Franck Courchamp, Department of Zoology, University of Cambridge, Downing Street, Cambridge, CB2 3EJ, UK. E-mail firstname.lastname@example.org. Tel: 01223-336643. Fax: 01223-336676.
1. Introduced predators account for a large part of the extinction of endemic insular species, which constitutes a major component of the loss of biodiversity among vertebrates. Eradication of alien predators from these ecosystems is often considered the best solution.
2. In some ecosystems, however, it can generate a greater threat for endemic prey through what is called the ‘mesopredator release’. This process predicts that, once superpredators are suppressed, a burst of mesopredators may follow which leads their shared prey to extinction.
3. This process is studied through a mathematical model describing a three species system (prey–mesopredator–superpredator). Analysis of the model, with and without control of meso- and superpredators, shows that this process does indeed exist and can drive shared prey to rapid extinction.
4. This work emphasizes that, although counter-intuitive, eradication of introduced superpredators, such as feral domestic cats, is not always the best solution to protect endemic prey when introduced mesopredators, such as rats, are also present.
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Most contemporary worldwide extinctions have occurred, or are currently occurring in island ecosystems. As an example, of the 30 species of reptiles and amphibians that have gone extinct since 1600, more than 90% are island forms (Honnegger 1981); 93% of 176 species or subspecies of birds (King 1985), and 81% of 65 mammal species extinctions (Ceballos & Brown 1995) that have occurred during this period have occurred on islands.
The introduction of vertebrate species is one of the most important threats to many endemic species in many islands (Moors & Atkinson 1984; Atkinson 1989). Numerous rare or endemic vertebrate species are currently endangered because of predation by introduced invertebrates, reptiles, birds or mammals, or by competition from or habitat destruction by introduced grazers such as rabbits (Oryctolagus cuniculus, Lilljeborg) or goats (Capra hircus, L.) (Moors & Atkinson 1984; Atkinson 1989; Williamson 1996). According to King (1985), predation by introduced animals has been a major cause of 42% of island bird extinctions in the past, and is a major factor endangering 40% of currently threatened island bird species. In particular, introduced feral cats (Felis catus, L.) are known to be a major threat to many island bird species. They are known to have been introduced into at least 65 island groups where they are responsible for the loss of many large land and seabird colonies, populations or even species (e.g. Jouventin et al. 1984; Rodriguez-Estrella et al. 1991; Monteiro, Ramos & Furness 1996); for example, a few cats (around five) were introduced to the Kerguelen Islands in the mid-century. They are now responsible for the decline or extinction of several bird populations in these islands, killing more than 3 million petrels per year (Chapuis 1995). Cats also constitute a major threat to many endemic reptile species or subspecies (e.g. Iverson 1978; Case & Bolger 1991; Arnaud et al. 1993) and mammals (Spencer 1991; Mellink 1992). Their impact has also been demonstrated through competition towards endemic mammalian predators such as island foxes (e.g. Urocyon littoralis dickeyi, Baird), which they replace when uncontrolled (Steve Kovach, personal communication).
Eradication of those alien cat populations is required in many cases, has often been tried, sometimes achieved (Rauzon 1985; Veitch 1985; Domm & Messersmith 1990; Cooper 1995) and several cat eradication programmes are currently underway. Paradoxically, in some particular situations, the presence of a controlled population of cats might be, at least temporarily, more beneficial to their endemic prey than its eradication. Such is the case on many islands where rodents have also been introduced.
The domestic cat is an opportunist predator (Fitzgerald 1988). When both bird and mammal prey are available, it is believed that the domestic cat diet will include mainly mammals (e.g. Konecny 1987; Nogales et al. 1992; Nogales & Medina 1996). In some island ecosystems, these cats maintain rodent populations at low levels. Although they also often prey upon endangered species, it is believed that, in some ecosystems at least, the beneficial effects of reducing the rodent population could outweigh the damage done to the endemic prey species (Fitzgerald et al. 1991; Tidemann, Yorkston & Russack 1994). The threat posed by introduced cats to the kakapo (Strigops habroptilus, Gray) on Stewart Island is a striking example. Here, cats prey lightly on this highly endangered bird species (remains were found in 5·1% of 118 collected scats, Karl & Best 1982), but even low predation pressure may be detrimental for fragile species (Rodriguez-Estrella et al. 1991). However, rat remains were found in 93·0% of these 118 scats (Karl & Best 1982), showing the indirect role cats might play in preserving native fauna through reduction of rat predation pressure on the kakapo. Moreover, the elimination of feral cat populations from such ecosystems could lead to a more severe negative impact on the endemic species, as a result of expansion of rodent populations once their predators are removed. Attempted reduction of the cat population of Amsterdam Island is alleged to have caused a compensating rise in the number of rats and mice, and so has been abandoned (Holdgate & Wace 1961). This process, termed ‘mesopredator release’, had been described in fragmented insular ecosystems (Souléet al. 1988) and applies well to many insular foodwebs (e.g. Schoener & Spiller 1987). Conversely, the eradication of rodents first (which has now proven feasible, even on relatively large islands, Taylor & Thomas 1989, 1993; Towns 1996) might induce cats to switch prey, resulting in a brutal increase in predation pressure on the threatened endemic species, as experienced for stoats and rats in New Zealand (Murphy & Bradfield 1992). Unfortunately (from the theoretical point of view), there is little field evidence from island management either of mesopredator release following superpredator eradication, or of predators switching prey following mesopredator eradication.
As the optimal control strategy is neither simple to find nor intuitive, it is studied through the analysis of a mathematical model which mimics the dynamics of the three species in this system. In the study reported here, the interactions of a prey species, such as a bird species, a threatening alien mesopredator, such as a rat, and an alien superpredator species, such as the feral domestic cat, were examined through their coupled dynamics. Through this model, the theoretical existence of ‘mesopredator release’ and the effect of the presence of a superpredator on the prey will be investigated. It is assumed that the superpredator preys both upon the prey and the mesopredator. For the sake of simplicity, reference will sometimes be made to them as bird, rat and cat, instead of prey, mesopredator and superpredator, respectively.
The preliminary models
For the sake of simplicity, models are first presented taking into account only two species, and only then is the third species added and its implied complications analysed. The construction and analysis of the models are based on previous work (Courchamp & Sugihara 1999), to which the reader can refer for additional details. The first two systems consist of two simple coupled differential equations, each representing the dynamics of one population. Each population is described by a simple logistic equation, modified to take into account its relationship with the other species. The other possible prey species populations are not taken into account; it is assumed that all the prey species form a single ‘bird’ population, with average characteristics. The realism of these assumptions has been discussed previously (Courchamp & Sugihara 1999). The number of individuals at time t in the prey, mesopredator and superpredator populations are B, R and C, respectively. The intrinsic growth rates of the prey, mesopredator and superpredator populations are rb, rr and rc, respectively. The predation rate of the superpredator is μb on the prey and μr on the mesopredator. The predation rate of the mesopredator is ηb on the prey and ηs on other food items (seeds, leaves, invertebrates). The carrying capacity of the environment for the prey population is Kb. The carrying capacities of the environment for the mesopredator and the superpredator populations are not constants, but depend partially (rats are omnivores) or totally (cats are carnivores) on the number of available individual prey on which their populations can feed at time t. For the mesopredator, the carrying capacity of the environment is the number of mesopredators that can live on food other than birds when there is no prey, to which is added the number of mesopredators that can be fed by the total of available prey at time t. The carrying capacity is thus the quantity of non-avian food S divided by the consumption rate ηs, plus the number of prey B divided by the mesopredator predation rate ηb: S/ηs + B/ηb, that is (ηbS + ηsB)/ηbηs. For the sake of simplicity, it is assumed that S is a constant (the carrying capacity in the absence of prey is kept under the form S/ηs instead of a constant, say Ks, to conserve homogeneous notation). Rats are opportunistic predators, and their diet (proportion of avian and non-avian food consumed) depends on relative availability of food items (Clark 1980). Accordingly, instead of ηbR, the predation rate of rats on birds is: BηbR/(S+B). In the two-species models, the rat/cat system is not presented, because it is the same as the bird/cat system.
The prey/mesopredator (bird/rat) system is given by:
There are several equilibrium points. The first two equilibrium states, extinction of both populations [0, 0] and extinction of the rat only [Kb, 0] are always unstable. The third state, extinction of the bird only, (0, S/ηs) is globally asymptotically stable if and only if rb ≤ ηb/ηs. These three points always exist. When both populations coexist, the system reaches (B*, S/ηs + B*/ηb). B* is given in the Appendix with the analysis of the system.
The prey/superpredator (bird/cat) system is given by:
It is the same for the mesopredator/superpredator (rat/cat) system (B is replaced by R and b indices are replaced by r indices). They have three stable equilibrium points only, as the predator cannot survive alone: [0, 0], [Kb, 0] and [Kb(1 − 1/rb), Kb(1 − 1/rb)/μb].
The complete model
To take the three species into account simultaneously, one needs to make further assumptions. First of all, like the rat, the domestic cat is an opportunist predator, which switches prey species according to their availability (Fitzgerald 1988). Accordingly, the number of birds and rats preyed upon by cats will depend on their respective numbers. Instead of μbC and μrC, one will find: μbBC/(B+R) and μrRC/(B+R) for the bird and the rat populations, respectively. Potential preferences of the cat for the indigenous prey over the introduced predator are not taken into account (see Courchamp, Langlais & Sugihara in press). The cat carrying capacity is: B/μb + R/μr. The compartmental representation of the model is given in Fig. 1.
One has the following system:
There are several states with this system: all populations go extinct [0, 0, 0], only the prey survives [Kb, 0, 0], only the mesopredator survives [0, S/ηs, 0], only the superpredator disappears [B*, S/ηs + B*/ηb, 0] (B* has the same value than in system (1–2)), only the prey disappears [0, S/ηs(1 − 1/rr), S/μrηs(1 − 1/rr)], only the mesopredator disappears [Kb (1 − 1/rb), 0, Kb(1−1rb)/μb], and, finally, no species disappears [B6*, R6*, C6*]. There can be between 0 and 5 stationary states with the equilibrium value of B between 0 and Kb, as solutions of an equation of the fifth degree. As its expression is very long and complex for the last point, it will not be presented here, but the authors will provide the Maple file upon request. For the same reason, the comparison between different points is too complex to be presented here analytically. The deterministic nature of the model allows, however, a numerical study.
It can be noted that the fourth, fifth and sixth equilibrium points are equivalent to some from the models 1–2 and 3–4. The conditions of existence of these points are described in the Appendix.
Superpredator presence and mesopredator release
The behaviour of the model is studied analytically (see Appendix), but simulations are presented for heuristic purposes. It can be seen from Fig. 2 that, under some conditions, while the rat alone would extirpate the bird (a) and the cat alone would not (b), the cat prevents extinction of the bird by the rat when both predators are present by controlling rat (c), or even by extirpating the rat (d).
One can see that, in some cases, the presence of the superpredator can indirectly protect a shared prey from a mesopredator. Even in circumstances where this is not the case, the elimination of the superpredator might be more harmful to the prey, through the ‘mesopredator release’, which arises when the pressure on mesopredators by superpredators is suppressed. To mimic this process, the effect of the suppression of the superpredator when the three species are present in equilibrium is studied. First one has to find values for which the bird population does not go extinct when the two other species are introduced. A range of parameters must be defined at which all three populations persist and reach an equilibrium state. This range of parameters implies high values of intrinsic growth rate of the prey (>1·3) if it is to survive the presence of the two predators. From this equilibrium state ([B6*, R6*, C6*]), the extinction of the superpredator population can be simulated and the outcome of the two remaining populations monitored: new simulations are run with the following initial conditions: [B6*, R6*, 0]. This is repeated for different values of sensible parameters, to obtain a pattern displayed Fig. 3. This shows that prey extinction through mesopredator release does occur in this system. Moreover, it is avoided only for higher values of prey intrinsic growth rate (>1·6). These correspond to values too high to be biologically realistic, and will not often be reached in natura by insular bird species. Indeed, in the absence of similar predation pressure, these endemic species seldom have developed antipredator life history traits, such as a high intrinsic growth rate. These results show that the only endemic species with intrinsic growth rates high enough to survive the introduction of both a superpredator and a mesopredator will normally be extirpated through mesopredator release if the control strategy implies only the superpredator eradication. The strategy of control therefore appears essential in these particular ecosystems.
Mesopredator release can occur following eradication of the superpredator and the presence of the superpredator also has negative effects on the prey population, therefore a study of control strategies is necessary to determine an optimal strategy. If one applies a control effort of λr on the rat population and of λc on the cat population, the model (5–7) becomes:
This provides the same number of equibrium points as model (5–7), although with different values. Values of these points are provided in the Appendix, together with their conditions of existence and stability.
Analysis of this model shows that when superpredator control is high enough (λc > rc), the superpredator disappears. As a result, the prey also disappears when the mesopredator control is not high enough [λr < rr(1 − rbηs/ηb) < rr), because of mesopredator release. In contrast, and surprisingly, the prey does not disappear when the superpredator control is below a certain threshold [i.e. rc(1 − rb) < λc < rc], provided the mesopredator control is high enough [λr > rr − μr(1 − λc/rc)/μb]. This illustrates the importance of the presence of the superpredator in the system, and the need to take it into account in control programmes.
This study examined, through a mathematical model, the fate of a prey species in an insular ecosystem into which both a mesopredator and a superpredator have been introduced. Although other species could have been considered, such as mongooses and reptiles (Case et al. 1992), the rat, feral cat and endemic bird species were taken as examples. Indeed, as Diamond (1989) stated, ‘rats and cats are the most notorious killers and island birds the most notorious victims, in this regard [extinctions due to introduced predators].’ This work shows several interesting features of direct concern to conservation biology. First, the result of this theoretical work shows that the presence of one predator only is sufficient to induce the extinction of the endemic prey. This is not new, as indicated by too many examples in natura. Second, when both the mesopredator and the superpredator are present, seven different situations may arise, among which is the case where the three species are present with stable dynamics. Interestingly, there is another case where both predator species can coexist indefinitely, even after the eradication of the prey species. Finally, and most interestingly, is the case where the superpredator causes the extinction of the mesopredator, but not of the prey.
It has been shown here that presence of a superpredator may have a global positive effect in insular ecosystems in which an introduced mesopredator threatens an endemic prey. Indeed, in the model presented here the presence of the superpredator may preclude the elimination of the prey by the mesopredator (or allow a larger prey population size). In addition, it has been demonstrated that superpredator eradication should be avoided, as a means to prevent what has been termed ‘mesopredator release’ (Souléet al. 1988): a sudden burst of mesopredators, once the superpredator pressure is suppressed. Rats have a lower predation rate on birds than cats, but they are much more numerous and can have a higher impact on the prey (Newman & McFadden 1990). Moreover, as they are omnivores, they can maintain a high population and a high predation pressure, even when the prey population size is low, which the cat cannot. This explains why rats alone eliminate the prey more easily than cats alone in the model (Fig. 2), and why the cat presence is sometimes beneficial to shared prey. In fact, over the last 400 years, rats and cats are said to be responsible for 54% and 26% of island extinctions caused by predators, respectively (King 1985). The study of the control strategies clearly shows that the fate of the prey will depend on the superpredator control level. Although counterintuitive, if the superpredator control is too high, the prey will disappear. This may be a further argument in favour of the use of biological control, especially with pathogens with a steady long-term impact (see Courchamp & Sugihara 1999), which are unlikely to be too brutal.
Despite its mathematical complexity, this model remains very simple in its representation of the biological reality. In particular, spatial and temporal population heterogeneities, which are important components of insular ecosystems, were not taken into account. Similarly, the fact that different prey species are present in these ecosystems has not been taken into account. Instead, only one species, which is supposed to represent the ‘average’ of all prey species, was considered. As the rats and the cats are opportunistic predators, which switch prey species according to their relative availability (Clark 1980; Fitzgerald 1988), it would be interesting to study the effect of the presence of several prey species. This model is robust enough to be extended to other ecosystems that can be well described by the prey–mesopredator–superpredator trophic web. It therefore holds for systems where several prey species are present (e.g. landbirds, seabirds and lizards) provided that they are prey of both the mesopredator and the superpredator. Some changes may occur in the solutions of the model according to the characterics of the species (e.g. if they have different antipredation response), but the general conclusions should remain the same. Similarly, the results will be the same if another superpredator, say a bird of prey, is present. Different efficiencies in hunting different preys would here again add more complexity. In contrast, the model results will undoubtedly be different if another species is present which does not fully fit into one of these three trophic levels, because it would describe a totally different system. It is the case, for example, if a prey species (e.g. the rabbit) is present which is not preyed upon by both the superpredator and the mesopredator: islands where cats, rats and rabbits have been introduced should therefore be described by a different model. Despite these possible improvements, this theoretical work shows that it is crucial to take into account the presence of other alien species when designing control programmes for one introduced species (see also Courchamp et al. in press for another example). Emphasis is placed on the following important distinction that should be understood from this work: in some ecosystems where rats are present, introduced cats might play a positive role (implying that their removal could have negative aspects on local fauna); however, their introduction cannot be recommended, whatever the circumstances.
Several points allow optimism about this particular area of conservation biology. First, the recovery and/or preservation of the ecosystems involved is not in conflict with local economic or politic interests, even if the governments of many concerned islands can hardly afford costly programmes such as mammal eradications. Second, eradication of introduced mammals such as domestic cats (Bloomer & Bester 1992), rabbits (Flux 1993), rats (Taylor & Thomas 1993), possums (Cowan 1992), foxes (Bailey 1992), goats (Parkes 1990) or others, once thought impossible, is now known to be feasible; for example, 120 successful ‘pest’ eradications have already been conducted on New Zealand islands (Veitch et al. 1992). Third, many studies show that, when the introduced species has been successfully removed from the whole ecosystem, the threatened species (plant or animal) generally recovers, sometimes rapidly, from the effects of these alien species (e.g. Cruz & Cruz 1987; Brothers & Copson 1988; Towns 1991, 1994; Newman 1994; Cooper et al. 1995). Fourth, in the cases where alien species induced the complete extinction of the population, local populations were often concerned; species or subspecies extinctions are less frequent (Moors & Atkinson 1984). Moreover, seabird populations are not dependent on the land for food, and breeding can continue on isolated islets and stacks free from predators (Atkinson 1985). In most cases of colony or population extinctions as a result of alien predators, there are, in nearby islets, populations able to recolonize the ecosystem once the predator eradication is achieved (Moors & Atkinson 1984; Case et al. 1992; Jouventin & Micol 1995). Last, in some cases (as in the well known case of the kakapo, Clout & Craig 1995; Powlesland et al. 1995) it is suspected that only a small number of individuals (cats as well as rats) have learned to kill the prey or eat the eggs (Grant, Pettit & Whittow 1981; Moors & Atkinson 1984). In these cases, when heavy programmes, such as complete eradication or long-term control, are not possible, selective control to eliminate these particular individuals may be sufficient in the short term, and should thus be implemented.
However, the situation is critical in many cases, and the media and scientific coverage of the situation on most islands does not seem to be proportionate to the problems faced by these often unique ecosystems (Crystal 1989). Despite numerous indications of catastrophic effects of introduced mammals on most oceanic islands, politically organized policies to resolve, or even prevent these effects remain comparatively few. Recent examples, and other older ones, show that solutions do exist. It is often the high financial constraints on eradication programmes that preclude them (Powlesland et al. 1995), or that impose undesired priorities in the order of their attainment (Brothers & Copson 1988). In these cases, a predictive study on the feasibility of the eradication and its potential effects on the ecosystem is needed, especially when there is more than one introduced species, in order to maximize the efficiency of eradication programmes. Theoretical studies, such as mathematical modelling could fulfil this task. This study shows that the intuitively evident need for feral cat eradication may not be the best solution in some specific cases, such as when introduced rodents are present. This is well illustrated by the case of Raoul Island, where it has been said that eradication of feral cats might bring little benefit to bird populations, because Norway rats are present on this island and constitute a major part of the diet of the cats (Fitzgerald et al. 1991). Empirical examples remain, however, scarce regarding the effects of eradication of predators (top predators or mesopredators) on population dynamics of coexisting species.
The idea that top predators may be important species for conservation biology is not new. It has been suggested that they have a disproportionate importance in food webs because their extinction can generate a cascade of unexpected secondary extinctions (Paine 1966; Pimm 1980). For example, the extinction of several prey species has been attributed to the increase of generalist and/or medium-sized predatory mammals because of the lack of top predators in different ecosystems (Terborgh & Winter 1980; Diamond & Case 1986; Wilcove, McLellan & Dobson 1986; Diamond 1989; Böhning-Gaese, Taper & Brown 1993; Goodrich & Buskirk 1995). With the example of lynx, mongooses and rabbits, it has been shown, both theoretically and empirically, that top predators actually benefit their prey through intraguild predation on other smaller predators which share the prey (Palomares et al. 1995). What is new here is rather the application of this idea to an already perturbed ecosystem, a rather counter-intuitive idea: in some cases the removal of one of the causes of perturbation may lead to increased damage. This is the case with the mesopredator release (Souléet al. 1988). In some systems, only the direct negative effects of the top predator on endemic threatened prey have been examined, while in some cases greater positive effects may be present on the same species. Obviously, outcomes of changes of these already perturbed trophic webs are not intuitive, and intervention as dramatic as species eradication should always be preceded by careful empirical and theoretical studies of the whole ecosystem. Indeed, in the present case, eradicating the rats before the cats (at first a seemingly sound strategy) might in fact lead to another case of mesopredator release, as shown by the recent example of Bird Island in the Seychelles. Here, a recent eradication of the introduced black rat population led to an explosion of the exotic crazy ant Anoplolepis longipes, Jerdon, which has been shown to be threatening the bird colonies which rat eradication was intended to protect (Feare 1998). As complete removal of those alien predators is most of the time (and often rightly) the only envisaged solution (Veitch 1985; Ashmole, Ashmole & Simmons 1994; Rainbolt & Coblentz 1997), this idea of possible positive effects of top predators should be kept in mind in conservation biology.
This work was supported by a Lavoisier fellowship from the French Foreign Office, by Biological Oceanography Grant ONR: N00014-95-1-0034, and endowment funds from the John Dove Isaacs Chair in Natural Philosophy, and was part of the IFRTP Program No. 279. We thank Professor T. Clutton-Brock, Doctors M. de L. Brooke, S. Lingle and P. Rohani for their helpful comments on the manuscript.
There are several stationary states for the first system (eqn 1,2). The first two equilibrium states [0, 0] and [Kb, 0] are always unstable. The third state [0, S/ηs] is globally asymptotically stable if and only if rb ≤ ηb/ηs. These three points always exist. When no populations go extinct, the system reaches [B*, S/ηs + B*/ηb], where B*, the bird population size at equilibrium, is the solution of a quadratic equation and therefore one can have 0, one or two equilibria with both populations present between 0 and Kb: B* = [−t + Ür1Übr(t2 − 4ηsrbu)]/2ηsrb, with t = ηsKb − ηsrb (Kb − S), and u = (ηb −ηsrb)(SKb). Numerically, when there are two points with admissible coexistence, one only, at most, is stable. If 0 < ηb ≤ ηs, then there are two cases: if ηb/ηs ≤ rb, the point is stable, otherwise it does not exist. If 0 < ηs ≤ ηb, then there are two cases: if ηb/ηs < rb, it is stable, if rb ≤ 1 < ηb/ηs it does not exist. If 1 < rb < ηb/ηs, then if ηb > ηb*, it does not exist, if ηb = ηb*, it is stable, if ηb < ηb* there are two values for this point. ηb* = (ηsBmax2 + rbηbSKb)/KbS, with Bmax = [(rb − 1)Kb − rbS]/2rb > 0.
The second system (eqns 3, 4) has 3 stationary states. P134: [0, 0], P234: [Kp, 0], and P334: [Kp(1 − 1/rp), (1/μp)Kp(1 − 1/rp)]. The dynamic behaviour of this system is given by four cases.
1 Case 1: rc > 1: if rp ≤ 1 the system reaches P134, otherwise it reaches P334.
2 Case 2: 0 < rp ≤ rc < 1: the system reaches P134.
3 Case 3: 0 < rc < 1, rc ≤ rp: in this case, the dynamics depend on the initial proportion of prey and predator. If C(0)/P(0) > μp(rp − rc)/(1 − rc), then the system reaches P134. If 0 < rc ≤ rp < 1, then the system reaches P134. If 1 < rp, then: (i) if rc + rp > 2, then the system reaches P334; (ii) if rc + rp = 2, then this stationary state becomes a centre, all nearby trajectory being periodical; (iii) if rc + rp < 2, then this point is unstable.
4 Case 4: rc = 1: if 0 < rp ≤ 1, the system reaches P134, otherwise it reaches P334.
There are several equilibrium points for this system. The two equilibrium points [Kb, 0, 0] and [0, S/ηs, 0] always exist but are never stable. There are 0, one or two equilibrium points where only the cat population disappears, since it is the solution of an equation of the second degree [B*, S/ηs + B*/ηb, 0]. These points are never stable when they exist. The point where only the prey disappears [0, S(1 − 1/rr)/ηs, (1/μr) (S/ηs) (1 − 1/rr)] is admissible if rr > 1 and is stable if rb < [μb/μr + ηb(1 − 1/rr)/ηs] and rc + rr > 2. The point where only the mesopredator disappears [Kb(1 − 1/rb), 0, (1/μb)Kb(1 − 1/rb)] is admissible if rb > 1 and is stable if rr < μr/μb and rc + rb > 2. In numerical simulations, at least one equilibrium point with persistence of the three species [B6*, R6*, C6*] is found if the above conditions are not fulfilled.
Several equilibirum points arise. Only the prey survive: [Kb, 0, 0]. This point is stable if λr > rr and λc > rc. Only the mesopredator survives: [0, (1 − λr/rr)S/ηs, 0]. This point is stable if λr < rr, λc > rc and λr < rr(1 −rbηs/ηb). Only the mesopredator disappears: [Kb(1 − 1/rb + λc/rbrc), 0, Kb(1 − 1/rb + λc/rbrc)(1−λc/rc)/μb]. This point is admissible if: rc > λc > rc(1 − rb). It is stable if rc > λc > rc(1 − rb), λr > rr − μr/μb(1 − λc/rc) and λc > (2 −rc − rb)/(2 − rc). The last condition implies rc < 2, which is fulfilled by studied cat populations in natura (e.g. rc = 0·43–0·55: Derenne 1976; or 0·233–1·171: van Aarde 1978, 1983). If not, one would have to replace it by: (2 − rc)λc > (2 − rc − rb). Only the prey disappears: [0, (1 − 1/rr + λc/rrrc − λr/rc)S/ηs, (1 − 1/rr + λc/rrrc − λr/rc)(1 − λc/rc)S/μrηs]. This point is admissible if: rc > λc > rc(1 − λrrr/rc − rr). It is stable if rc > λc > rc(1 − λrrr/rc − rr), rb < μb(1 − λc/rc)/μc + ηb(1 − 1/rr + λc/rrrc − λr/rc)/ηs and λc > rc(λr+2 − rc − rr)/(2 − rc). The same comment holds for the last condition (rc > 2). Only the superpredator disappears: as in the case without control, there are 0, 1 or 2 points, given by the solutions of an equation of the second degree. [−t ± Ür1Übr(t2 − 4rrηsηbu)/2rrηsηb, S/ηs + B*/ηb, 0], with here t = λrηsKb + rr[ηsKb − ηsrb (Kb − S)], and u = λrηbSKb + rr (ηb − ηsrb)(SKb). The point where the superpredator disappears is unstable if λc < rc (new introductions lead to other states). Numerically, this point can be stabilized by increasing the value of λc (increasing the control of the superpredator). In numerical simulations, at least one equilibrium point with persistence of the three species [B6*, R6*, C6*] is found if the above conditions are not fulfilled.