# The Ricker Competition Model of Two Species: Dynamic Modes and Phase Multistability

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{n}and y

_{n}are the fish population sizes of competing species in the nth year; A and B characterize the fish fecundity on the spawning ground, the fry survival rate, as well as the fraction of the population going to spawn. Coefficients α′ and β′ describe the competitiveness of species and mortality of individuals due to lack of food. ${\chi}_{n}$ is the competition intensity that is determined as follows:

_{n}tends to zero. The sequence x

_{n}, being limited, can be divergent; it can fluctuate with a period of 2 or 4, or be irregular. The papers [49,60,68] study model (1) and focus on the stability area of its nontrivial solution in the coordinate plane of the fixed point. In particular, under constant environmental conditions, two competitors are shown to coexist if intraspecific competition is higher than interspecific [60]. If interspecific competition is greater, then one of the species is eliminated [60]. Based on the eigenvalues analysis of system (1) Jacobian, work [63] has found the stability conditions for fixed points, which partially coincides with the results of previous studies [48,60,68]. Furthermore, paper [62] demonstrates the possibility of the existence of an in-phase 2-periodic point. An antiphase two-periodic orbit is shown by simulation to emerge in the irregular dynamics region.

_{1}= K

_{2}= 1 was investigated in paper [26]. A special case of model (3) with A′ = B′, β = γ, α = δ = A′, K

_{1}= K

_{2}= 1 is studied in [69] that analyzes bifurcations of fixed and periodic points. H. Smith has researched the asymptotic behavior of planar order-preserving difference equations with particular attention to those arising from models of two-species competition [70].

## 2. Model (1) Study for Local Stability

- A trivial fixed point that corresponds to the extinction of both populations:

- 2.
- Two semitrivial solutions that correspond to the extinction of one of two species:

- 3.
- A nontrivial fixed point corresponding to the sustainable existence of both species in the community

- $\phi \rho <1$ (Model (4) has four fixed points);
- $\phi \rho =1$ (The system degenerates and has a non-simple nontrivial solution [73]);
- $\phi \rho >1$ (The nontrivial fixed point is negative, i.e., species coexistence is not possible). It corresponds to a situation where interspecific competition between species is greater than their self-limitation, i.e., $\phi \rho >1$ (αδ–βγ < 0). In [34,48,62,68], stable coexistence of two competing populations is shown to be impossible with $\phi \rho >1$.

^{2}+ pλ + q = 0 belong to the circle |λ| < 1 if and only if

_{1}λ

_{2}= 1, and on the segment (−2 < p < 2), limiting the “stability triangle”; they are also conjugated as follows: λ

_{1,2}= exp(±iφ).

#### 2.1. The Stability Area of Trivial Solution (5)

#### 2.2. The Stability Areas of Semitrivial Solutions

^{ρ}; (2) λ = –1, B = e

^{2}; (3) q = 1, A = B

^{ρ}/(1 − ln B).

^{φ}; (2) λ = –1, A = e

^{2}; (3) q = 1, B = A

^{φ}/(1 − ln A).

^{2}

^{ρ}, e

^{2}) and (e

^{2}, e

^{2}

^{φ}), respectively. Figure 1 shows the stability areas of semitrivial solutions. The semitrivial fixed points lose their stability due to period-doubling bifurcation or transcritical bifurcation. The Neimark–Sacker bifurcation lines do not bound the stability domains of semitrivial fixed points.

^{ρ}(Figure 1c); for φρ > 1, their stability regions overlap. Figure 2 shows the overlapping of stability areas of solutions (6) and (7) with φρ > 1 when φ = 2.5 and ρ > 0.4.

^{2}(B > e

^{2}). If φρ = 1, then bistability as the coexistence of stable solutions (6) and (7) becomes impossible. The structure of the attraction basins of coexisting dynamic modes in the areas with Figure 3, Figure 4 and Figure 5 is depicted in Figure 3.

#### 2.3. The Stability Area of Nontrivial Solution with φρ < 1

^{2ρ}, B = e

^{2}) and (A = e

^{2}, B = e

^{2φ}) that are also the intersection points of the lines λ = 1 and λ = –1 for semitrivial fixed points of system (4). Therefore, codimension 2 bifurcations arise at these points. The stability area of nontrivial solution (8) has no Neimark-Sacker bifurcation boundary, which coincides with the findings of R. Luis et al. [34]. Then, system (4) reveals no transition from stationary dynamics to quasi-periodic oscillations.

## 3. Model (4) Dynamics Modes

_{0}and y

_{0}. Let us consider features and emergence mechanisms of multistable dynamic modes from the shaded areas in Figure 5.

## 4. Periodic Fixed Points of Model (4): Phase Multistability

**X**

_{n}under the mapping F iterated N times. The operator ${F}^{N}$ is cumbersome because of the transcendence of the right-hand side of system (4). Therefore, the periodic points can only be found numerically solving the following system of equations:

#### 4.1. Symmetric Case with $A=B$ and $\rho =\phi $, When Both Species Have the Same Growth Rates and Competition Parameters

_{n}relative to y

_{n}. $\tau $ is determined by the formula: $\begin{array}{cc}\tau =\underset{0\le \tau <T}{\mathrm{arg}\mathrm{min}}\left|{x}_{n}-{y}_{n+\tau}\right|,& n=1,2,3\dots \end{array}$. When the difference is zero, the oscillations of variables x and y are in phase. For example, Figure 6a shows in-phase 2-cycle with $\tau =0$. For the 4-cycle, its trajectories with $\tau =0$ is depicted in Figure 7a, where one can see the in-phase dynamics of variables x and y. With $\tau =2$, variable x oscillations will almost coincide with those of y, if y trajectory is shifted by two iterations (Figure 7c). The trajectories in Figure 7a–d correspond to periodic points in Figure 6 which are marked by circles in the phase plane. Unstable points are designated as ${\overline{T}}_{\tau}$, and their location is shown by white circles.

_{0}occurs when the first eigenvalue of system (14) with $N=2$ passes through –1, and the second one lies in the unit circle. The new four stable points 4

_{2}corresponding to the out-of-phase 4-cycle are located at some distance from x = y. After that, as for fixed point (8), the periodic point 2

_{0}bifurcates into ${\overline{4}}_{0}$-cycle (4 saddle points) since the second eigenvalue passes through –1 (Figure 6b). These bifurcations give out-of-phase 4-cycles that differ in the initial phase, but do not change the dynamics and the attraction basins fundamentally. Figure 6b shows the results of these two bifurcations.

_{0}becomes stable. Here the synchronous mode ‘captures’ a part of the attraction basin of out-of-phase 4-cycle along the quadrant bisector, as well as the areas located in a checkerboard pattern. These attraction basins are shown in black in Figure 6c. Initial conditions from the black areas can give four different 4

_{0}-cycles with different initial phases, but they are not shown in Figure 6c, so as not to overload it. For more information, Figure 7a–d shows some model (4) trajectories of regular dynamics with different phase shifts and amplitude of oscillations.

^{4}appear around points 4

_{2}; and they coexist with the in-phase cycles emerging due to the cascade of period-doubling bifurcations of point 4

_{0}(Figure 8a).

_{2}loses its stability, and the invariant curve Q

^{4}takes its attraction basins. The attraction basins of the in-phase chaotic attractor are those of point 4

_{0}(Figure 8b).

_{1}coexists with the in-phase chaotic attractor C

_{0}and the Shilnikov-shape attractor C

^{4}having two-dimensional basins of attraction (Figure 8b). Further two components of C

^{4}merge with each other and their stable manifolds are tangent to each other along line y = x at A = B ≈ 15.36. Therefore, the four-component attractor C

^{4}becomes a two-component attractor coexisting with the out-of-phase 2-cycle (${2}_{1}$). At the same time, the basin of attraction of in-phase chaotic attractor C

_{0}collapses to segments lying on the one-dimensional manifold $y=x$. In this case, in-phase dynamics is stable only with small perturbations along the bisector, but it is transversally unstable under perturbations in the perpendicular direction to the bisector [85].

^{4}merge at A = B ≈ 17.06, and a hyperchaotic attractor filling a large part of the phase space occurs. However, it quickly breaks down, and system (4) reveals only the out-of-phase 2-cycle (periodic point 2

_{1}) in a certain range of parameters. The attraction basins of each phase of this 2-cycle are quite fragmented, especially in the attraction basin of the breaking chaotic attractor. With starting points from these areas, for example, about the bisector plane, the transitional dynamics to the out-of-phase 2-cycle can be quite long and complicated.

_{4}) appearance in the vicinity of the out-of-phase 2-cycle (2

_{1}) elements. In the phase space, the out-of-phase 6-cycle (6

_{4}) takes a part of the 2-cycle basin of attraction; they, along with dynamic modes emerging due to their bifurcations, coexist. In addition, with changes to the values of parameters A and B, the in-phase 3-cycle quickly bifurcates, which gives a new 6-cycle that is partially synchronous (Figure 8c). The 6-cycle elements lie quite close to the line $y=x$, and the phase shift of variable x

_{n}relative to y

_{n}is 3 (6

_{3}in Figure 8c). As a result, there are three different dynamic modes.

_{4}-cycle, and only then the partially synchronous 6-cycle (6

_{3}) bifurcates. Therefore, at a sufficiently wide area of parameter values, system (4) demonstrates two different types of quasi-periodic dynamics that are simultaneously possible and are periodic as synchronous or partially synchronous.

_{3}leads to the fact that emerging closed invariant curves tangent each other along the stable manifold of saddle point ${\overline{3}}_{0}$, which gives two connected homoclinic orbits around the right ($y<x$) and left ($y>x$)points of cycle ${\overline{6}}_{3}$. As a result, a three-component discrete Lorentz-shape attractor C

^{3}arises; one of its components is shown in the third column of Figure 8d.

_{n}is depicted in the second graph of Figure 7h. The trajectory is seen to turn many times around the right point 6

_{3}. When the trajectory is close to saddle ${\overline{3}}_{0}$, it is attracted to the left point vicinity, where the trajectory also continues to turn for an indefinite time. In this case, there are only two closed invariant curves Q

^{2}corresponding to the out-of-phase dynamic modes, and periodic point 6

_{4}does not exist.

^{3}breaks up with $A=B\approx 23.57$, and system (4) reveals only two closed invariant curves Q

^{2}. They merge at $A=B\approx 24.6473$, which gives a hyperchaotic attractor that densely fills the phase plane. Here, the dynamics of phase variables become non-synchronous.

#### 4.2. Non-Symmetric Case with $A\ne B$ or $\rho \ne \phi $, When Both Species Have Different Growth Rates or Competition Parameters

_{1}) is a result of saddle-node bifurcation (SN) instead of supercritical period-doubling bifurcation of cycle ${\overline{2}}_{1}$ emerging with $A=B$ and $\rho =\phi $. The periodic points are not symmetrical with respect to each other, as well as to line $x=y$ (Figure 6f). With $A\ne B$ or $\rho \ne \phi $, the periodic point ${\overline{2}}_{1}$ is away from the saddle-node points and not strictly between them, as in the case with $A=B$ and $\rho =\phi $. Scenarios of stability loss of this periodic point are also different. For example, if $\rho =\phi $ and the difference $A-B$ is small, then periodic point 2

_{1}loses its stability according to the Neimark–Sacker scenario. If the difference $A-B$ is large enough, then we observe the period-doubling scenario (Figure 5c). If interspecific competition coefficients differ significantly, then the multistability domain with these two scenarios shifts depending on the values of parameters ρ and φ (Figure 5e,f).

_{n}and y

_{n}are values of system (2) phase variables after transients. We use the last 500 steps from the sequence representing 10,000 model (4) iterations calculated to build the dynamic mode maps. If the value σ = 0 then dynamic modes are fully synchronous; if σ→1 then non-synchronous modes are observed. We use the last values of x

_{n}and y

_{n}(n = 10,000) as the starting point to calculate the following 10,000 system (4) iterations in order to find the Lyapunov exponents.

_{2}) due to the period-doubling bifurcation of the 2-cycle. This occurs before the “simple” period-doubling of 2-cycle (2

_{0}) that brings about the appearance of stable in-phase 4-cycle (4

_{0}). With crossing PR and then PD as the dashed line, only a subcritical period-doubling bifurcation occurs and an unstable cycle ${\overline{4}}_{0}$ appears. It becomes stable crossing the solid line PD

^{+}(2

_{0}), when the period doubles again. Figure 6b,c shows these bifurcations. It can be assumed that the occurrence of out-of-phase cycles precedes in-phase cycles’ appearance. Thus, between PD(2

_{0}) and PR lines, there are other stable dynamics modes to the left of line SN and no in-phase ones.

^{−}and PR

^{+}, which start at the right intersection point of PD(2

_{0}) and PR lines. Periodic points of PR

^{−}correspond to the appearance of an unstable out-of-phase 4-cycle (${\overline{4}}_{2}$) that becomes stable when passing through the line PR

^{+}. In the area above the line PR

^{+}, the in-phase dynamic modes (2

_{0}, 4

_{0}, 8

_{0}, etc.) coexist with the out-of-phase ones (4

_{2}, 8

_{2}, etc.). A decrease in the parameter ρ values leads to a transition through the boundary PD

^{−}at which an unstable out-of-phase 2-cycle (${\overline{2}}_{1}$) appears due to the occurrence of a subcritical period-doubling bifurcation of the nontrivial fixed point. Further, when the competitive impact of species x and y on each other becomes comparable, a saddle-node bifurcation occurs passing through the line SN, which gives the emergence of a stable periodic point 2

_{1}with lower values of A = B. The curve SN has two branches starting at the point ρ = φ, where codimension-two pitchfork bifurcation occurs. The following stable cycles of system (4) are simultaneously possible between the lines SN and $\rho $= 0: in-phase 2

_{0}, 4

_{0}, 8

_{0}- periodic points, etc., out-of-phase 2

_{1}, 4

_{1}-periodic points, etc. (Figure 9a), as well as 4

_{2}, 8

_{2}ones, etc. If the value of ρ is close to the value of φ, then Neimark–Sacker bifurcation occurs (Figure 9a,b), and two invariant curves appear around elements of point 2

_{1}. Between the lines PD(2

_{0}) and PR, to the left of SN, there are only 2

_{1}and 4

_{2}cycles.

_{0}and its bifurcations lead to the appearance of in-phase cycles 2

_{0}, 4

_{0}, etc.

_{0}, 8

_{0}, etc. cycles and unstable out-of-phase four-periodic point ${\overline{4}}_{2}$.

_{0}, 8

_{0}, etc. coexist with out-of-phase ones 4

_{2}, 8

_{2}, etc.

_{0}, 8

_{0}, etc., stable out-of-phase periodic points 4

_{2}, 8

_{2}, etc., and an unstable out-of-phase ${\overline{2}}_{1}$ cycle. Domain IVʹ corresponds to stable 2

_{0}-cycle and unstable ${\overline{2}}_{1}$ cycle.

_{2}cycle. Domain Vʹ corresponds to the coexistence of 4

_{2}with ${\overline{2}}_{1}$; domain V″ stands for existence 4

_{2}and stable out-of-phase cycle 2

_{1}. There is an unstable ${\overline{4}}_{0}$ cycle above PD(2

_{0}) but below PD

^{+}(2

_{0}) in domains V, Vʹ, and V″.

_{0}, 4

_{0}, etc. cycles, the out-of-phase 4

_{2}, 8

_{2}, etc. periodic points, and the out-of-phase 2

_{1}cycle that bifurcates via period-doubling or Neimark–Sacker scenario. Domain VIʹ shows the existence of the in-phase 2

_{0}and the out-phase 2

_{1}cycles.

_{1}) with the passing of which cycle 4

_{1}occurs. The upper part of the PD(2

_{1}) line separates the domain Q with two closed invariant curves emerging around the cycle 2

_{1}elements from the area with out-of-phase high period cycles and chaos. The remaining subareas differ in the number of additional periodic saddle points that occur due to the large number of inflections and branches of system (14) nullclines. In Figure 9c, the figures indicate the total number of intersection points of nullclines, excluding semi-trivial points. One can see, higher values of parameter A = B and lower ones of ρ increase this number.

_{n}relative to y

_{n}. The scenario of appearance of these dynamic modes significantly depends on the intensity of competitive interaction of both species in the community.

## 5. Discussion and Conclusions

_{n}relative to y

_{n}. The appearance scenario of these dynamic modes significantly depends on the intensity of competitive interaction of both species in the community. In other words, the value variation in parameters characterizing the competitive interaction of species can result in a change in the community development scenario.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Stability domains of semi-trivial fixed points (6) and (7) with different values of ρ and φ.

**Figure 2.**Areas of existence and coexistence of fixed points (5)–(7) for different values of ρ and φ with ρφ > 1.

**Figure 5.**Dynamic mode maps of model (4) for different values of parameters φ and ρ: (

**a**) ρ = φ = 0.5; (

**b**) ρ = φ = 0.1; (

**c**) ρ = φ = 0.03; (

**d**) ρ = 0.01, φ = 0.2; (

**e**) ρ = 0.01, φ = 0.1; (

**f**) ρ = 0.01, φ = 0.05. Figures correspond to the period of observed cycles. Q stands for quasiperiodic dynamics, C is chaotic dynamics and 0 corresponds to the community extinction. The subscript and superscript 0 correspond to the extinction of species x and y, respectively. Several coexisting modes are possible in the shaded area.

**Figure 6.**(

**1, 3 rows**) System (14) nullclines show the mechanism of periodic point emergence for system (4); (

**2, 4 rows**) Attraction basins of stable points are the black circles above. (

**a**–

**c**) Symmetric case with A = B and φ = 0.1; (

**d**–

**f**) Asymmetric case at A≠B and φ =0.03 demonstrates the emergence of an out-of-phase 2-cycle through a saddle-node bifurcation (SN). Stable/unstable fixed points are designated as ${T}_{\tau}$/${\overline{T}}_{\tau}$ and are marked by black/white circles. T is period, subscript τ is the phase shift of variable x

_{n}relative to y

_{n}. Attraction basins of different colors, marked in the same way, correspond to oscillations with the different initial phases.

**Figure 7.**Examples of regular (

**a**–

**d**) and irregular (

**e**–

**h**) dynamics of system (4). 2

_{0}and 4

_{0}is in-phase oscillations x

_{n}and y

_{n}with period 2 and 4, 2

_{1}and 4

_{2}is out-of-phase oscillations. Q and C are quasiperiodic and chaotic out-phase dynamics.

**Figure 8.**The first column is attraction basins of stable dynamics modes. The second one (except for (

**d**)) is system (14) nullclines. The third one presents coexisting attractors with their enlarged fragments. (

**a**,

**c**) are the closed invariant curves. The discrete Shilnikov-shape attractor is (

**b**) and the Lorentz-shape one, (

**d**). Stable fixed points are designated as ${T}_{\tau}$ and are marked by black circles. Unstable points are designated as ${\overline{T}}_{\tau}$ and shown by white circles or crosses. T is the period and subscript τ is the phase shift of variable x

_{n}relative to y

_{n}. Attraction basins of different colors, marked in the same way, correspond to oscillations with the different initial phases.

**Figure 9.**(

**a**,

**b**) Dynamic mode maps (

**left**), maps of sums of Lyapunov exponents (

**center**) and synchronization index maps (

**right**) for asymptotic dynamic modes of system (4) at φ = 0.03 and starting point (

**a**) x

_{0}= 1.7, y

_{0}= 5 and (

**b**) x

_{0}= 0.8, y

_{0}= 4.3 with changing A = B and ρ. The line highlights the approximate boundary between dynamics modes with different values of the synchronization index. Figures correspond to the period of observed cycles. Q stands for quasiperiodic dynamics. C is chaotic dynamics. Subscript is the phase shift of variable x

_{n}relative to y

_{n}. (

**c**) Bifurcation diagram of system (4), where domains I–IV differ in multistable dynamics modes. Figures correspond to the number of system (16) fixed points (right). A numerator is fixed points’ amount between PD(2

_{0}) and PD

^{+}(2

_{0}) lines, the denominator corresponds to those between PR and PD(2

_{0}) lines. Period-doubling bifurcations are shown by the dashed lines or designated as PD(${T}_{\tau}$), where ${T}_{\tau}$ is a periodic point that doubles its period.

Intervals for the Parameter Values | Development Scenario | |
---|---|---|

$\rho \phi <1$ | $A<{B}^{\rho}$$,B1$ | species y displaces species x |

$B<{A}^{\phi}$$,A1$ | species x displaces species y | |

$AB>{A}^{\phi}{B}^{\rho}$$,A1$$,B1$ | species x and y coexist | |

$\rho \phi >1$ | $B>{A}^{\phi}$$,B1$ | species y displaces species x |

$A>{B}^{\rho}$$,A1$ | species x displaces species y | |

$AB<{A}^{\phi}{B}^{\rho}$$,A1$$,B1$ | the displacement of a species by another one depends on the values of initial conditions |

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**MDPI and ACS Style**

Kulakov, M.; Neverova, G.; Frisman, E.
The Ricker Competition Model of Two Species: Dynamic Modes and Phase Multistability. *Mathematics* **2022**, *10*, 1076.
https://doi.org/10.3390/math10071076

**AMA Style**

Kulakov M, Neverova G, Frisman E.
The Ricker Competition Model of Two Species: Dynamic Modes and Phase Multistability. *Mathematics*. 2022; 10(7):1076.
https://doi.org/10.3390/math10071076

**Chicago/Turabian Style**

Kulakov, Matvey, Galina Neverova, and Efim Frisman.
2022. "The Ricker Competition Model of Two Species: Dynamic Modes and Phase Multistability" *Mathematics* 10, no. 7: 1076.
https://doi.org/10.3390/math10071076