the impala, which graze and browse, are able to compete positively with nyala, and lately also tend to be overpopulated. According to the Mentis step-point method1 that is used to calculate the various carrying capacities, the reserve is capable of sustaining approximately 2000 nyala and 1500 impala. Population estimates for Ndumo show that these two species became too abundant, threatening smaller species, and cropping programmes have been running continuously. Despite the rigorous cropping of nyala over an extended period of time, reserve managers have been unable to control their numbers. Although carrying capacity for red duiker has not been calculated for Ndumo (because they do not have a significant impact on any other grazer species), carrying capacities calculated for similar reserves suggest that Ndumo can easily sustain 1000–1200 red duikers. Table 1 provides the population estimates of nyala, impala and red duiker for the period 1983–2003, as well as the numbers of nyala and impala cropped during this period. From the population estimates it is clear that the nyala were never under the estimated carrying capacity during the
period under discussion, despite the massive effort to control their numbers. A more severe cropping quota for the impala population was introduced, yet the population is increasing while red duiker numbers are decreasing steadily, causing concern about their future survival. Large fluctuations in population estimates in consecutive years raise serious concerns about the quality of the available data. The impala population, for example, was reported to have almost trebled from 1992 to 1993 and quadrupled from 1996 to 1997, which is counterintuitive. According to field workers at Ndumo, these discrepancies could be attributed to good rainfall during the period 1987–1989, followed by a dry season and an epidemic in 1991–1992, affecting mainly the nyala population. Two excellent wet seasons followed during the period 1995–1996, which could account for the almost unbounded growth of the impala and nyala populations during this period. The authors are of the opinion that discrepancies in population estimates may also be the result of different counting methods employed over the decades, and impenetrable vegetation in large areas of the reserve that could affect the accuracy of the game counts. For example, prior to 1999 cropping quotas were based on managerial opinions of veld and animal conditions, with the specific objective of controlling the population numbers. Since 1999 a more modern predator–prey simulation program has been implemented to determine cropping quotas, yet from a mathematical point of view these quotas still seem to be calculated in a random fashion. A constant cropping term is certainly not involved, and neither a linear nor a quadratic cropping system is evident. At best it can be stated that, over the past two decades, approximately 22% of the nyala population and 14% of the impala population have been removed per annum.
3. 1
For a full discussed of this method, consult Bothma-Du Preez (1996).
A three competing species model
In a preceding study, Fay and Greeff (1999a,b) apply a deterministic approach to model the dynamics of two com-
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peting species, nyala and bushbuck, and discuss the possible outcomes of different cropping strategies as applicable in
but are not applicable when we consider the coexistence of species. Depending on the values of the parameters, the eighth critical point at
ce − bg − cj + gj + bk − ek −cf + ag + ch − gh − ak + fk −ae + bf − bh + eh + aj − fj ; ; ceh − bgh − cfj + agj − aek + bfk ceh − bgh − cfj + agj − aek + bfk ceh − bgh − cfj + agj − aek + bfk
the Ndumo Game Reserve. The literature on multi-species interaction is primarily based on hypothetical situations to investigate features such as stability, bifurcations and possible chaos, as discussed by Gilpen (1975), Chi et al. (1998), Li and York (1975), May (1973), and May and Leonard (1975). However, practical examples of models for three interacting species applied to a specific real-world situation could not be found in the literature, although some hypothetical examples are discussed by Overton (1984). Consider an ecosystem consisting of only three species, X1 , X2 and X3 , competing for food in a small, predator-free enclosed area, where migration is ruled out. Assume that species X3 is physically small and a browser, species X2 is larger and a mixed feeder (browser and grazer), while species X1 represents the largest browser species. If species X1 and X2 tend to be overpopulated, they will compete aggressively with members of their own species as well as with the other two species, respectively. The system dynamics, ignoring all other factors, could then be represented by a simplistic system comprising three nonlinear differential equations: dX1 = X1 (b1 − a11 X1 − a12 X2 − a13 X3 ), dt dX2 = X2 (b2 − a21 X1 − a22 X2 − a23 X3 ), dt dX3 = X3 (b3 − a31 X1 − a32 X2 − a33 X3 ) dt
(3.1)
where dXi /dt indicates the rate of change and bi represents the growth coefficient of population Xi , while the aij ’s represent interspecies competition (i = j) or intraspecies competition (i = j). Mutualism is not present in this system, and since there are no predators, logistic growth could be expected. It is assumed that all species are similarly exposed to related exterior factors, which are assumed to be stable over the period of investigation. The extravagance of such assumptions over a period of time in nature is acknowledged, but the sentiment is that droughts, fires, floods or disease will affect the growth and dynamics of competing species on the same trophic level in an enclosed environment in the same way. To simplify matters, the above system can be normalized and represented by x˙ = x(1 − ax − by − cz), z˙ = z(1 − hx − jy − kz)
y˙ = y(1 − ex − fy − gz), (3.2)
where, as usual, x˙ denotes the first derivative dx/dt. In the analysis of systems of differential equations it is useful to consider solutions that do not change over time, that is, for which simultaneously x˙ = 0, y˙ = 0 and z˙ = 0. Such solutions are called equilibrium or critical points of the system. For system (3.2), eight critical points can be identified: seven of these critical points are located on the boundaries of the nonnegative population octant and indicate extinction of one, two or all three species. These points could be stable or unstable,
is the only one that could be located in the positive population octant and would be of interest if it could be classified as being stable. Coexistence of the species, indicated by asymptotic stability of the system at the critical point in the positive population quadrant, can be ascertained if all the eigenvalues associated with this critical point are negative. The analytical analysis above is independent of initial conditions, but it should be borne in mind that eigenvalue stability analysis was originally designed by and for engineers who are able to initialize their systems in the neighbourhood of the critical points, while the ecologist cannot be assured of these conditions. Therefore, conditions for global stability should also be considered. Gilpen (1975) states that information about the global stability of an ecological system can be deduced from the behaviour of solution trajectories when one (or more) population densities is (are) low. Gilpen argues that, if all populations have positive growth at low densities, the community will retain all of its species. The system is then regarded as being ecologically stable. May (1971) and Van der Hoff (2000) concur that if any of the two species sub-communities related to the three species community are unstable, then it does not necessarily imply instability of the three species community. However, the criteria tend to be similar: usually stability in the two species sub-communities goes hand in hand with stability in the three species community. The converse applies to instability.
4.
A model for nyala, impala and red duiker
The arbitrary model developed to describe the nyala, impala and red duiker interaction is based on the traditional LotkaVolterra model (Lotka, 1956), and is subject to certain assumptions and other factors of importance: • It is assumed that only three species are present in the interaction situation. • All exterior factors that may affect the dynamics of these species are assumed to be stable for the period under discussion, and catastrophes such as droughts, epidemics and fires are ruled out or affect species sharing the same habitat in a similar fashion. • A continuous model is preferred to a discrete model since the populations are large enough and no distinction is made between the sex and age groups of a species. Continuous models in this case are more accurate in describing the population variations over time than their discrete counterparts, since the calving seasons are not restricted to a specific period in the tropical climate of Ndumo. Stochastic processes that may be present are taken into account by variation of parameters when discussing the sensitivity and stability of the “final” model. • The model is offered as a convenient vehicle to explore complex patterns that can be expected from nonlinear
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deterministic models, when applied to a wide range of ecological problems. For the competing species in Ndumo, let x˙ describe the rate of change in the nyala population, y˙ the rate of change in the impala population and z˙ the rate of red duiker population changes over time. All populations are measured in thousands, and since the populations are non-negative, we restrict our attention to the non-negative population octant in three-dimensional space. The model proposed, without cropping or other external interference, is a system of nonlinear, first-order differential equations given by x˙ = x(1.24 − 0.2x − 0.1y − 0.02z), y˙ = y(1.22 − 0.18x − 0.12y − 0.02z), z˙ = z(1.15 − 0.2x − 0.1y − 0.02z)
(4.1)
According to officials at Ndumo, nyala had a growth rate of approximately 24%, impala 22% and the red duiker 15% in the last two decades, hence the parameters b1 , b2 and b3 in system (3.1) are replaced with numerical values b1 = 1.24, b2 = 1.22 and b3 = 1.15. The parameters that indicate inter- and intraspecies competition in system (3.1) are extremely difficult to estimate, and the values chosen above in Eq. (4.1) are based on educated speculation and reasonable estimates as per ratio impact expected and interpreted by field workers.2 The following facts, and consequent assumptions, were taken into consideration in the choice of these parameter estimates: • Since the nyala density has been much higher than that of the other two species and they have been overpopulated over an extended period of time, it is assumed that their inter- and intraspecies competition effect will play a dominant role in the dynamics of the three species. The intraspecies competition parameter value for nyala would be the highest of all, as a result of overpopulation over time, thus the value of a11 = 0.2 is taken as the reference value for pro rata estimates of the other competition parameters. • Because of high density, nyala would have almost the same negative effect on impala, which is a mixed feeder, hence a12 = 0.18 seems a reasonable guess. On the other hand, duiker use a lower browse line than nyala and will suffer in the same way as the young nyala, so we assume the value a13 = 0.2. • Because impala are browsers and grazers, while both nyala and duiker are browsers only, they do not share the exact same feeding niche with their competitors. With the estimated value of a22 = 0.12 (lower intraspecies competition than nyala) as a reference, the impala’s effect on nyala and duiker estimated as a12 = a32 = 0.1 would be reasonable. • Duiker do not exercise significant inter- or intraspecies competition because of the relative absence of carry-
2
An alternative mathematical method, based on time integration to determine the unknown parameters of such a dynamical system, is discussed by Greeff and Shatalov (submitted for publication).
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ing capacity restrictions, and the low browse line shared only with the young of its competing counterparts, hence a13 = a23 = a33 = 0.02. • Initial population densities used for the numerical analyses are as in 1983: 5000 nyala, 500 impala and 860 red duiker, indicated in 1000s in all the following calculations and figures. • The carrying capacities for the two larger species, nyala and impala, are calculated at 2000 and 1500, respectively, and are based on an evaluation of veld conditions, rainfall patterns and vegetation biodiversity, and the total area of the reserve, based on the widely used Mentis step-point method. From reports of similar reserves in the same region, the permissible density for red duiker can be estimated at 1000. • It is assumed that the carrying capacities have already been incorporated into the estimated growth rates and the competition parameters of these species, therefore logistic growth has already been accounted for in the initial model. The model (without any cropping terms) is an attempt to fit the trend of population densities over two decades as if no cropping has taken place. To project population estimates for each year in the absence of annual cropping, nyala and impala offtakes for the previous year are multiplied by 1.24 and 1.22, respectively to reflect the species growth rates applied to numbers cropped, and the results are then added to the population estimates for the following year to represent natural population growth without cropping. When these adapted population estimates and the solution curves of the model are plotted together as population–time graphs over 20 years, a reasonable fit of population trends is achieved. This is shown in Fig. 1, with time indicated on the horizontal axis and population densities (all measured in 1000s) on the vertical axis. A perfect fit of the model to the data is improbable. It must be emphasized that no population estimate gives absolute results—a standard error of up to 20% in census data is not uncommon, as reported by Smuts (1982) and Starfield and Bleloch (1986), and may be even worse for Ndumo. However, the global trends, if not the time-dependent fluctuations of the three competing species over a period of 20 years, are clearly mimicked by model output: the nyala population tends to overshoot the carrying capacity and decreases slightly as impala numbers steadily increase, while the red duiker may face localized extinction over time. System (4.1) has six critical points, all located on the boundaries of the non-negative octant. With eigenvalue analysis, five of these critical points prove to be unstable while the sixth, located at (4.467; 3.467; 0), is stable. The model therefore predicts that, over time and with no management interference, the three competing species would settle at a population pair of approximately 4667 nyala, 3467 impala and eventual extinction of the red duiker species. This corresponds to all indications of population estimates, and confirms the fears of Ndumo Game Reserve management. Would the eventual result over time be different, should the initial densities of the three populations be changed to extremes of, say, 1000 nyala, 4000 impala and 100 red duiker? The global stability of the proposed system (4.1), with respect to varying initial population sizes, is illustrated by the threedimensional phase diagram in Fig. 2.
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Fig. 1 – Population–time graphs of the model solutions calculated without any cropping, compared with projected census numbers if no cropping were implemented.
To produce the phase diagram, 10 sets of random initial population pairs were generated from extreme but probable density ranges, namely x ∈ [1; 7], y ∈ [0.2; 4] and z ∈ [0.1; 2], all measured in 1000s. This means that the initial nyala density can randomly take on any value from 1000 to 7000, the impala density can range from 200 to 4000 and the duiker can start off with any initial population between 100 and 2000 inclusive. The trajectories with such randomly chosen initial population pairs were plotted over a time interval of 50 years, and all tend towards the stable node at (4.4667; 3.4667; 0), indicating global stability (in a mathematical sense) at this point. However, since one of the species becomes extinct, this is an unstable system in an ecological sense. Assuming the validity of the initial model (4.1), it would seem that the officials at Ndumo were correct in their decision to implement a cropping programme to decrease the nyala and impala populations to carrying capacities, in an attempt to save the red duiker population from extinction.
5. A mathematical investigation into the effect of cropping
• Cropping at a constant rate. • Cropping proportional to the size of the population, also called linear cropping. • Cropping proportional to the square of the size of the population, referred to as quadratic cropping.
Each of these strategies, implemented on the nyala and impala populations, is considered below.
5.1.
Constant cropping
Starfield and Bleloch (1986) suggest that constant cropping is fundamentally unsound, because in practice, nobody would continue to crop at a constant rate once it was obvious that the cropped species was in a major decline. Consider the proposed model (4.1) for the three competing species, with constant cropping terms g and h now added to the equations representing the nyala and impala population dynamics: x˙ = x(1.24 − 0.2x − 0.1y − 0.02z) − g, y˙ = y(1.22 − 0.18x − 0.12y − 0.02z) − h,
According to Starfield and Bleloch (1986), the following possible cropping strategies to obtain equilibrium in ecological situations could be considered:
z˙ = z(1.15 − 0.2x − 0.1y − 0.02z)
(5.1)
During the past two decades, on average, 22% of the nyala and 14% of the impala population at Ndumo have been removed, although the cropping rates for each year have been widely random. If management had decided to implement constant cropping in 1983, with a fixed rate calculated as 22% and 14% of the population densities at that stage (5000 nyala and 500 impala), then g = 1.100 and h = 0.70 would have been the constant cropping rate for the ensuing years. For these values, the system has six critical points, all of them unsuitable. Two of them contain complex numbers, which does not make biological sense, while the remaining four have either negative or zero z-solutions. Smaller (or larger) constant cropping terms, and randomly chosen initial populations, yield similar results. For the three competing species in Ndumo it is clearly illogical to apply constant cropping to nyala and impala, resulting in global instability of the model.
5.2. Fig. 2 – Three-dimensional phase portrait of system (4.1), with random initial populations, shows trajectories tending to the global stable point at (4.4667; 3.4667; 0).
Linear cropping
Cropping proportional to the size of the population may be effective under certain conditions, but in this case proves to be unsuccessful too. The governing system for linear cropping
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applied to nyala and impala is
5.3.
x˙ = x(1.24 − 0.2x − 0.1y − 0.02z) − gx, y˙ = y(1.22 − 0.18x − 0.12y − 0.02z) − hy, z˙ = z(1.15 − 0.2x − 0.1y − 0.02z)
(5.2)
Officials at Ndumo assumed, quite reasonably, that reducing nyala and impala densities would encourage the recovery of red duiker numbers. However, their strategy of removing approximately 22% and 14% of nyala and impala in a more or less linear fashion has failed to stabilize the populations at an acceptable equilibrium point. A mathematical investigation into the properties of the above system, with linear cropping terms set at 0.22x and 0.14y (representing linear cropping of 22% and 14%, respectively), explains management’s inability to control the nyala and impala numbers. When solving this system for x˙ = y˙ = z˙ = 0, we obtain six critical points, all situated on the boundaries of the non-negative population octant, predicting extinction of one or more species. The question, whether any other ratio of the cropping parameters g and h would achieve the desired effect, comes to mind. Linear cropping, in practice and mathematically, has the same effect as reducing the growth rate of a species in model (4.1), both being of the first-order. The above system can therefore be simplified to
x˙ = x(k − 0.2x − 0.1y − 0.02z), z˙ = z(1.15 − 0.2x − 0.1y − 0.02z)
˙ y=y(m − 0.18x − 0.12y − 0.02z), (5.3)
with k = (1.24 − g) and m = (1.22 − h). This allows an analytical investigation to determine whether there exists a combination of linear cropping values g and h that will yield an acceptable stable equilibrium for the three species in the population quadrant. The six critical points of the system (5.3) are (0; 0; 0), (0; 0; 57.5), (0; 0.83333m; 0), (0; −57.5 + 50m; 345 − 250m), (5k; 0; 0) and (20k − 16.6667m; −30k + 33.333m; 0), each indicating extinction of one or more species. Linear cropping does not offer a solution to this specific problem. Starfield and Bleloch (1986) advance the following reason for the failure of their broad interpretation of cropping strategies: “If a population doubles over a period of time, do not double the cropping quota (which may seem a reasonable action to take) but quadruple it. As a general principle, a linear response to population changes is inadequate.”
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Quadratic cropping
Quadrupling the cropping quota seems to be a dangerous option, but in real terms means that the population densities are closing in on a stable equilibrium on a parabolic path and not in a linear fashion. When quadratic cropping terms are introduced to model (4.1), the system to be analysed mathematically is x˙ = x(1.24 − 0.2x − 0.1y − 0.02z) − gx2 , y˙ = y(1.22 − 0.18x − 0.12y − 0.02z) − hy2 , z˙ = z(1.15 − 0.2x − 0.1y − 0.02z)
(5.4)
The effect of quadratic cropping would be similar to mathematically increasing the intraspecies competition parameter values for nyala and impala, both terms being quadratic. System (5.4) can therefore be written as x˙ = x(1.24 − kx − 0.1y − 0.02z), y˙ = y(1.22 − 0.18x − my − 0.02z), z˙ = z(1.15 − 0.2x − 0.1y − 0.02z)
(5.5)
with k = (0.2 + g) and m = (0.12 + h) in order to simplify the eigenvalue stability analysis. For this system there is only one critical point in the population quadrant, of which the coordinate values and stability will depend on the numerical values of k and m. The intention is to apply a deductive procedure to determine if, for any combination of reasonable values for k and m, it is possible to identify a stable critical point close to (2; 1.5; 1), representing the carrying capacities of the three species. Using an iterative simulation procedure, realistic values of k and m, chosen between 0 and 1 were substituted into (5.5) and an eigenvalue analysis performed on the resultant models. In each case the critical value (2; 1.5; 10) was unstable. As an example, the resulting population–time graphs for parameter values k = 0.2 + 0.3 = 0.5 and m = 0.12 + 0.38 = 0.5 substituted in system (5.5) are given in Fig. 3. Initial populations are taken as in 2003, namely 5692 nyala, 2344 impala and 179 red duiker. These estimated values for k and m do not yield any stable critical point, but encourage swift recovery of red duiker numbers, while the other two populations are decreased to carrying capacities within approximately 5 years. Using this as a reference, a suitable cropping strategy may be developed to stabilize the situation in the Ndumo Game Reserve.
Fig. 3 – The effect of suggested quadratic cropping of nyala and impala: immediate and drastic decrease in their numbers, with a sharp increase in duiker numbers when realistic combinations of cropping parameters are introduced.
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Fig. 4 – Stability is obtained at (2.107; 1.636; 1.129) with quadratic cropping of all three species, with initial populations taken as (3; 2; 0.1). Similar results are obtained for initial population sizes and cropping parameter values randomly varied within a 20% deviation.
6. Development of a cropping strategy for Ndumo With nyala and impala densities as high as in 2003, it is clear that immediate and drastic measures should be taken to bring the situation under control. Furthermore, since predators are limited in the Ndumo Game Reserve, human interference (cropping) on a continuous basis is inevitable to maintain some stability over time. However, it is evident that with cropping of two species, whether constant, linear or quadratic, no stability of the system is possible without extinction of one or more species. This corresponds closely to Gilpen’s observation (1975), confirmed by Chi et al. (1998), regarding odd-numbered competing species systems: They concur that a three species competition community may exhibit population oscillations, and eventual stable limit cycles at best, if the intraspecies competition is more or less equal, and the interspecies competition is relatively low in comparison. However, stability is improbable if the weakest species does not possess higher growth potential than the others when at low density, which is the case for the red duiker in Ndumo. May and Leonard (1975) add that, for an evennumbered competition system, stable limit cycle solutions may be obtained more often, and under less stringent conditions of competition. Keeping these general statements in mind, the development of a suitable cropping strategy may be possible. Since quadratic cropping, in practice and mathematically, has the same effect as increasing the intraspecies competition of a species, consider the option of equalizing the intraspecies competition for the three species described by system (4.1). In other words, if we add a quadratic cropping term to each species’ equation of such magnitude that each species similarly competes with its own, at a level of, say, 0.5x2 , 0.5y2 and 0.5z2 . The resulting system is
initial population points, model (6.1) ultimately produces a stable critical point at (2.107; 1.636; 1.131) over time, which is sufficiently close to the desired population densities for the three species in Ndumo.
6.1.
Proposed cropping strategy for Ndumo
With population numbers as in 2000 in mind (4963 nyala, 2731 impala and 357 red duiker), various strategies may be considered. In order to “force” populations closer to their carrying capacities (2000 nyala, 1500 impala and 1000 duiker) as quickly as possible, and then sustain stability over time, the following strategy is proposed:
• Reduce nyala and impala numbers immediately and drastically in the shortest possible time to 3000 and 2000, respectively, with cropping or the exporting of animals. • Then introduce quadratic cropping to nyala and impala according to system (5.4) for the following 4–5 years, to bring their numbers within the carrying capacity range of each species and, because of the lower interspecies competition, allow the duiker population to increase. • When the duiker population is also more or less at carrying capacity, introduce quadratic cropping to all three species over time as described by system (6.1).
This strategy of eventual quadratic cropping of all three species with initial populations close to their respective carrying capacity mathematically yields a stable and desired equilibrium state. As an example, system (6.1) with initial population point (3; 2; 0.1), reaches the only stable critical point at (2.107; 1.636; 1.129) after approximately 8 years, as depicted by Fig. 4.
6.2. An example of the numerical interpretation of the proposed cropping strategy
x˙ = x(1.24 − 0.2x − 0.1y − 0.02z) − 0.3x2 , y˙ = y(1.22 − 0.18x − 0.12y − 0.02z) − 0.38y2 , z˙ = z(1.15 − 0.2x − 0.1y − 0.02z) − 0.48z2
(6.1)
Intuition predicts that cropping of red duiker would lead to localized extinction of the species that is already in decline, and should not even be considered. However, using various
The proposed strategy can be represented in table format, to simplify the task of managers not necessarily equipped to do the mathematical calculations for numbers of respective species to be cropped. Under the assumption that management has agreed to the drastic initial reduction of nyala numbers, the following phases are introduced:
149
Impala
– 969 937 927 921 908 – 1646 1369 1291 1258 1168 100 156 264 454 782 1342 2000 1710 1658 1649 1645 1637 3000 2358 2208 2159 2137 2188 100 212 216 377 652 1122 2000 2679 2595 2576 2566 2545 3000 4004 3577 3450 3395 3356
Impala
Numbers to be removed
Nyala Nyala
Duiker
Desired densities as described by system (5.4)
Duiker Impala
Projected numbers to be removed each year are indicated in the last two columns.
Projected population densities and estimated removals for each year, starting with various initial population sizes, can be calculated in the same way to assist management to realize their objectives.
T T+1 T+2 T+3 T+4 T+5
3
Expected densities as predicted by system (4.1)
Suppose it were possible to introduce a large enough predator population into the reserve that would feed on all three competing species. According to Holling (1965) and Murray (1993) the following scenario could be expected. While nyala and impala numbers are high, the predator would use them as its primary source of food. As soon as the duiker population picks
Nyala
6.3. Ecological motivation for the proposed management strategy
End of year
• Phase 1: Assume, for example,3 population sizes of 3000 nyala, 2000 impala and 100 duiker as initial conditions for the end of year T. Referring to Table 2, initial quadratic removals of nyala and impala for the following 5 years are calculated as follows: ◦ A projection is made of expected population sizes at the end of year T + 1 without any cropping, using model (4.1). ◦ The desired population sizes with quadratic cropping of only nyala and impala over a 5-year period are calculated with system (5.4). ◦ The number of nyala and impala to be removed during year T + 1 is then calculated as the difference between the expected and desired population sizes at the end of year T + 1. ◦ At the beginning of year T + 2, the desired population sizes (assuming successful cropping during year T + 1) are taken as the new initial conditions to calculate expected growth for year T + 2, and the numbers to be removed during year T + 2. This procedure is repeated to calculate removals for the following years. ◦ After 4 or 5 years of quadratic cropping of nyala and impala, and with constant monitoring of actual census numbers compared with the projected numbers in Table 2, the situation should be re-evaluated before progressing to the next phase. Suppose the population estimates at the end of year T + 5 confirm that the populations are just above the carrying capacities of 2000 nyala, 1500 impala and 1000 duiker. Cropping may not be discontinued, and stability will have to be maintained over time by continuous cropping of all three species, since the original system (4.1) predicts no mathematical stability for the three competing species ecosystem in the absence of cropping or predation. However, it is essential to monitor all species densities closely, especially that of red duiker, before moving on to the next phase. • Phase 2: Assuming that all three populations are now close to their respective carrying capacities, introduce quadratic cropping to the red duiker population as well. Following the same reasoning as before, the projected removals of the respective species are represented in Table 3. However, will it be possible to convince a good field ranger to even consider cropping a species that, historically, tends towards localized extinction? In ecological terms, the above strategy is not unreasonable, because a predator’s behaviour is represented by the proposed actions of management.
Table 2 – An example of expected population densities with no cropping (with desired densities of previous year taken as initial point), compared with desired population densities when quadratic cropping of nyala and impala is implemented
e c o l o g i c a l m o d e l l i n g 2 1 1 ( 2 0 0 8 ) 142–152
150
Duiker
– 676 587 544 522
Impala
– 906 909 910 912
x˙ = x(1.24 − 0.2x − 0.1y − 0.02z) − 0.3x2 p, y˙ = y(1.22 − 0.18x − 0.12y − 0.02z) − 0.38y2 p,
– 1202 1199 1201 1202
Nyala
Numbers to be removed
up sufficiently, the predator would naturally include in its diet the smaller and more vulnerable duiker, and at a higher frequency than nyala and impala. If the density of this fictitious predator (man) is sustained at an almost constant level by sources practically independent of its prey, its quadratic predation on the three competing prey species can be modelled as an even-numbered species system:
z˙ = z(1.15 − 0.2x − 0.1y − 0.02z) − 0.48z2 p,
1342 1240 1129 1164 1149 1637 1632 1632 1633 1634 Projected numbers to be removed each year are indicated in the last three columns.
2118 2107 2106 2106 2106 1637 2538 2541 2543 2546 T+5 T+6 T+7 T+8 T+9
2188 3309 3305 3307 3308
1342 1916 1777 1708 1671
Duiker Impala Nyala Duiker Impala Nyala
Expected densities as predicted by system (4.1)
Desired densities as described by system (5.4)
p˙ = p(−0.0028 + 0.0003x2 + 0.00038y2 + 0.00048z2 )
End of year
Table 3 – An example of expected population densities if no cropping is done, compared with desired population densities if quadratic cropping of nyala, impala and red duiker is implemented with populations at (2.188; 1.637; 1.342)
e c o l o g i c a l m o d e l l i n g 2 1 1 ( 2 0 0 8 ) 142–152
(6.2)
The values assumed for quadratic predation parameters, based mainly on intuition and statements by Gilpen (1975), Pimm (1991) and Chi et al. (1998) with respect to equal intraspecies competition, can be justified for the specific situation: the nyala population should lose more individuals because of predation as apposed to the effect of intraspecies competition, while impala and duiker will suffer pro rata more from predation than nyala because of their smaller sizes. The predator’s feeding parameters are in the same ratio as the predation parameters, taking into consideration that in the model, all prey species are measured in thousands, while the “predator” is taken as one unit,4 represented by the team of field rangers. System (6.2) yields 26 critical points, of which 25 predict either extinction of one or more species, or indicate population densities as complex numbers. The one remaining critical point located at (2.047; 1.589; 1.102; 0.001) has negative, real eigenvalues, indicating global stability of the system at this point. The population–time graphs in Fig. 5 shows that, with quadratic cropping to mimic predation on all three species, stable coexistence close to the carrying capacities is possible. Uncertainty surrounding the assumed parameter values of the proposed model poses the ever-present and concerning question: Will model behaviour be as predictable if all the parameter values are changed randomly and simultaneously, allowing a standard error of up to 20%? Stochastic behaviour of the assumed parameter values was investigated, and the graphic results of 100 independent simulations are given in Fig. 6. Model sensitivity is illustrated by the vertical lines at t = 0, 1, 2, . . ., 20 when all parameters are varied simultaneously and randomly with a possible 20% deviation in a uniform distribution. Resulting deviation from the model output is inscribed by a smooth behavioural band, without unexpected tendencies depicting extinctions or uncontrolled growth of a species over future time. Statistically speaking, this implies a high level of confidence regarding the predictive qualities and stability of the model. Further statistical tests with respect to model efficiency and hypothesis testing, as discussed by Fay et al. (2006), confirm this observation.
4 According to Estes (1992), predation of approximately 100 lion can be simulated by the organized cropping activities of one team of field rangers.
e c o l o g i c a l m o d e l l i n g 2 1 1 ( 2 0 0 8 ) 142–152
151
Fig. 5 – Densities of the four species are maintained close to their respective carrying capacities over time, with predator-related cropping.
Fig. 6 – Model sensitivity to stochastic behaviour of parameter values is illustrated by the vertical lines at t = 0,1,2, . . ., 20 when all parameter values are varied simultaneously and randomly for 100 independent simulations.
It is important to remember that all exterior factors are assumed to be stable for the period under discussion, which is hardly probable for 20 years. In a dynamic ecological system it is necessary to re-evaluate all the factors involved on a continuous basis, to compare model projections with real-world data, and adapt cropping strategies as necessary.
7.
Conclusion
The model derived here fits the classical pattern of populations growing initially in a logistic fashion, then reaching a restraint as they exhaust the habitat, while endangering the existence of smaller species. To some extent it explains why previous cropping strategies in the Ndumo Game Reserve have failed, suggesting scientifical-based alternatives. The authors are of the opinion that the validity of the model should also be tested in practice. In the first instance, continued monitoring of the population densities, without any cropping, will provide data comparable with the projected instability indicated in the model. Secondly, the experimental cropping of the respective species should be instituted with caution, and the response of the populations compared with that predicted by the model before moving on to the next phase of the proposed strategy.
Acknowledgements This research project was supported by the National Research Foundation, South Africa (grant no. 2054454) since 1999. The authors thank the Technikon Pretoria (now partner in the merged Tshwane University of Technology), the University of Southern Mississippi, and personnel of the Ndumo Game Reserve for the data supplied, their prolonged support and useful suggestions over an extended period of time.
references
Bothma-Du Preez, B.L.J., 1996. Game Range Management. J. L. v. Schaik, Pretoria. Chi, C., Hsu, S., Wu, L., 1998. On the assymetric May–Leonard model of three competing species. SIAM J. Appl. Math. 58, 1. Estes, R.D., 1992. Behaviour Guide to African Mammals. University of California Press, California. Fay, T.H., Greeff, J.C., 1999a. Nyala and bushbuck: a competing species model I. Math. Comput. Educ. 33 (2), 17–26. Fay, T.H., Greeff, J.C., 1999b. Nyala and bushbuck: a competing species model II. Math. Comput. Educ. 33 (2), 26–35. Fay, T.H., Greeff, J.C., Groeneveld, H.T., Eisenberg, B.E., 2006. Testing the model for one predator and two mutualistic prey species. Ecol. Model. 196, 245–255.
152
e c o l o g i c a l m o d e l l i n g 2 1 1 ( 2 0 0 8 ) 142–152
Gilpen, M.E., 1975. Limit cycles in competition communities. Am. Nat. 109, 51–60. Holling, C.S., 1965. The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Ent. Soc. Can. 45, 1–60. Li, T.Y., York, J.A., 1975. Period times three implies chaos. Am. Math. Monthly 82, 985–992. Lotka, A.J., 1956. Elements of Physical Biology. Williams and Wilkens, Maryland (reprinted as Elements of Mathematical Biology, 1924, Dover, NY). May, R.M., 1973. Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton, NJ. May, R.M., Leonard, W.J., 1975. Nonlinear aspects of competition between three species. SIAM J. Appl. Math. 29, 2.
Murray, J.D., 1993. Mathematical Biology. Springer-Verlag, New York. Overton, W.S., 1984. In: Hall, C.A.S. (Ed.), A Strategy of Model Construction. Ecosystem Modelling in Theory and in Practice, vol. 5. Wiley and Sons, New York, pp. 52–53. Pimm, S.L., 1991. The Balance of Nature? University of Chicago Press, Chicago. Smuts, G.L., 1982. Lion. Macmillan, South Africa. Starfield, A.M., Bleloch, A.L., 1986. Building Models for Conservation and Wildlife Management. Macmillan Publishing Company, Brisbane. Van der Hoff, Q., 2000. Stability of Three competing Species Models. M.S. thesis, University of Southern Mississippi, Hattiesburg, USA.
