Periodic orbits when ca bmath.bd.psu.edu/faculty/jprevite/REU05/Scavenger2.pdfecology since their introduction in the early 20th century ([12], [20]). Lotka-Volterra models lack the

IntroductionLotka-Volterra systems of differential equations have played an important role in the development of mathematical

ecology since their introduction in the early 20th century ([12], [20]). Lotka-Volterra models lack the complexity andrealism of more recently developed population models, but their generality makes them a convenient starting point foranalyzing ecological systems. Tilman remarks that, "Because the purpose of ecology is to understand the causes of patternsin nature, we should start by studying the largest, most general, and most repeatable patterns," and "General patterns providea framework within less general patterns can be more effectively studied" [11]. Indeed, in [14] Berryman describes how theLotka-Volterra systems have served as a basis for the development of more realistic models that involve ratio-dependentfunctional responses. When new concepts in population modelling are to be studied, it is logical to begin with the mostgeneral and simplistic mathematical framework. Thus the Lotka-Volterra models provide a foundation for new modellingideas to build upon.

The classical Lotka-Volterra systems model the quantities (or densities) of species in an ecosystem under the assump-tion that the per-capita growth rate of each species is a linear function of the abundances of the species in the system [12].This framework allows for the modelling of a wide range of pairwise interactions between species, including predator-prey,competitive, and symbiotic relationships. Unfortunately, such pairwise interactions do not encapsulate the diversity ofinterspecies relationships present in observed ecosystems. In particular, the classical Lotka-Volterra systems fail to modelthe presence of scavenger species. Scavenger species subsist on carrion (the dead remains of other creatures). When ascavenger species is present in a system with a predator species and a prey species, the scavenger will benefit when thepredator kills its prey. This is not the type of pairwise interaction that can be modelled by the classical Lotka-Volterraequations, so the inclusion of scavengers requires a modification of these systems.

The important role that scavengers play in ecosystems has long been recognized. In 1877, naturalist SebastianTenney enjoined his colleagues to devote more study to scavengers [15]. Dr. Tenney would be disappointed to learn thatscavengers have long been neglected in population ecology, relative to herbivores and predators. In recent years, the field ofsystems ecology has placed great emphasis on the study of detritus and detritrivores. Several mathematical models have beendeveloped to model detritus and detritrivore abundance in ecosystems ([9], [10], [11], [12], [13]). These models all treat theamount of detritus as a distinct state-variable. The detritrivores under consideration are primarily decomposers (includingbacteria and fungi) and microinverterbrates. The analysis we put forth differs from these studies in two important respects.First, we avoid explicitly including the detritus as a variable, which allows for easier analysis and system visualization.Second, we concentrate primarily on vertebrate carrion-feeders (e.g. hyenas, vultures, and ravens) rather than decomposers.A recent review by DeValt, Rhodes, and Shivik indicates that carrion use by terrestrial vertebrates is much more prevalentthan previously thought, and that the significance of terrestrial scavenging has been underestimated by the ecological commu-nity [31]. They conclude that "a clearer perspective on carrion use by terrestrial vertebrates will improve our understandingof critical ecological processes, particularly those associated with energy flow and trophic interactions" [31].

A variety of species-specific studies of scavenger and detritrivore populations have been conducted. A thoroughlisting can be found in [7], but we will site three examples here. In [21], the impact of fisheries discards on benthic scaven-ger populations is studied. In [22], the impact of elk carrion on large carnivore populations is studied. In [23], a model ofwolf-elk population dynamics that addresses the amount of carrion available to scavengers is developed. This model ishighly specific to the wolf and elk of Yellowstone Park. It is a complicated discrete model using Leslie matrices, and theonly solutions available are numerical. We attempt to model scavengers in a far more general format, and we use analyticrather than numeric methods.

Stephen Heard [7] developed the notion of processing chain ecology. Heard introduced compartmental models weredifferent species alter the state of resources, thereby affecting the type of resources available to other species. This type ofmodelling can be applicable to scavenger population modelling because a predator could be viewed as altering the state of aresource (prey) to a state that is useful to a scavenger (carrion). Heard's model was primarily directed at the processing offauna and non-living resources (as opposed to the processing of a prey animal), as is apparent in his extensive table ofapplications. Heard's model, like the detritus models mentioned above, also suffers from the drawbacks of including resourcestates (carrion) as explicit variables, thereby increasing the dimensionality of the system.

Lotka-Volterra background In general, an nth order Lotka-Volterra system takes the form:

dxiÅÅÅÅÅÅÅÅÅdt = li xi + xi ⁄ j=1n Mij xj , i = 1, 2, …, nH1L

where " i, j li and Mi j are real constants [20]. Ecologists frequently use an nth order Lotka-Volterra system to model thepopulations of n different species interacting in a food web. The n ä n matrix M with entries Mij is called the interactionmatrix of the system. For predator-prey Lotka-Volterra systems, the interaction matrix M has a special form:" Hi, jL œ 81, 2, …, n< such that i j, Mi j Mji § 0 [19]. The classical two-species predator-prey Lotka-Volterra model isgiven by:

dxÅÅÅÅÅÅÅdt = ax - bxydyÅÅÅÅÅÅÅdt = -cy + dxy H2L

where x is the population of a prey species, y is the population of a predator species, and a, b, c, and d are positive constants.When analyzing the qualitative behavior of systems such as (2), it is convenient to rescale variables so that some of theconstants involved equal one. The number of constants that may be eliminated by variable rescaling will generally dependon the number of variables in the system and the structure of the differential equations. For the remainder of this paper,differential equations will be written in a rescaled format so that the maximum possible number of constants are unity. Forexample, (2) may be rescaled as:

dxÅÅÅÅÅÅÅdt = x - xydyÅÅÅÅÅÅÅdt = -ay + xy H3L

This set of equations has equilibria at H0, 0L and Ha, 1L. Lotka and Volterra independently demonstrated that the trajectoriesof (3) in the positive quadrant are closed curves surrounding the non-zero equilibrium [20]. Furthermore, they showed thatthe time averages of x and y over these trajectories are simply the x and y coordinates of the non-zero equilibrium:1ÅÅÅÅÅT Ÿ0

T x „ t = a 1ÅÅÅÅÅT Ÿ0T y „ t = 1

where T is the period of the trajectory under consideration. Integrating both sides of the first equation in H3L from 0 to Tshows that:1ÅÅÅÅÅT Ÿ0

T xy „ t = a.

Introduction of the Scavenger SystemThe standard nth order Lotka-Volterra predator prey system is constructed under the assumption that the per capita

growth rate of species xi can be written as the linear function li + ⁄ j=1n Mij x j , where li is the intrinsic growth rate of xi , and

Mij describes the impact of species xj on the per capita growth rate of species xi . The term Mii is usually zero or negative,and the term Mii xi takes into account the impact of intraspecific competition on the per capita growth rate of xi [19]. Theterms Mij xj , j = 1, 2, …, i - 1, i + 1, …, n, take into account the predator prey relationships between xi and the otherspecies in the community.

We propose a system of differential equations, similar to the Lotka-Volterra predator prey model, that allows scaven-ger species to be modeled in addition to the usual predator and prey species. We begin with the most basic such model.Consider a prey species x and a predator species y that obey the system of differential equations (3). Now we add a scaven-ger species z that feeds on the carrion produced when an animal from the population y kills an animal of the population x.For the moment, we ignore the obvious possibility that z would also feed on carrion produced from natural death. Thenumber of carrion produced by y preying on x should be proportional to the number of interactions between members of thepopulations y and x, and hence proportional to the product xy. The per capita growth rate of z should be a combination of theintrinsic growth rate of z and a term accounting for the number of carrion in the environment. Hence we arrive at the model:

dxÅÅÅÅÅÅÅdt = x - xydyÅÅÅÅÅÅÅdt = -ay + xydzÅÅÅÅÅÅdt = -bz + cxyz

H4LProposition: Let HxHtL, yHtL, zHtLL be a solution to (4) with initial condition HxH0L, yH0L, zH0LL = Hx0 , y0 , z0 L in the positiveoctant. If ca = b, then Ha, 1, sL 0 < s < ¶ is a line of fixed points, and if HxHtL, yHtL, zHtLL is not on this line, then it is periodic.If ca > b, then limtØ¶ zHtL = ¶. If ca < b, then limtØ¶ zHtL = 0.

Proof: Let HxHtL, yHtL, zHtLL be a solution to H4L with initial condition in the positive octant. It is easily seen that the coordinateplanes are invariant with respect to (4). For the xy-plane, the upward pointing unit normal vector is H0, 0, 1L.H0, 0, 1L ÿ I dxÅÅÅÅÅÅÅdt , dyÅÅÅÅÅÅÅdt , dzÅÅÅÅÅÅdt M …z=0 = 0, so the plane is invariant. Similar statements hold for the yz and xz planes. The invariance ofthe coordinate planes implies that HxHtL, yHtL, zHtLL must lie in the positive octant for all t > 0.

Let p : !3 Ø !2 be the projection given by pHx, y, zL = Hx, yL. Note that the first two equations in (4) are simply the classical2-dimensinonal Lotka-Volterra predator-prey equations. Neither of these equations depend on z, so pHxHtL, yHtL, zHtLL isindependent of zHtL. Recall that solutions to the classical 2-dimensional Lotka-Volterra predator prey equations are closedcurves in the positive quadrant. Then pHxHtL, yHtL, zHtLL must coincide with one of the closed curves, which implies that ,thesolutions to H4L are confined to cylinders parallel to the z-axis. This also means that pHxHtL, yHtL, zHtLL must be periodic withsome period T > 0.

Define a return map n Ø zn by zn = zHnTL.From the Lotka-Volterra background section, recall that 1ÅÅÅÅÅT Ÿ0

T xHtL yHtL „ t = a. xHtL and yHtL are periodic, so " n œ ",1ÅÅÅÅÅT Ÿ0

T xHtL yHtL „ t = 1ÅÅÅÅÅT ŸnTHn+1L T xHtL yHtL „ t = a. In the positive octant, dÅÅÅÅÅÅdt lnHzHtLL = z' HtLÅÅÅÅÅÅÅÅÅÅzHtL = -b + c xHtL yHtL, so for any n œ ",

1ÅÅÅÅÅT ŸnTHn+1L T

-b + c xHtL yHtL „ t = 1ÅÅÅÅÅT ŸnTHn+1L T dÅÅÅÅÅÅdt lnHzHtLL „ t = 1ÅÅÅÅÅT lnI zHHn+1L T LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅzHnTL M = 1ÅÅÅÅÅT lnI zn+1ÅÅÅÅÅÅÅÅÅÅzn

M.If ca > b, then 1ÅÅÅÅÅT lnI zn+1ÅÅÅÅÅÅÅÅÅÅzn

M = 1ÅÅÅÅÅT ŸnTHn+1L T

-b + c xHtL yHtL „ t = -b + ca > 0, so zn+1 > zn . Therefore zn , n = 1, 2, … is a monotoni-cally increasing sequence. Suppose, for contradiction, that limnØ¶ zn = z* . Then there would have to exist a periodicsolution to (4), HxHtL, yHtL, zHtLL, with initial condition Hx0 , y0 , z* L. Defining a return map n Ø zn for this solution by zn = zHnTLand proceeding as before, we see that zn+1 > zn " n œ ", violating the periodicity of HxHtL, yHtL, zHtLL. Hence limnØ¶ zn = ¶.

If ca < b, then 1ÅÅÅÅÅT lnI zn+1ÅÅÅÅÅÅÅÅÅÅznM = 1ÅÅÅÅÅT ŸnT

Hn+1L T-b + c xHtL yHtL „ t = -b + ca < 0, so zn+1 < zn . Therefore zn , n = 1, 2, … is a monotoni-

cally decreasing sequence. Suppose, for contradiction, that limnØ¶ zn = z* > 0. Then there would have to exist a periodicsolution to (4), HxHtL, yHtL, zHtLL, with initial condition Hx0 , y0 , z* L. Defining a return map n Ø zn for this solution by zn = zHnTLand proceeding as before, we see that zn+1 < zn " n œ ", violating the periodicity of HxHtL, yHtL, zHtLL. Hence limnØ¶ zn = 0.

If ca = b, then 1ÅÅÅÅÅT lnI zn+1ÅÅÅÅÅÅÅÅÅÅznM = 1ÅÅÅÅÅT ŸnT

Hn+1L T-b + c xHtL yHtL „ t = -b + ca = 0, so zn+1 = zn . ThereforeHxHHn + 1LTL, yHHn + 1LTL, zHHn + 1LTLL = HxHnTL, yHnTL, zHnTLL " n œ " , so HxHtL, yHtL, zHtLL is periodic with period T.

If x0 = a and y0 = 1, then dxÅÅÅÅÅÅÅdt = dyÅÅÅÅÅÅÅdt = dzÅÅÅÅÅÅdt = 0, so Hx0 , y0 , z0 L is a fixed point, regardless of the initial value z0 . HenceHa, 1, sL 0 < s < ¶ is a line of fixed points.

Periodic orbits when ca = b

Unbounded orbits when ca > b

Trajectories appoach xy plane when ca < b.

Adding intraspecific competitionThe above model has the obvious flaw that the species z will go extinct or grow unbounded unless ca = b. The set of

parameter values such that ca = b is a set of measure zero, so the probability of this occurring in nature is zero. We seek amodel that gives bounded, non-zero trajectories without having to use a degenerate set of parameter values, so we modify (4)in the following way:

dxÅÅÅÅÅÅÅdt = x - xydyÅÅÅÅÅÅÅdt = -ay + xydzÅÅÅÅÅÅdt = -bz + cxyz - z2

H5LIn this case, the term z2 accounts for the intraspecific competition of the scavengers. This system has equilibria atH0, 0, 0L, Ha, 1, 0L, H0, 0, -bL, and Ha, 1, -b + acL. The third of these equilibria occurs in negative space; hence we shallignore it. We are particularly interested in the case where an equillibria occurs in positive space, so we restrict our attentionto the case ac > b. H0, 0, 0L is a saddle point. H1, a, 0L has a two-dimensional center manifold tangent to the xy-plane. Thispoint has an unstable manifold tangent to the line Ha, 1, sL 0 < s < ¶.

Proposition: Let HxHtL, yHtL, zHtLL be a solution to (5) with initial condition HxH0L, yH0L, zH0LL = Hx0 , y0 , z0 L in the positiveoctant and suppose ca > b. If x0 = a and y0 = 1, then limtØ¶ zHtL = ac - b. Otherwise, HxHtL, yHtL, zHtLL approaches a uniquelimit cycle in the positive octant.

Proof: Let HxHtL, yHtL, zHtLL be a solution to H5L with initial condition in the positive octant. As in the basic scavenger model,the coordinate planes are invariant with respect to (5), and hence a solution with initial point in the positive octant cannotleave the positive octant.

In the special case x0 = a and y0 = 1, then xHtL = a, yHtL = 1 and dzÅÅÅÅÅÅdt = zH-b + ac - zHtLL. dzÅÅÅÅÅÅdt < 0 for zHtL > ac - b and dzÅÅÅÅÅÅdt > 0for zHtL > ac - b, so limtØ¶ zHtL = ac - b.

Suppose x0 a or y0 1. Let p :!3 Ø !2 be the projection given by pHx, y, zL = Hx, yL. Note that the first two equations in(4) are simply the classical 2-dimensinonal Lotka-Volterra predator-prey equations. Neither of these equations depend on z,so pHxHtL, yHtL, zHtLL is independent of zHtL. Recall that solutions to the classical 2-dimensional Lotka-Volterra predator preyequations are closed curves in the positive quadrant. Then pHxHtL, yHtL, zHtLL must coincide with one of the closed curves. Inother words, the solutions to H5L are confined to cylinders parallel to the z-axis. A choice of initial point Hx0 , y0 L completelydetermines the invariant cylinder that HxHtL, yHtL, zHtLL will remain on for all t > 0. Also from the 2-dimensional Lotka--Volt-erra equations, it is clear that pHxHtL, yHtL, zHtLL must be periodic with some period T > 0.

Define a return map n Ø zn by zn = zHnTL.We shall break the remainder of the proof down into several Lemmas.

Lemma #1: Any return sequence 8zn < for a solution HxHtL, yHtL, zHtLL of (5) must be monotonic.Proof: For contradiction, suppose that HxHtL, yHtL, zHtLL is a solution with non-monotonic return sequence 8zn <. Without lossof generality, suppose that there exists a k œ " such that zk < zk+1 and zk+1 > zk+2 . Parameterize the section of zHtL from zk tozk+1 by g1 HtL so that g1 H0L = zk and g1 HTL = zk+1 . Parameterize the section of zHtL from zk+1 to zk+2 by g2 HtL so thatg2 H0L = zk+1 and g2 HTL = zk+2 . Then the section of zHtL from zk to zk+2 can be parameterized by:

g3 HtL =9 g1 HtL 0 § t < Tg2 Ht - TL T § t § 2T .

Let wHtL = g2 HtL - g1 HtL and note that wH0L > 0 and wHTL < 0. By the intermediate value theorem, there exists a t`, 0 <t` < Tsuch that wHt`L = 0 and hence g2 Ht`L = g1 Ht`L . This implies that g3 Ht`L = g1 Ht`L = g2 Ht`L = g3 Ht` + TL, so g3 HtL, and hence zHtL, areperiodic with period T . But this implies HxHtL, yHtL, zHtLL is periodic with period T , and hence z1 = z2 = …, violating thehypothesis that zk < zk+1 and zk+1 > zk+2 .Thus any return sequence 8zn < for any solution HxHtL, yHtL, zHtLL must be monotonic.

Lemma #2: Every invariant cylinder of the system (5) must have at least one non-zero limit cycle.Proof: Consider a solution of (5), HxHtL, yHtL, zHtLL with initial point Hx0 , y0 , 0L, so that zHtL = 0 for all t > 0. Let T be theperiod of HxHtL, yHtLL. Let K be the invariant cylinder that contains HxHtL, yHtL, zHtLL for all t > 0. Clearly Ÿ0

T zHtL „ t = 0. Definiteintegrals are continuously dependent on their initial conditions, so there exists a d > 0 such that, if z0 < d and HxHtL, yHtL, zHtLLis a solution of (5) with initial point Hx0 , y0 , z0 L, then Ÿ0

T zHtL „ t < TH-b + acL. Define a return map n Ø zn by zHnTL = zn .Then lnI z1ÅÅÅÅÅÅz0

M = Ÿ0T H 1ÅÅÅÅz L H dzÅÅÅÅÅÅdt L „ t = Ÿ0

T-b + cxy - z „ t = -bT + caT - Ÿ0

T z „ t > 0, so z1 > z0 and hence 8zn < is a monotonicallyincreasing sequence. Note that HxHtL, yHtL, zHtLL is contained in K for all t > 0.

dzÅÅÅÅÅÅdt < 0 if zHtL > -b + xHtL yHtL and dzÅÅÅÅÅÅdt > 0 if zHtL < -b + xHtL yHtL. Let k = max 8-b + cxy » Hx, y, zL œ K<. zHtL has initial pointbelow the surface -b + xHtL yHtL and zHtL is decreasing above the surface -b + cxHtL yHtL, so zHtL. Let W be the intersection of Kand the surface -b + cxy. Then, if zHtL intersects W, the point of intersection must be a local maximum for zHtL. k is thelargest of all such possible maxima, so k is an upper bound for zHtL. In particular, this means that 8zn< is a bounded sequence.8zn < is a bounded monotonic sequence, so there exists a z* > 0 such that limnØ¶ zn = z* . Then the solution to (5) with initialcondition Hx0 , y0 , z* L is a limit cycle on K.

Intersecstion of the surface z = -b + cxy and an invariant cylinder K.Lemma #3: Each invariant cylinder has a unique non-zero limit cycle.Proof: Suppose, for contradiction, that K is an invariant cylinder of H5L and that there are two limit cycles on K. LetHx0 , y0 , z0 L be a point on the first limit cycle, and let Hx0 , y0 , zè0 L be a point on the second limit cycle. Let HxHtL, yHtL, zHtLL be asolution with initial point Hx0 , y0 , z0 L and let HxèHtL, yèHtL, zèHtLL be a solution with initial point Hx0 , y0 , zè0 L. The first two equa-tions in H5L are independent of z, so by the uniqueness of solutions to initial value problems, HxHtL, yHtLL = HxèHtL, yèHtLL " t > 0.HxHtL, yHtL, zHtLL and HxèHtL, yèHtL, zèHtLL are limit cycles, so they are periodic with some period T > 0. Thus, zHTL = zH0L andzèHTL = zèH0L and hence lnI zHT LÅÅÅÅÅÅÅÅÅÅzH0L M = lnI zèHTLÅÅÅÅÅÅÅÅÅÅzèH0L M = 0. Then Ÿ0T I 1ÅÅÅÅz M dzÅÅÅÅÅÅÅdt „ t = lnI zHTLÅÅÅÅÅÅÅÅÅÅzH0L M = 0, so Ÿ0T -b + cx y - z „ t = 0. Ÿ0T x y „ t = Ta,so Ÿ0T z „ t = THac - bL. A similar set of calculations show that Ÿ0T zè „ t = THac - bL. But this implies that there exists some t*such that zHt* L = zèHt* L. Then the two solutions HxHtL, yHtL, zHtLL and HxèHtL, yèHtL, zèHtLL would pass through the same point, whichis impossible. Thus each invariant cylinder contains a unique limit cycle.Lemma #4: Every solution HxHtL, yHtL, zHtLL with initial point Hx0 , y0 , z0 L in the positive octant is bounded and does notintersect the coordinate planes.Proof: From the 2-dimensional Lotka-Volterra equations, we know that HxHtL, yHtL, zHtLL lies on an invariant cylinder thatdoes not intersect the xz or yz planes. From the proof of Lemma #2, HxHtL, yHtL, zHtLL cannot decrease to intersect the xy plane.If z0 < k, then zHtL is bounded above by k. If z0 > k, then zHtL is decreasing until zHtL § -b + cxHtL yHtL. Thus zHtL is decreasinguntil zHtL § k, after which time it is bounded above by k.Consider a solution HxHtL, yHtL, zHtLL to (5) confined to an invariant cylinder K determined by Hx0 , y0 L. From the four Lemma'sabove, and applying the Ponicare-Bendixson theorem to the 2-dimensinoal cylinder K, HxHtL, yHtL, zHtLL must approach theunique non-zero limit cycle on K as t Ø ¶.Limit cycle on one invariant cylinderMore realistic generalizationScavenging species do not solely subsist on carrion produced by predation of one species on another. Scavengersalso consume carrion produced by natural death (e.g. starvation, disease, accidents). A suitable modification of our model, inwhich the scavenger species z benefits from carrion produced from both natural death of x and y and predation is:dxÅÅÅÅÅÅÅdt = x - xydyÅÅÅÅÅÅÅdt = -ay + xydzÅÅÅÅÅÅdt = -bz + cxyz + dxz + eyz - z2 H6LWe are interested in the case when ac + ad + e > b, because when this occurs there is an equilibrium of (6) in positive space.In this case there are four total equilibria;, one of which is H0, 0, -bL, which has no physical meaning and will be discarded.H0, 0, 0L is a saddle point. Ha, 1, 0L is an equilibrium with a 2-dimensional center manifold tangent to the xy-plane and anunstable manifold tangent to the line Ha, 1, sL 0 < s < ¶. H1, a, ac + ad + e - bL is an equilibrium with a two dimensionalcenter manifold and a stable manifold tangent to the line Ha, 1, sL 0 < s < ¶.The same arguments used for the system H5L show that the trajectories of H6L are confined to invariant cylinders and that eachcylinder has a unique attracting non-zero limit cycle.Limit cycle on one invariant cylinderEcological notesThe scavenger systems have several interesting connections to other ecological concepts. First, it should be notedthat the above models conform to the general criteria in [16] and [17] for acceptable population models. We do not claimthat these scavenger systems are completely realistic, but we feel that they serve as a starting point for more complicated andaccurate models.A notable feature of (4), (5), and (6) is that these systems include higher-order interactions. Although the term"higher -order interaction" has been used in several different ways in the ecological literature, it is usually defined as "aninteraction that occurs whenever one species effects the nature of the direct interaction between two other species" [2]. Ineach of our systems dzÅÅÅÅÅÅÅdx depends on y and dzÅÅÅÅÅÅÅdy depends on x, so our model contains higher order interactions. The presence ofhigher order interactions in ecosystems has sparked much debate in the ecological community [1], [2], [3], and [4]. Higherorder interaction models significantly broaden the range of possible ecosystem dynamics from the classical models that onlytake into account pairwise interaction. The scavenger models proposed here provide dynamically simple examples of higherorder interaction systems, and they would make an excellent pedagogical tool for the introduction of this ecological concept.The product form of the carrion term cxyz in (5) makes this system somewhat similar to ecosystem models of"interactively essential resources", which are described in [5], [6], and [8]. In ecology, Von Liebig's Law of the Minimumstates that the growth rate of a species is determined by the most limiting resource [6]. Resources that obey Von Liebig'sLaw are referred to as perfectly essential; those that do not are referred to as interactively essential. Analysis of systemsinvolving interactively essential resources have mainly focused on plants that depend on different chemical combinations.For example, phytoplankton depends on the interaction between iron and nitrogen [6]. In our models (4) and (5), the predatorspecies y and the prey species x could be viewed as interactively essential resources for the scavenger species z.The scavenger systems proposed here will no doubt need to be greatly enhanced before being directly applicable toreal populations. One important direction for enhancement involves an ecological concept discussed in [31]: the competitivebalance between scavengers and decomposers. Decomposers, most notably bacteria and fungi, directly compete with scaven-gers for carrion. Many types of bacteria produce toxic substances that spoil the carrion and prevent scavengers from takingadvantage of the resource. To overcome this, scavengers have evolved in two manners: first they have become more efficientat quickly finding and consuming carrion before decomposers can spoil it, and second they have developed a limited abilityto "detoxify" spoiled carrion. The competitive balance between decomposers and scavengers is critical to the structure offood webs; resources consumed by scavengers keep energy and nutrients at higher trophic levels, while resources consumedby decomposers recycle energy and resources to low trophic levels. An important addition to the scavenger models presentedhere would be the inclusion of competitive decomposers.Representation of quasi-polynomial systems as Lotka-Volterra equationsThe scavenger models introduced above are not of the form (1) and hence are not Lotka-Volterra systems. They are,however, members of a much broader class of differential equations known as quasi-polynomial systems. In this section, weshow how the work of Hernandez-Bermejo, Fairen, and Brenig [29] and Gleria, Figueiredo, and Filho [30] concerningquasi-polynomial systems can be used to analyze our scavenger models. In general, an nth-order quasi-polynomial (QP)system has the form:x° i = li xi + xi ‚ j=1m Aij ‰k=1n xkBjk i = 1, 2, …, n H7LHere l is an n-dimensional column vector, A is a n äm matrix and B is a män matrix. The entries of l, A, and B can be anyarbitrary real numbers. When B is the identity matrix, the system is a Lotka-Volterra system. n is the order of the system andm is the number of different monomial expressions under consideration. H7L is form invariant under the transformationxi = ‰ j=1n y jDij , i = 1, 2, …, n H8Lwhere Di j are the entries of a n ä n invertible matrix D. The new system takes the formy° i = li yi + yi ‚ j=1m Ai j ‰k=1n ykB jk i = 1, 2, …, n H9Lwhere l = D-1 l, A = D-1 A, and B = BD.Brenig, Herndandez-Bermejo, and Fairen [29] have shown that the solutions to (7) are topologically equivalent tothose of (9). The set of QP systems are partitioned into different equivalence classes, where two systems belong to the sameequivalence class if and only if there is a transformation of the form H8L that relates them. The qualitative behavior of allsystems in an equivalence class are essentially the same. Finding the most useful representation of a quasi-polynomialsystem is not always immediately apparent, but the references [24], [25], [26], [27], [28], [29], and [30] give some guidance.In [22], is shown that every QP system is equivalent to one in which m ¥ n and A and B have maximal rank. In [30], particu-lar attention is paid to transforming a QP system into an equivalent Lotka-Volterra system. This method requires the introduc-tion of embedding variables to obtain a situation in which m = n. Thus the system becomes more tractable because thedegree of nonlinearity is decreased, but more complicated because the dimensionality is increased. The true dynamics of thesystem are contained in manifold of dimension n embedded in the dimension m space. The solution trajectories on thismanifold will be topologically equivalent to the original system. When the general procedure is applied to our scavengersystem (1), we obtain:y° 1 = y1 - y1 y2y° 2 = -ay2 + y1 y2y° 3 = -by3 + cy3 y4y° 4 = H1 - aL y4 + y1 y4 - y2 y4The dynamics of the original system take place on the manifold defined by y1 -1 y2 -1 y4 = 1. This transformation wassomewhat trivial, but more complicated possibilities exist as well. In [31], it is shown how a differential time reparameteriza-tion, dt = ¤k=1m xk -Brk dt, can lead to a wide variety of representations for a given QP system. The QP formalism is notnecessary for the analysis of the preceding scavenger systems, but more complicated scavenger systems require such tech-niques.Four-species systemWe now turn our attention to a system of four interacting species. The species x, y, and z comprise a simple three-spe-cies food chain. Chauvet, Paullet, Prevtie, and Walls showed that a three species food chain without intraspecific competi-tion terms has stability in only a degenerate case. Thus we add an intraspecific competition term to the population x to makethe system stable, and hence physically realistic. We next add a scavenger species w that survives by consuming the carrionfrom the predation of z on y and the predation of y on x. To make the system stable, we add an intraspecific competition termto the scavenger species w. Our model takes the following form:x° = x - bxy - x2y° = -ay + dxy - yzz° = -cz + yzw° = -ew + f wyz + gwxy - w2We are interested in the case where a positive equilibrium occurs, so we restrict our attention to the parameter values thatsatisfy: bc < 1, d > aÅÅÅÅÅÅÅÅÅÅÅÅ1-bc , and cHd f + gL H1 - bcL > e + ac f , in which caseH1 - bc, c, d - a - bcd, -e - a c f + c d f - b c2 d f + c g - b c2 gL is an equilibrium in positive space. The Jacobian of thesystem evaluated at this equilibrium has a positive determinant and a negative trace, indicating that the sum of the eigenval-ues is negative and the product is positive. Furthermore, one of the eigenvalues can be solved for algebraically and isnegative. Thus there are three possible combinations of eigenvalues: 1) there are four real negative eigenvalues, 2) there arefour real eigenvalues, two of which are negative and two of which are positive, or 3) there are two negative real eigenvaluesand two complex conjugate eigenvalues. These three possibilities leave open the question of local stability at the equilib-rium. To solve the question of stability, we us an quasi-monomial transformation to obtain the following system:u° 1 = u1 - u1 2 - b u1 u2u° 2 = -a u2 + d u1 u2 - u2 u3u° 3 = -c u3 + u3 u2u° 4 = -e u4 - u4 2 + g u4 u5 + f u4 u6u° 5 = H1 - aL u5 + Hd - 1L u1 u5 - b u2 u5 - u3 u5u° 6 = -Ha + cL u6 + d u1 u6 + u2 u6 - u3 u6Here we are interested in the subspace defined by u5ÅÅÅÅÅÅÅÅÅÅÅÅÅu1 u2 = 1 and u6ÅÅÅÅÅÅÅÅÅÅÅÅÅu2 u3 = 1.Using the system above, it is possible to construct an Lyapunov functionf Hu1 , u2 , u3 , u4 , u5 , u6 L = ⁄i=16 ui - ui* LnI uiÅÅÅÅÅÅÅui* M - ui* , whereu1* = 1 - b c, u2* = c, u3* = d - a - bcd, u4* = -e - a c f + c d f - b c2 d f + c g - b c2 g, u5* = cH1 - bcL, u6* = cHd - a - bcdLThis Lyapunov function, restricted to the subspace determined by u5ÅÅÅÅÅÅÅÅÅÅÅÅÅu1 u2 = 1 and u6ÅÅÅÅÅÅÅÅÅÅÅÅÅu2 u3 = 1, and written in terms of our originalcoordinates, gives an Lyapunov function for the original sys-tem:gHx, y, z, wL = x - x* LnH xÅÅÅÅÅÅÅx* L - x* + y - y* LnI yÅÅÅÅÅÅÅy* M - y* + z - z* LnH zÅÅÅÅÅÅz* L -z* + w - w* LnH wÅÅÅÅÅÅÅÅw* L - w* + x y - x* y* LnI x yÅÅÅÅÅÅÅÅÅÅÅÅx* y* M - x* y* + y z - y* z* LnI y zÅÅÅÅÅÅÅÅÅÅÅÅy* z* M - y* z*where x* = 1 - bc, y* = c, x* = d - a - bcd, x* = - e - a c f + c d f - b c2 d f + c g - b c2 g.Some really nasty algebra shows that this is, indeed, an Lyapunov function, and hence the equilibrium in positive space isglobally stable.

Scavenger.nb 1

Printed by Mathematica for Students
















T xy „ t = a.





































Let wHtL = g2 HtL - g1 HtL and note that wH0L > 0 and wHTL < 0. By the intermediate value theorem, there exists a t`, 0 <t` < Tsuch that wHtL = 0 and hence g2 HtL = g1 HtL . This implies that g3 HtL = g1 HtL = g2 HtL = g3 Ht + TL, so g3 HtL, and hence zHtL, areperiodic with period T . But this implies HxHtL, yHtL, zHtLL is periodic with period T , and hence z1 = z2 = …, violating thehypothesis that zk < zk+1 and zk+1 > zk+2 .Thus any return sequence 8zn < for any solution HxHtL, yHtL, zHtLL must be monotonic.





T-b + cxy - z „ t = -bT + caT - Ÿ0



Intersecstion of the surface z = -b + cxy and an invariant cylinder K.

Lemma #3: Each invariant cylinder has a unique non-zero limit cycle.Proof: Suppose, for contradiction, that K is an invariant cylinder of H5L and that there are two limit cycles on K. LetHx0 , y0 , z0 L be a point on the first limit cycle, and let Hx0 , y0 , zè0 L be a point on the second limit cycle. Let HxHtL, yHtL, zHtLL be asolution with initial point Hx0 , y0 , z0 L and let HxèHtL, yèHtL, zèHtLL be a solution with initial point Hx0 , y0 , zè0 L. The first two equa-tions in H5L are independent of z, so by the uniqueness of solutions to initial value problems, HxHtL, yHtLL = HxèHtL, yèHtLL " t > 0.HxHtL, yHtL, zHtLL and HxèHtL, yèHtL, zèHtLL are limit cycles, so they are periodic with some period T > 0. Thus, zHTL = zH0L andzèHTL = zèH0L and hence lnI zHT LÅÅÅÅÅÅÅÅÅÅzH0L M = lnI zèHTLÅÅÅÅÅÅÅÅÅÅzèH0L M = 0. Then Ÿ0

T I 1ÅÅÅÅz M dzÅÅÅÅÅÅÅdt „ t = lnI zHTLÅÅÅÅÅÅÅÅÅÅzH0L M = 0, so Ÿ0T

-b + cx y - z „ t = 0. Ÿ0T x y „ t = Ta,

so Ÿ0T z „ t = THac - bL. A similar set of calculations show that Ÿ0

T zè „ t = THac - bL. But this implies that there exists some t*

such that zHt* L = zèHt* L. Then the two solutions HxHtL, yHtL, zHtLL and HxèHtL, yèHtL, zèHtLL would pass through the same point, whichis impossible. Thus each invariant cylinder contains a unique limit cycle.

Lemma #4: Every solution HxHtL, yHtL, zHtLL with initial point Hx0 , y0 , z0 L in the positive octant is bounded and does notintersect the coordinate planes.Proof: From the 2-dimensional Lotka-Volterra equations, we know that HxHtL, yHtL, zHtLL lies on an invariant cylinder thatdoes not intersect the xz or yz planes. From the proof of Lemma #2, HxHtL, yHtL, zHtLL cannot decrease to intersect the xy plane.If z0 < k, then zHtL is bounded above by k. If z0 > k, then zHtL is decreasing until zHtL § -b + cxHtL yHtL. Thus zHtL is decreasinguntil zHtL § k, after which time it is bounded above by k.

Consider a solution HxHtL, yHtL, zHtLL to (5) confined to an invariant cylinder K determined by Hx0 , y0 L. From the four Lemma'sabove, and applying the Ponicare-Bendixson theorem to the 2-dimensinoal cylinder K, HxHtL, yHtL, zHtLL must approach theunique non-zero limit cycle on K as t Ø ¶.

Limit cycle on one invariant cylinderMore realistic generalizationScavenging species do not solely subsist on carrion produced by predation of one species on another. Scavengersalso consume carrion produced by natural death (e.g. starvation, disease, accidents). A suitable modification of our model, inwhich the scavenger species z benefits from carrion produced from both natural death of x and y and predation is:dxÅÅÅÅÅÅÅdt = x - xydyÅÅÅÅÅÅÅdt = -ay + xydzÅÅÅÅÅÅdt = -bz + cxyz + dxz + eyz - z2 H6LWe are interested in the case when ac + ad + e > b, because when this occurs there is an equilibrium of (6) in positive space.In this case there are four total equilibria;, one of which is H0, 0, -bL, which has no physical meaning and will be discarded.H0, 0, 0L is a saddle point. Ha, 1, 0L is an equilibrium with a 2-dimensional center manifold tangent to the xy-plane and anunstable manifold tangent to the line Ha, 1, sL 0 < s < ¶. H1, a, ac + ad + e - bL is an equilibrium with a two dimensionalcenter manifold and a stable manifold tangent to the line Ha, 1, sL 0 < s < ¶.The same arguments used for the system H5L show that the trajectories of H6L are confined to invariant cylinders and that eachcylinder has a unique attracting non-zero limit cycle.Limit cycle on one invariant cylinderEcological notesThe scavenger systems have several interesting connections to other ecological concepts. First, it should be notedthat the above models conform to the general criteria in [16] and [17] for acceptable population models. We do not claimthat these scavenger systems are completely realistic, but we feel that they serve as a starting point for more complicated andaccurate models.A notable feature of (4), (5), and (6) is that these systems include higher-order interactions. Although the term"higher -order interaction" has been used in several different ways in the ecological literature, it is usually defined as "aninteraction that occurs whenever one species effects the nature of the direct interaction between two other species" [2]. Ineach of our systems dzÅÅÅÅÅÅÅdx depends on y and dzÅÅÅÅÅÅÅdy depends on x, so our model contains higher order interactions. The presence ofhigher order interactions in ecosystems has sparked much debate in the ecological community [1], [2], [3], and [4]. Higherorder interaction models significantly broaden the range of possible ecosystem dynamics from the classical models that onlytake into account pairwise interaction. The scavenger models proposed here provide dynamically simple examples of higherorder interaction systems, and they would make an excellent pedagogical tool for the introduction of this ecological concept.The product form of the carrion term cxyz in (5) makes this system somewhat similar to ecosystem models of"interactively essential resources", which are described in [5], [6], and [8]. In ecology, Von Liebig's Law of the Minimumstates that the growth rate of a species is determined by the most limiting resource [6]. Resources that obey Von Liebig'sLaw are referred to as perfectly essential; those that do not are referred to as interactively essential. Analysis of systemsinvolving interactively essential resources have mainly focused on plants that depend on different chemical combinations.For example, phytoplankton depends on the interaction between iron and nitrogen [6]. In our models (4) and (5), the predatorspecies y and the prey species x could be viewed as interactively essential resources for the scavenger species z.The scavenger systems proposed here will no doubt need to be greatly enhanced before being directly applicable toreal populations. One important direction for enhancement involves an ecological concept discussed in [31]: the competitivebalance between scavengers and decomposers. Decomposers, most notably bacteria and fungi, directly compete with scaven-gers for carrion. Many types of bacteria produce toxic substances that spoil the carrion and prevent scavengers from takingadvantage of the resource. To overcome this, scavengers have evolved in two manners: first they have become more efficientat quickly finding and consuming carrion before decomposers can spoil it, and second they have developed a limited abilityto "detoxify" spoiled carrion. The competitive balance between decomposers and scavengers is critical to the structure offood webs; resources consumed by scavengers keep energy and nutrients at higher trophic levels, while resources consumedby decomposers recycle energy and resources to low trophic levels. An important addition to the scavenger models presentedhere would be the inclusion of competitive decomposers.Representation of quasi-polynomial systems as Lotka-Volterra equationsThe scavenger models introduced above are not of the form (1) and hence are not Lotka-Volterra systems. They are,however, members of a much broader class of differential equations known as quasi-polynomial systems. In this section, weshow how the work of Hernandez-Bermejo, Fairen, and Brenig [29] and Gleria, Figueiredo, and Filho [30] concerningquasi-polynomial systems can be used to analyze our scavenger models. In general, an nth-order quasi-polynomial (QP)system has the form:x° i = li xi + xi ‚ j=1m Aij ‰k=1n xkBjk i = 1, 2, …, n H7LHere l is an n-dimensional column vector, A is a n äm matrix and B is a män matrix. The entries of l, A, and B can be anyarbitrary real numbers. When B is the identity matrix, the system is a Lotka-Volterra system. n is the order of the system andm is the number of different monomial expressions under consideration. H7L is form invariant under the transformationxi = ‰ j=1n y jDij , i = 1, 2, …, n H8Lwhere Di j are the entries of a n ä n invertible matrix D. The new system takes the formy° i = li yi + yi ‚ j=1m Ai j ‰k=1n ykB jk i = 1, 2, …, n H9Lwhere l = D-1 l, A = D-1 A, and B = BD.Brenig, Herndandez-Bermejo, and Fairen [29] have shown that the solutions to (7) are topologically equivalent tothose of (9). The set of QP systems are partitioned into different equivalence classes, where two systems belong to the sameequivalence class if and only if there is a transformation of the form H8L that relates them. The qualitative behavior of allsystems in an equivalence class are essentially the same. Finding the most useful representation of a quasi-polynomialsystem is not always immediately apparent, but the references [24], [25], [26], [27], [28], [29], and [30] give some guidance.In [22], is shown that every QP system is equivalent to one in which m ¥ n and A and B have maximal rank. In [30], particu-lar attention is paid to transforming a QP system into an equivalent Lotka-Volterra system. This method requires the introduc-tion of embedding variables to obtain a situation in which m = n. Thus the system becomes more tractable because thedegree of nonlinearity is decreased, but more complicated because the dimensionality is increased. The true dynamics of thesystem are contained in manifold of dimension n embedded in the dimension m space. The solution trajectories on thismanifold will be topologically equivalent to the original system. When the general procedure is applied to our scavengersystem (1), we obtain:y° 1 = y1 - y1 y2y° 2 = -ay2 + y1 y2y° 3 = -by3 + cy3 y4y° 4 = H1 - aL y4 + y1 y4 - y2 y4The dynamics of the original system take place on the manifold defined by y1 -1 y2 -1 y4 = 1. This transformation wassomewhat trivial, but more complicated possibilities exist as well. In [31], it is shown how a differential time reparameteriza-tion, dt = ¤k=1m xk -Brk dt, can lead to a wide variety of representations for a given QP system. The QP formalism is notnecessary for the analysis of the preceding scavenger systems, but more complicated scavenger systems require such tech-niques.Four-species systemWe now turn our attention to a system of four interacting species. The species x, y, and z comprise a simple three-spe-cies food chain. Chauvet, Paullet, Prevtie, and Walls showed that a three species food chain without intraspecific competi-tion terms has stability in only a degenerate case. Thus we add an intraspecific competition term to the population x to makethe system stable, and hence physically realistic. We next add a scavenger species w that survives by consuming the carrionfrom the predation of z on y and the predation of y on x. To make the system stable, we add an intraspecific competition termto the scavenger species w. Our model takes the following form:x° = x - bxy - x2y° = -ay + dxy - yzz° = -cz + yzw° = -ew + f wyz + gwxy - w2We are interested in the case where a positive equilibrium occurs, so we restrict our attention to the parameter values thatsatisfy: bc < 1, d > aÅÅÅÅÅÅÅÅÅÅÅÅ1-bc , and cHd f + gL H1 - bcL > e + ac f , in which caseH1 - bc, c, d - a - bcd, -e - a c f + c d f - b c2 d f + c g - b c2 gL is an equilibrium in positive space. The Jacobian of thesystem evaluated at this equilibrium has a positive determinant and a negative trace, indicating that the sum of the eigenval-ues is negative and the product is positive. Furthermore, one of the eigenvalues can be solved for algebraically and isnegative. Thus there are three possible combinations of eigenvalues: 1) there are four real negative eigenvalues, 2) there arefour real eigenvalues, two of which are negative and two of which are positive, or 3) there are two negative real eigenvaluesand two complex conjugate eigenvalues. These three possibilities leave open the question of local stability at the equilib-rium. To solve the question of stability, we us an quasi-monomial transformation to obtain the following system:u° 1 = u1 - u1 2 - b u1 u2u° 2 = -a u2 + d u1 u2 - u2 u3u° 3 = -c u3 + u3 u2u° 4 = -e u4 - u4 2 + g u4 u5 + f u4 u6u° 5 = H1 - aL u5 + Hd - 1L u1 u5 - b u2 u5 - u3 u5u° 6 = -Ha + cL u6 + d u1 u6 + u2 u6 - u3 u6Here we are interested in the subspace defined by u5ÅÅÅÅÅÅÅÅÅÅÅÅÅu1 u2 = 1 and u6ÅÅÅÅÅÅÅÅÅÅÅÅÅu2 u3 = 1.Using the system above, it is possible to construct an Lyapunov functionf Hu1 , u2 , u3 , u4 , u5 , u6 L = ⁄i=16 ui - ui* LnI uiÅÅÅÅÅÅÅui* M - ui* , whereu1* = 1 - b c, u2* = c, u3* = d - a - bcd, u4* = -e - a c f + c d f - b c2 d f + c g - b c2 g, u5* = cH1 - bcL, u6* = cHd - a - bcdLThis Lyapunov function, restricted to the subspace determined by u5ÅÅÅÅÅÅÅÅÅÅÅÅÅu1 u2 = 1 and u6ÅÅÅÅÅÅÅÅÅÅÅÅÅu2 u3 = 1, and written in terms of our originalcoordinates, gives an Lyapunov function for the original system:gHx, y, z, wL = x - x* LnH xÅÅÅÅÅÅÅx* L - x* + y - y* LnI yÅÅÅÅÅÅÅy* M - y* + z - z* LnH zÅÅÅÅÅÅz* L -z* + w - w* LnH wÅÅÅÅÅÅÅÅw* L - w* + x y - x* y* LnI x yÅÅÅÅÅÅÅÅÅÅÅÅx* y* M - x* y* + y z - y* z* LnI y zÅÅÅÅÅÅÅÅÅÅÅÅy* z* M - y* z*where x* = 1 - bc, y* = c, x* = d - a - bcd, x* = - e - a c f + c d f - b c2 d f + c g - b c2 g.Some really nasty algebra shows that this is, indeed, an Lyapunov function, and hence the equilibrium in positive space isglobally stable.

Scavenger.nb 2

















T xy „ t = a.





































Let wHtL = g2 HtL - g1 HtL and note that wH0L > 0 and wHTL < 0. By the intermediate value theorem, there exists a t, 0 <t < Tsuch that wHt`L = 0 and hence g2 Ht`L = g1 Ht`L . This implies that g3 Ht`L = g1 Ht`L = g2 Ht`L = g3 Ht` + TL, so g3 HtL, and hence zHtL, areperiodic with period T . But this implies HxHtL, yHtL, zHtLL is periodic with period T , and hence z1 = z2 = …, violating thehypothesis that zk < zk+1 and zk+1 > zk+2 .Thus any return sequence 8zn < for any solution HxHtL, yHtL, zHtLL must be monotonic.





T-b + cxy - z „ t = -bT + caT - Ÿ0






-b + cx y - z „ t = 0. Ÿ0T x y „ t = Ta,






Limit cycle on one invariant cylinder

More realistic generalizationScavenging species do not solely subsist on carrion produced by predation of one species on another. Scavengers

also consume carrion produced by natural death (e.g. starvation, disease, accidents). A suitable modification of our model, inwhich the scavenger species z benefits from carrion produced from both natural death of x and y and predation is:

dxÅÅÅÅÅÅÅdt = x - xydyÅÅÅÅÅÅÅdt = -ay + xydzÅÅÅÅÅÅdt = -bz + cxyz + dxz + eyz - z2

H6LWe are interested in the case when ac + ad + e > b, because when this occurs there is an equilibrium of (6) in positive space.In this case there are four total equilibria;, one of which is H0, 0, -bL, which has no physical meaning and will be discarded.H0, 0, 0L is a saddle point. Ha, 1, 0L is an equilibrium with a 2-dimensional center manifold tangent to the xy-plane and anunstable manifold tangent to the line Ha, 1, sL 0 < s < ¶. H1, a, ac + ad + e - bL is an equilibrium with a two dimensionalcenter manifold and a stable manifold tangent to the line Ha, 1, sL 0 < s < ¶.The same arguments used for the system H5L show that the trajectories of H6L are confined to invariant cylinders and that eachcylinder has a unique attracting non-zero limit cycle.Limit cycle on one invariant cylinderEcological notesThe scavenger systems have several interesting connections to other ecological concepts. First, it should be notedthat the above models conform to the general criteria in [16] and [17] for acceptable population models. We do not claimthat these scavenger systems are completely realistic, but we feel that they serve as a starting point for more complicated andaccurate models.A notable feature of (4), (5), and (6) is that these systems include higher-order interactions. Although the term"higher -order interaction" has been used in several different ways in the ecological literature, it is usually defined as "aninteraction that occurs whenever one species effects the nature of the direct interaction between two other species" [2]. Ineach of our systems dzÅÅÅÅÅÅÅdx depends on y and dzÅÅÅÅÅÅÅdy depends on x, so our model contains higher order interactions. The presence ofhigher order interactions in ecosystems has sparked much debate in the ecological community [1], [2], [3], and [4]. Higherorder interaction models significantly broaden the range of possible ecosystem dynamics from the classical models that onlytake into account pairwise interaction. The scavenger models proposed here provide dynamically simple examples of higherorder interaction systems, and they would make an excellent pedagogical tool for the introduction of this ecological concept.The product form of the carrion term cxyz in (5) makes this system somewhat similar to ecosystem models of"interactively essential resources", which are described in [5], [6], and [8]. In ecology, Von Liebig's Law of the Minimumstates that the growth rate of a species is determined by the most limiting resource [6]. Resources that obey Von Liebig'sLaw are referred to as perfectly essential; those that do not are referred to as interactively essential. Analysis of systemsinvolving interactively essential resources have mainly focused on plants that depend on different chemical combinations.For example, phytoplankton depends on the interaction between iron and nitrogen [6]. In our models (4) and (5), the predatorspecies y and the prey species x could be viewed as interactively essential resources for the scavenger species z.The scavenger systems proposed here will no doubt need to be greatly enhanced before being directly applicable toreal populations. One important direction for enhancement involves an ecological concept discussed in [31]: the competitivebalance between scavengers and decomposers. Decomposers, most notably bacteria and fungi, directly compete with scaven-gers for carrion. Many types of bacteria produce toxic substances that spoil the carrion and prevent scavengers from takingadvantage of the resource. To overcome this, scavengers have evolved in two manners: first they have become more efficientat quickly finding and consuming carrion before decomposers can spoil it, and second they have developed a limited abilityto "detoxify" spoiled carrion. The competitive balance between decomposers and scavengers is critical to the structure offood webs; resources consumed by scavengers keep energy and nutrients at higher trophic levels, while resources consumedby decomposers recycle energy and resources to low trophic levels. An important addition to the scavenger models presentedhere would be the inclusion of competitive decomposers.Representation of quasi-polynomial systems as Lotka-Volterra equationsThe scavenger models introduced above are not of the form (1) and hence are not Lotka-Volterra systems. They are,however, members of a much broader class of differential equations known as quasi-polynomial systems. In this section, weshow how the work of Hernandez-Bermejo, Fairen, and Brenig [29] and Gleria, Figueiredo, and Filho [30] concerningquasi-polynomial systems can be used to analyze our scavenger models. In general, an nth-order quasi-polynomial (QP)system has the form:x° i = li xi + xi ‚ j=1m Aij ‰k=1n xkBjk i = 1, 2, …, n H7LHere l is an n-dimensional column vector, A is a n äm matrix and B is a män matrix. The entries of l, A, and B can be anyarbitrary real numbers. When B is the identity matrix, the system is a Lotka-Volterra system. n is the order of the system andm is the number of different monomial expressions under consideration. H7L is form invariant under the transformationxi = ‰ j=1n y jDij , i = 1, 2, …, n H8Lwhere Di j are the entries of a n ä n invertible matrix D. The new system takes the formy° i = li yi + yi ‚ j=1m Ai j ‰k=1n ykB jk i = 1, 2, …, n H9Lwhere l = D-1 l, A = D-1 A, and B = BD.Brenig, Herndandez-Bermejo, and Fairen [29] have shown that the solutions to (7) are topologically equivalent tothose of (9). The set of QP systems are partitioned into different equivalence classes, where two systems belong to the sameequivalence class if and only if there is a transformation of the form H8L that relates them. The qualitative behavior of allsystems in an equivalence class are essentially the same. Finding the most useful representation of a quasi-polynomialsystem is not always immediately apparent, but the references [24], [25], [26], [27], [28], [29], and [30] give some guidance.In [22], is shown that every QP system is equivalent to one in which m ¥ n and A and B have maximal rank. In [30], particu-lar attention is paid to transforming a QP system into an equivalent Lotka-Volterra system. This method requires the introduc-tion of embedding variables to obtain a situation in which m = n. Thus the system becomes more tractable because thedegree of nonlinearity is decreased, but more complicated because the dimensionality is increased. The true dynamics of thesystem are contained in manifold of dimension n embedded in the dimension m space. The solution trajectories on thismanifold will be topologically equivalent to the original system. When the general procedure is applied to our scavengersystem (1), we obtain:y° 1 = y1 - y1 y2y° 2 = -ay2 + y1 y2y° 3 = -by3 + cy3 y4y° 4 = H1 - aL y4 + y1 y4 - y2 y4The dynamics of the original system take place on the manifold defined by y1 -1 y2 -1 y4 = 1. This transformation wassomewhat trivial, but more complicated possibilities exist as well. In [31], it is shown how a differential time reparameteriza-tion, dt = ¤k=1m xk -Brk dt, can lead to a wide variety of representations for a given QP system. The QP formalism is notnecessary for the analysis of the preceding scavenger systems, but more complicated scavenger systems require such tech-niques.Four-species systemWe now turn our attention to a system of four interacting species. The species x, y, and z comprise a simple three-spe-cies food chain. Chauvet, Paullet, Prevtie, and Walls showed that a three species food chain without intraspecific competi-tion terms has stability in only a degenerate case. Thus we add an intraspecific competition term to the population x to makethe system stable, and hence physically realistic. We next add a scavenger species w that survives by consuming the carrionfrom the predation of z on y and the predation of y on x. To make the system stable, we add an intraspecific competition termto the scavenger species w. Our model takes the following form:x° = x - bxy - x2y° = -ay + dxy - yzz° = -cz + yzw° = -ew + f wyz + gwxy - w2We are interested in the case where a positive equilibrium occurs, so we restrict our attention to the parameter values thatsatisfy: bc < 1, d > aÅÅÅÅÅÅÅÅÅÅÅÅ1-bc , and cHd f + gL H1 - bcL > e + ac f , in which caseH1 - bc, c, d - a - bcd, -e - a c f + c d f - b c2 d f + c g - b c2 gL is an equilibrium in positive space. The Jacobian of thesystem evaluated at this equilibrium has a positive determinant and a negative trace, indicating that the sum of the eigenval-ues is negative and the product is positive. Furthermore, one of the eigenvalues can be solved for algebraically and isnegative. Thus there are three possible combinations of eigenvalues: 1) there are four real negative eigenvalues, 2) there arefour real eigenvalues, two of which are negative and two of which are positive, or 3) there are two negative real eigenvaluesand two complex conjugate eigenvalues. These three possibilities leave open the question of local stability at the equilib-rium. To solve the question of stability, we us an quasi-monomial transformation to obtain the following system:u° 1 = u1 - u1 2 - b u1 u2u° 2 = -a u2 + d u1 u2 - u2 u3u° 3 = -c u3 + u3 u2u° 4 = -e u4 - u4 2 + g u4 u5 + f u4 u6u° 5 = H1 - aL u5 + Hd - 1L u1 u5 - b u2 u5 - u3 u5u° 6 = -Ha + cL u6 + d u1 u6 + u2 u6 - u3 u6Here we are interested in the subspace defined by u5ÅÅÅÅÅÅÅÅÅÅÅÅÅu1 u2 = 1 and u6ÅÅÅÅÅÅÅÅÅÅÅÅÅu2 u3 = 1.Using the system above, it is possible to construct an Lyapunov functionf Hu1 , u2 , u3 , u4 , u5 , u6 L = ⁄i=16 ui - ui* LnI uiÅÅÅÅÅÅÅui* M - ui* , whereu1* = 1 - b c, u2* = c, u3* = d - a - bcd, u4* = -e - a c f + c d f - b c2 d f + c g - b c2 g, u5* = cH1 - bcL, u6* = cHd - a - bcdLThis Lyapunov function, restricted to the subspace determined by u5ÅÅÅÅÅÅÅÅÅÅÅÅÅu1 u2 = 1 and u6ÅÅÅÅÅÅÅÅÅÅÅÅÅu2 u3 = 1, and written in terms of our originalcoordinates, gives an Lyapunov function for the original system:gHx, y, z, wL = x - x* LnH xÅÅÅÅÅÅÅx* L - x* + y - y* LnI yÅÅÅÅÅÅÅy* M - y* + z - z* LnH zÅÅÅÅÅÅz* L -z* + w - w* LnH wÅÅÅÅÅÅÅÅw* L - w* + x y - x* y* LnI x yÅÅÅÅÅÅÅÅÅÅÅÅx* y* M - x* y* + y z - y* z* LnI y zÅÅÅÅÅÅÅÅÅÅÅÅy* z* M - y* z*where x* = 1 - bc, y* = c, x* = d - a - bcd, x* = - e - a c f + c d f - b c2 d f + c g - b c2 g.Some really nasty algebra shows that this is, indeed, an Lyapunov function, and hence the equilibrium in positive space isglobally stable.

Scavenger.nb 3

















T xy „ t = a.










































T-b + cxy - z „ t = -bT + caT - Ÿ0






-b + cx y - z „ t = 0. Ÿ0T x y „ t = Ta,










H6LWe are interested in the case when ac + ad + e > b, because when this occurs there is an equilibrium of (6) in positive space.In this case there are four total equilibria;, one of which is H0, 0, -bL, which has no physical meaning and will be discarded.H0, 0, 0L is a saddle point. Ha, 1, 0L is an equilibrium with a 2-dimensional center manifold tangent to the xy-plane and anunstable manifold tangent to the line Ha, 1, sL 0 < s < ¶. H1, a, ac + ad + e - bL is an equilibrium with a two dimensionalcenter manifold and a stable manifold tangent to the line Ha, 1, sL 0 < s < ¶.The same arguments used for the system H5L show that the trajectories of H6L are confined to invariant cylinders and that eachcylinder has a unique attracting non-zero limit cycle.


Ecological notesThe scavenger systems have several interesting connections to other ecological concepts. First, it should be noted

that the above models conform to the general criteria in [16] and [17] for acceptable population models. We do not claimthat these scavenger systems are completely realistic, but we feel that they serve as a starting point for more complicated andaccurate models.A notable feature of (4), (5), and (6) is that these systems include higher-order interactions. Although the term"higher -order interaction" has been used in several different ways in the ecological literature, it is usually defined as "aninteraction that occurs whenever one species effects the nature of the direct interaction between two other species" [2]. Ineach of our systems dzÅÅÅÅÅÅÅdx depends on y and dzÅÅÅÅÅÅÅdy depends on x, so our model contains higher order interactions. The presence ofhigher order interactions in ecosystems has sparked much debate in the ecological community [1], [2], [3], and [4]. Higherorder interaction models significantly broaden the range of possible ecosystem dynamics from the classical models that onlytake into account pairwise interaction. The scavenger models proposed here provide dynamically simple examples of higherorder interaction systems, and they would make an excellent pedagogical tool for the introduction of this ecological concept.The product form of the carrion term cxyz in (5) makes this system somewhat similar to ecosystem models of"interactively essential resources", which are described in [5], [6], and [8]. In ecology, Von Liebig's Law of the Minimumstates that the growth rate of a species is determined by the most limiting resource [6]. Resources that obey Von Liebig'sLaw are referred to as perfectly essential; those that do not are referred to as interactively essential. Analysis of systemsinvolving interactively essential resources have mainly focused on plants that depend on different chemical combinations.For example, phytoplankton depends on the interaction between iron and nitrogen [6]. In our models (4) and (5), the predatorspecies y and the prey species x could be viewed as interactively essential resources for the scavenger species z.The scavenger systems proposed here will no doubt need to be greatly enhanced before being directly applicable toreal populations. One important direction for enhancement involves an ecological concept discussed in [31]: the competitivebalance between scavengers and decomposers. Decomposers, most notably bacteria and fungi, directly compete with scaven-gers for carrion. Many types of bacteria produce toxic substances that spoil the carrion and prevent scavengers from takingadvantage of the resource. To overcome this, scavengers have evolved in two manners: first they have become more efficientat quickly finding and consuming carrion before decomposers can spoil it, and second they have developed a limited abilityto "detoxify" spoiled carrion. The competitive balance between decomposers and scavengers is critical to the structure offood webs; resources consumed by scavengers keep energy and nutrients at higher trophic levels, while resources consumedby decomposers recycle energy and resources to low trophic levels. An important addition to the scavenger models presentedhere would be the inclusion of competitive decomposers.Representation of quasi-polynomial systems as Lotka-Volterra equationsThe scavenger models introduced above are not of the form (1) and hence are not Lotka-Volterra systems. They are,however, members of a much broader class of differential equations known as quasi-polynomial systems. In this section, weshow how the work of Hernandez-Bermejo, Fairen, and Brenig [29] and Gleria, Figueiredo, and Filho [30] concerningquasi-polynomial systems can be used to analyze our scavenger models. In general, an nth-order quasi-polynomial (QP)system has the form:x° i = li xi + xi ‚ j=1m Aij ‰k=1n xkBjk i = 1, 2, …, n H7LHere l is an n-dimensional column vector, A is a n äm matrix and B is a män matrix. The entries of l, A, and B can be anyarbitrary real numbers. When B is the identity matrix, the system is a Lotka-Volterra system. n is the order of the system andm is the number of different monomial expressions under consideration. H7L is form invariant under the transformationxi = ‰ j=1n y jDij , i = 1, 2, …, n H8Lwhere Di j are the entries of a n ä n invertible matrix D. The new system takes the formy° i = li yi + yi ‚ j=1m Ai j ‰k=1n ykB jk i = 1, 2, …, n H9Lwhere l = D-1 l, A = D-1 A, and B = BD.Brenig, Herndandez-Bermejo, and Fairen [29] have shown that the solutions to (7) are topologically equivalent tothose of (9). The set of QP systems are partitioned into different equivalence classes, where two systems belong to the sameequivalence class if and only if there is a transformation of the form H8L that relates them. The qualitative behavior of allsystems in an equivalence class are essentially the same. Finding the most useful representation of a quasi-polynomialsystem is not always immediately apparent, but the references [24], [25], [26], [27], [28], [29], and [30] give some guidance.In [22], is shown that every QP system is equivalent to one in which m ¥ n and A and B have maximal rank. In [30], particu-lar attention is paid to transforming a QP system into an equivalent Lotka-Volterra system. This method requires the introduc-tion of embedding variables to obtain a situation in which m = n. Thus the system becomes more tractable because thedegree of nonlinearity is decreased, but more complicated because the dimensionality is increased. The true dynamics of thesystem are contained in manifold of dimension n embedded in the dimension m space. The solution trajectories on thismanifold will be topologically equivalent to the original system. When the general procedure is applied to our scavengersystem (1), we obtain:y° 1 = y1 - y1 y2y° 2 = -ay2 + y1 y2y° 3 = -by3 + cy3 y4y° 4 = H1 - aL y4 + y1 y4 - y2 y4The dynamics of the original system take place on the manifold defined by y1 -1 y2 -1 y4 = 1. This transformation wassomewhat trivial, but more complicated possibilities exist as well. In [31], it is shown how a differential time reparameteriza-tion, dt = ¤k=1m xk -Brk dt, can lead to a wide variety of representations for a given QP system. The QP formalism is notnecessary for the analysis of the preceding scavenger systems, but more complicated scavenger systems require such tech-niques.Four-species systemWe now turn our attention to a system of four interacting species. The species x, y, and z comprise a simple three-spe-cies food chain. Chauvet, Paullet, Prevtie, and Walls showed that a three species food chain without intraspecific competi-tion terms has stability in only a degenerate case. Thus we add an intraspecific competition term to the population x to makethe system stable, and hence physically realistic. We next add a scavenger species w that survives by consuming the carrionfrom the predation of z on y and the predation of y on x. To make the system stable, we add an intraspecific competition termto the scavenger species w. Our model takes the following form:x° = x - bxy - x2y° = -ay + dxy - yzz° = -cz + yzw° = -ew + f wyz + gwxy - w2We are interested in the case where a positive equilibrium occurs, so we restrict our attention to the parameter values thatsatisfy: bc < 1, d > aÅÅÅÅÅÅÅÅÅÅÅÅ1-bc , and cHd f + gL H1 - bcL > e + ac f , in which caseH1 - bc, c, d - a - bcd, -e - a c f + c d f - b c2 d f + c g - b c2 gL is an equilibrium in positive space. The Jacobian of thesystem evaluated at this equilibrium has a positive determinant and a negative trace, indicating that the sum of the eigenval-ues is negative and the product is positive. Furthermore, one of the eigenvalues can be solved for algebraically and isnegative. Thus there are three possible combinations of eigenvalues: 1) there are four real negative eigenvalues, 2) there arefour real eigenvalues, two of which are negative and two of which are positive, or 3) there are two negative real eigenvaluesand two complex conjugate eigenvalues. These three possibilities leave open the question of local stability at the equilib-rium. To solve the question of stability, we us an quasi-monomial transformation to obtain the following system:u° 1 = u1 - u1 2 - b u1 u2u° 2 = -a u2 + d u1 u2 - u2 u3u° 3 = -c u3 + u3 u2u° 4 = -e u4 - u4 2 + g u4 u5 + f u4 u6u° 5 = H1 - aL u5 + Hd - 1L u1 u5 - b u2 u5 - u3 u5u° 6 = -Ha + cL u6 + d u1 u6 + u2 u6 - u3 u6Here we are interested in the subspace defined by u5ÅÅÅÅÅÅÅÅÅÅÅÅÅu1 u2 = 1 and u6ÅÅÅÅÅÅÅÅÅÅÅÅÅu2 u3 = 1.Using the system above, it is possible to construct an Lyapunov functionf Hu1 , u2 , u3 , u4 , u5 , u6 L = ⁄i=16 ui - ui* LnI uiÅÅÅÅÅÅÅui* M - ui* , whereu1* = 1 - b c, u2* = c, u3* = d - a - bcd, u4* = -e - a c f + c d f - b c2 d f + c g - b c2 g, u5* = cH1 - bcL, u6* = cHd - a - bcdLThis Lyapunov function, restricted to the subspace determined by u5ÅÅÅÅÅÅÅÅÅÅÅÅÅu1 u2 = 1 and u6ÅÅÅÅÅÅÅÅÅÅÅÅÅu2 u3 = 1, and written in terms of our originalcoordinates, gives an Lyapunov function for the original system:gHx, y, z, wL = x - x* LnH xÅÅÅÅÅÅÅx* L - x* + y - y* LnI yÅÅÅÅÅÅÅy* M - y* + z - z* LnH zÅÅÅÅÅÅz* L -z* + w - w* LnH wÅÅÅÅÅÅÅÅw* L - w* + x y - x* y* LnI x yÅÅÅÅÅÅÅÅÅÅÅÅx* y* M - x* y* + y z - y* z* LnI y zÅÅÅÅÅÅÅÅÅÅÅÅy* z* M - y* z*where x* = 1 - bc, y* = c, x* = d - a - bcd, x* = - e - a c f + c d f - b c2 d f + c g - b c2 g.Some really nasty algebra shows that this is, indeed, an Lyapunov function, and hence the equilibrium in positive space isglobally stable.

Scavenger.nb 4

















T xy „ t = a.





































Let wHtL = g2 HtL - g1 HtL and note that wH0L > 0 and wHTL < 0. By the intermediate value theorem, there exists a t`, 0 <t` < Tsuch that wHtL = 0 and hence g2 HtL = g1 HtL . This implies that g3 HtL = g1 HtL = g2 HtL = g3 Ht + TL, so g3 HtL, and hence zHtL, areperiodic with period T . But this implies HxHtL, yHtL, zHtLL is periodic with period T , and hence z1 = z2 = …, violating thehypothesis that zk < zk+1 and zk+1 > zk+2 .Thus any return sequence 8zn < for any solution HxHtL, yHtL, zHtLL must be monotonic.





T-b + cxy - z „ t = -bT + caT - Ÿ0






-b + cx y - z „ t = 0. Ÿ0T x y „ t = Ta,













that the above models conform to the general criteria in [16] and [17] for acceptable population models. We do not claimthat these scavenger systems are completely realistic, but we feel that they serve as a starting point for more complicated andaccurate models.

A notable feature of (4), (5), and (6) is that these systems include higher-order interactions. Although the term"higher -order interaction" has been used in several different ways in the ecological literature, it is usually defined as "aninteraction that occurs whenever one species effects the nature of the direct interaction between two other species" [2]. Ineach of our systems dzÅÅÅÅÅÅÅdx depends on y and dzÅÅÅÅÅÅÅdy depends on x, so our model contains higher order interactions. The presence ofhigher order interactions in ecosystems has sparked much debate in the ecological community [1], [2], [3], and [4]. Higherorder interaction models significantly broaden the range of possible ecosystem dynamics from the classical models that onlytake into account pairwise interaction. The scavenger models proposed here provide dynamically simple examples of higherorder interaction systems, and they would make an excellent pedagogical tool for the introduction of this ecological concept.

The product form of the carrion term cxyz in (5) makes this system somewhat similar to ecosystem models of"interactively essential resources", which are described in [5], [6], and [8]. In ecology, Von Liebig's Law of the Minimumstates that the growth rate of a species is determined by the most limiting resource [6]. Resources that obey Von Liebig'sLaw are referred to as perfectly essential; those that do not are referred to as interactively essential. Analysis of systemsinvolving interactively essential resources have mainly focused on plants that depend on different chemical combinations.For example, phytoplankton depends on the interaction between iron and nitrogen [6]. In our models (4) and (5), the predatorspecies y and the prey species x could be viewed as interactively essential resources for the scavenger species z.

The scavenger systems proposed here will no doubt need to be greatly enhanced before being directly applicable toreal populations. One important direction for enhancement involves an ecological concept discussed in [31]: the competitivebalance between scavengers and decomposers. Decomposers, most notably bacteria and fungi, directly compete with scaven-gers for carrion. Many types of bacteria produce toxic substances that spoil the carrion and prevent scavengers from takingadvantage of the resource. To overcome this, scavengers have evolved in two manners: first they have become more efficientat quickly finding and consuming carrion before decomposers can spoil it, and second they have developed a limited abilityto "detoxify" spoiled carrion. The competitive balance between decomposers and scavengers is critical to the structure offood webs; resources consumed by scavengers keep energy and nutrients at higher trophic levels, while resources consumedby decomposers recycle energy and resources to low trophic levels. An important addition to the scavenger models presentedhere would be the inclusion of competitive decomposers.

Representation of quasi-polynomial systems as Lotka-Volterra equationsThe scavenger models introduced above are not of the form (1) and hence are not Lotka-Volterra systems. They are,

however, members of a much broader class of differential equations known as quasi-polynomial systems. In this section, weshow how the work of Hernandez-Bermejo, Fairen, and Brenig [29] and Gleria, Figueiredo, and Filho [30] concerningquasi-polynomial systems can be used to analyze our scavenger models. In general, an nth-order quasi-polynomial (QP)system has the form:x° i = li xi + xi ‚ j=1m Aij ‰k=1n xkBjk i = 1, 2, …, n H7LHere l is an n-dimensional column vector, A is a n äm matrix and B is a män matrix. The entries of l, A, and B can be anyarbitrary real numbers. When B is the identity matrix, the system is a Lotka-Volterra system. n is the order of the system andm is the number of different monomial expressions under consideration. H7L is form invariant under the transformationxi = ‰ j=1n y jDij , i = 1, 2, …, n H8Lwhere Di j are the entries of a n ä n invertible matrix D. The new system takes the formy° i = li yi + yi ‚ j=1m Ai j ‰k=1n ykB jk i = 1, 2, …, n H9Lwhere l = D-1 l, A = D-1 A, and B = BD.Brenig, Herndandez-Bermejo, and Fairen [29] have shown that the solutions to (7) are topologically equivalent tothose of (9). The set of QP systems are partitioned into different equivalence classes, where two systems belong to the sameequivalence class if and only if there is a transformation of the form H8L that relates them. The qualitative behavior of allsystems in an equivalence class are essentially the same. Finding the most useful representation of a quasi-polynomialsystem is not always immediately apparent, but the references [24], [25], [26], [27], [28], [29], and [30] give some guidance.In [22], is shown that every QP system is equivalent to one in which m ¥ n and A and B have maximal rank. In [30], particu-lar attention is paid to transforming a QP system into an equivalent Lotka-Volterra system. This method requires the introduc-tion of embedding variables to obtain a situation in which m = n. Thus the system becomes more tractable because thedegree of nonlinearity is decreased, but more complicated because the dimensionality is increased. The true dynamics of thesystem are contained in manifold of dimension n embedded in the dimension m space. The solution trajectories on thismanifold will be topologically equivalent to the original system. When the general procedure is applied to our scavengersystem (1), we obtain:y° 1 = y1 - y1 y2y° 2 = -ay2 + y1 y2y° 3 = -by3 + cy3 y4y° 4 = H1 - aL y4 + y1 y4 - y2 y4The dynamics of the original system take place on the manifold defined by y1 -1 y2 -1 y4 = 1. This transformation wassomewhat trivial, but more complicated possibilities exist as well. In [31], it is shown how a differential time reparameteriza-tion, dt = ¤k=1m xk -Brk dt, can lead to a wide variety of representations for a given QP system. The QP formalism is notnecessary for the analysis of the preceding scavenger systems, but more complicated scavenger systems require such tech-niques.Four-species systemWe now turn our attention to a system of four interacting species. The species x, y, and z comprise a simple three-spe-cies food chain. Chauvet, Paullet, Prevtie, and Walls showed that a three species food chain without intraspecific competi-tion terms has stability in only a degenerate case. Thus we add an intraspecific competition term to the population x to makethe system stable, and hence physically realistic. We next add a scavenger species w that survives by consuming the carrionfrom the predation of z on y and the predation of y on x. To make the system stable, we add an intraspecific competition termto the scavenger species w. Our model takes the following form:x° = x - bxy - x2y° = -ay + dxy - yzz° = -cz + yzw° = -ew + f wyz + gwxy - w2We are interested in the case where a positive equilibrium occurs, so we restrict our attention to the parameter values thatsatisfy: bc < 1, d > aÅÅÅÅÅÅÅÅÅÅÅÅ1-bc , and cHd f + gL H1 - bcL > e + ac f , in which caseH1 - bc, c, d - a - bcd, -e - a c f + c d f - b c2 d f + c g - b c2 gL is an equilibrium in positive space. The Jacobian of thesystem evaluated at this equilibrium has a positive determinant and a negative trace, indicating that the sum of the eigenval-ues is negative and the product is positive. Furthermore, one of the eigenvalues can be solved for algebraically and isnegative. Thus there are three possible combinations of eigenvalues: 1) there are four real negative eigenvalues, 2) there arefour real eigenvalues, two of which are negative and two of which are positive, or 3) there are two negative real eigenvaluesand two complex conjugate eigenvalues. These three possibilities leave open the question of local stability at the equilib-rium. To solve the question of stability, we us an quasi-monomial transformation to obtain the following system:u° 1 = u1 - u1 2 - b u1 u2u° 2 = -a u2 + d u1 u2 - u2 u3u° 3 = -c u3 + u3 u2u° 4 = -e u4 - u4 2 + g u4 u5 + f u4 u6u° 5 = H1 - aL u5 + Hd - 1L u1 u5 - b u2 u5 - u3 u5u° 6 = -Ha + cL u6 + d u1 u6 + u2 u6 - u3 u6Here we are interested in the subspace defined by u5ÅÅÅÅÅÅÅÅÅÅÅÅÅu1 u2 = 1 and u6ÅÅÅÅÅÅÅÅÅÅÅÅÅu2 u3 = 1.Using the system above, it is possible to construct an Lyapunov functionf Hu1 , u2 , u3 , u4 , u5 , u6 L = ⁄i=16 ui - ui* LnI uiÅÅÅÅÅÅÅui* M - ui* , whereu1* = 1 - b c, u2* = c, u3* = d - a - bcd, u4* = -e - a c f + c d f - b c2 d f + c g - b c2 g, u5* = cH1 - bcL, u6* = cHd - a - bcdLThis Lyapunov function, restricted to the subspace determined by u5ÅÅÅÅÅÅÅÅÅÅÅÅÅu1 u2 = 1 and u6ÅÅÅÅÅÅÅÅÅÅÅÅÅu2 u3 = 1, and written in terms of our originalcoordinates, gives an Lyapunov function for the original system:gHx, y, z, wL = x - x* LnH xÅÅÅÅÅÅÅx* L - x* + y - y* LnI yÅÅÅÅÅÅÅy* M - y* + z - z* LnH zÅÅÅÅÅÅz* L -z* + w - w* LnH wÅÅÅÅÅÅÅÅw* L - w* + x y - x* y* LnI x yÅÅÅÅÅÅÅÅÅÅÅÅx* y* M - x* y* + y z - y* z* LnI y zÅÅÅÅÅÅÅÅÅÅÅÅy* z* M - y* z*where x* = 1 - bc, y* = c, x* = d - a - bcd, x* = - e - a c f + c d f - b c2 d f + c g - b c2 g.Some really nasty algebra shows that this is, indeed, an Lyapunov function, and hence the equilibrium in positive space isglobally stable.

Scavenger.nb 5

















T xy „ t = a.










































T-b + cxy - z „ t = -bT + caT - Ÿ0






-b + cx y - z „ t = 0. Ÿ0T x y „ t = Ta,


















however, members of a much broader class of differential equations known as quasi-polynomial systems. In this section, weshow how the work of Hernandez-Bermejo, Fairen, and Brenig [29] and Gleria, Figueiredo, and Filho [30] concerningquasi-polynomial systems can be used to analyze our scavenger models. In general, an nth-order quasi-polynomial (QP)system has the form:

x° i = li xi + xi ‚j=1

mAij ‰

k=1

nxk

Bjk i = 1, 2, …, n H7LHere l is an n-dimensional column vector, A is a n äm matrix and B is a män matrix. The entries of l, A, and B can be anyarbitrary real numbers. When B is the identity matrix, the system is a Lotka-Volterra system. n is the order of the system andm is the number of different monomial expressions under consideration. H7L is form invariant under the transformation

xi = ‰j=1

ny j

Dij , i = 1, 2, …, n H8Lwhere Dij are the entries of a n ä n invertible matrix D. The new system takes the form

y° i = l`

i yi + yi ‚j=1

mA`

i j ‰k=1

nyk

B`

jk i = 1, 2, …, n H9Lwhere l

`= D-1 l, A` = D-1 A, and B` = BD.

Brenig, Herndandez-Bermejo, and Fairen [29] have shown that the solutions to (7) are topologically equivalent tothose of (9). The set of QP systems are partitioned into different equivalence classes, where two systems belong to the sameequivalence class if and only if there is a transformation of the form H8L that relates them. The qualitative behavior of allsystems in an equivalence class are essentially the same. Finding the most useful representation of a quasi-polynomialsystem is not always immediately apparent, but the references [24], [25], [26], [27], [28], [29], and [30] give some guidance.In [22], is shown that every QP system is equivalent to one in which m ¥ n and A and B have maximal rank. In [30], particu-lar attention is paid to transforming a QP system into an equivalent Lotka-Volterra system. This method requires the introduc-tion of embedding variables to obtain a situation in which m = n. Thus the system becomes more tractable because thedegree of nonlinearity is decreased, but more complicated because the dimensionality is increased. The true dynamics of thesystem are contained in manifold of dimension n embedded in the dimension m space. The solution trajectories on thismanifold will be topologically equivalent to the original system. When the general procedure is applied to our scavengersystem (1), we obtain:

y° 1 = y1 - y1 y2

y° 2 = -ay2 + y1 y2

y° 3 = -by3 + cy3 y4

y° 4 = H1 - aL y4 + y1 y4 - y2 y4The dynamics of the original system take place on the manifold defined by y1

-1 y2-1 y4 = 1. This transformation was

somewhat trivial, but more complicated possibilities exist as well. In [31], it is shown how a differential time reparameteriza-tion, dt = ¤k=1

m xk-Brk dt, can lead to a wide variety of representations for a given QP system. The QP formalism is not

necessary for the analysis of the preceding scavenger systems, but more complicated scavenger systems require such tech-niques.Four-species systemWe now turn our attention to a system of four interacting species. The species x, y, and z comprise a simple three-spe-cies food chain. Chauvet, Paullet, Prevtie, and Walls showed that a three species food chain without intraspecific competi-tion terms has stability in only a degenerate case. Thus we add an intraspecific competition term to the population x to makethe system stable, and hence physically realistic. We next add a scavenger species w that survives by consuming the carrionfrom the predation of z on y and the predation of y on x. To make the system stable, we add an intraspecific competition termto the scavenger species w. Our model takes the following form:x° = x - bxy - x2y° = -ay + dxy - yzz° = -cz + yzw° = -ew + f wyz + gwxy - w2We are interested in the case where a positive equilibrium occurs, so we restrict our attention to the parameter values thatsatisfy: bc < 1, d > aÅÅÅÅÅÅÅÅÅÅÅÅ1-bc , and cHd f + gL H1 - bcL > e + ac f , in which caseH1 - bc, c, d - a - bcd, -e - a c f + c d f - b c2 d f + c g - b c2 gL is an equilibrium in positive space. The Jacobian of thesystem evaluated at this equilibrium has a positive determinant and a negative trace, indicating that the sum of the eigenval-ues is negative and the product is positive. Furthermore, one of the eigenvalues can be solved for algebraically and isnegative. Thus there are three possible combinations of eigenvalues: 1) there are four real negative eigenvalues, 2) there arefour real eigenvalues, two of which are negative and two of which are positive, or 3) there are two negative real eigenvaluesand two complex conjugate eigenvalues. These three possibilities leave open the question of local stability at the equilib-rium. To solve the question of stability, we us an quasi-monomial transformation to obtain the following system:u° 1 = u1 - u1 2 - b u1 u2u° 2 = -a u2 + d u1 u2 - u2 u3u° 3 = -c u3 + u3 u2u° 4 = -e u4 - u4 2 + g u4 u5 + f u4 u6u° 5 = H1 - aL u5 + Hd - 1L u1 u5 - b u2 u5 - u3 u5u° 6 = -Ha + cL u6 + d u1 u6 + u2 u6 - u3 u6Here we are interested in the subspace defined by u5ÅÅÅÅÅÅÅÅÅÅÅÅÅu1 u2 = 1 and u6ÅÅÅÅÅÅÅÅÅÅÅÅÅu2 u3 = 1.Using the system above, it is possible to construct an Lyapunov functionf Hu1 , u2 , u3 , u4 , u5 , u6 L = ⁄i=16 ui - ui* LnI uiÅÅÅÅÅÅÅui* M - ui* , whereu1* = 1 - b c, u2* = c, u3* = d - a - bcd, u4* = -e - a c f + c d f - b c2 d f + c g - b c2 g, u5* = cH1 - bcL, u6* = cHd - a - bcdLThis Lyapunov function, restricted to the subspace determined by u5ÅÅÅÅÅÅÅÅÅÅÅÅÅu1 u2 = 1 and u6ÅÅÅÅÅÅÅÅÅÅÅÅÅu2 u3 = 1, and written in terms of our originalcoordinates, gives an Lyapunov function for the original system:gHx, y, z, wL = x - x* LnH xÅÅÅÅÅÅÅx* L - x* + y - y* LnI yÅÅÅÅÅÅÅy* M - y* + z - z* LnH zÅÅÅÅÅÅz* L -z* + w - w* LnH wÅÅÅÅÅÅÅÅw* L - w* + x y - x* y* LnI x yÅÅÅÅÅÅÅÅÅÅÅÅx* y* M - x* y* + y z - y* z* LnI y zÅÅÅÅÅÅÅÅÅÅÅÅy* z* M - y* z*where x* = 1 - bc, y* = c, x* = d - a - bcd, x* = - e - a c f + c d f - b c2 d f + c g - b c2 g.Some really nasty algebra shows that this is, indeed, an Lyapunov function, and hence the equilibrium in positive space isglobally stable.

Scavenger.nb 6

















T xy „ t = a.










































T-b + cxy - z „ t = -bT + caT - Ÿ0






-b + cx y - z „ t = 0. Ÿ0T x y „ t = Ta,




















mAij ‰

k=1

nxk


xi = ‰j=1

ny j


y° i = l`

i yi + yi ‚j=1

mA` i j ‰

k=1

nyk

B`


`= D-1 l, A

`= D-1 A, and B` = BD.


y° 1 = y1 - y1 y2

y° 2 = -ay2 + y1 y2

y° 3 = -by3 + cy3 y4





necessary for the analysis of the preceding scavenger systems, but more complicated scavenger systems require such tech-niques.

Four-species systemWe now turn our attention to a system of four interacting species. The species x, y, and z comprise a simple three-spe-

cies food chain. Chauvet, Paullet, Prevtie, and Walls showed that a three species food chain without intraspecific competi-tion terms has stability in only a degenerate case. Thus we add an intraspecific competition term to the population x to makethe system stable, and hence physically realistic. We next add a scavenger species w that survives by consuming the carrionfrom the predation of z on y and the predation of y on x. To make the system stable, we add an intraspecific competition termto the scavenger species w. Our model takes the following form:

x° = x - bxy - x2

y° = -ay + dxy - yzz° = -cz + yzw° = -ew + f wyz + gwxy - w2

We are interested in the case where a positive equilibrium occurs, so we restrict our attention to the parameter values thatsatisfy: bc < 1, d > aÅÅÅÅÅÅÅÅÅÅÅÅ1-bc , and cHd f + gL H1 - bcL > e + ac f , in which caseH1 - bc, c, d - a - bcd, -e - a c f + c d f - b c2 d f + c g - b c2 gL is an equilibrium in positive space. The Jacobian of thesystem evaluated at this equilibrium has a positive determinant and a negative trace, indicating that the sum of the eigenval-ues is negative and the product is positive. Furthermore, one of the eigenvalues can be solved for algebraically and isnegative. Thus there are three possible combinations of eigenvalues: 1) there are four real negative eigenvalues, 2) there arefour real eigenvalues, two of which are negative and two of which are positive, or 3) there are two negative real eigenvaluesand two complex conjugate eigenvalues. These three possibilities leave open the question of local stability at the equilib-rium. To solve the question of stability, we us an quasi-monomial transformation to obtain the following system:

u° 1 = u1 - u12 - b u1 u2

u° 2 = -a u2 + d u1 u2 - u2 u3u° 3 = -c u3 + u3 u2u° 4 = -e u4 - u4

2 + g u4 u5 + f u4 u6u° 5 = H1 - aL u5 + Hd - 1L u1 u5 - b u2 u5 - u3 u5u° 6 = -Ha + cL u6 + d u1 u6 + u2 u6 - u3 u6

Here we are interested in the subspace defined by u5ÅÅÅÅÅÅÅÅÅÅÅÅÅu1 u2= 1 and u6ÅÅÅÅÅÅÅÅÅÅÅÅÅu2 u3

= 1.Using the system above, it is possible to construct an Lyapunov functionf Hu1 , u2 , u3 , u4 , u5 , u6 L = ⁄i=1

6 ui - ui* LnI uiÅÅÅÅÅÅÅui

* M - ui* , where

u1* = 1 - b c, u2

* = c, u3* = d - a - bcd, u4

* = -e - a c f + c d f - b c2 d f + c g - b c2 g, u5* = cH1 - bcL, u6

* = cHd - a - bcdLThis Lyapunov function, restricted to the subspace determined by u5ÅÅÅÅÅÅÅÅÅÅÅÅÅu1 u2

= 1 and u6ÅÅÅÅÅÅÅÅÅÅÅÅÅu2 u3= 1, and written in terms of our original

coordinates, gives an Lyapunov function for the original system:gHx, y, z, wL = x - x* LnH xÅÅÅÅÅÅÅx* L - x* + y - y* LnI yÅÅÅÅÅÅÅy* M - y* + z - z* LnH zÅÅÅÅÅÅz* L -

z* + w - w* LnH wÅÅÅÅÅÅÅÅw* L - w* + x y - x* y* LnI x yÅÅÅÅÅÅÅÅÅÅÅÅx* y* M - x* y* + y z - y* z* LnI y zÅÅÅÅÅÅÅÅÅÅÅÅy* z* M - y* z*

where x* = 1 - bc, y* = c, x* = d - a - bcd, x* = - e - a c f + c d f - b c2 d f + c g - b c2 g.Some really nasty algebra shows that this is, indeed, an Lyapunov function, and hence the equilibrium in positive space isglobally stable.

Scavenger.nb 7

















T xy „ t = a.










































T-b + cxy - z „ t = -bT + caT - Ÿ0






-b + cx y - z „ t = 0. Ÿ0T x y „ t = Ta,




















mAij ‰

k=1

nxk


xi = ‰j=1

ny j


y° i = l`

i yi + yi ‚j=1

mA` i j ‰

k=1

nyk

B`


`= D-1 l, A

`= D-1 A, and B` = BD.


y° 1 = y1 - y1 y2

y° 2 = -ay2 + y1 y2

y° 3 = -by3 + cy3 y4








x° = x - bxy - x2



u° 1 = u1 - u12 - b u1 u2






* M - ui* , where

u1* = 1 - b c, u2

* = c, u3* = d - a - bcd, u4







Scavenger.nb 8


IntroductionLotka-Volterra systems of differential equations have played an important role in the development of mathematicalecology since their introduction in the early 20th century ([12], [20]). Lotka-Volterra models lack the complexity andrealism of more recently developed population models, but their generality makes them a convenient starting point foranalyzing ecological systems. Tilman remarks that, "Because the purpose of ecology is to understand the causes of patternsin nature, we should start by studying the largest, most general, and most repeatable patterns," and "General patterns providea framework within less general patterns can be more effectively studied" [11]. Indeed, in [14] Berryman describes how theLotka-Volterra systems have served as a basis for the development of more realistic models that involve ratio-dependentfunctional responses. When new concepts in population modelling are to be studied, it is logical to begin with the mostgeneral and simplistic mathematical framework. Thus the Lotka-Volterra models provide a foundation for new modellingideas to build upon.The classical Lotka-Volterra systems model the quantities (or densities) of species in an ecosystem under the assump-tion that the per-capita growth rate of each species is a linear function of the abundances of the species in the system [12].This framework allows for the modelling of a wide range of pairwise interactions between species, including predator-prey,competitive, and symbiotic relationships. Unfortunately, such pairwise interactions do not encapsulate the diversity ofinterspecies relationships present in observed ecosystems. In particular, the classical Lotka-Volterra systems fail to modelthe presence of scavenger species. Scavenger species subsist on carrion (the dead remains of other creatures). When ascavenger species is present in a system with a predator species and a prey species, the scavenger will benefit when thepredator kills its prey. This is not the type of pairwise interaction that can be modelled by the classical Lotka-Volterraequations, so the inclusion of scavengers requires a modification of these systems.The important role that scavengers play in ecosystems has long been recognized. In 1877, naturalist SebastianTenney enjoined his colleagues to devote more study to scavengers [15]. Dr. Tenney would be disappointed to learn thatscavengers have long been neglected in population ecology, relative to herbivores and predators. In recent years, the field ofsystems ecology has placed great emphasis on the study of detritus and detritrivores. Several mathematical models have beendeveloped to model detritus and detritrivore abundance in ecosystems ([9], [10], [11], [12], [13]). These models all treat theamount of detritus as a distinct state-variable. The detritrivores under consideration are primarily decomposers (includingbacteria and fungi) and microinverterbrates. The analysis we put forth differs from these studies in two important respects.First, we avoid explicitly including the detritus as a variable, which allows for easier analysis and system visualization.Second, we concentrate primarily on vertebrate carrion-feeders (e.g. hyenas, vultures, and ravens) rather than decomposers.A recent review by DeValt, Rhodes, and Shivik indicates that carrion use by terrestrial vertebrates is much more prevalentthan previously thought, and that the significance of terrestrial scavenging has been underestimated by the ecological commu-nity [31]. They conclude that "a clearer perspective on carrion use by terrestrial vertebrates will improve our understandingof critical ecological processes, particularly those associated with energy flow and trophic interactions" [31].A variety of species-specific studies of scavenger and detritrivore populations have been conducted. A thoroughlisting can be found in [7], but we will site three examples here. In [21], the impact of fisheries discards on benthic scaven-ger populations is studied. In [22], the impact of elk carrion on large carnivore populations is studied. In [23], a model ofwolf-elk population dynamics that addresses the amount of carrion available to scavengers is developed. This model ishighly specific to the wolf and elk of Yellowstone Park. It is a complicated discrete model using Leslie matrices, and theonly solutions available are numerical. We attempt to model scavengers in a far more general format, and we use analyticrather than numeric methods.











T xy „ t = a.










































T-b + cxy - z „ t = -bT + caT - Ÿ0






-b + cx y - z „ t = 0. Ÿ0T x y „ t = Ta,




















mAij ‰

k=1

nxk


xi = ‰j=1

ny j


y° i = l`

i yi + yi ‚j=1

mA` i j ‰

k=1

nyk

B`


`= D-1 l, A

`= D-1 A, and B` = BD.


y° 1 = y1 - y1 y2

y° 2 = -ay2 + y1 y2

y° 3 = -by3 + cy3 y4








x° = x - bxy - x2



u° 1 = u1 - u12 - b u1 u2






* M - ui* , where

u1* = 1 - b c, u2

* = c, u3* = d - a - bcd, u4







Scavenger.nb 9


IntroductionLotka-Volterra systems of differential equations have played an important role in the development of mathematicalecology since their introduction in the early 20th century ([12], [20]). Lotka-Volterra models lack the complexity andrealism of more recently developed population models, but their generality makes them a convenient starting point foranalyzing ecological systems. Tilman remarks that, "Because the purpose of ecology is to understand the causes of patternsin nature, we should start by studying the largest, most general, and most repeatable patterns," and "General patterns providea framework within less general patterns can be more effectively studied" [11]. Indeed, in [14] Berryman describes how theLotka-Volterra systems have served as a basis for the development of more realistic models that involve ratio-dependentfunctional responses. When new concepts in population modelling are to be studied, it is logical to begin with the mostgeneral and simplistic mathematical framework. Thus the Lotka-Volterra models provide a foundation for new modellingideas to build upon.The classical Lotka-Volterra systems model the quantities (or densities) of species in an ecosystem under the assump-tion that the per-capita growth rate of each species is a linear function of the abundances of the species in the system [12].This framework allows for the modelling of a wide range of pairwise interactions between species, including predator-prey,competitive, and symbiotic relationships. Unfortunately, such pairwise interactions do not encapsulate the diversity ofinterspecies relationships present in observed ecosystems. In particular, the classical Lotka-Volterra systems fail to modelthe presence of scavenger species. Scavenger species subsist on carrion (the dead remains of other creatures). When ascavenger species is present in a system with a predator species and a prey species, the scavenger will benefit when thepredator kills its prey. This is not the type of pairwise interaction that can be modelled by the classical Lotka-Volterraequations, so the inclusion of scavengers requires a modification of these systems.The important role that scavengers play in ecosystems has long been recognized. In 1877, naturalist SebastianTenney enjoined his colleagues to devote more study to scavengers [15]. Dr. Tenney would be disappointed to learn thatscavengers have long been neglected in population ecology, relative to herbivores and predators. In recent years, the field ofsystems ecology has placed great emphasis on the study of detritus and detritrivores. Several mathematical models have beendeveloped to model detritus and detritrivore abundance in ecosystems ([9], [10], [11], [12], [13]). These models all treat theamount of detritus as a distinct state-variable. The detritrivores under consideration are primarily decomposers (includingbacteria and fungi) and microinverterbrates. The analysis we put forth differs from these studies in two important respects.First, we avoid explicitly including the detritus as a variable, which allows for easier analysis and system visualization.Second, we concentrate primarily on vertebrate carrion-feeders (e.g. hyenas, vultures, and ravens) rather than decomposers.A recent review by DeValt, Rhodes, and Shivik indicates that carrion use by terrestrial vertebrates is much more prevalentthan previously thought, and that the significance of terrestrial scavenging has been underestimated by the ecological commu-nity [31]. They conclude that "a clearer perspective on carrion use by terrestrial vertebrates will improve our understandingof critical ecological processes, particularly those associated with energy flow and trophic interactions" [31].A variety of species-specific studies of scavenger and detritrivore populations have been conducted. A thoroughlisting can be found in [7], but we will site three examples here. In [21], the impact of fisheries discards on benthic scaven-ger populations is studied. In [22], the impact of elk carrion on large carnivore populations is studied. In [23], a model ofwolf-elk population dynamics that addresses the amount of carrion available to scavengers is developed. This model ishighly specific to the wolf and elk of Yellowstone Park. It is a complicated discrete model using Leslie matrices, and theonly solutions available are numerical. We attempt to model scavengers in a far more general format, and we use analyticrather than numeric methods.Stephen Heard [7] developed the notion of processing chain ecology. Heard introduced compartmental models weredifferent species alter the state of resources, thereby affecting the type of resources available to other species. This type ofmodelling can be applicable to scavenger population modelling because a predator could be viewed as altering the state of aresource (prey) to a state that is useful to a scavenger (carrion). Heard's model was primarily directed at the processing offauna and non-living resources (as opposed to the processing of a prey animal), as is apparent in his extensive table ofapplications. Heard's model, like the detritus models mentioned above, also suffers from the drawbacks of including resourcestates (carrion) as explicit variables, thereby increasing the dimensionality of the system.Lotka-Volterra background In general, an nth order Lotka-Volterra system takes the form:dxiÅÅÅÅÅÅÅÅÅdt = li xi + xi ⁄ j=1n Mij xj , i = 1, 2, …, nH1Lwhere " i, j li and Mi j are real constants [20]. Ecologists frequently use an nth order Lotka-Volterra system to model thepopulations of n different species interacting in a food web. The n ä n matrix M with entries Mij is called the interactionmatrix of the system. For predator-prey Lotka-Volterra systems, the interaction matrix M has a special form:" Hi, jL œ 81, 2, …, n< such that i j, Mi j Mji § 0 [19]. The classical two-species predator-prey Lotka-Volterra model isgiven by:dxÅÅÅÅÅÅÅdt = ax - bxydyÅÅÅÅÅÅÅdt = -cy + dxy H2Lwhere x is the population of a prey species, y is the population of a predator species, and a, b, c, and d are positive constants.When analyzing the qualitative behavior of systems such as (2), it is convenient to rescale variables so that some of theconstants involved equal one. The number of constants that may be eliminated by variable rescaling will generally dependon the number of variables in the system and the structure of the differential equations. For the remainder of this paper,differential equations will be written in a rescaled format so that the maximum possible number of constants are unity. Forexample, (2) may be rescaled as:dxÅÅÅÅÅÅÅdt = x - xydyÅÅÅÅÅÅÅdt = -ay + xy H3LThis set of equations has equilibria at H0, 0L and Ha, 1L. Lotka and Volterra independently demonstrated that the trajectoriesof (3) in the positive quadrant are closed curves surrounding the non-zero equilibrium [20]. Furthermore, they showed thatthe time averages of x and y over these trajectories are simply the x and y coordinates of the non-zero equilibrium:1ÅÅÅÅÅT Ÿ0T x „ t = a 1ÅÅÅÅÅT Ÿ0T y „ t = 1where T is the period of the trajectory under consideration. Integrating both sides of the first equation in H3L from 0 to Tshows that:1ÅÅÅÅÅT Ÿ0T xy „ t = a.Introduction of the Scavenger SystemThe standard nth order Lotka-Volterra predator prey system is constructed under the assumption that the per capitagrowth rate of species xi can be written as the linear function li + ⁄ j=1n Mij x j , where li is the intrinsic growth rate of xi , andMij describes the impact of species xj on the per capita growth rate of species xi . The term Mii is usually zero or negative,and the term Mii xi takes into account the impact of intraspecific competition on the per capita growth rate of xi [19]. Theterms Mij xj , j = 1, 2, …, i - 1, i + 1, …, n, take into account the predator prey relationships between xi and the otherspecies in the community.We propose a system of differential equations, similar to the Lotka-Volterra predator prey model, that allows scaven-ger species to be modeled in addition to the usual predator and prey species. We begin with the most basic such model.Consider a prey species x and a predator species y that obey the system of differential equations (3). Now we add a scaven-ger species z that feeds on the carrion produced when an animal from the population y kills an animal of the population x.For the moment, we ignore the obvious possibility that z would also feed on carrion produced from natural death. Thenumber of carrion produced by y preying on x should be proportional to the number of interactions between members of thepopulations y and x, and hence proportional to the product xy. The per capita growth rate of z should be a combination of theintrinsic growth rate of z and a term accounting for the number of carrion in the environment. Hence we arrive at the model:






































T-b + cxy - z „ t = -bT + caT - Ÿ0






-b + cx y - z „ t = 0. Ÿ0T x y „ t = Ta,




















mAij ‰

k=1

nxk


xi = ‰j=1

ny j


y° i = l`

i yi + yi ‚j=1

mA` i j ‰

k=1

nyk

B`


`= D-1 l, A

`= D-1 A, and B` = BD.


y° 1 = y1 - y1 y2

y° 2 = -ay2 + y1 y2

y° 3 = -by3 + cy3 y4








x° = x - bxy - x2



u° 1 = u1 - u12 - b u1 u2






* M - ui* , where

u1* = 1 - b c, u2

* = c, u3* = d - a - bcd, u4







Scavenger.nb 10


IntroductionLotka-Volterra systems of differential equations have played an important role in the development of mathematicalecology since their introduction in the early 20th century ([12], [20]). Lotka-Volterra models lack the complexity andrealism of more recently developed population models, but their generality makes them a convenient starting point foranalyzing ecological systems. Tilman remarks that, "Because the purpose of ecology is to understand the causes of patternsin nature, we should start by studying the largest, most general, and most repeatable patterns," and "General patterns providea framework within less general patterns can be more effectively studied" [11]. Indeed, in [14] Berryman describes how theLotka-Volterra systems have served as a basis for the development of more realistic models that involve ratio-dependentfunctional responses. When new concepts in population modelling are to be studied, it is logical to begin with the mostgeneral and simplistic mathematical framework. Thus the Lotka-Volterra models provide a foundation for new modellingideas to build upon.The classical Lotka-Volterra systems model the quantities (or densities) of species in an ecosystem under the assump-tion that the per-capita growth rate of each species is a linear function of the abundances of the species in the system [12].This framework allows for the modelling of a wide range of pairwise interactions between species, including predator-prey,competitive, and symbiotic relationships. Unfortunately, such pairwise interactions do not encapsulate the diversity ofinterspecies relationships present in observed ecosystems. In particular, the classical Lotka-Volterra systems fail to modelthe presence of scavenger species. Scavenger species subsist on carrion (the dead remains of other creatures). When ascavenger species is present in a system with a predator species and a prey species, the scavenger will benefit when thepredator kills its prey. This is not the type of pairwise interaction that can be modelled by the classical Lotka-Volterraequations, so the inclusion of scavengers requires a modification of these systems.The important role that scavengers play in ecosystems has long been recognized. In 1877, naturalist SebastianTenney enjoined his colleagues to devote more study to scavengers [15]. Dr. Tenney would be disappointed to learn thatscavengers have long been neglected in population ecology, relative to herbivores and predators. In recent years, the field ofsystems ecology has placed great emphasis on the study of detritus and detritrivores. Several mathematical models have beendeveloped to model detritus and detritrivore abundance in ecosystems ([9], [10], [11], [12], [13]). These models all treat theamount of detritus as a distinct state-variable. The detritrivores under consideration are primarily decomposers (includingbacteria and fungi) and microinverterbrates. The analysis we put forth differs from these studies in two important respects.First, we avoid explicitly including the detritus as a variable, which allows for easier analysis and system visualization.Second, we concentrate primarily on vertebrate carrion-feeders (e.g. hyenas, vultures, and ravens) rather than decomposers.A recent review by DeValt, Rhodes, and Shivik indicates that carrion use by terrestrial vertebrates is much more prevalentthan previously thought, and that the significance of terrestrial scavenging has been underestimated by the ecological commu-nity [31]. They conclude that "a clearer perspective on carrion use by terrestrial vertebrates will improve our understandingof critical ecological processes, particularly those associated with energy flow and trophic interactions" [31].A variety of species-specific studies of scavenger and detritrivore populations have been conducted. A thoroughlisting can be found in [7], but we will site three examples here. In [21], the impact of fisheries discards on benthic scaven-ger populations is studied. In [22], the impact of elk carrion on large carnivore populations is studied. In [23], a model ofwolf-elk population dynamics that addresses the amount of carrion available to scavengers is developed. This model ishighly specific to the wolf and elk of Yellowstone Park. It is a complicated discrete model using Leslie matrices, and theonly solutions available are numerical. We attempt to model scavengers in a far more general format, and we use analyticrather than numeric methods.Stephen Heard [7] developed the notion of processing chain ecology. Heard introduced compartmental models weredifferent species alter the state of resources, thereby affecting the type of resources available to other species. This type ofmodelling can be applicable to scavenger population modelling because a predator could be viewed as altering the state of aresource (prey) to a state that is useful to a scavenger (carrion). Heard's model was primarily directed at the processing offauna and non-living resources (as opposed to the processing of a prey animal), as is apparent in his extensive table ofapplications. Heard's model, like the detritus models mentioned above, also suffers from the drawbacks of including resourcestates (carrion) as explicit variables, thereby increasing the dimensionality of the system.Lotka-Volterra background In general, an nth order Lotka-Volterra system takes the form:dxiÅÅÅÅÅÅÅÅÅdt = li xi + xi ⁄ j=1n Mij xj , i = 1, 2, …, nH1Lwhere " i, j li and Mi j are real constants [20]. Ecologists frequently use an nth order Lotka-Volterra system to model thepopulations of n different species interacting in a food web. The n ä n matrix M with entries Mij is called the interactionmatrix of the system. For predator-prey Lotka-Volterra systems, the interaction matrix M has a special form:" Hi, jL œ 81, 2, …, n< such that i j, Mi j Mji § 0 [19]. The classical two-species predator-prey Lotka-Volterra model isgiven by:dxÅÅÅÅÅÅÅdt = ax - bxydyÅÅÅÅÅÅÅdt = -cy + dxy H2Lwhere x is the population of a prey species, y is the population of a predator species, and a, b, c, and d are positive constants.When analyzing the qualitative behavior of systems such as (2), it is convenient to rescale variables so that some of theconstants involved equal one. The number of constants that may be eliminated by variable rescaling will generally dependon the number of variables in the system and the structure of the differential equations. For the remainder of this paper,differential equations will be written in a rescaled format so that the maximum possible number of constants are unity. Forexample, (2) may be rescaled as:dxÅÅÅÅÅÅÅdt = x - xydyÅÅÅÅÅÅÅdt = -ay + xy H3LThis set of equations has equilibria at H0, 0L and Ha, 1L. Lotka and Volterra independently demonstrated that the trajectoriesof (3) in the positive quadrant are closed curves surrounding the non-zero equilibrium [20]. Furthermore, they showed thatthe time averages of x and y over these trajectories are simply the x and y coordinates of the non-zero equilibrium:1ÅÅÅÅÅT Ÿ0T x „ t = a 1ÅÅÅÅÅT Ÿ0T y „ t = 1where T is the period of the trajectory under consideration. Integrating both sides of the first equation in H3L from 0 to Tshows that:1ÅÅÅÅÅT Ÿ0T xy „ t = a.Introduction of the Scavenger SystemThe standard nth order Lotka-Volterra predator prey system is constructed under the assumption that the per capitagrowth rate of species xi can be written as the linear function li + ⁄ j=1n Mij x j , where li is the intrinsic growth rate of xi , andMij describes the impact of species xj on the per capita growth rate of species xi . The term Mii is usually zero or negative,and the term Mii xi takes into account the impact of intraspecific competition on the per capita growth rate of xi [19]. Theterms Mij xj , j = 1, 2, …, i - 1, i + 1, …, n, take into account the predator prey relationships between xi and the otherspecies in the community.We propose a system of differential equations, similar to the Lotka-Volterra predator prey model, that allows scaven-ger species to be modeled in addition to the usual predator and prey species. We begin with the most basic such model.Consider a prey species x and a predator species y that obey the system of differential equations (3). Now we add a scaven-ger species z that feeds on the carrion produced when an animal from the population y kills an animal of the population x.For the moment, we ignore the obvious possibility that z would also feed on carrion produced from natural death. Thenumber of carrion produced by y preying on x should be proportional to the number of interactions between members of thepopulations y and x, and hence proportional to the product xy. The per capita growth rate of z should be a combination of theintrinsic growth rate of z and a term accounting for the number of carrion in the environment. Hence we arrive at the model:dxÅÅÅÅÅÅÅdt = x - xydyÅÅÅÅÅÅÅdt = -ay + xydzÅÅÅÅÅÅdt = -bz + cxyz H4LProposition: Let HxHtL, yHtL, zHtLL be a solution to (4) with initial condition HxH0L, yH0L, zH0LL = Hx0 , y0 , z0 L in the positiveoctant. If ca = b, then Ha, 1, sL 0 < s < ¶ is a line of fixed points, and if HxHtL, yHtL, zHtLL is not on this line, then it is periodic.If ca > b, then limtØ¶ zHtL = ¶. If ca < b, then limtØ¶ zHtL = 0.Proof: Let HxHtL, yHtL, zHtLL be a solution to H4L with initial condition in the positive octant. It is easily seen that the coordinateplanes are invariant with respect to (4). For the xy-plane, the upward pointing unit normal vector is H0, 0, 1L.H0, 0, 1L ÿ I dxÅÅÅÅÅÅÅdt , dyÅÅÅÅÅÅÅdt , dzÅÅÅÅÅÅdt M …z=0 = 0, so the plane is invariant. Similar statements hold for the yz and xz planes. The invariance ofthe coordinate planes implies that HxHtL, yHtL, zHtLL must lie in the positive octant for all t > 0.Let p : !3 Ø !2 be the projection given by pHx, y, zL = Hx, yL. Note that the first two equations in (4) are simply the classical2-dimensinonal Lotka-Volterra predator-prey equations. Neither of these equations depend on z, so pHxHtL, yHtL, zHtLL isindependent of zHtL. Recall that solutions to the classical 2-dimensional Lotka-Volterra predator prey equations are closedcurves in the positive quadrant. Then pHxHtL, yHtL, zHtLL must coincide with one of the closed curves, which implies that ,thesolutions to H4L are confined to cylinders parallel to the z-axis. This also means that pHxHtL, yHtL, zHtLL must be periodic withsome period T > 0.Define a return map n Ø zn by zn = zHnTL.From the Lotka-Volterra background section, recall that 1ÅÅÅÅÅT Ÿ0T xHtL yHtL „ t = a. xHtL and yHtL are periodic, so " n œ ",1ÅÅÅÅÅT Ÿ0T xHtL yHtL „ t = 1ÅÅÅÅÅT ŸnTHn+1L T xHtL yHtL „ t = a. In the positive octant, dÅÅÅÅÅÅdt lnHzHtLL = z' HtLÅÅÅÅÅÅÅÅÅÅzHtL = -b + c xHtL yHtL, so for any n œ ",1ÅÅÅÅÅT ŸnTHn+1L T -b + c xHtL yHtL „ t = 1ÅÅÅÅÅT ŸnTHn+1L T dÅÅÅÅÅÅdt lnHzHtLL „ t = 1ÅÅÅÅÅT lnI zHHn+1L T LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅzHnTL M = 1ÅÅÅÅÅT lnI zn+1ÅÅÅÅÅÅÅÅÅÅzn M.If ca > b, then 1ÅÅÅÅÅT lnI zn+1ÅÅÅÅÅÅÅÅÅÅzn M = 1ÅÅÅÅÅT ŸnTHn+1L T -b + c xHtL yHtL „ t = -b + ca > 0, so zn+1 > zn . Therefore zn , n = 1, 2, … is a monotoni-cally increasing sequence. Suppose, for contradiction, that limnØ¶ zn = z* . Then there would have to exist a periodicsolution to (4), HxHtL, yHtL, zHtLL, with initial condition Hx0 , y0 , z* L. Defining a return map n Ø zn for this solution by zn = zHnTLand proceeding as before, we see that zn+1 > zn " n œ ", violating the periodicity of HxHtL, yHtL, zHtLL. Hence limnØ¶ zn = ¶.If ca < b, then 1ÅÅÅÅÅT lnI zn+1ÅÅÅÅÅÅÅÅÅÅzn M = 1ÅÅÅÅÅT ŸnTHn+1L T -b + c xHtL yHtL „ t = -b + ca < 0, so zn+1 < zn . Therefore zn , n = 1, 2, … is a monotoni-cally decreasing sequence. Suppose, for contradiction, that limnØ¶ zn = z* > 0. Then there would have to exist a periodicsolution to (4), HxHtL, yHtL, zHtLL, with initial condition Hx0 , y0 , z* L. Defining a return map n Ø zn for this solution by zn = zHnTLand proceeding as before, we see that zn+1 < zn " n œ ", violating the periodicity of HxHtL, yHtL, zHtLL. Hence limnØ¶ zn = 0.If ca = b, then 1ÅÅÅÅÅT lnI zn+1ÅÅÅÅÅÅÅÅÅÅzn M = 1ÅÅÅÅÅT ŸnTHn+1L T -b + c xHtL yHtL „ t = -b + ca = 0, so zn+1 = zn . ThereforeHxHHn + 1LTL, yHHn + 1LTL, zHHn + 1LTLL = HxHnTL, yHnTL, zHnTLL " n œ " , so HxHtL, yHtL, zHtLL is periodic with period T.If x0 = a and y0 = 1, then dxÅÅÅÅÅÅÅdt = dyÅÅÅÅÅÅÅdt = dzÅÅÅÅÅÅdt = 0, so Hx0 , y0 , z0 L is a fixed point, regardless of the initial value z0 . HenceHa, 1, sL 0 < s < ¶ is a line of fixed points.




















T-b + cxy - z „ t = -bT + caT - Ÿ0






-b + cx y - z „ t = 0. Ÿ0T x y „ t = Ta,




















mAij ‰

k=1

nxk


xi = ‰j=1

ny j


y° i = l`

i yi + yi ‚j=1

mA`

i j ‰k=1

nyk

B`


`= D-1 l, A` = D-1 A, and B` = BD.


y° 1 = y1 - y1 y2

y° 2 = -ay2 + y1 y2

y° 3 = -by3 + cy3 y4








x° = x - bxy - x2



u° 1 = u1 - u12 - b u1 u2






* M - ui* , where

u1* = 1 - b c, u2

* = c, u3* = d - a - bcd, u4







Scavenger.nb 11


IntroductionLotka-Volterra systems of differential equations have played an important role in the development of mathematicalecology since their introduction in the early 20th century ([12], [20]). Lotka-Volterra models lack the complexity andrealism of more recently developed population models, but their generality makes them a convenient starting point foranalyzing ecological systems. Tilman remarks that, "Because the purpose of ecology is to understand the causes of patternsin nature, we should start by studying the largest, most general, and most repeatable patterns," and "General patterns providea framework within less general patterns can be more effectively studied" [11]. Indeed, in [14] Berryman describes how theLotka-Volterra systems have served as a basis for the development of more realistic models that involve ratio-dependentfunctional responses. When new concepts in population modelling are to be studied, it is logical to begin with the mostgeneral and simplistic mathematical framework. Thus the Lotka-Volterra models provide a foundation for new modellingideas to build upon.The classical Lotka-Volterra systems model the quantities (or densities) of species in an ecosystem under the assump-tion that the per-capita growth rate of each species is a linear function of the abundances of the species in the system [12].This framework allows for the modelling of a wide range of pairwise interactions between species, including predator-prey,competitive, and symbiotic relationships. Unfortunately, such pairwise interactions do not encapsulate the diversity ofinterspecies relationships present in observed ecosystems. In particular, the classical Lotka-Volterra systems fail to modelthe presence of scavenger species. Scavenger species subsist on carrion (the dead remains of other creatures). When ascavenger species is present in a system with a predator species and a prey species, the scavenger will benefit when thepredator kills its prey. This is not the type of pairwise interaction that can be modelled by the classical Lotka-Volterraequations, so the inclusion of scavengers requires a modification of these systems.The important role that scavengers play in ecosystems has long been recognized. In 1877, naturalist SebastianTenney enjoined his colleagues to devote more study to scavengers [15]. Dr. Tenney would be disappointed to learn thatscavengers have long been neglected in population ecology, relative to herbivores and predators. In recent years, the field ofsystems ecology has placed great emphasis on the study of detritus and detritrivores. Several mathematical models have beendeveloped to model detritus and detritrivore abundance in ecosystems ([9], [10], [11], [12], [13]). These models all treat theamount of detritus as a distinct state-variable. The detritrivores under consideration are primarily decomposers (includingbacteria and fungi) and microinverterbrates. The analysis we put forth differs from these studies in two important respects.First, we avoid explicitly including the detritus as a variable, which allows for easier analysis and system visualization.Second, we concentrate primarily on vertebrate carrion-feeders (e.g. hyenas, vultures, and ravens) rather than decomposers.A recent review by DeValt, Rhodes, and Shivik indicates that carrion use by terrestrial vertebrates is much more prevalentthan previously thought, and that the significance of terrestrial scavenging has been underestimated by the ecological commu-nity [31]. They conclude that "a clearer perspective on carrion use by terrestrial vertebrates will improve our understandingof critical ecological processes, particularly those associated with energy flow and trophic interactions" [31].A variety of species-specific studies of scavenger and detritrivore populations have been conducted. A thoroughlisting can be found in [7], but we will site three examples here. In [21], the impact of fisheries discards on benthic scaven-ger populations is studied. In [22], the impact of elk carrion on large carnivore populations is studied. In [23], a model ofwolf-elk population dynamics that addresses the amount of carrion available to scavengers is developed. This model ishighly specific to the wolf and elk of Yellowstone Park. It is a complicated discrete model using Leslie matrices, and theonly solutions available are numerical. We attempt to model scavengers in a far more general format, and we use analyticrather than numeric methods.Stephen Heard [7] developed the notion of processing chain ecology. Heard introduced compartmental models weredifferent species alter the state of resources, thereby affecting the type of resources available to other species. This type ofmodelling can be applicable to scavenger population modelling because a predator could be viewed as altering the state of aresource (prey) to a state that is useful to a scavenger (carrion). Heard's model was primarily directed at the processing offauna and non-living resources (as opposed to the processing of a prey animal), as is apparent in his extensive table ofapplications. Heard's model, like the detritus models mentioned above, also suffers from the drawbacks of including resourcestates (carrion) as explicit variables, thereby increasing the dimensionality of the system.Lotka-Volterra background In general, an nth order Lotka-Volterra system takes the form:dxiÅÅÅÅÅÅÅÅÅdt = li xi + xi ⁄ j=1n Mij xj , i = 1, 2, …, nH1Lwhere " i, j li and Mi j are real constants [20]. Ecologists frequently use an nth order Lotka-Volterra system to model thepopulations of n different species interacting in a food web. The n ä n matrix M with entries Mij is called the interactionmatrix of the system. For predator-prey Lotka-Volterra systems, the interaction matrix M has a special form:" Hi, jL œ 81, 2, …, n< such that i j, Mi j Mji § 0 [19]. The classical two-species predator-prey Lotka-Volterra model isgiven by:dxÅÅÅÅÅÅÅdt = ax - bxydyÅÅÅÅÅÅÅdt = -cy + dxy H2Lwhere x is the population of a prey species, y is the population of a predator species, and a, b, c, and d are positive constants.When analyzing the qualitative behavior of systems such as (2), it is convenient to rescale variables so that some of theconstants involved equal one. The number of constants that may be eliminated by variable rescaling will generally dependon the number of variables in the system and the structure of the differential equations. For the remainder of this paper,differential equations will be written in a rescaled format so that the maximum possible number of constants are unity. Forexample, (2) may be rescaled as:dxÅÅÅÅÅÅÅdt = x - xydyÅÅÅÅÅÅÅdt = -ay + xy H3LThis set of equations has equilibria at H0, 0L and Ha, 1L. Lotka and Volterra independently demonstrated that the trajectoriesof (3) in the positive quadrant are closed curves surrounding the non-zero equilibrium [20]. Furthermore, they showed thatthe time averages of x and y over these trajectories are simply the x and y coordinates of the non-zero equilibrium:1ÅÅÅÅÅT Ÿ0T x „ t = a 1ÅÅÅÅÅT Ÿ0T y „ t = 1where T is the period of the trajectory under consideration. Integrating both sides of the first equation in H3L from 0 to Tshows that:1ÅÅÅÅÅT Ÿ0T xy „ t = a.Introduction of the Scavenger SystemThe standard nth order Lotka-Volterra predator prey system is constructed under the assumption that the per capitagrowth rate of species xi can be written as the linear function li + ⁄ j=1n Mij x j , where li is the intrinsic growth rate of xi , andMij describes the impact of species xj on the per capita growth rate of species xi . The term Mii is usually zero or negative,and the term Mii xi takes into account the impact of intraspecific competition on the per capita growth rate of xi [19]. Theterms Mij xj , j = 1, 2, …, i - 1, i + 1, …, n, take into account the predator prey relationships between xi and the otherspecies in the community.We propose a system of differential equations, similar to the Lotka-Volterra predator prey model, that allows scaven-ger species to be modeled in addition to the usual predator and prey species. We begin with the most basic such model.Consider a prey species x and a predator species y that obey the system of differential equations (3). Now we add a scaven-ger species z that feeds on the carrion produced when an animal from the population y kills an animal of the population x.For the moment, we ignore the obvious possibility that z would also feed on carrion produced from natural death. Thenumber of carrion produced by y preying on x should be proportional to the number of interactions between members of thepopulations y and x, and hence proportional to the product xy. The per capita growth rate of z should be a combination of theintrinsic growth rate of z and a term accounting for the number of carrion in the environment. Hence we arrive at the model:dxÅÅÅÅÅÅÅdt = x - xydyÅÅÅÅÅÅÅdt = -ay + xydzÅÅÅÅÅÅdt = -bz + cxyz H4LProposition: Let HxHtL, yHtL, zHtLL be a solution to (4) with initial condition HxH0L, yH0L, zH0LL = Hx0 , y0 , z0 L in the positiveoctant. If ca = b, then Ha, 1, sL 0 < s < ¶ is a line of fixed points, and if HxHtL, yHtL, zHtLL is not on this line, then it is periodic.If ca > b, then limtØ¶ zHtL = ¶. If ca < b, then limtØ¶ zHtL = 0.Proof: Let HxHtL, yHtL, zHtLL be a solution to H4L with initial condition in the positive octant. It is easily seen that the coordinateplanes are invariant with respect to (4). For the xy-plane, the upward pointing unit normal vector is H0, 0, 1L.H0, 0, 1L ÿ I dxÅÅÅÅÅÅÅdt , dyÅÅÅÅÅÅÅdt , dzÅÅÅÅÅÅdt M …z=0 = 0, so the plane is invariant. Similar statements hold for the yz and xz planes. The invariance ofthe coordinate planes implies that HxHtL, yHtL, zHtLL must lie in the positive octant for all t > 0.Let p : !3 Ø !2 be the projection given by pHx, y, zL = Hx, yL. Note that the first two equations in (4) are simply the classical2-dimensinonal Lotka-Volterra predator-prey equations. Neither of these equations depend on z, so pHxHtL, yHtL, zHtLL isindependent of zHtL. Recall that solutions to the classical 2-dimensional Lotka-Volterra predator prey equations are closedcurves in the positive quadrant. Then pHxHtL, yHtL, zHtLL must coincide with one of the closed curves, which implies that ,thesolutions to H4L are confined to cylinders parallel to the z-axis. This also means that pHxHtL, yHtL, zHtLL must be periodic withsome period T > 0.Define a return map n Ø zn by zn = zHnTL.From the Lotka-Volterra background section, recall that 1ÅÅÅÅÅT Ÿ0T xHtL yHtL „ t = a. xHtL and yHtL are periodic, so " n œ ",1ÅÅÅÅÅT Ÿ0T xHtL yHtL „ t = 1ÅÅÅÅÅT ŸnTHn+1L T xHtL yHtL „ t = a. In the positive octant, dÅÅÅÅÅÅdt lnHzHtLL = z' HtLÅÅÅÅÅÅÅÅÅÅzHtL = -b + c xHtL yHtL, so for any n œ ",1ÅÅÅÅÅT ŸnTHn+1L T -b + c xHtL yHtL „ t = 1ÅÅÅÅÅT ŸnTHn+1L T dÅÅÅÅÅÅdt lnHzHtLL „ t = 1ÅÅÅÅÅT lnI zHHn+1L T LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅzHnTL M = 1ÅÅÅÅÅT lnI zn+1ÅÅÅÅÅÅÅÅÅÅzn M.If ca > b, then 1ÅÅÅÅÅT lnI zn+1ÅÅÅÅÅÅÅÅÅÅzn M = 1ÅÅÅÅÅT ŸnTHn+1L T -b + c xHtL yHtL „ t = -b + ca > 0, so zn+1 > zn . Therefore zn , n = 1, 2, … is a monotoni-cally increasing sequence. Suppose, for contradiction, that limnØ¶ zn = z* . Then there would have to exist a periodicsolution to (4), HxHtL, yHtL, zHtLL, with initial condition Hx0 , y0 , z* L. Defining a return map n Ø zn for this solution by zn = zHnTLand proceeding as before, we see that zn+1 > zn " n œ ", violating the periodicity of HxHtL, yHtL, zHtLL. Hence limnØ¶ zn = ¶.If ca < b, then 1ÅÅÅÅÅT lnI zn+1ÅÅÅÅÅÅÅÅÅÅzn M = 1ÅÅÅÅÅT ŸnTHn+1L T -b + c xHtL yHtL „ t = -b + ca < 0, so zn+1 < zn . Therefore zn , n = 1, 2, … is a monotoni-cally decreasing sequence. Suppose, for contradiction, that limnØ¶ zn = z* > 0. Then there would have to exist a periodicsolution to (4), HxHtL, yHtL, zHtLL, with initial condition Hx0 , y0 , z* L. Defining a return map n Ø zn for this solution by zn = zHnTLand proceeding as before, we see that zn+1 < zn " n œ ", violating the periodicity of HxHtL, yHtL, zHtLL. Hence limnØ¶ zn = 0.If ca = b, then 1ÅÅÅÅÅT lnI zn+1ÅÅÅÅÅÅÅÅÅÅzn M = 1ÅÅÅÅÅT ŸnTHn+1L T -b + c xHtL yHtL „ t = -b + ca = 0, so zn+1 = zn . ThereforeHxHHn + 1LTL, yHHn + 1LTL, zHHn + 1LTLL = HxHnTL, yHnTL, zHnTLL " n œ " , so HxHtL, yHtL, zHtLL is periodic with period T.If x0 = a and y0 = 1, then dxÅÅÅÅÅÅÅdt = dyÅÅÅÅÅÅÅdt = dzÅÅÅÅÅÅdt = 0, so Hx0 , y0 , z0 L is a fixed point, regardless of the initial value z0 . HenceHa, 1, sL 0 < s < ¶ is a line of fixed points. Periodic orbits when ca = b



















T-b + cxy - z „ t = -bT + caT - Ÿ0






-b + cx y - z „ t = 0. Ÿ0T x y „ t = Ta,




















mAij ‰

k=1

nxk


xi = ‰j=1

ny j


y° i = l`

i yi + yi ‚j=1

mA`

i j ‰k=1

nyk

B`


`= D-1 l, A` = D-1 A, and B` = BD.


y° 1 = y1 - y1 y2

y° 2 = -ay2 + y1 y2

y° 3 = -by3 + cy3 y4








x° = x - bxy - x2



u° 1 = u1 - u12 - b u1 u2






* M - ui* , where

u1* = 1 - b c, u2

* = c, u3* = d - a - bcd, u4







Scavenger.nb 12


Documents

Periodic orbits when ca bmath.bd.psu.edu/faculty/jprevite/REU05/Scavenger2.pdfecology since their introduction in the early 20th century ([12], [20]). Lotka-Volterra models lack the