Interesting Features of Three-Dimensional Discrete Lotka-Volterra Dynamics

Discrete Lotka-Volterra systems in one dimension (the logistic equation) and two dimensions have been studied extensively, revealing a wealth of complex dynamical regimes. We show that three-dimensional discrete Lotka-Volterra dynamical systems exhibit all of the dynamics of the lower dimensional systems and a great deal more. In fact and in particular, there are dynamical features including analogs of flip bifurcations, Neimark-Sacker bifurcations and chaotic strange attracting sets that are essentially three-dimensional. Among these are new generalizations of Neimark-Sacker bifurcations and novel chaotic strange attractors with distinctive candy cane type shapes. Several of these dynamical are investigated in detail using both analytical and simulation techniques.


Introduction
We shall consider a natural discrete analog of the famous and ubiquitous Lotka-Volterra model, developed independently by Alfred Lotka [1] and Vito Volterra [2] for the dynamics of a population comprising several interacting species, say ( )  (2) where the birth-rate parameters i λ and interaction coefficients ij γ are usually nonnegative constants as we shall assume in the sequel, with 1 ii γ = for all 1 i m ≤ ≤ , 1 j n ≤ ≤ . Analogous to our remark above, we note that the 1-dimensional version of (2) is the discrete logistic equation that, in contrast to (1) for 1 2 m ≤ ≤ , can exhibit chaotic dynamics.
Although our focus here will be on the 3-dimensional version of (2), we first describe a few useful details for the m-dimensional system, which has dynamics defined by iterates of the map : m m F →   comprising a discrete (semi-) dynamical system and is represented as where the coefficients are as described in (2). Here, the λ 's and γ 's represent the birth-rates and interaction strengths of and among the populations, respectively. Inasmuch as we are primarily thinking of x as a population vector, it is natural to consider the map F restricted to x  X X : (4) and the admissible i λ ranges have to be adjusted accordingly.
It is clear from (3) that a necessary and sufficient condition for X is that the following system of inequalities be satisfied:  One can by lengthy but straightforward calculations find the maximizers and associated maximum values of each of the i ϕ , which yield a rather complicated system of inequalities determining the optimal admissible values of λ . However, the system is complex to the point of being very unwieldy, we shall opt for a cruder but much easier to use system of inequalities described in the following result. Lemma 1. The birth-rate parameters satisfying the system of inequalities ≤ ≤ ∑ (6) are admissible in X for the discrete dynamical system generated by the iterates of the map (3).
Proof. It follows from (4) and (5)  λ which implies that (5) follows from (6), thereby completing the proof. The generalities discussed above aside, our almost exclusive focus shall be on the discrete dynamical system generated by the iterates of the map 3 3 : 1  12  13  2  21  23  3  31  32   , , : , , , , Clearly, the x, y-, x, zand y, z -planes as well as the x-, yand z-axes are F-invariant, so all of the interesting dynamics for the 1-dimensional and 2-dimensional cases of (2)-(3) including flip bifurcations, period-doubling cascades (leading to chaos), Neimark-Sacker bifurcations, (2-dimensional) transverse heteroclinic orbit induced chaos and chaotic strange attractors of Hausdorff dimension between one and two such as found in [5] [6] [7] [8] are subsumed under the dynamics defined by (7). Naturally then, one can expect even more complex and interesting higher dimensional analogs of such dynamics in the 3-dimensional case, which shall be elucidated in the sequel. In particular, we shall find and analyze flip bifurcations, certain higher dimensional Neimark-Sacker type bifurcations to be described in the sequel, several 3-dimensional chaotic regimes and a few unusual chaotic strange attractors corresponding to long-term population dynamic states. This includes an apparently new type of chaotic strange attractor shaped like a candy cane. As one might expect, there have been several studies of the dynamics of higher dimensional systems of the Lotka-Volterra (L-V) and related types such as [9]- [15], but our investigation is novel in a number of respects, especially with regard to the Neimark-Sacker bifurcation generalizations and the candy cane chaotic strange attractors. Nevertheless, our work shares some common elements with the dynamics literature. For example, Bischi and Tramontana [10] show that their 3-dimensional Lotka-Volterra type model for industrial clusters exhibits standard flip and Neimark-Sacker bifurcations along with interesting chaotic attractors, which are not candy canes. Another example in Yousef et al. [15], although fundamentally different from our model owing to the delay in one of the growth rates, also is shown to exhibit flip and Neimark-Sacker bifurcations along with Marotto chaos.
The investigation is organized as follows: In Section 2, we describe some basic features of the discrete dynamical system generated by the map (7) such as fixed points and invariant manifolds. One of our foci will be various types of bifurcations that are fully 3-dimensional versions of those found in lower dimensional systems, and one of them-the flip bifurcation shall be investigated in some depth in Section 3. This is followed in Section 4 with a definition of certain higher dimensional analogs of Neimark-Sacker bifurcations and the identification of various coefficient/parameter ranges for which they are exhibited for (7). In addition, several simulations are included to illustrate these bifurcations. Next, in Section 5 the focus is on chaotic strange attractors. We show simulation examples indicating the wide variety of such steady-state sets. In particular, we provide examples where the nature of the steady-state attractor is strongly dependent on the parameters and auxiliary data associated to the dynamical system. Most importantly, however, we introduce new types of chaotic strange attractors, which we analyze in detail. Finally, in Section 6, we present conclusions and plans for future related research including an in depth analysis of the higher dimensional Neimark-Sacker type bifurcations and candy cane attractors, which appear to be quite common in population dynamics.

Basic Dynamical Properties
It is clear from its definition that the map (7) enjoys the following invariance properties concerning the coordinate axes denoted as ( ) { } : ,0,0 : . If any of the coordinates of the initial point for the discrete dynamics for (4) is zero, it follows from Theorem 1 that the whole process reduces to one of two dimensions or less, where it has been shown that the system can exhibit flip bifurcations, period-doubling cascades to chaos, horseshoe type chaos and a variety of other dynamical phenomena, so we shall focus on initial points with no zero coordinates, which are sometimes referred to as coexistence points, mainly in the context of population dynamics. Thus, it may be said that we are restricting our attention to fully three-dimensional analogs of such dynamical behavior. In par-ticular, we shall be most interested in fixed points of the map (7) that are also coexistence points. These fixed points can be found by solving the system of linear equations   1  12  13  1   2  21  23  2   3  31  32  3   1 ,   1 , 1 , which has a unique solution if the matrix Γ is nonsingular. We note that it is sometimes convenient to recast the discrete dynamical system in terms of (translated coordinates) with respect to a fixed point ( ) , , x y z * * * with , x y * * and z * positive, namely with :

Flip Bifurcations and Period-Doubling Cascades
Flip bifurcations and associated period-doubling cascades to chaos are well known types of bifurcations and behaviors thoroughly explicated in such texts as [16] [17], which have been shown to exist in discrete Lotka-Volterra (L-V) dynamical systems associated with the map (7) and its higher and lower dimensional versions.
There are numerous parameter sets for which (7) exhibits flip bifurcations and subsequent period-doubling cascades leading to one-dimensional logistic map type chaos, several of which are described in this section. We begin with some rather simple examples and then analyze a less obvious case.

Some Simple Flip and Period-Doubling Examples
Consider the special case chosen for its ease of analysis of the family of maps (7) given as : , , , This class of L-V maps has the bifurcation behavior described in the following result, which has a straightforward proof that is left to the reader. The dynamics are also illustrated in Figure 1.
Theorem 3. The discrete dynamical system generated by the map (10)  λ ∈ : i) The system F λ has a flip bifurcation at λ p for 3 λ = at which the restric- ii) The restriction f has a period-doubling cascade leading to chaos along L λ as λ increases from 3 to a limit of approximately 3.57 following that of the standard 1-dimensional logistic map.
iii) The fixed point λ p is a sink for 2 3 λ < < and is hyperbolic for X . It should be noted that by permuting the coordinates, two analogs of (10) with flips along lines parallel to the xand y-axes are obtained. Also observe that these examples can be readily extended to higher-dimensional systems comprising an arbitrary number of populations.

More Complicated Flip and Period-Doubling Examples
There are numerous other flip bifurcation/period-doubling cascades to chaos that can occur in the L-V discrete dynamical systems generated by maps of the form (4), but which are not nearly as recognizable as those described above. We shall confine our attention to just the one example : G λ → X X , with equal birth rates serving as the bifurcation parameter (with 2 3.6 λ < ≤ ), shown below. The dynamics shall be analyzed in detail for prescribed ranges of variation of the interaction coefficients for For the coexistence fixed point, we simply solve the following system of equations in which we impose the initial constraints In order to better understand the flip bifurcations that can occur for the map (12), we first consider the following special case: It follows from Lemma 1 that 0 3.7 λ ≤ ≤ is an admissible range for this map. Then, from (13) we find that the unique coexistence fixed point is The type of this fixed point is determined by the eigenvalues of the derivative matrix ( ) which are  , , for all λ , comprises the stable manifold ( ) s W λ p over the same parameter interval. Moreover, It follows of course, that there is a flip bifurcation at 3 λ = at which value the linearized center manifold is , , : , In this parameter range, the attractor is a 2-cycle on The flip bifurcation and period-doubling cascade converging to a Milnor attractor is determined by the restriction of g λ to the continuation of ( ) u W λ p as a 1-dimensional g λ -invariant manifold, which we now analyze.
The determination of the center and unstable manifolds is more conveniently handled by translating the coordinates so that the fixed point is always at the origin as described in (9). In particular, we define the λ -dependent translation 3 3 : , , : where the coexistence point for the original map in the translated coordinates now corresponds to ( ) ( ) , , 0, 0, 0 : . We assume that the invariant 1-dimensional invariant manifold that begins as (super) attracting for 2 3 λ < < , becomes the center manifold with coefficients that are analytic functions of the parameter, converge on a ζ -interval containing the origin and large enough so that the restriction For invariance, we must have 3, λ containing in its interior a λ ∞ such that f λ ∞ has a Milnor attractor that is the period-doubling cascade limit for f λ as λ λ ∞ → . Substituting (20) in (21) and (22) and equating coefficients of like powers of ζ , we obtain the values of the leading order coefficients and the following recurrence relations for the higher order terms and ( ) and [ ] is the usual greatest integer function. Several terms in the series (20) computed using the above recursion formulas are as follows: Now, it a straightforward matter to prove, for example using the method of majorants, that the power series (27)  Lemma 4. The discrete dynamical system generated by the iterates of g λ defined by (15) has the following properties for , , : , : :  such that f λ ∞ has a Milnor attractor. It follows from continuity and smoothness considerations that the qualitative behavior characterized in Lemma 4 should persist for sufficiently small additional (nonzero) interaction coefficients in (12). In order to lend some precision to this observation, a decidedly non-optimal example of this is formulated and a proof is sketched in what follows. In addition, the bifurcation behavior of the this system is illustrated by the simulations in Figure 2.
Theorem 5. Suppose the discrete dynamical system generated by the map (12), which is represented in translated form taking the varying coexistence fixed point into the origin by the map (18) and d differ from zero, satisfies the following properties: Then the dynamical system generated by the map (12) has a flip bifurcation at λ p for a value of the bifurcation parameter λ λ * = near 3 along what is initially a center and then an unstable manifold followed by a smooth invariant extension ( ) W λ * p containing a period-doubling cascade to a (Milnor) attractor ∞  as λ increases to a value of about 3.6.
Proof Sketch. We describe the foundational elements of the proof and leave the verification of some of the details which tends to be rather lengthy, albeit routine to the reader. First, it follows from (13) ; , , , , , : ; , , , , , and We shall compare these with the coefficients of (31), which are readily found (in terms of the fixed point (29) ; , , , , , : It is a simple, but rather tedious, matter to use the formulas above to verify that the assumptions (i)-(iv) imply that the following properties obtain: (p1) , where we have used a form of parametrization analogous to that in our analysis of (15), can be shown to have the same basic unimodular form as that of (15). Hence, it follows that this is an analog of ( )

Higher Dimensional Neimark-Sacker Type Bifurcations
The type of higher dimensional analog of the Neimark-Sacker bifurcation that we have in mind is one of codimension 1 for a discrete dynamical system in m  , 3 m ≥ , with a fixed point that bifurcates from a sink to a source, giving birth to an invariant homeomorph of an ( ) 1 m − -sphere (which is typically as smooth as the system except on a lower dimensional subset) that grows in size as a particular parameter increases. Although we concentrate mainly on 3 m = , the following is a nice smooth example of the type of bifurcation we interested in for 2 m ≥ : Consider the smooth (actually real analytic) map where ⋅ denotes the Euclidean norm on m  and the parameter ( ) 0, λ ∈ ∞ . The origin is a fixed point for all λ , which is a sink for 0 1 λ < < and a source when 1 λ > . We note that the origin is the only fixed point for 0 1 λ < ≤ , but  , , , , , , , which is homeomorphic and piecewise-linearly diffeomorphic to the 2-sphere 2  and is an F-invariant, locally attracting set for the dynamical system having a diameter that increases with λ , as indicated by the simulation in Fig. 4.1. This behavior is somewhat like a piecewise-linear analog of Neimark-Sacker bifurcations in 2  (see [19] [20]); one in fact that has several natural variations. For example, if we consider the following minor modification of (36): where 0 1 ε < < , it is not difficult to verify that this system first has a flip bifurcation at 3 λ = , followed by the fixed point of interest changing from a saddle to a source (at 3 λ ε = + ) and generating a locally attracting, F-invariant piecewise-linear isomorph of 2  like that of (36), only elongated in the x-direction.
It is also rather easy to envisage even higher dimensional generalization of the bifurcations described above for dynamical system far more varied than discrete Lotka-Volterra types. However, here we shall confine our attention to such bifurcations in 3-dimensional Lotka-Volterra systems, reserving a more extensive, general treatment for a later investigation.

Definition of Neimark-Sacker Analogs and Another L-V Example
We first define the analogs of Neimark-Sacker bifurcations for discrete 3-dimensional dynamical systems briefly described above and then state an existence theorem for certain Lotka-Volterra systems. For this, we consider a parameter-dependent 1 C map of the form Observe that the key elements, such as the 2-cycles and invariant 2-cycle pairs of planes, of the NST bifurcation for the simple map (36) are all hyperbolic or normally hyperbolic and therefore locally structurally stable. Consequently, one expects that small changes in the interaction coefficients (which are zero) should still produce NST bifurcations, which is confirmed by our next result. Theorem 6. If the Lotka-Volterra map 3 3 : There is really no loss of generality in this since the general argument is essentially the same as this simpler one, only the calculations are considerably more complicated, albeit routine. Therefore, we assume that It is easy to show that the coexistence fixed point of (42) is λ > , which is 1-dimensional and transverse to the plane z z λ = at the coexistence fixed point. One should also note that the plane z z λ = is F-invariant. Our focus is actually going to be on F 2 and we shall find it convenient to use the translated form of the map described in (9) for which the coexistence fixed point is moved to the origin. The translated version of (42) is , , : , , : , , , , , , , 2  2  2  1  2  3   1  1  1  2  3  2  2  2  1  3   3  3   , , :  , ,   : , , ,  : , , : As a first step in completing the proof, we note that it follows from our previous analysis that there exists a positive ε such that when 0 , , which is such that every point in a sufficiently small neighborhood, with the exception of 0, is attracted to the four pairs of diagonal 2-cycles comprising the vertices of ( ) R λ ∂ . Naturally, one expects that curvilinear analogs of the above results to hold for ε sufficiently small, which what we shall now confirm to complete the proof. We first seek an attracting  2  2  3  1  2 , , , Upon substituting (52) in the above equation, it is straightforward to inductively show that if ε is sufficiently small and   . More precisely, the convergence is to the union of the four 2-cycle sinks comprising the diagonal vertex pairs. Thus, the proof is complete.
It should also to be possible to prove the above theorem using the implicit function theorem, but neither this nor the method used in the proof appears to be feasible for higher dimensions and more general parameter ranges. However, it appears some of the iterative image techniques employed in [21] can adapted to extending the results here and we plan to demonstrate this in a forthcoming paper.
The NST bifurcation for (42) in the case where 0.03 a b = = is illustrated in Figure 3

Steady-State Chaotic Strange Attracting Sets
There are numerous types of chaotic strange attracting sets that can occur for the dynamics of the map (1). Naturally, there are the well-known lower dimensional examples embedded in the coordinate lines and planes such as in [4] [5] [8]. More to the point for the purposes here are fully 3-dimensional examples such as those indicated by simulation and shown in  In this section we shall focus our attention on a detailed analysis of types of  3-dimensional Lotka-Volterra maps exhibiting steady-state behavior manifesting itself in chaotic strange attracting sets (cf. [22] [23]).
We consider maps of the form 1  12  13  2  21  23  3  31  32   , , : , , , , Furthermore, we assume that In what follows we shall demonstrate just how complicated the attracting sets and attractors can be and how wildly and dramatically they can change as certain parameters are varied. However, an inkling of the variety and complexity leading to chaotic attractors can readily be obtained by considering the simple uncoupled map . Moreover, S λ ∞ has a (Milnor) attractor that is the cartesian product 3 :    of the period-doubling cascade limit attractor for each of the coordinate functions and 4 S has a dense chaotic (Milnor) attractor

Bifurcation of an Attractor from a Sink to an Attracting Cycle to a Chaotic Strange Attractor
We shall now analyze a L-V map family with a codimension-1 bifurcation sequence beginning with a sink attractor followed by cycle attractors and ending in a true horseshoe type chaotic strange attractor. In particular, for ease of computation we consider the family of maps : It should be noted that 0 4 λ ≤ ≤ is an admissible parameter range for this family of maps.

Types of Fixed Points of the Map (56)
The fixed points of (56) are easily computed to be We can then determine the types of these fixed points from the spectral properties of the derivative matrix As the above matrix is upper triangular, the eigenvalues at the fixed points are just the diagonal elements evaluated at the fixed ( ) , , which determine the types of the fixed points.

Dynamics in
In this case, it follows from the above that the only fixed point in X is the origin 0 p , which is a hyperbolic sink when 0 1 λ < < and still a sink when 1 λ = .

Dynamics in
, where F X is as in (55) , where K is compact. Proof. Obviously, the map depends smoothly on the parameters. Moreover, as we shall see, the principal features of our proof, both analytical and qualitative, are smoothly persistent under small perturbations of the interaction coefficients. Therefore, owing to the manner in which the theorem is formulated, it suffices to verify the result for the rather simple system just studied above; namely, Let Q be the solid prism having the following vertices in the 0 z = plane: 3 5 , ,0 20 20  Figure 7. We note that both the upper and lower faces of Q are mapped into the , x y -plane ( 0 z = ). Moreover, it is easy to verify that Q is a trapping set for the map for 3.7 a λ λ < ≤ ; i.e.,      the horseshoe of the above theorem is analogous to the twisted horseshoe for the 2-dimensional Lotka-Volterra map [5].

Conclusions and Suggestions
In this study of the discrete Lotka-Volterra dynamical system model, we have shown not only that there is an abundance of fully 3-dimensional flip bifurcations and related period-doubling cascades to chaos, but some rather novel analogs of Neimark-Sacker bifurcations and chaotic strange attractors. These bifurcations have been analyzed in considerable detail exploiting, for example, the identification of the strange chaotic attractors as generalized attracting horseshoes, which are shaped like candy canes. The focus in this investigation was on 3-dimensional systems, but in future research we plan to identify and analyze higher dimensional analogs of the Neimark-Sacker-type bifurcations, candy cane attractors and related dynamical phenomena. Moreover, we shall make an effort to find other examples of interesting and useful discrete dynamical system models, besides those of Lotka-Volterra type, that exhibit these and possibly other novel, unusual or unexpected types of behavior.