Multiattractors in a several-predators-one-prey system

Nonlinear Phenomena in Complex Systems, vol. 28, no. 3 (2025), pp. 289 - 297 Multiattractors in a Several-Predators-One-Prey System Sergey Kryzhevich1, ∗and Gunnar S¨oderbacka2, † 1Institute of Applied Mathematics, Faculty of Applied Physics and Mathematics, Gda´nsk University of Technology, ul. Narutowicza 11/12, 80-233 Gda´nsk, POLAND 2˚ Abo Akademi, Turku FI-20500, FINLAND (Received 13 May, 2025) We give a review of results, mostly latest, on a several-predators-one-prey system, giving conditions for the extinction of one predator and for different types of coexistence of the predators. We mainly consider the case of two predators and the problem of multistability. We pay a lot of attention to the case when a one-dimensional discrete model is giving a good approximation to the behaviour. In this case, we have no more than two attractors. We note that this one-dimensional model also has applications in other fields. We also give a short review of the case when the one-dimensional model is not working, giving more attractors and leaving a lot of interesting unsolved problems related to bifurcations that arise. PACS numbers: 05.45.-a, 05.45.Ac Keywords: predator-prey,population dynamics, period bubbling, shrimp structures, EOS maps, period doubling bifurcations, multistability DOI: https://doi.org/10.5281/zenodo.17236371 1. Introduction We consider a general system of ncompeting predators feeding on the same prey, introduced by Hsu and Waltman, and first analyzed in [1] and in references therein. X0 i=piϕi(S)Xi−diXi, i = 1, ..., n, (1a) S0=H(S)− n X i=1 qiϕi(S)Xi,(1b) where •the non-negative variable Srepresents the prey populations, •the non-negative variables Xirepresent the predator populations, •ϕiis non-decreasing, •parameters pi, qi, diare positive. ∗E-mail: [email protected] †E-mail: [email protected] In the main case, we consider the functions Hand ϕidefined by H(S) = r S 1−S K, ϕi(S) = S S+Ai ,(2) A more general choice of these functions is introduced in [2, 3]. We assume pi> di, otherwise at least one predator will go extinct and the system dimension is reduced. Using a time change and linear variable transformations given in [4] we can convert equations (1) with functions (2) to the system x0 i=mi s−λi s+ai xi,(3a) s0= 1−s− n X i=1 xi s+ai!s, (3b) Here the new variables are normalized populations xiand sand the parameters are positive. In this system there are three parameters for each predator (ai, λi, and mi). We will mainly 289 290 Sergey Kryzhevich and Gunnar S¨oderbacka consider the case of two predators, thus having six positive parameters. Also, we will assume all λi<1, because otherwise again one predator becomes extinct. In this work, we make a review of results for this system, with priority to the newest results. We give more details on the case when we can use a one-dimensional discrete model and add some news in a special section. The system is one of the first counterexamples to the ecological exclusion principle (see references in [5]) according to which predators feeding on the same prey cannot coexist. Here in non-degenerate situations, the predators cannot coexist with constant populations at an equilibrium, but there are possibilities for cyclic and more complicated coexistence including chaos. We here give a review of these types of coexistence and give priority to the problem of multistability. 2. Review of main results of the behaviour of the system 2.1. The case with only one predator and behaviour near coordinate planes It is important to understand the behaviour in each of the coordinate planes with only one predator to understand the general behaviour. In this case, we have the Rosenzweig-McArtur model which is analyzed everywhere and some short information is given in [6]. If the predator survives, the predator and prey coexist at an equilibrium or a limit cycle, which are globally attracting (in the space with positive variables, and except an equilibrium inside the limit cycle). It is important to remember that in the phase portraits the behaviour in time is not seen unless the time direction is known. Thus we have to keep in mind that if the cycles are big (in the sense far from equilibrium), the populations can be small for long time intervals and easy to observe only for short times. Some estimates for the size of the limit cycle are given in [7–10]. We now look at the behaviour of the twopredator-one-prey system near the coordinate planes, where one predator is absent. Suppose that λ1, λ2<1and also assume λ1> λ2 (otherwise we can change the order of the predators). We always have an equilibrium O0at the origin and at the point O1= (0,0,1). The equilibrium O0is a saddle with a two-dimensional stable manifold in the predator plane, where prey is absent, and a one-dimensional unstable manifold along the prey axis between O0and O1. The latter point is also a saddle, but admits a one-dimensional stable manifold along the s-axis. If we have a globally attractor at an equilibrium P1in the x1s-coordinate plane and a limit cycle in the x2s-coordinate plane with an equilibrium P2inside, then the equilibrium P1 will be a saddle with a two-dimensional stable manifold inside the coordinate plane and a onedimensional unstable one going out from the coordinate plane, and the equilibrium P2will be a saddle with a two-dimensional unstable manifold inside the limit cycle in the coordinate plane and a one-dimensional stable manifold one coming from out of the coordinate plane. If we have limit cycles in both the x1sand the x2s-coordinate planes with equilibria P1and P2correspondingly inside the limit cycles, then the equilibrium P1repels in any direction and the equilibrium P2will be a saddle of the same type as in the previous case. The zero-isoclines for the predator equations in the inner coordinate planes are the lines s=λi and the zero-isocline for the prey equation is given by the quadratic expression xi= (1 − s)(s+ai), s ≤1. These isoclines intersect at the equilibria Piand the trajectories rotate around these equilibria. This implies that it is natural to define the Poincar´e map on a part of these isoclines in the coordinate planes and here a part of the lines s=λiworks well. In the space it is a bit more complicated but still we have the planes s=λias zero-isoclines for the predator equations and for the prey equation we have a surface in s≤1given by straight lines for the predators, Нелинейные явления в сложных системах Т. 28, № 3, 2025 Multiattractors in a Several-Predators-One-Prey System 291 for example, in the predator plane it is given by x1 a1+x2 a2= 1. The main dynamics is also here a rotation around the intersections of these isoclines and if the attractor is far from these intersections we can define a Poincar´e map on a plane s=const for examining it. This is anyhow often not true and then it is better to define a Poincar´e map on a part of the surface s0= 0. This often happens when the one-dimensional model is not working. Both types of Poincar´e maps are used in the references we have given in this overview. 2.2. Extinction conditions for one predator The following is proved in [4]: Suppose λi> λj. If aj>ai L+ai(L−1), L =λi(1−λj) λj(1−λi)the predator igoes extinct. More extinction results are found in [3]. It is known that the probabality of coexistence decreases when the number of predators increases, see [5]. The inequality above indicates the same. These conditions are sufficient but not necessary for extinction. To prove these results, only the equations for the predators are used. Thus there is hope to use results like those from [7–10] to take into account also the prey equation and get essentially better sufficient conditions and estimates of the parameter regions where one predator is going extinct or not. 2.3. Different types of coexistence We here and further only discuss the case with two predators. All equilibria are on the boundaries of the phase space in some predator-prey coordinate plane, and thus there are no inner equilibria where the predators coexist. An exception is the degenerate case, when all λiare equal. Anyhow, they can coexist in a cyclic or chaotic way, and there can be more than one attractor. We have found examples with up to four attractors [11]. In some case, the dynamics can be welldescribed by a one-dimensional bimodal Poincar´e map, which can be analyzed also analytically, as we will see below. In this case, we can have no more than two attractors, and we can see there are two attractors only for a small parameter region, hard to find without using this simple onedimensional model map. In the case this one dimensional model is not working, the situation is more complicated and not well understood. Sometimes it resembles the Shilnikov type spiral chaos, but still we have not discovered any contours from which we can use the Shilnikov theorems in [12]. Attempts to explain it from different types of bifurcations have also not been successful. A detailed explanation is given below. 2.4. The one dimensional model map In some cases, the motion on the attractors looks like moving on a two-dimensional tube around the zero-isoclines of the system and the contraction in one direction (the sum of the predators) is very strong. Thus we have managed to construct a one-dimensional Poincar´e map defined on a subset of the set defined by s= λ2, s0<0. The way to obtain this map is explained in [11]. The map can be written in the form f(v) = fk,b(v) = v−b+k 1 + ev,(4) where v= ln(x2/x1). and the essential variable here is the ratio of the predator populations. Conditions for existence of such models can be found in [2]. One condition is that there are quite big limit cycles in both predator-prey coordinate planes. The map is bimodal and the graph of such a function we see in the figure 1. It has two critical points, the local maximum at −xcand the local minimum at xc>0. The function is increasing on both IL= (−∞,−xc]and IR= [xc,∞)with the derivative being between 0and 1and decreasing in IC= [−xc, xc]. Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 292 Sergey Kryzhevich and Gunnar S¨oderbacka FIG. 1: Graph of the model map with the asymptotes y=x−b(lower for x→ ∞) and y=x+k−b(upper for x→ −∞). We notice that to have a fixed point meaning possibility for coexistence of predators we need k > b > 0. Such a map can be useful in many other applications and is interesting itself. Some central properties we give here, more to be found in [13]. It is sometimes called the EOS-map [14] and possesses many interesting properties. We briefly list them here and give the details in Section 3 below. •Symmetry: Any map for k < 2bis equivalent to one for 2b<k<b. •For k < b2 b−2the fixed point x0= ln k b−1 is globally attracting. •The Schwarizian derivative is negative for k > 4and it has at most two attractors. •There can be at most one two-periodic orbit. •The interval [f(xc), f(−xc)] is globally absorbing. •The lowest periods of coexisting attractors equal three and four. Our numerical studies demonstrate the evidence of coexisting attractors with all periods mand n, where m+n > 6and m, n > 2. Figure 2 summarizes some results from numerical and analytical examinations. We use the notations: b1=−2xc+1+exc, b2= 2 xc+1+e−xc.(5) There is a k-value k∗at which b1=b2can be found from the equation sinh(xc)=2xc. For k≤k∗, either the global attractor is a fixed point or there is a two-periodic orbit attracting everything except preimages of the fixed point. For k > k∗we obtained the following. •For b < b2there is a globally absorbing interval containing only the critical point xc, where the map is unimodal. •For b > b1there is a globally absorbing interval containing only the critical point −xc, where the map is unimodal. •b2< b < b1the absorbing interval [f(xc), f(−xc)] contains both critical points and sometimes there are two attractors. Since the phenomena of multiattractors is observed only for a very small region of parameters, it is necessary to ask whether this is preserved for the original continuous system. Such problems with using one-dimensional models are examined in [15]. Whatsoever, numerical experiments show that there is a good correspondence with the original system of differential equation. We examine the behaviour of the mentionned one-dimensional mapping in the parameters space. In [11] we have found an example for the original system with one 3-periodic attractor coexisting with one 4-periodic one. Of course, in general, the coexisting attractors look more like chaotic. Нелинейные явления в сложных системах Т. 28, № 3, 2025 Multiattractors in a Several-Predators-One-Prey System 293 FIG. 2: Regions in parameter plane. Curves correspond to period doubling, to b=b1, to b=b2and to k=band k= 2 b. Regions for two-periodic attractors is shown in red, for three-periodic attractors in yellow, 4-periodic attractors in magenta, for 5-periodic attractors in green, for 6-periodic attractors in cyan and for higher periods or expected chaos in blue. The black rings are parameters, where numerical experiments suggest existence of two attractors. 2.5. More complicated coexistence If the model 1D map is not working, we can have much more complicated behaviour. One example of parameter regions where there sometimes are four coexisting attractors is discussed in [11]. There have been early works studying bifurcations from which cyclic coexistence can arise. One such is found in [16]. Even if the attractor looks like spiral-chaos, we have not managed to prove any Shilinikov-type case arising from different contours generalizing Shilnikov theorems. One attempt is seen in [4], where only one simple cycle can appear from the bifurcating contour. Recently in [17] it is proved that if there is a globally stable equilibrium in one predator-prey coordinate plane and a limit cycle in the other, then bifurcations from the degenerate case when λ1=λ2create a one-periodic cycle of coexistence. Studying numerically the Poincar´e map around this cycle we have noticed a rotation immediately after the bifurcation increasing one λ, but after some time this rotation disappears and there is a usual period doubling bifurcation. Often there is also observed a 3-periodic orbit, but this appears somewhere from outside, thus leaving an open question from where it arises. 3. Shrimp structures and periodbubbling in EOS mappings We study the one-dimensional map of the form (4). Except population dynamics these maps also have some applications in machine learning and Game Theory. It is known that those maps admit at most two attracting periodic orbits and typically have one. We study the mechanisms by which bistability appears and the structures that the bistability regions form in the space of parameters. Shrimp structures, coexistence of two periodic attractors as well as period-bubbling routes to chaos have been discussed in many papers, first introduced in [18] (see [19] and references therein for the more recent survey). The existence of Arnold tongues for prey-predator models has also been observed in [20]. Numerical evidence of period-bubbling scenarios for some ecological models has been discussed in [21]. By definition [22] (see also [23]) "Shrimps are formed by a regular set of adjacent windows centered around the main pair of intersecting superstable parabolic arcs. A shrimp is a doubly infinite mosaic of stability domains composed of an innermost main domain plus all the adjacent stability domains arising from two period-doubling cascades together with their corresponding domains of chaos." The numerical experiments performed for the considered mapping confirm the presence of all the mentioned phenomena in our model. This Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 294 Sergey Kryzhevich and Gunnar S¨oderbacka will be discussed below together with a detailed review of recently obtained results. 3.1. 1D models Now we return to the mapping (4). We assume that k > b > 0, then the considered map admits a single fixed point x0:= ln k b−1. In the paper [24] the map y y+ (1 −y)eα(y−β)(6) (α > 0,β∈(0,1)) having certain applications in machine learning was studied. Remarkably, the transformation x=h(y) = ln 1−y y,y=1 ex+ 1 converts the mapping (6) to the map gα,β(y) := y+α ey+ 1 −αβ that is definitely of the type (4). Therefore, we can apply the results obtained for the map (6) to our mapping (4). Some of obtained properties we have listed in previous section. We list more properties below. Remark. We have already mentioned the symmetry from [13]. The map fk,b has the following symmetry property: fk,b(x) = −fk,k−b(−x). Therefore, it suffices to consider the parameters from the domain 0≤b≤k/2(the cases b < 0or b>kare trivial). For the sake of convenience, denote a=b/k. Remark. The map (6) has the following property: for any Borel probability invariant measure νwith ν({0,1})=0, we have Z1 0 ydν =β. In terms of our mapping, this means that for any Borel probability invariant measure µwe have Z∞ −∞ 1 ex+ 1dµ =b k.(7) In particular, the above formula is true for the average values of the function 1/(ex+ 1) over any periodic orbit of the map fk,b. The case b=k/2is quite trivial [25, Theorem 3.10]. Only two subcases are possible. 1. k≤8, the fixed point x0is stable and attracts all other points. 2. k > 8, the fixed point is unstable. There exists a period-2 orbit that attracts all the points of except the pedigree of the point 0(the point itself and all its iterative preimages). This periodic orbit does not intersect the segment IC. In general, period-2 orbits play a special role for the considered mapping. The following result about stable periodic points is proven in [25]. Theorem 1. Consider the mapping fk,ak with a∈(0,1). If a=p/q, where p, q ∈N are coprime, then there exists a k0∈R+such that for every k≥k0the map fk,ak admits an attracting periodic orbit of period qattracting Lebesgue almost all points of R. This attracting periodic orbit lies in ILSIR∩[a−1, a]. Remark. Moreover, ILcontains q−ppoints of the periodic orbit and IRcontains ppoints. This follows from the property (7), more precisely, from its analog for the map (6). In fact, the statement of Theorem 1 demonstrates the existence of Arnold tongues for the mapping fk,b. So we call the domains where the map admits periodic attractors of the given period p/q. 3.2. Chaotic invariant set Meanwhile, the family fk,b admits a chaotic invariant set with a non-classical symbolic dynamics. Нелинейные явления в сложных системах Т. 28, № 3, 2025 Multiattractors in a Several-Predators-One-Prey System 295 Let Σbe the set of all one-sided infinite sequences of symbols ’0’ and ’1’ endowed with the standard metrics: Σ = {σ= (σj∈ {0,1}:j∈N∪ {0})}; dΣ(σ, θ) = ∞ X j=0 |σj−θj| 2j. Consider a subset Σ0⊂Σgiven by the following formula: Σ0={σ∈Σ : σjσj+1 = 0,∀j∈N∪ {0}}. In other words, the set Σ0consists of segments without neighboring symbols ’1’. We list some evident properties of Σ0. 1. Σ0is an infinite closed subset of Σand, hence, compact. This is because the set of all sequences of Σwith ’1’ at positions j and j+ 1 is open for any j; 2. Σ0is invariant with respect to the standard shift mapping Son Σ(erasing the first symbol of a sequence). 3. The periodic points of the map Sare dense in Σ0. This is because the periodic points may be obtained by an infinite repetition of finite sequences of digits (if this does not result in entries ’11’), e.g. 010 010 . . . or 10100 10100 . . .. 4. For any m∈Nthe set Σ0contains at least one point of period m(for instance, the point corresponding to the repeating sequence 10 . . . 0for m > 1and the sequence of zeros if m= 1). 5. The set Σ0is transitive with respect to S. The point with a dense orbit may be obtained by writing down all the finite admissible sequences of ’0’ and ’1’, separated by ’0’: σ∗= 0 0 1 0 00 0 01 0 10 0 000 0 001 0 010 0 100 0 101 0 . . . Theorem 2. For any a∈(0,1),a6= 1/2, there is k0=k0(a)>0such that for any k > k0 there exists the compact set Q⊂[fmin, fmax], invariant with respect to fk,ak and such that 1. |(f2 k,k)0(x)|>1for any x∈Q; 2. for any m≥2, the mapping fm k,ak|Qis topologically conjugated to the Bernoulli shift on the set Σ0. We conjecture that, given a fixed value of aand a big parameter k, all the nonwandering points of the mapping fk,ak consists of one or two attractors and a chaotic invariant set given by Theorem 2. 3.3. Bifurcations It was noticed in the paper [13] that the bistability of the system can be observed in neighborhoods of intersections of period-doubling bifurcation curves (the evidence of such transverse intersections has been obtained numerically). Here we suggest another way to detect the bistability regions (they are quite difficult to find by sprinkling the points of the phase space). First of all, notice that for any admissible kand band any x∈ILSIR, we have 0≤ f0 k,b(x)<1. Therefore, the periodic orbits, that do not intersect with IC, do not bifurcate. This is why any Arnold tongue, existing due to Theorem 1, must contain a curve of the parametric values that corresponds to a critical point (one of two) of the period q. The intersections of such curves, corresponding to distinct critical points, do also imply bistability. Moreover, in this case, a shrimp structure corresponding to the intersection of two Arnold tongues may be observed. Finally, let us fix the parameter k(big enough) and start varying the parameter bfrom 0to k/2. After a stable periodic solution passes through a critical point (in the direction of increasing/decreasing of b), there might be three options: Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 296 Sergey Kryzhevich and Gunnar S¨oderbacka 1. the periodic solution doubles the period; 2. it passes via a critical point once again and then halves the period or disappears in a saddle-node bifurcation; 3. a bifurcation of codimension 2 or higher is observed. Although we do not see any obstacle to meet the second and the third scenario, it is only the first one that is detected numerically. The perioddoubling and period-halving are both observable that correspond to the period-bubbling scenario, similarly to the example of [18]. Meanwhile, the presence of the big parameter makes it difficult to study the complete cascade of bifurcations and its properties. 4. Conclusion We made a review on recently published or later observed results on the behaviour of several-predators-one-prey system. We have mostly considered the case of two predators and examined different types of possible coexistence. We observe two cases: when a one-dimensional discrete model is applicable, and when this model does not work. The latter case often resembles the spiral-chaos. We studied the map generating the onedimensional model in details. The map can have two attractors for a small region of parameters, which is also observed in the original system. However, those attractors are very hard to find unless taking help of the in the simpler onedimensional discrete model. Still there are some interesting open questions. If the 1D model does not work, we can have more attractors. We give a review of publications, where these are observed. We also discuss some bifurcations of these attractors. In this case when it is not possible to use a one dimensional approximating model and numerically often something looking like spiral-chaos is observed, the situation is much more complicated and the dynamics is very rich. Up to now we have no good model for a two dimensional Poincar´e map to be analyzed. To plot bifurcation curves in parameter planes we need to develop new numerical algorithms, as those from known packages often do not work. Even if some bifurcations are known they usually only explain simpler cyclic behaviour and the questions from where more complicated behaviour arises are open. Acknowledgments We wish to thank S. V. Gonchenko, A. Kazakov, T. Lindstr¨om, N. Lundstr¨om and A. V. Osipov for useful discussions during work on this subject. 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