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21Динамические системы, 2019, том 9(37), №3, 273-288 MSC 2010: 34C60, 92D25, 34C25, 37C05
Review on the behaviour of a many predator-one prey system
G. J. Soderbacka*, A. S. Petrov**
* Abo Akademi,
Turku FI-20500, Finland E-mail: [email protected] **Lobachevsky State University,
Nizhny Novgorod 603022 E-mail: [email protected]
Abstract. We consider a known predator-prey system, where more than one predator compete for the same prey. Mainly the case with two predators is considered. A review of general results is given, among them conditions for the extinction of one predator and an investigation of the different types of coexistence of predators. In non-degenerate cases the predators in this model cannot coexist at an equilibrium, but there can be a cyclic or more complicated coexistence. Many numerical results are presented. A model map for a Poincare map is given under some conditions. But the most interesting case where there can arise "spiral-like" attractors is not well known here, and we pose open questions. We discuss some bifurcations and the existence of systems with several attractors. Keywords: bifurcation, chaos, predator-prey.
1. Introduction
In this work we make a review of the behaviour of a system of two predators and one prey. We discuss extinction and possible types of coexistence. The coexistence can be cyclic or chaotic of different types. In some cases the chaos is well described by a model map, in other cases it seems to be a spiral-like attractor. We conjecture that at least some of the spiral chaos comes from bifurcations from a contour described here. There is an open question whether there are even more types of chaos. We also give examples, where the system has more than one attractor. Some of the results are more general for the case of more predators, like dissipativity and extinction.
The general system of n competing predators feeding on the same prey is considered to be of the type
X{ = )Xi - diXi, i = 1,... ,n,
n (1 1) S' = H (S) -J] qm(S )Xi, { • ;
i=1
where the variable S represents the prey populations and the variables Xi represent the predator populations. They are, of course, non-negative. The function pi is assumed non-decreasing. The function H describes the behaviour of the prey without predators and is usually of logistic type. An exception is the Lotka-Volterra system, the behaviour
© G. J. SODERBACKA, A. S. PETROV
of which is simple in this case. These systems were introduced by Hsu and Waltman [6, 7, 8]. We mainly consider the case where
H(S) = rS^l - K), Pi(S) = J+J-, i = 1,...,n. (1.2)
A. V. Osipov [12] introduced a family defined by some conditions on the functions of system (1.1) given below.
We assume H(0) = H(K) = 0 for some K > 0, H'(K) < 0, H''(s) < 0 and <£¿(0) = 0, <i(s) > 0. The functions <i and H are of the class C2[0, to) and the variables
S Qi
xi and s are non-negative: xi > 0, s > 0. The change of variables s = — and xi = Xi
KK
gives the system
xi = 4>i(s)Xi, i = 1,... ,n,
n
xi
:i.3)
s' =h(s) Ms) i=1
where ^
h(s) = kH(sK), ^¿(s) = <i(sK), 0i(s) = pi^i(s) - di.
The following conditions A1 — A5 are the main conditions introduced by Osipov [12]. Here and further we will assume that i takes values from the set {1, 2,... ,n}.
A1. All the considered functions are of the class C2[0, to) and the variables xi and s are non-negative: xi > 0, s > 0.
A2. ^(0) = 0, ^i(s) > 0 for s > 0.
A3. ^i(s) > 0 for s > 0 and there exists a Xi > 0 such that ^i(Xi) = 0.
A4. h(0) = h(1) = 0, h'(1) < 0 and h''(s) < 0 for s > 0.
A5. 0 < \n < ••• < \2 < \1 < 1.
If Xi in (A3) are all different we can always reorder the equations so that (A5) is satisfied. We observe that if Xi > 1 for some i, then the corresponding predator cannot survive.
The most standard example of system (1.3), which we will consider now, is obtained from the functions in (1.2). We assume pi > di. If not, the corresponding predator will not survive. Using the time change t = rt, where t is the new time, and the variable S Qi
changes s = — ,xi = Xi, we get the simplified equations K rK
/ _ s — Xi
Xi mi Xi
s + ai
s' = (1 — s — У^ Xi I s,
V 1=1s + ai)
:i.4)
where
A,
ai =
m,
Pi - di
Xi
diAi
K r ' " K(Pi - di)'
In earlier works [4, 5, 13, 14], for n = 2, we have discussed the behaviour of the system, boundaries for extinction, different types of coexistence of the predators, cyclic and chaotic. In some cases we have conjectured the existence of spiral chaos [15, 16, 17, 11]. In this work we present an overview of earlier results and some new results from numerical experiments of this system. It is a serious update of the review in [14], but without elementary introduction. Before we consider the properties of this systems, we shortly look at the Lotka-Volterra system, where in (1.1) the functions are
H (S ) = rS, Vi(S ) = S,
and thus the system becomes
X' = (piS - di)Xi, г = 1, S' :
n
r ViXi) S,
i=l
A change of the time т = rt and the variables xi =
QiXi
gives the system
xi = mi(s - Xi)xi, i s1 = - ^ x^ s,
1,... ,n,
where mi = P,\i = d. Further, suppose that Xn < r Pi
< X2 < Xl. Then all predators
except the population xn go extinct. To see that, use the Lyapunov function ln
x
mi xn
The main system we consider is anyhow rich in the behaviour of coexistence of the predators and it is one of the first systems, where the known Ecological Principle of Exclusion does not hold in general.
The outline of our work is the following. We start with discussing the dissipativity and extinction problem for any number of predators. Then we restrict ourselves to two predators and after shortly mentioning the chaos got from a model map (details can be found in earlier works) we give some typical results from numerical examination of the system. We look for extinction boundaries and for boundaries between different cyclic and chaotic behaviour. We continue with discussing a contour from which the spiral-like chaos might bifurcate. Then we show a bifurcation diagram from which we immediately conclude the existence of, at least, two attractors. At the end we give an example of the dynamics of a Poincare map and mention about modifications in order to get more realistic systems from a biological point of view.
r
n
2. Dissipativity
We consider system (1.3). We find a positively invariant set for the system. More results on dissipativity are found in [12]. Let
~ , ^ ~ / \ &(s) + S ,r x1 . x2 . . xn .
Wi = sup Wi(s),Wi(s) =-—--,V =--1---1----+---+ s.
0<s<1 Wi (s) W1 W2 Wn
The value of wi is finite and positive, because wlji(s) ^ — to for s ^ 0 + 0 and wlji(s) ^ (1) = Qi > 0 for s ^ 1 — 0. We now claim the following:
Statement 1. The set formed by the inequalities s,xi > 0 and V < 1 is positively invariant for system 1.3 satisfying conditions A1 — A5 and all trajectories of the system, with positive initial values enter the set in finite time.
Proof. It can be checked directly, that if s = 1 then V' < 0 except at (0,..., 0,1) where V' = 0. In other points we get
V = h(s) + J]
i=l
'Ш — Ш
Wi
Xi =
h(s) (1 — V) +M- (V — s) + £ W Ms) — WiШ]
— « f J in■
1 s 1 s W
i=1
h(s) (1 — v) + £ w
1 — s Wi
i=1
h(s)
+ 0i(s) — WiWi(s)
< 0.
1—s
From here the statment follows. □
Let us examine the case where
sbi _ x • sbi
h(s) = (1 — s)s,Ms) = ,Wi (s) = -b—. (2.1)
sbi + ai sbi + ai
If 0 < bi < 1 then wlji(s) increases and wi = Qi. If bi > 1 then it is possible that wi > Qi. For example, bi = 2ai = 1,Xi = 0.1 gives Wt;i(0.5) = 3.1 and Qi = 2.9.
3. Extinction
We here look for the competition between predators i and j for system (1.4). We find sufficient conditions for the extinction of one of them. More general results on extinction are found in [12]. We assume there is some j such that Xj < Xi, that is i > j.
Statement 2. Let L = -Xj) and Xi > Xj. If aj > --^-— then the
predator i goes extinct.
X3 (1 — Xi) i 3' J 3 L + ai(L — 1)
Because L > 1 the predator i always goes extinct when aj > —-. We observe
L — 1
that condition a
aj > / n—7T L + ai(L — 1)
follows from aj > a and aj > ai. L
Proof. We look at the function n defined by
n(s) = j) = m(s)n2(s), Pi{s)
where
mj (s - Xj) s + ai
ni(s) = —,-j ,V2(s)
mi(s - Xi)> s + aj
We use notations 7 = n(0),a = n(l). We notice that 7 < a is equivalent to a'
a. >-y--. We introduce two numbers
j L + ai(L - 1)
k0 = max n(s),K1 = min n(s).
se[o,xj ^ se[Ai,i]
They exist and are positive. We prove k0 < k1. We start with the case where ai < aj. From Xj < Xi follows n1(s) < 1 for 0 < s < Xj and n1(s) > 1 for s > Xi. Thus we get n(s) < n2(s) for 0 < s < Xj and n(s) > n2(s) for s > Xi. Because is increasing or constant for a1 < a2 we conclude that n(s1) > n2(s1) ^ V2(so) > n(s0) for so < Xj and s1 > Xi from which follows k0 < k1 and (3.2).
We now consider the case where aj < ai. We observe that in this case both n1 and n2 are decreasing and thus also n Then k0 = n(0) = 7 and k1 = n(1) = a. We observe that if k is a number such that k0 < k < k1 then
K^i(s) - fa(s) < 0 (3.2)
for all s E (0,1).
Really, for s E (0, Xj], fai(s) is negative and faj(s) is non-positive and from n(s) < k0 we get faj(s) > K0fai(s) > Kfai(s) implying (3.2). For s E (XjXi], fai(s) < 0 < faj(s) implying (3.2). Finally for s E (Xi, 1) both fai(s) and faj(s) are positive and from n(s) > k1 we get faj(s) > K1fai(s) > Kfai(s) implying (3.2).
Consider now the function U defined by U(x, y) = ln ( — ). For the time derivative
\xj J
we get U' = Kfa(s) — faj(s) < 0 and predator xi goes extinct. □
Remark. We notice that the proof can easily be modified for the functions in (2.1), so that the statement holds also if these functions are chosen for system (1.3).
It is well known [3] that when the number of predators increases, the probability for coexistence of all of them tends to zero. This is quite natural also from our estimates
showing that for coexistence we must have a > Lai-i, where L > 1 is defined as in statement 2 for j = i — 1.
4. General behaviour
We shortly decribe the main local behaviour of the three-dimensional system. According to assumption (A5) we suppose Ai > A2. The system always has two equilibria: (0,0,0), which always is a saddle with two-dimensional stable manifold in the plane s = 0, and (0,0,1), which is a saddle with one-dimensional stable manifold on the axis xi = x2 = 0 if Ai; A2 < 1 (in other cases at least one predator goes extinct and the system reduces to smaller dimension). When Ai;A2 < 1 there are two more equilibria:
a) b)
Ai = 0.45, Л2 = 0.3, ai = a2 = 0.3, mi = m-2 = 1 Ai = 0.3, A2 = 0.2, ai = a2 = 0.3, mi = m-2 = 1
Fig. 1. (a) In plane x2 =0 the equilibrium P1 is a global attractor, in the whole space a saddle with one-dimensional unstable manifold, in plane x1 =0 there is a limit cycle, the equilibrium in this plane is a saddle with one-dimensional stable manifold. (b) Plane x1 =0 there is a limit cycle, the equilibrium in this plane is a saddle with one-dimensional stable manifold. In plane x2 = 0 there is a limit cycle, the equilibrium in this plane is a sink.
• Pi = ((1 — Ai)(1 + ai)), 0,Ai) which is a saddle with one-dimensional unstable
1 — ai 1 — ai 1 — ai
manifold for Ai > —^— and a source if Ai < —^— • In the case Ai > —^—,
Pi is a global attractor in the plane x2 = 0-
• P2 = (0, (1 — A2)(1 + a2)),A2) which is a saddle with one-dimensional stable manifold for A2 < —^^ and a sink if A2 > —^^ • In the case A2 > —^^, P2 is a global attractor in the plane xi = 01_ai
There is a unique globally attracting limit cycle in the plane x2 = 0 if Ai < —^—
1 — a2
and a unique globally attracting stable limit cycle in the plane xi = 0 if A2 < —^— • ISSN 0203-3755 Динамические системы, 2019, том 9(37), №3
This was first proved by Cheng [2]. The uniqueness of limit cycles for this and similar systems can also be proved using the known Zhang Zhi-fen theorems [18]. Estimates for the size of the cycles for critically small ai and Xi are given in [9, 10]. The size of the cycle we determine by the maximal and minimal populations on the cycle. The cycle is called big if at least one population sometimes gets small. The behaviour around the coordinate planes is shown in figure 1.
There is no equilibrium for X1 = X2 or a1 = a2, where the predators coexist, anyhow they can coexist in a cyclic or chaotic way. Conditions for construction of some well-defined Poincare maps on s = const, s' < 0 are obtained in [13].
In the case where the Poincare map is well-defined, very often there is a strong contraction in the (x1 + x2)-direction and it is shown by numerical experiments and theoretical estimating arguments that the one dimensional model map given by
ff \ «i ki + k2eV f (v) = p + v----u,
J y ' 1 + ev
where p,u and ki are constants and v = \n(x2/x1) gives a good approximation. This map is derived and analyzed for simple behaviour in [4, 5].
4.1. Charts of dynamical regimes
We now present some result of numerical two-parametric analysis.
In figures 2 - 4 we see the results of numerical investigations of the behaviour of the system for fixed Xi and mi where i = 1, 2. We have examined the behaviour for five different random initial conditions for a grid of parameter values of a1 and a2. A predator is considered to go extinct if the populations becomes less than e-100. We call the attractor n-cyclic if the intersection with s = X2, s' < 0 is n-periodic under the Poincare map defined on this surface. We have denoted regions with x if the first predator x1 goes extinct and with y if the second predator x2 goes extinct. Regions, where there is observed only simple one-periodic cycles, are coloured cyan. Regions, where there is observed a 2-cyclic attractor, are coloured magenta. Regions, where there is observed a 3-cyclic attractor, are coloured dark yellow. Sometimes also regions where the second predator x2 goes extinct can be seen in green. Regions, where there is observed chaos, but no 3-cyclic attractor, are coloured blue. We observe, that for some parameter values two different types of attractors have been detected.
In figures 2 - 3 we have added a figure calculating the boundaries for extinction of the predator x1 for different values of mi and compared with the theoretical estimate (the curve to the left).
We make the observation that the behaviour depends on m1 and m2 and strongly on the difference of X1 and X2. When X1 ^ X2 we roughly have the following situation for a1 increasing: for a1 small, the predator x1 goes extinct, for a little bigger, the predator x2 goes extinct and for even bigger a1 there is a simple cyclic coexistence. The predator x2 can go extinct only if a1 < 1 — 2X1, because if a1 > 1 — 2X1 then there is saddle equilibrium in the coordinate plane x2 = 0 and there is more likely to
Fig. 2. (a),(b),(c) Charts of dynamical regimes for the system 1.4 on the parameter plane (a1, a2) for mi = 0.1, mi = 1, and mi = 5, respectively (Ai = 0.35, Ai = 0.2). Cyan color (1) corresponds to simple periodic regimes; magenta color (2) — 2-periodic regimes; dark yellow (3) — 3-periodic regimes; blue (> 3) — periodic regimes with period > 3 and a chaotic regimes; red color (x1e) — regime corresponding to the extinction of the first predator x1; and green color (x2e) — the extinction of the second predator x2. (d) Curves of the extinction of the predator x1: G theoretical estimate (3.1), green, blue, and red curves are given by compared (3.1) for mi = 0.1, mi = 1, and mi = 5, respectively.
be some kind of spiral chaos if it is not cyclic. Of this reason there is no extinction of the second predator in figure 3.
We have not included pictures for small Ai and A2. In this case there is a hope to obtain theoretical estimates for the stability of the limit cycles in the coordinate planes. If both are unstable the predators coexist.
Open problem. Find out good reasons for the different behaviour observed in the
Fig. 3. (a),(b),(c) Charts of dynamical regimes for the system 1.4 on the parameter plane (a1, a2) for mi = 0.1, mi = 1, and mi = 5, respectively (Ai = 0.5, Ai = 0.2). Cyan color (1) corresponds to simple periodic regimes; magenta color (2) — 2-periodic regimes; dark yellow (3) — 3-periodic regimes;blue (> 3) — periodic regimes with period > 3 and a chaotic regimes; red color (x1e) — regime corresponding to the extinction of the first predator x1. (d) Curves of the extinction of the predator x1: G theoretical estimate (3.1), green, blue, and red curves are given by compared (3.1) for mi = 0.1, mi = 1, and mi = 5, respectively.
figures and find approximate expressions for different bifurcation lines. Find bifurcation curves for attractors by numerical methods. How many attractors can we have in the same system for different parameters?
The cases where Ai = 0.3 and Ai = 0.2 and mi > 0.2 are very interesting even if we do not include a figure here. But the existence of three different attractors is frequent in these cases.
In figure 5 we can see a case with three attractors.
Fig. 4. (a), (b), (c), (d) Charts of dynamical regimes for the system 1.4 on the parameter plane (a1, a2) for (a),(b) Ai = 0.3 and Ai = 0.2 and mi = 0.1, (c) Ai = 0.25 and Ai = 0.2 and mi = 0.1, (d) Ai = 0.35 and A1 = 0.2 and m1 = 5 and m2 = 0.1. Cyan color (1) corresponds to simple periodic regimes; magenta color (2) — 2-periodic regimes; dark yellow (3) — 3-periodic regimes; blue (> 3) — periodic regimes with period > 3 and a chaotic regimes; red color (xie) — regime corresponding to the extinction of the first predator xi; and green color (x2e) — the extinction of the second predator X2.
We now analyze the bifurcation diagram in [14]. This was produced for a2 = 0.02, Ai = 0.35, A2 = 0.2, mi = m2 = 1 and for ai as bifurcation parameter. New versions of this bifurcation diagram are produced in figure 6.
In figure 6 (a) we produce a bifurcation diagram looking at the value of ln ( — )
\XiJ
on the intersection of the attractor with s = A2,s' < 0. The initial values for a1 = 0.1 were taken as x1 = x2 = s = 0.5 and increasing a1 for the next value of a1 we take the
Fig. 5. Attractors for parameters ai = 0.855, Ai = 0.3, a2 = 0.0154, A2 = 0.2, mi = m2 = 0.5. There is a simple cycle (cyan) a 3-cyclic (green) and a 15-cyclic (red) attractor.
Fig. 6. (a) bifurcation diagram for a2 = 0.02, Ai = 0.35, A2 = 0.2, mi = m2 = 1. The range of bifurcation parameter a1 goes from 0.1 to 2. (b) bifurcation diagram for a2 = 0.02, A1 = 0.35, A2 = 0.2, mi = m2 = 1. The range of bifurcation parameter a1 goes from 1.5 to 1.6.
initial values to be the last point on the attractor calculated for the previous value of a\. In figure 6(b) the bifurcation diagram is produced for the same fixed parameters
as in the previous bifurcation diagram, but the value of a1 goes from 1.6 to 1.5. The initial values for ai = 1.6 are chosen as x1 = 0.34, x2 = 0.22, s = 0.35 and decreasing ai for the next value of a1 we take the initial values to be the last point on the attractor calculated for the previous value of a1.
Comparing the bifurcation diagrams we easily see two attractors for values of a1 around 1.6.
5. Spiral chaos from a contour?
Finally, we discuss the nature of "spiral" attractors observed in the system for parameters a1 = 0.5, A1 = 0.33, a2 = 0.001^, A2 = 0.01^, m1 = 1, m2 = 0.2, see figure 7. Usually, such attractors appear due to a Shilnikov homoclinic orbit to the saddle-focus equilibrium [15, 16, 17, 11]. In the problem under consideration all equilibria are located in the invariant planes and, thus, we cannot have a homoclinic orbit. We suppose that in our problem another scenario is possible, when spiral chaos appears from a heteroclinic cycle.
Xl
Fig. 7. Attractors for parameters a1 = 0.5, Ai = 0.33, a2 = 0.001 v, A2 = 0.01v, mi = l,m2 = 0.2, in the cases v = 1,4,10,12.
1. The unstable separatrix of the equilibrium at ((1 — Ai)(Ai + a1), 0, Ai) until it hits s = 0 at P* = (x1,x2, 0) in the case a2 = A2 = 0.
2. The curve along x2 = Cx{, where y = m2AA2ai and C = x2(x*)-Y from P* to
m1A1a2
(0,0, 0).
3. The line segment x1 = x2 = 0, 0 < s < 1.
4. The unstable separatrix of the equilibrium (0, 0,1) in the plane x2 = 0 reaching the equilibrium, where part 1 starts.
We suppose that the spiral chaos arises from this contour. We support the conjecture by showing some attractors developing from the contour changing a parameter. We look at some attractors for a1 = 0.5, A1 = 0.33, a2 = 0.001v, A2 = 0.01v,m1 = 1,m2 = 0.2. We can see them in figures 7-8. The attractor for v < 1 is so near to the contour that we can not see the differences. Increase in v shows us a series of attractors, where we can observe some spiral-like chaos development.
Fig. 8. Attractors for parameters ai = 0.5, Ai = 0.33, a2 = 0.001^, A2 = 0.01v, = 1,m2 = 0.2, in the cases v = 15, 22, 25, 29.
We also find out how the intersection of an attractor with a Poincare section looks like. In figure 9 we see the attractor and intersection with s' = 0 in the part, where s' is increasing. It is an open problem to find some kinds of model maps in such cases.
Fig. 9. Attractor and intersection with a Poincaré section for parameters ai = 0.5, Ai = 0.33, a2 = 0.015, A2 = 0.15, mi = 1,m2 = 1. The sequence of numbered points are iterates of the corresponding Poincare map.
6. Conclusion.
We have given an overview of results of system (1.1) starting from dissipativity and extinction. The results of dissipativity could be improved to get a smaller positively invariant set using non-linear upper boundary like it was done for the system with only one predator in [10]. The results of extinction use only the equations for the predators, they can be improved by also using the equation for the prey. Our numerical results try to include the general picture of the behaviour of the system when we assume Xi and mi constant and change the parameters ai. We have discussed the open problem of how many attractors can be found. We give numerical results which argument for the bifurcation leading to chaos starting from a contour got for some parameters tending to zero. A typical behaviour of the dynamics on an attractor on a Poincare section is shown in figure 9.
In most of the chaotic behaviour studied here one population can get very low, and there is the question whether this can be realistic. Some modifications were suggested in [14]. Anyhow, there is also complicated behaviour in systems, where the populations are not getting too low. Such an example was also given in [14]. Another realistic case is the example we gave with three attractors. Yet another such interesting example we get for ai = 1.5, Xi = 0.3, a2 = 0.01, X2 = 0.2, mi = m2 = 1, where easily three attractors can be observed. One simple cyclic, another 3-periodic and a 4-periodic one. Studying the bifurcations of these attractors changing parameter 0.5 < a1 < 2 we
see that the simple periodic one always exists, while the branches from the 3- and 4-periodic attractors exist for intervals which overlap on a smaller interval.
The standard system has cycles with very low populations for small a and A. In nature this does not occur because the predator changes behaviour to feeding on other preys, where however it cannot survive for ever. Because this change is sudden in Arctic regions (stochastic in Middle Europe) we there get a system with switches. Such a system is given in [14].
References
1. Afraimovich V. S, Gonchenko S. V., LermanL.M, Shilnikov A. L, TuraevD. V. Scientific Heritage of L.P. Shilnikov, Regular and Chaotic Dynamics 19, No.4, 435-460 (2014).
2. Cheng K.S. Uniqueness of limit cycle for a predator-prey system, SIAM J. Appl. Anal. 12, No.4, 541-548 (1981).
3. Coste J. Dynamical regression in many species ecosystems: The case of many predators competing for several preys, SIAM J. Appl. Math. 45, 555-564 (1985).
4. Eirola T, OsipovA. V., Soderbacka G. Chaotic regimes in a dynamical system of the type many predators one prey. Research reports A 386, Helsinki University of Technology, (1996).
5. EirolaT., OsipovA. V., SoderbackaG. On the appearance of chaotic regimes in one dynamical system of type two perdators — one prey, Actual Problems of Modern Mathematics, Boxitogorsk, 1, 39-70 (1996).
6. HsuB., HubellS.P., WaltmanP. Competing predators, SIAM J. Appl. Math. 35, No.4, 617-625 (1978).
7. HsuS. B. Limiting behaviour for competing species, SIAM J. Appl. Math. 34, No.4, 760763 (1978).
8. HsuB., HubellS.P., WaltmanP. A contribution to the theory of competing predators, Ecological Monographs 48, No.3, 337-349 (1978).
9. HsuS.-B., Shi J. Relaxation oscillation profile of limit cycle in predator-prey system, Discrete and Continuous Dynamical Systems Series B 11, No.4, 893-911 (2009).
10. LundstromN. L. P., Soderbacka G. Estimates of size of cycle in a predator-prey system, Differential Equations and Dynamical Systems, DOI: 10.1007/s12591-018-0422-x (2018).
11. Ovsyannikovl. M., Shilnikov L. P. On systems with a saddle-focus homoclinic curve. Matematicheskii Sbornik 172, No.4, 552-570 (1986).
12. OsipovA. V., SoderbackaG. Extinction and coexistence of predators, Dinamicheskie Sistemy 6(34), No.1, 61-70 (2016).
13. OsipovA. V., Soderbacka G. Poincare map construction for some classic two predators-one prey systems. Int. J. Bifurcation Chaos Appl. Sci. Eng. 27, No.8, Article ID 1750116, 9 p. (2017).
14. OsipovA. V., Soderbacka G. Review of results on a system of the type many predators - one prey, Nonlinear Systems and Their Remarkable Mathematical Structures, CRC, Vol.1, 520-540 (2018).
15. Shil'nikovL. P. A case of the existence of a countable number of periodic motions. Sov. Math., Dokl. 6, 163-166 (1965).
16. Shilnikov L. P. The existence of a countable set of periodic motions in the neighborhood of a homoclinic curve. Sov. Math., Dokl. 8, 102-106 (1967).
17. Shilnikov L. P. On the question of the structure of an extended neighborhood of a structurally stable state of equilibrium of saddle-focus type. Mat. Sb., N. Ser. 81(123), 92-103 (1970).
18. Zhang Z-f., DingT-r, Huang W-z, Dong Z-x. Qualitative Theory of Differential Equations, AMS, Providence, Rhode Island, (1991).
Получена 13.06.2019 Переработана 25.11.2019
