Multirhythmicity, Synchronization, and Noise-Induced Dynamic Diversity in a Discrete Population Model with Competition

Abstract

The problem of mathematical modeling and analysis of stochastic phenomena in population systems with competition is considered. This problem is investigated based on a discrete system of two populations modeled by the Ricker map. We study the dependence of the joint dynamic behavior on the parameters of the growth rate and competition intensity. It is shown that, due to multistability, random perturbations can transfer the population system from one attractor to another, generating stochastic P-bifurcations and transformations of synchronization modes. The effectiveness of a mathematical approach, based on the stochastic sensitivity technique and the confidence domain method, in the parametric analysis of these stochastic effects is demonstrated. For monostability zones, the phenomenon of stochastic generation of the phantom attractor is found, in which the system enters the trigger mode with alternating transitions between states of almost complete extinction of one or the other population. It is shown that the noise-induced effects are accompanied by stochastic D-bifurcations with transitions from order to chaos.

1. Introduction

Currently, mathematical models in the form of differential and difference equations are widely used in the study of complex population dynamics processes [1,2,3]. Mathematical analysis of deformations in dynamic regimes in these nonlinear models is based on modern bifurcation theory [4,5,6]. In these circumstances, answering important biological questions involves studying bifurcations such as period doubling, Neimark–Sacker, transcritical, crisis, and flip. In these studies, transitions from order to chaos are of particular interest [7].

Population models that account for different biological factors, such as various prey–predator functional responses, age structure, delay, migrations, and infection spread, have been the subject of extensive research (see, e.g., [8,9,10,11,12]). Interspecific competition between coexisting populations for common resources is an important biological factor that deserves careful study. The effect of competition on the cooperative dynamics of coexisting populations has been actively studied by biologists both in the laboratory and in the wild (see, e.g., [13,14,15,16,17,18,19,20] and the references therein). In studying the complex processes of population dynamics with competition, simple mathematical models play a key role in helping to cut through this complexity and provide general insights. The conceptual deterministic Ricker-type model of two species with competition was introduced by R. May in [21] and has been studied in detail in many successive papers (see, e.g., [11,22]). Other mathematical models have also been used to analyze the effects of competition, among which the most famous is the Lotka–Volterra competition model [23,24,25]. In populations coupled by competition, investigating synchronization regimes [26] is an attractive subject from both empirical and theoretical perspectives [27,28,29,30,31].

It is known that a population, like any living system, is susceptible to random disturbances that vary in nature. Therefore, along with deterministic analysis, the study of stochastic effects in mathematical models of population dynamics is of significant interest to researchers (see, e.g., [1,28,32,33,34,35,36]). Stochastic population models, mostly continuous-time models with interspecific competition, have been studied by many authors (see, e.g., [37,38,39,40,41,42,43] and the references therein). The aim of this paper is to study the constructive role of random perturbations in population dynamics with competition based on a stochastic version of the discrete-time model [11,21].

When analyzing stochastic phenomena in map-based population models, the main technique is still the method of direct numerical modeling of random solutions. However, in parametric analysis, these methods are rather time-consuming. So, it is important to develop analytical approaches for the stochastic analysis of dynamic models. For discrete-time systems, the Frobenius–Perron equations give a full mathematical description of the stochastic dynamics [44,45]. However, the direct use of these functional equations presents serious difficulties, even in the case of one-dimensional models. To approximate probabilistic distributions near regular and chaotic attractors of map-based systems, a constructive method for determining confidence domains using stochastic sensitivity analysis has been proposed. In [46], the details of this new mathematical theory are presented. This theory and the constructive method for confidence domains were successfully applied to parametric studies of diverse, frequently counterintuitive, noise-induced phenomena in population models (see, e.g., [8,10,47]).

The novelty of the presented paper lies in answering the important question of how random perturbations deform the behavior of a population system with competition under a high degree of multistability. Here, the probabilistic mechanisms generating transitions between coexisting attractors are discussed in detail. We show that this population model exhibits complex behavior even in the case of monostability. Here, the phenomenon of the noise-induced generation of a phantom attractor is observed. The stochastic phenomenon of a phantom attractor was first discovered in a formal mathematical model defined by the system of differential equations with cubic nonlinearity [48]. The essence of this phenomenon is a noise-induced concentration of random states in the zone of the phase plane, where the initial unperturbed deterministic system has no attractors. This counterintuitive phenomenon was also discovered in the Langford model, defined by three differential equations [49], and in a neuronal model [50]. In this paper, we first show the phenomenon of the noise-induced generation of phantom attractors in discrete population dynamics and discuss its non-trivial biological meaning. Information on the location of the phantom attractor allows one to identify new variants of the behavior of population systems that have no analogs in deterministic theory.

The remainder of this paper is organized as follows. Section 2 introduces the original deterministic two-dimensional model of populations coupled by interspecific competition and gives a brief overview of the variability of attractors, their basins, and multistability. Section 3 explores the effects of parametric random disturbances on the competition intensity. Here, the dispersion of random states around attractors and noise-induced transitions are analyzed both numerically and analytically with the help of confidence domains. Section 4 introduces and studies the phenomenon of the stochastic generation of phantom attractors, which describe trigger regimes with the near-total extinction of one or the other competing population. In Section 5, stochastic D-bifurcations, with transitions from order to chaos, are discussed.

2. Deterministic Model: Multistability, Regular and Chaotic Attractors, and Synchronization

Consider the population model with competition for resources

$$\begin{array}{c}{x}_{t+1}=Axexp(-x_t-\rho y_t)\hfill \\ y_{t+1}=Byexp(-y_t-\phi x_t),\hfill \end{array}$$

(1)

where x and y are the population densities of competing species; A and B are the growth rates of species x and y, respectively; and the parameters $\rho$ and $\phi$ characterize the intensity of competitive relationships between species x and y. The dynamic regimes of an isolated population $(\rho =0)$ are shown in the bifurcation diagram in Figure 1, where the growth rate A varies.

In what follows, we consider a symmetric case, where $A=B$ and $\rho =\phi$, so that system (1) becomes

$$\begin{array}{c}{x}_{t+1}=Axexp(-x_t-\rho y_t)\hfill \\ y_{t+1}=Ayexp(-y_t-\rho x_t).\hfill \end{array}$$

(2)

We focus on the study of the behavior of system (2) in terms of its dependence on the parameters $A>0$ and $\rho \ge 0$.

System (2) has four equilibria: the trivial equilibrium $M_0(0,0),$ semi-trivial equilibria $M_1(0,lnA),$$M_2(lnA,0)$, and one non-trivial equilibrium $M_3\left({\displaystyle \frac{lnA}{1+\rho }},{\displaystyle \frac{lnA}{1+\rho }}\right)$. The equilibrium $M_0$ represents a regime of complete extinction of both populations. The equilibria $M_1$ and $M_2$ correspond to the extinction of one population only. The equilibrium $M_3$ describes a non-degenerate interaction between both populations. The equilibria $M_1,M_2$, and $M_3$ make biological sense if $A>1$.

The Jacobi matrix of system (2) is written as

$$F(x,y)=\left[\begin{array}{cc}A(1-x)exp(-x-\rho y)& -A\rho xexp(-x-\rho y)\\ -A\rho yexp(-y-\rho x)& A(1-y)exp(-y-\rho x)\end{array}\right].$$

For the trivial equilibrium $M_0$, the Jacobi matrix

$$F_0=\left(\begin{array}{cc}A& 0\\ 0& A\end{array}\right)$$

has eigenvalues ${\lambda }_1={\lambda }_2=A$, so $M_0$ is stable for $A<1$ and unstable otherwise.

For the semi-trivial equilibria $M_1$ and $M_2$, their Jacobi matrices

$$F_1=\left(\begin{array}{cc}{A}^{1-\rho }& 0\\ -\rho lnA& 1-lnA\end{array}\right),F_2=\left(\begin{array}{cc}{A}^{1-\rho }& -\rho lnA\\ 0& 1-lnA\end{array}\right)$$

have eigenvalues ${\lambda }_1=1-lnA,{\lambda }_2=A^{1-\rho }$, so $M_1$ and $M_2$ are stable for $1<A<e^2,\rho >1$.

For the non-trivial equilibrium $M_3$, the Jacobi matrix

$$F_3=\left(\begin{array}{cc}{\displaystyle 1-\frac{lnA}{1+\rho }}& {\displaystyle -\rho \frac{lnA}{1+\rho }}\\ {\displaystyle -\rho \frac{lnA}{1+\rho }}& {\displaystyle 1-\frac{lnA}{1+\rho }}\end{array}\right)$$

has eigenvalues

$${\lambda }_1=1-lnA,{\lambda }_2=1+{\displaystyle \frac{\rho -1}{\rho +1}}lnA.$$

Thus, the equilibrium $M_3$ is stable if and only i $1<A<e^2,\rho <1$. In what follows, we focus on the case $A>1,\rho <1$. Note that in this parameter range, the equilibria $M_0,M_1$, and $M_2$ are unstable. As A increases and passes the bifurcation point $A_1=e^2$, the eigenvalue ${\lambda }_1$ of the matrix $F_3$ becomes less than −1, which means the loss of stability of $M_3$ and the birth of a stable 2-cycle.

The variety of dynamic regimes of system (2) is illustrated in Figure 2. In Figure 2a, the x-coordinates of system (2) attractors that lie on the diagonal $y=x$ are plotted against the parameter A for $\rho =0$, $\rho =0.5$, and $\rho =0.99$. These diagrams are structurally similar and demonstrate the well-known Feigenbaum scenario of period doubling leading to chaos. Note that the amplitude of these diagonal attractors decreases as the competition intensity $\rho$ increases. Such an arrangement of oscillatory attractors indicates complete synchronization between subpopulations x and y, both in phase and amplitude.

Along with such diagonal attractors, system (2) can exhibit other coexisting attractors that do not belong to the diagonal $y=x$. So, multistability is an important feature of the model under consideration. From a biological perspective, multistability means that even with all biological parameters fixed, the behavior of a population can change qualitatively depending on its initial state.

In Figure 2b, we present an $(A,\rho )$-parametric diagram, where zones of mono- and multistability are shown. A more wide zone of multistability is observed for small values of the competition intensity $\rho$.

Let us fix $\rho =0.03$. In Figure 2c, bifurcation diagrams for various coexisting attractors are shown in different colors. As can be seen, in some A-parameter zones, this system is four-stable. In the enlarged fragment in Figure 2d, some bifurcation values are marked. The largest Lyapunov exponents of the coexisting attractors are plotted with corresponding colors in Figure 2e. To calculate the largest Lyapunov exponents, we used the standard Benettin algorithm.

As one can see, for the small competition intensity $\rho =0.03$, an increase in A causes complications of dynamic regimes. The details of such transformations are shown in Figure 3, where attractors and their basins are plotted for several values of the parameter A corresponding to regular and chaotic oscillations.

For $A=8$ (see Figure 3a), system (2) is monostable and has a 2-cycle $\mathcal{R}$ (red) that lies on the diagonal $y=x$. Thus, regardless of the initial state, the solutions of system (2) tend to this cycle, and complete synchronization of the oscillatory activity of both populations is observed. When the parameter A passes the bifurcation value $A_2=8.927$, another attractor appears, the non-diagonal 2-cycle $\mathcal{B}$ (blue). For $A=10$, in Figure 3b, the cycles $\mathcal{R}$ and $\mathcal{B}$ are plotted along with their basins in white and light blue, respectively. In this case of bistability, the behavior of the system depends on the initial point: the solutions starting in the basin of $\mathcal{R}$ exhibit complete synchronization, while the solutions starting in the basin of $\mathcal{B}$ exhibit anti-phase synchronization.

When the parameter A passes the next bifurcation value $A_3=12.306,$ a third attractor appears: the 4-cycle $\mathcal{G}$ (green) (see Figure 3c for $A=12.45$). Note that the basin (white) of the 2-cycle $\mathcal{R}$ transforms into the diagonal $y=x>0$. The basin (light green) of the 4-cycle $\mathcal{G}$ consists of the initial states corresponding to the regime of in-phase synchronization. Thus, in system (2) with $A=12.45$, three attractors and three qualitatively different regimes of synchronization are observed: complete for $\mathcal{R}$, in-phase for $\mathcal{G}$, and anti-phase for $\mathcal{B}$.

With a further increase in A, the 2-cycle $\mathcal{R}$ turns into a 4-cycle. Attractors and basins of such a tri-rhythmic system are shown in Figure 3d for $A=13$. Note that the basin of the diagonal attractor $\mathcal{R}$ becomes non-degenerate again.

In Figure 3e, attractors and basins are plotted for $A=15$. Here, the diagonal attractor $\mathcal{R}$ is a 4-piece chaos. Another 4-piece chaotic attractor $\mathcal{P}$ (purple) is located close to $\mathcal{R}$. One more 4-piece chaotic attractor $\mathcal{G}$ (green) also exhibits an in-phase regime of synchronization. Along with these in-phase chaotic attractors, the system (2) with $A=15$ exhibits an anti-phase regular attractor $\mathcal{B}$ in the form of the 2-torus.

Figure 3f for $A=16$ shows the coexistence of the 6-cycle $\mathcal{R}$, 2-piece chaotic attractor $\mathcal{P}$, and almost-destroyed 2-torus $\mathcal{B}$. Only two attractors, 1-piece chaos $\mathcal{R}$ and 2-piece chaos $\mathcal{B}$, are observed for $A=18$ (see Figure 3g). For $A=20$, one can see 1-piece chaos $\mathcal{R}$ and 1-piece chaos $\mathcal{B}$ (see Figure 3h). Note that in this case $\mathcal{R}$ represents the mode of complete synchronization with the basin $y=x>0$, and $\mathcal{B}$ represents a new regime with temporal intermittency of in-phase and anti-phase synchronization modes with the basin $y\ne x>0$.

Thus, population systems coupled with competition exhibit a wide variety of regular and chaotic regimes with different types of synchronization. These regimes are determined not only by the combination of parameters A and $\rho$ but also by the choice of initial data.

The influence of random perturbations, inevitably present in any living system, causes an additional variety of modes of behavior of the population model under consideration. The constructive role of such perturbations is investigated in the next sections. In our stochastic analysis, the multistability found above in the original deterministic model is an important factor.

3. Stochastic Model:Analysis of Dispersion of Random States Around Attractors and Noise-Induced Transitions

Random perturbations can cause changes on various scales in the behavior of a system. These can be local changes caused by transitions between close attractors or global changes that dramatically alter the behavior of the system. In the model under consideration, global changes include the transition from anti-phase synchronization to in-phase synchronization or noise-induced temporary near-extinction of the population.

In this paper, we study stochastic effects caused by random disturbances in parameters $\rho$ and $\phi$ characterizing the intensity of competitive relationships between species x and y: $\rho \to {\rho }_{\epsilon }={[\rho +\epsilon {\xi }_t]}^{+},\phi \to {\phi }_{\epsilon }={[\phi +\epsilon {\eta }_t]}^{+}$. Here, ${\xi }_t$ and ${\eta }_t$ represent uncorrelated standard white Gaussian noise with intensity $\epsilon$. In this stochastic case, we use the following truncation: ${\left[\alpha \right]}^{+}=\alpha$ for $\alpha \ge 0$ and ${\left[\alpha \right]}^{+}=0$ for $\alpha <0$. This truncation is used to make biological sense of the competition parameters: ${\rho }_{\epsilon }\ge 0$ and ${\phi }_{\epsilon }\ge 0.$ So, a stochastic version of the population model (1) is

$$\begin{array}{c}{x}_{t+1}=Axexp(-x_t-{[\rho +\epsilon {\xi }_t]}^{+}{y}_t)\hfill \\ y_{t+1}=Byexp(-y_t-{[\phi +\epsilon {\eta }_t]}^{+}{x}_t).\hfill \end{array}$$

In the symmetric case under consideration, where $A=B$ and $\rho =\phi$, the stochastic variant of system (2) is written as

$$\begin{array}{c}{x}_{t+1}=Axexp(-x_t-{[\rho +\epsilon {\xi }_t]}^{+}{y}_t)\hfill \\ y_{t+1}=Ayexp(-y_t-{[\rho +\epsilon {\eta }_t]}^{+}{x}_t).\hfill \end{array}$$

(3)

To model Gaussian noise in system (3), we use the standard Box–Muller transform of uniformly distributed random sequences.

In the presence of random perturbations, the solutions of the stochastic system depart from deterministic attractors and form probability distributions around them. For weak noise, these distributions concentrate near the original deterministic attractors. In Figure 4, random states of system (3) with $\epsilon =0.01$ are shown near deterministic attractors for three values of the parameter A. For $A=7$, the deterministic attractor is the stable equilibrium $M_3$ (red solid circle in Figure 4a). For $A=8$ (see Figure 4b), the deterministic attractor is the stable 2-cycle. For $A=10$, there are two coexisting stable 2-cycles (see Figure 4c). The random states in these figures are indicated by dots of the corresponding colors. As can be seen, the geometry of the dispersion of random states changes as A varies. Even for the same A, the dispersions around coexisting attractors can be different (compare the dispersions of the blue and pink dots in Figure 4c).

To analyze this dispersion, it is necessary to take into account the stochastic sensitivity of attractors. The stochastic sensitivity technique is a constructive tool for quantifying the variance of random solutions and describing the geometry of probability distributions in terms of confidence domains. For equilibria and discrete cycles, confidence domains take the form of ellipses constructed based on the eigenvalues and eigenvectors of the stochastic sensitivity matrices of the corresponding attractors. The mathematical details of the stochastic sensitivity theory can be found in [46]. It is worth noting that the mathematical technique of stochastic sensitivity and the confidence domain method avoid the very expensive and time-consuming analysis of noise-induced phenomena based on direct numerical simulation of random solutions.

In Figure 4, the confidence ellipses around the attractors, equilibria, and cycles under consideration are plotted using dashed lines. As can be seen, these ellipses accurately describe the main geometrical features of the distribution of random states in solutions obtained through direct numerical simulation.

Multistability and Noise-Induced Transitions

In the case of multistability with the coexistence of several attractors, increasing noise can cause stochastic transitions between the basins of these attractors. Let us consider how such noise-induced transitions occur in system (3) with $A=10$. The deterministic system (2) with $A=10$ exhibits the coexistence of two 2-cycles. One of them, $\mathcal{R}$, shown in red in Figure 3b, represents the complete synchronization mode, while the other, $\mathcal{B}$ (blue in Figure 3b), represents the anti-phase synchronization mode. In biological terms, the $\mathcal{R}$ regime means that the fluctuating densities of both populations increase and decrease simultaneously. In contrast, the $\mathcal{B}$ regime means that the growth of one population is accompanied by a decrease in the other population.

In Figure 5, we show the u-coordinates of random states $(x_{1001},y_{1001}),\dots ,(x_{1100},y_{1100})$ of system (3) solutions starting at the deterministic anti-phase 2-cycle $\mathcal{B}$ (Figure 5a) and the in-phase 2-cycle $\mathcal{R}$ (Figure 5b). Here and throughout, the coordinate u$\left(u=(x-y)/\sqrt{2}\right)$ represents the projection of the point $(x,y)$ onto the line $x+y=0$, which is orthogonal to the diagonal line $y=x$. Figure 5a shows that the dispersion around the states of the anti-phase 2-cycle $\mathcal{B}$ first gradually increases, then abruptly changes, and localizes near $u=0$. This jump occurs due to the transition of random solutions from the basin of the anti-phase 2-cycle $\mathcal{B}$ to a small neighborhood of the in-phase 2-cycle $\mathcal{R}$. As can be seen in Figure 5b, within this $\epsilon$-interval, the dispersion of random solutions starting at the in-phase 2-cycle $\mathcal{R}$ grows gradually. This stochastic effect can be summarized as the noise-induced destruction of the regime of anti-phase synchronization in system (3).

For a description of the transitions between in-phase and anti-phase synchronization modes, it is convenient to use the synchronization indicator $z_t=\mathrm{sgn}\left((x_{t+1}-x_t)(y_{t+1}-y_t)\right).$ The value $z_t=1$ corresponds to in-phase oscillations, and $z_t=-1$ reflects anti-phase oscillations mode. This synchronization indicator is shown in green in Figure 5. Using this indicator, one can localize the threshold $\epsilon$-zone where transitions from anti-phase ($z=-1$) to in-phase ($z=1$) oscillatory regimes occur. As can be seen, when $\epsilon >0.1$, the stochastic system exhibits fast switching between values $z=-1$ and $z=1$. This means that random disturbances destroy the stable in-phase synchronization regime, and the system transits to the regime of alternation of anti-phase and in-phase oscillations.

In Figure 6a, time series of system (3) solutions starting at the anti-phase cycle $\mathcal{B}$ are plotted for $A=10$ and $\epsilon =0.02$. In the enlarged fragment in Figure 6b, the details of the “anti-phase → in-phase” transition are clearly seen. In the parametric analysis of these noise-induced transitions, the method of confidence ellipses can be effectively used. In Figure 6c, confidence ellipses around states of coexisting in-phase 2-cycle $\mathcal{R}$ and anti-phase 2-cycle $\mathcal{B}$ are plotted using dashed lines for $\epsilon =0.01$ and solid lines for $\epsilon =0.02$. For $\epsilon =0.01$, these ellipses completely belong to the corresponding basins of attraction. This means that random solutions are localized near states of corresponding cycles; therefore, noise-induced transitions do not occur (see also Figure 4c and Figure 5). With increasing noise, the expanding ellipses can intersect the separatrix and capture points of the basins of the coexisting attractors. This scenario is observed in system (3) for $\epsilon =0.02$: blue solid ellipses partially occupy the basin of the 2-cycle $\mathcal{R}$. The intersection of the confidence ellipse with the separatrix signals the onset of noise-induced “anti-phase→in-phase” transitions. It should be noted that even for $\epsilon =0.02$, the red solid confidence ellipse lies in the basin of $\mathcal{R}$, so reverse “in-phase→anti-phase” transitions are not observed. The prediction presented here, based on the method of confidence ellipses, agrees well with the results of direct numerical simulations (compare Figure 5 and Figure 6).

As shown in Figure 2c,d, an increase in A implies an increase in the number of coexisting attractors. Indeed, in Figure 3c,d, for $A=12.45$ and $A=13$, along with the anti-phase 2-cycle $\mathcal{B}$ and in-phase cycle $\mathcal{R}$, one more in-phase 4-cycle $\mathcal{G}$ (green) exists. Noise-induced effects in the tri-stability cases are illustrated in Figure 7 for $A=12.45$ and Figure 8 for $A=13$.

For $A=12.45$ (see Figure 2c), the basin of the completely synchronized 2-cycle $\mathcal{R}$ is degenerate: $y=x>0.$ So, for any small disturbances, the solution starting at $\mathcal{R}$ immediately falls into the basin of the 4-cycle $\mathcal{G}$. Consider the behavior of random solutions starting at $\mathcal{G}$ and $\mathcal{B}$. In Figure 7a, the u-coordinates of solutions starting at the anti-phase 2-cycle $\mathcal{B}$ (top) and in-phase 4-cycle $\mathcal{G}$ (bottom) are plotted. Here, the synchronization indicator z is shown in green.

In the top panel, one can see an abrupt deformation of the dispersion of random states corresponding to the noise-induced transition from $\mathcal{B}$ to $\mathcal{G}$, accompanied by the “anti-phase → in-phase” transformation indicated by z. The confidence ellipses shown in Figure 7c accurately predict this phenomenon.

In the bottom panel in Figure 7a, random states of solutions starting at the in-phase 4-cycle $\mathcal{G}$ are plotted. As can be seen, even for weak noise, distributions begin to merge. The details of these transformations of the probability density $p\left(u\right)$ are illustrated in Figure 7b. Here, the four-peak plot of $p\left(u\right)$ for $\epsilon =0.001$ transforms into a two-peak form for $\epsilon =0.006$ and becomes unimodal for $\epsilon =0.01$. Such qualitative changes in the form of the probability density function can be interpreted as stochastic P-bifurcations (phenomenological bifurcations) [51]. So, in this case, increasing noise generates a cascade $4\to 2\to 1$ of stochastic P-bifurcations.

Now, consider $A=13$. Interestingly, in this case, the basin of the diagonal attractor $\mathcal{R}$ is non-degenerate (see white areas in Figure 8c). The deformations of the dispersion of random states for solutions starting at this diagonal in-phase 4-cycle $\mathcal{R}$ are shown in Figure 8a. Here, the scenario of the qualitative transformations of the probability density $p\left(u\right)$ becomes more complicated (see Figure 8b). For $\epsilon =0.003$, the function $p\left(u\right)$ (blue) has a narrow high peak. This means that the random states are concentrated in a small neighborhood of the cycle $\mathcal{R}$. With increasing noise, the single-peak plot transforms into a four-peak one (see red curve in Figure 8b for $\epsilon =0.01$). This means that random solutions begin to localize near the states of the 4-cycle $\mathcal{G}$. Note that this transition $\mathcal{R}\to \mathcal{G}$ is accurately predicted by the confidence ellipses shown in Figure 8c. With a further increase in noise, the four-peak plot of $p\left(u\right)$ turns into a single-peak plot (see green curve for $\epsilon =0.04$). Such stochastic mixing entails the destruction of synchronization (see the jumping synchronization indicator shown in green in Figure 8a).

For $A=15$, system (2) is four-stable (see Figure 3e) with the following attractors: anti-phase two-torus $\mathcal{B}$ (blue), four-piece completely synchronized chaos $\mathcal{R}$ with the degenerate basin $y=x>0$, and two four-piece chaotic in-phase attractors, $\mathcal{P}$ (purple) and $\mathcal{G}$ (green). Stochastic transformations of random solutions starting at $\mathcal{G}$, $\mathcal{P}$, and $\mathcal{B}$ are illustrated in Figure 9. Here, along with the effect of the noise-induced mixing of solutions, the phenomenon of the destruction of anti-phase synchronization of solutions starting at $\mathcal{B}$ is shown.

The results of the extended parametric analysis of noise-induced transitions between in-phase and anti-phase synchronization modes are presented in Figure 10. In this analysis, we consider system (3) solutions starting at the anti-phase state $x_0=4,y_0=1$ and estimate a mean value $\langle z\rangle$ of the synchronization indicator z using samples $(x_{1001},y_{1001}),\dots ,$$(x_{2000},y_{2000})$. In Figure 10a, plots of $\langle z\rangle$ are shown for several values of the parameter A versus the noise intensity $\epsilon$. Here, we choose examples with qualitatively different behavior. For $A=7.5$, the system (2) has an in-phase 2-cycle as a single attractor. In this case, random solutions quickly transit to the small vicinity of the cycle $\mathcal{R}$, so $\langle z\rangle$ becomes equal to one. With a further increase in $\epsilon$, the regime of in-phase synchronization is preserved until the stochastic alternation of in-phase and anti-phase synchronization begins. The value $\langle z\rangle$ monotonically decreases and stabilizes.

For $A=9$ and $A=10$, more complex three-stage transformations of $\langle z\rangle$ are observed. First, one can see the anti-phase synchronization. Then, in the second stage, an abrupt transition to the in-phase synchronization mode is observed. This change in the synchronization mode occurs for $A=9$ at smaller noise intensities than for $A=10$. The third stage is the monotonous decrease in $\langle z\rangle$ corresponding to the mixed-mode regime. For $A=18$, the two-stage scenario is observed: first, we observe the anti-phase synchronization and then the stochastic mixing. Qualitatively, another type of behavior is observed for $A=20$. The initial deterministic attractor is chaotic and exhibits mixed-mode oscillations, even without any stochastic forcing $\left(\langle z\rangle =0\right)$. In the presence of random disturbances, this behavior is preserved in a rather large $\epsilon$-interval. With a further increase in $\epsilon$, the value of $\langle z\rangle$ only slightly increases. Note that in the stabilization zone, the value of $\langle z\rangle$ hardly depends on A. In Figure 10b, the value of $\langle z\rangle$ is plotted in color in the $(A,\epsilon )$-plane. Here, complex processes of the noise-induced transformation of synchronization regimes can be seen.

To summarize, the random forcing in system (3) causes various phenomena associated with transitions between coexisting attractors and qualitative changes in synchronization modes. However, along with these effects, the system can exhibit phenomena that are not connected with transitions between attractors.

4. Phantom Attractors and Noise-Induced Trigger Regimes

In this section, we fix $\rho =0.99$. It should be emphasized that for this value of the parameter, the deterministic system (2) is monostable for any A (see Figure 2b). The attractors of this system are shown in green in the bifurcation diagram (Figure 2a). However, even in this case of monostability, random forcing can cause qualitative changes in the behavior of the stochastic system (3).

In Figure 11, we present the stochastic effects for $A=6$ and $A=10$. For $A=6$, the deterministic system (2) possesses a stable equilibrium as a single attractor. In Figure 11a, random states and time series of system (3) with $A=6$ are shown for different values of the noise intensity. For small values of $\epsilon$, random states have a Gaussian-type distribution, with the maximal concentration at the stable equilibrium. With an increase in noise, the distribution changes qualitatively and becomes bimodal, with concentration near the axes $x=0$ and $y=0$ (see pink dots for $\epsilon =0.2$). The corresponding time series in Figure 11a (right) show trigger-type behavior with temporal stabilization near $x=0$ or $y=0$. Such stabilization has an important biological meaning: for sufficiently strong noise, the system goes into a regime of alternating temporary near-extinction of one or another population. In Figure 11b, a similar phenomenon is shown for the case where $A=10$. The deterministic attractor here is a 2-cycle.

The details of the qualitative changes in the corresponding probability density function $p\left(u\right)$ are shown in Figure 12. In Figure 12a, for $A=6$, one can see how the unimodal plot of $p\left(u\right)$ with the maximum at the equilibrium ($u=0$) is transformed into the bimodal plot with symmetric peaks on the borders. In Figure 12b, for $A=10$, it is shown how the single-peak plot of $p\left(u\right)$ transforms into a four-peak form. So, in this case, stochastic P-bifurcations are observed.

It should be emphasized that the new peaks in the probability density function appear in the zone of the phase plane where the deterministic system (1) does not possess any attractors. Such a phenomenon, called a “phantom attractor”, was studied in [49,50]. In system (3), the appearance of peaks of the probability density function of the phantom attractor signals the transition to the trigger regime with the temporary near-extinction of coexisting species.

This stochastic phenomenon of the temporary near-extinction of subpopulations is quite general and is also observed at other values of A. We illustrate this phenomenon in Figure 13 for $A=15.5,A=16,$ and $A=20$.

The results of the statistical analysis of the probability of temporary near-extinction are presented in Figure 14. Here, for different values of A and $\rho$, we consider solutions of the stochastic system (3) starting at the deterministic attractors lying at $y=x$ and calculate the probability of falling into the domain $\left\{\right(x,y\left)\right|x<\delta$ or $y<\delta \}$ for time T. In our analysis, we used $\delta =0.01$ and $T=1000$. It can be seen that the higher the intensity $\rho$ of the competition, the smaller the noise that causes the temporary near-extinction. An increase in A slightly decreases the corresponding threshold noise intensity.

Let us consider intervals $\tau$ of the time residence of system (3) in the mode of near-extinction. These intervals for some stochastic solutions are plotted in red in Figure 15 (left) for the threshold $\delta =0.01$. Obviously, these intervals are random. In Figure 15 (right), we present the mean values $\langle \tau \rangle$ of these random intervals for $\rho =0.99$ and different values of A versus the parameter $\epsilon$. As can be seen, for any fixed A, an increase in $\epsilon$ implies an increase in the mean value $\langle \tau \rangle$. Note that for fixed $\epsilon$, the mean value $\langle \tau \rangle$ decreases as A increases.

5. Noise-Induced Transitions from Order to Chaos

Interestingly, these transformations of distributions of random states in the phase plane are accompanied by changes in the internal dynamics. The largest Lyapunov exponent $\Lambda$ is a standard quantitative characteristic of these changes. In Figure 16, plots of $\Lambda \left(\epsilon \right)$ are shown for $\rho =0.99$ and several values of the parameter A. For chosen values of A, the deterministic system (2) has regular attractors (stable cycles). Note that $A=14$ and $A=14.5$ represent cycles from the period doubling zone, whereas $A=16$ and $A=18.5$ were taken from narrow-order windows lying between chaos (see Figure 2). All these cycles belong to the diagonal $y=x$.

At these values of A, the dynamics of system (3), under the influence of weak noise, preserve regularity: $\Lambda \left(\epsilon \right)<0.$ However, at a certain value of $\epsilon$, the largest Lyapunov exponent starts to grow and becomes positive. Such a change, stochastic D-bifurcation [51], marks the onset of the transition from order to chaos. Note that these transitions to chaos are observed at noise intensities for which the stochastic attractor is still localized near the diagonal of the first quadrant of the phase plane. The sharp decrease in the Lyapunov exponent is related to the beginning of the formation of the phantom attractor. Recall that the peculiarity of phantom attractors in the model under consideration is the temporary stabilization of the system solutions near the coordinate axes, signaling the near-extinction of one of the competing populations.

6. Conclusions

This paper analyzed the impact of interspecific competition for common limited resources in systems of interacting populations. This problem was investigated based on a discrete system of two populations modeled by the Ricker map. We considered the symmetric case and studied the dependence of the joint dynamic behavior on the parameters of the growth rate and competition intensity. Mono- and multistability zones were identified, and attractors (regular and chaotic) and their basins were described. The main content of this paper was devoted to the study of the variability of the joint behavior of competing populations in the presence of random perturbations. It was shown that, in the presence of multistability, the population system, under the influence of random perturbations, can move from one attractor to another, generating stochastic P-bifurcations and transformations of synchronization modes. It was revealed that the dominant direction of these transformations is the transition from anti-phase to in-phase synchronization. For the parametric analysis of these stochastic effects, the method of confidence domains was applied. For monostable regimes, the phenomenon of the stochastic generation of the phantom attractor was found, in which the system enters a trigger mode with alternating transitions between states of near-total extinction of one or the other population. It was shown that noise-induced effects are accompanied by stochastic D-bifurcations with transitions from order to chaos. The developed methodology for analyzing stochastic phenomena using the stochastic sensitivity technique and confidence domains was presented in this paper for the Ricker-type model with competition. It should be emphasized that our approach can be successfully applied to other population models, both discrete and continuous.